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+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "[nltk_data] Downloading package punkt to\n",
+      "[nltk_data]     /home/pranav_kirsur/nltk_data...\n",
+      "[nltk_data]   Package punkt is already up-to-date!\n",
+      "[nltk_data] Downloading package averaged_perceptron_tagger to\n",
+      "[nltk_data]     /home/pranav_kirsur/nltk_data...\n",
+      "[nltk_data]   Package averaged_perceptron_tagger is already up-to-\n",
+      "[nltk_data]       date!\n",
+      "[nltk_data] Downloading package punkt to\n",
+      "[nltk_data]     /home/pranav_kirsur/nltk_data...\n",
+      "[nltk_data]   Package punkt is already up-to-date!\n"
+     ]
+    }
+   ],
+   "source": [
+    "import nltk\n",
+    "import pranav\n",
+    "import ahish\n",
+    "import discourse_connector\n",
+    "# import SimToSumm\n",
+    "import SimToTitle\n",
+    "import csv\n",
+    "import textprocessing"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "textDict = textprocessing.getTextDict()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[{'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.7 Solution Let the original charge on sphere A be q and that on B be q′', 'At a distance r between their centres, the magnitude of the electrostatic force on each is given by neglecting the sizes of spheres A and B in comparison to r', 'When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2', 'Similarly, after D touches B, the redistributed charge on each is q′/2', 'Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus the electrostatic force on A, due to B, remains unaltered', '7 Forces between Multiple Charges The mutual electric force between two charges is given by Coulomb’s law', 'How to calculate the force on a charge where there are not one but several charges around? Consider a system of n stationary charges q1, q2, q3, ..., qn in vacuum', 'What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question', 'Recall that forces of mechanical origin add according to the parallelogram law of addition', 'Is the same true for forces of electrostatic origin?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time', 'The individual forces are unaffected due to the presence of other charges', 'This is termed as the principle of superposition', 'To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in (a)', 'The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges', 'Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq', '(1.3) even though other charges are present', 'Thus, F12 In the same way, the force on q1 due to q3, denoted by F13, is given by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which again is the Coulomb force on q1 due to q3, even though other charge q2 is present', 'Thus the total force F1 on q1 due to the two charges q2 and q3 is given as (1.4) The above calculation of force can be generalised to a system of charges more than three, as shown in (b)', 'The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn', 'The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n: i.e.,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.8 A system of (a) three charges (b) multiple charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.5) The vector sum is obtained as usual by the parallelogram law of addition of vectors', 'All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle', 'Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l', 'What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in ?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.9 Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC, AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l', 'By symmatry AO = BO = CO', 'Thus, Force F1 on Q due to charge q at A = along AO Force F2 on Q due to charge q at B = along BO Force F3 on Q due to charge q at C = along CO The resultant of forces F2 and F3 is along OA, by the parallelogram law', 'Therefore, the total force on Q = = 0, where is the unit vector along OA', 'It is clear also by symmetry that the three forces will sum to zero', 'Suppose that the resultant force was non-zero but in some direction', 'Consider what would happen if the system was rotated through 60° about O', 'Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in 0', 'What is the force on each charge?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.10 Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in 0', 'By the parallelogram law, the total force F1 on the charge q at A is given by F1 = F where is a unit vector along BC', 'The force of attraction or repulsion for each pair of charges has the same magnitude The total force F2 on charge q at B is thus F2 = F 2, where 2 is a unit vector along AC', 'Similarly the total force on charge –q at C is F3 = F , where is the unit vector along the direction bisecting the ∠BCA', 'It is interesting to see that the sum of the forces on the three charges is zero, i.e., F1 + F2 + F3 = 0 The result is not at all surprising', 'It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law', 'The proof is left to you as an exercise', '1.8 Electric Field Let us consider a point charge Q placed in vacuum, at the origin O', 'If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law', 'We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P', 'In order to answer such questions, the early scientists introduced the concept of field', 'According to this, we say that the charge Q produces an electric field everywhere in the surrounding', 'When another charge q is brought at some point P, the field there acts on it and produces a force', 'The electric field produced by the charge Q at a point r is given as (1.6) where r/r, is a unit vector from the origin to the point r', 'Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r', 'The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position', 'The effect of the charge has been incorporated in the existence of the electric field', 'We obtain the force F exerted by a charge Q on a charge q, as (1.7) Note that the charge q also exerts an equal and opposite force on the charge Q', 'The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa', 'If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q', 'Thus, F(r) = q E(r) (1.8) Equation (1.8) defines the SI unit of electric field as N/C*', 'Some important remarks may be made here: From Eq', '(1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it', 'Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point', 'The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge', 'Note that the source charge Q must remain at its original location', 'However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move', 'A way out of this difficulty is to make q negligibly small', 'The force F is then negligibly small but the ratio F/q is finite and defines the electric field: (1.9)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice', 'When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet', 'Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q', 'This is because F is proportional to q, so the ratio F/q does not depend on q', 'The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q', 'Thus, the electric field E due to Q is also dependent on the space coordinate r', 'For different positions of the charge q all over the space, we get different values of electric field E', 'The field exists at every point in three-dimensional space', 'For a positive charge, the electric field will be directed radially outwards from the charge', 'On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards', '(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r', 'Thus at equal distances from the charge Q, the magnitude of its electric field E is same', 'The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry', '1.8.1 Electric field due to a system of charges Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O', 'Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn', 'We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r', 'Electric field E1 at r due to q1 at r1 is given by E1 = where is a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P', 'In the same manner, electric field E2 at r due to q2 at r2 is E2 = where is a unit vector in the direction from q2 to P and r2P is the distance between q2 and P', 'Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn', 'By the superposition principle, the electric field E at r due to the system of charges is (as shown in 2) E(r) = E1 (r) + E2 (r) + … + En(r) = E(r) (1.10) E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges', '8.2 Physical significance of electric field You may wonder why the notion of electric field has been introduced here at all', 'After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq', '(1.5)]', 'Why then introduce this intermediate quantity called the electric field? For electrostatics, the concept of electric field is convenient, but not really necessary', 'Electric field is an elegant way of characterising the electrical environment of a system of charges', 'Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system)', 'Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field', 'The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point', 'Electric field is a vector field, since force is a vector quantity.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena', 'Suppose we consider the force between two distant charges q1, q2 in accelerated motion', 'Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light', 'Thus, the effect of any motion of q1 on q2 cannot arise instantaneously', 'There will be some time delay between the effect (force on q2) and the cause (motion of q1)', 'It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful', 'The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2', 'The notion of field elegantly accounts for the time delay', 'Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs', 'They have an independent dynamics of their own, i.e., they evolve according to laws of their own', 'They can also transport energy', 'Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy', 'The concept of field was first introduced by Faraday and is now among the central concepts in physics', 'Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1', 'The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance', 'Compute the time of fall in each case', 'Contrast the situation with that of ‘free fall under gravity’', 'Figure 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Solution In 3(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field', 'The acceleration of the electron is ae = eE/me where me is the mass of the electron', 'Starting from rest, the time required by the electron to fall through a distance h is given by For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m, te = 2.9 × 10–9s In 3 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE', 'The acceleration of the proton is ap = eE/mp where mp is the mass of the proton; mp = 1.67 × 10–27 kg', 'The time of fall for the proton is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus, the heavier particle (proton) takes a greater time to fall through the same distance', 'This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body', 'Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall', 'To see if this is justified, let us calculate the acceleration of the proton in the given electric field:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity', 'The acceleration of the electron is even greater', 'Thus, the effect of acceleration due to gravity can be ignored in this example', 'Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart', 'Calculate the electric fields at points A, B and C shown in 4']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.14 Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude', 'Hence the magnitude of the total electric field EA at A is EA = E1A + E2A = 7.2 × 104 N C–1 EA is directed toward the right', 'The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude = 4 × 103 N C–1 The magnitude of the total electric field at B is EB = E1B – E2B = 3.2 × 104 N C–1 EB is directed towards the left', 'The magnitude of each electric field vector at point C, due to charge q1 and q2 is = 9 × 103 N C–1 The directions in which these two vectors point are indicated in 4', 'The resultant of these two vectors is = 9 × 103 N C–1 EC points towards the right', '1.9 Electric Field Lines We have studied electric field in the last section', 'It is a vector quantity and can be represented as we represent vectors', 'Let us try to represent E due to a point charge pictorially', 'Let the point charge be placed at the origin', 'Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point', 'Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward', 'Figure 1.15 shows such a picture', 'In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow', 'Connect the arrows pointing in one direction and the resulting figure represents a field line', 'We thus get many field lines, all pointing outwards from the point charge', 'Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No', 'Now the magnitude of the field is represented by the density of field lines', 'E is strong near the charge, so the density of field lines is more near the charge and the lines are closer', 'Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines', 'Another person may draw more lines', 'But the number of lines is not important', 'In fact, an infinite number of lines can be drawn in any region', 'It is the relative density of lines in different regions which is important', 'We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions', 'So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines', 'Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge', 'We started by saying that the field lines carry information about the direction of electric field at different points in space', 'Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points', 'The field lines crowd where the field is strong and are spaced apart where it is weak', 'Figure 1.16 shows a set of field lines', 'We can imagine two equal and small elements of area placed at points R and S normal to the field lines there', 'The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points', 'The picture shows that the field at R is stronger than at S', 'To understand the dependence of the field lines on the area, or rather the solid angle subtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions', 'Recall how a (plane) angle is defined in two-dimensions', 'Let a small transverse line element ∆l be placed at a distance r from a point O', 'Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r', 'Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2', 'We know that in a given solid angle the number of radial field lines is the same', 'In 6, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is ∆Ω at P1 and an element of area ∆Ω at P2, respectively', 'The number of lines (say n) cutting these area elements are the same', 'The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively', 'Since n and ∆Ω are common, the strength of the field clearly has a 1/r2 dependence.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.15 Field of a point charge.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations', 'Faraday called them lines of force', 'This term is somewhat misleading, especially in case of magnetic fields', 'The more appropriate term is field lines (electric or magnetic) that we have adopted in this book', 'Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges', 'An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point', 'An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve', 'A field line is a space curve, i.e., a curve in three dimensions.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 shows the field lines around some simple charge configurations', 'As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane', 'The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward', 'The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges', 'The field lines follow some important general properties: Field lines start from positive charges and end at negative charges', 'If there is a single charge, they may start or end at infinity', 'In a charge-free region, electric field lines can be taken to be continuous curves without any breaks', 'Two field lines can never cross each other', '(If they did, the field at the point of intersection will not have a unique direction, which is absurd.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 Field lines due to some simple charge configurations.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(iv) Electrostatic field lines do not form any closed loops', 'This follows from the conservative nature of electric field (Chapter 2)', '10 Electric Flux Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface', 'The rate of flow of liquid is given by the volume crossing the area per unit time v dS and represents the flux of liquid flowing across the plane', 'If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ', 'Therefore, the flux going out of the surface dS is v.dS', 'For the case of the electric field, we define an analogous quantity and call it electric flux', 'We should, however, note that there is no flow of a physically observable quantity unlike the case ofliquid flow', 'In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point', 'This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S', 'Now suppose we tilt the area element by angle θ', 'Clearly, the number of field lines crossing the area element will be smaller', 'The projection of the area element normal to E is ∆S cosθ', 'Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ', 'When θ = 90°, field lines will be parallel to ∆S and will not cross it at all .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.18 Dependence of flux on the inclination θ between E and .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The orientation of area element and not merely its magnitude is important in many contexts', 'For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring', 'If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation', 'This shows that an area element should be treated as a vector', 'It has a magnitude and also a direction', 'How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane', 'Thus the direction of a planar area vector is along its normal', 'How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements', 'Each small area element may be treated as planar and a vector associated with it, as explained before', 'Notice one ambiguity here', 'The direction of an area element is along its normal', 'But a normal can point in two directions', 'Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context', 'For the case of a closed surface, this convention is very simple', 'The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal', 'This is the convention used in 9', 'Thus, the area element vector ∆S at a point on a closed surface equals ∆S where ∆S is the magnitude of the area element and is a unit vector in the direction of outward normal at that point', 'We now come to the definition of electric flux', 'Electric flux ∆φ through an area element ∆S is defined by ∆φ = E.∆S = E ∆S cosθ (1.11) which, as seen before, is proportional to the number of field lines cutting the area element', 'The angle θ here is the angle between E and ∆S', 'For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element', 'Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element', 'The unit of electric flux is N C–1 m2', 'The basic definition of electric flux given by Eq', '(1.11) can be used, in principle, to calculate the total flux through any given surface', 'All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up', 'Thus, the total flux φ through a surface S is φ ~ Σ E.∆S (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element', 'This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq', '(1.12) is written as an integral', '11 Electric Dipole An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a', 'The line connecting the two charges defines a direction in space', 'By convention, the direction from –q to q is said to be the direction of the dipole', 'The mid-point of locations of –q and q is called the centre of the dipole.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.19 Convention for defining normal and ∆S.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The total charge of the electric dipole is obviously zero', 'This does not mean that the field of the electric dipole is zero', 'Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out', 'However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out', 'The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q)', 'These qualitative ideas are borne out by the explicit calculation as follows: 1.11.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle', 'The results are simple for the following two cases: when the point is on the dipole axis, and when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre', 'The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+q due to the charge q, by the parallelogram law of vectors', 'For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in 0(a)', 'Then [1.13(a)] where is the unit vector along the dipole axis (from –q to q)', 'Also [1.13(b)] The total field at P is (1.14) For r >> a (r >> a) (1.15) For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by [1.16(a)] [1.16(b)] and are equal', 'The directions of E+q and E–q are as shown in 0(b)', 'Clearly, the components normal to the dipole axis cancel away', 'The components along the dipole axis add up', 'The total electric field is opposite to', 'We have E = – (E +q + E –q) cosθ (1.17) At large distances (r >> a), this reduces to (1.18) From Eqs', '(1.15) and (1.18), it is clear that the dipole field at large distances does not involve q and a separately; it depends on the product qa', 'This suggests the definition of dipole moment', 'The dipole moment vector p of an electric dipole is defined by p = q × 2a (1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q', 'In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis (r >> a) (1.20)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole', 'p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['At a point on the equatorial plane (r >> a) (1.21) Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3', 'Further, the magnitude and the direction of the dipole field depends not only on the distance r but also on the angle between the position vector r and the dipole moment p', 'We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite', 'Such a dipole is referred to as a point dipole', 'For a point dipole, Eqs', '(1.20) and (1.21) are exact, true for any r', '1.11.2 Physical significance of dipoles In most molecules, the centres of positive charges and of negative charges* lie at the same place', 'Therefore, their dipole moment is zero', 'CO2 and CH4 are of this type of molecules', 'However, they develop a dipole moment when an electric field is applied', 'But in some molecules, the centres of negative charges and of positive charges do not coincide', 'Therefore they have a permanent electric dipole moment, even in the absence of an electric field', 'Such molecules are called polar molecules', 'Water molecules, H2O, is an example of this type', 'Various materials give rise to interesting properties and important applications in the presence or absence of electric field', 'Example 1.10 Two charges ±10 µC are placed 5.0 mm apart', 'Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in 1(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in 1(b)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['fIGURE 1.21 Solution (a) Field at P due to charge +10 µC = = 4.13 × 106 N C–1 along BP Field at P due to charge –10 µC = 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP', 'In this example, the ratio OP/OB is quite large (= 60)', 'Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole', 'For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude (r/a >> 1) where p = 2a q is the magnitude of the dipole moment', 'The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q)', 'Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, E = = 2.6 × 105 N C–1 along the dipole moment direction AB, which is close to the result obtained earlier', '(b) Field at Q due to charge + 10 µC at B = = 3.99 × 106 N C–1 along BQ Field at Q due to charge –10 µC at A = = 3.99 × 106 N C–1 along QA', 'Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA', 'Therefore, the resultant electric field at Q due to the two charges at A and B is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Centre of a collection of positive point charges is defined much the same way as the centre of mass: .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 2 × along BA = 1.33 × 105 N C–1 along BA', 'As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole: (r/a >> 1)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 1.33 × 105 N C–1', 'The direction of electric field in this case is opposite to the direction of the dipole moment vector', 'Again, the result agrees with that obtained before', '1.12 Dipole in a Uniform External Field Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in 2', '(By permanent dipole, we mean that p exists irrespective of E; it has not been induced by E.) There is a force qE on q and a force –qE on –q', 'The net force on the dipole is zero, since E is uniform', 'However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole', 'When the net force is zero, the torque (couple) is independent of the origin', 'Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces)', 'Magnitude of torque = q E × 2 a sinθ = 2 q a E sinθ Its direction is normal to the plane of the paper, coming out of it', 'The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it', 'Thus, τ = p × E (1.22) This torque will tend to align the dipole with the field E', 'When p is aligned with E, the torque is zero', 'What happens if the field is not uniform? In that case, the net force will evidently be non-zero', 'In addition there will, in general, be a torque on the system as before', 'The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E', 'In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform', 'Figure 1.23 is self-explanatory', 'It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field', 'When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field', 'In general, the force depends on the orientation of p with respect to E', 'This brings us to a common observation in frictional electricity', 'A comb run through dry hair attracts pieces of paper', 'The comb, as we know, acquires charge through friction', 'But the paper is not charged', 'What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field', 'Further, the electric field due to the comb is not uniform', 'In this situation, it is easily seen that the paper should move in the direction of the comb! 13 Continuous Charge Distribution We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn', 'One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus', 'For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions', 'For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents', 'It is more feasible to consider an area element ∆S on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element', 'We then define a surface charge density σ at the area element by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.22 Dipole in a uniform electric field.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.23) We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density', 'The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*', 'σ represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically', 'The units for σ are C/m2', 'Similar considerations apply for a line charge distribution and a volume charge distribution', 'The linear charge density λ of a wire is defined by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.24) where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element', 'The units for λ are C/m', 'The volume charge density (sometimes simply called charge density) is defined in a similar manner: (1.25) where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents', 'The units for ρ are C/m3', 'The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics', 'When we refer to the density of a liquid, we are referring to its macroscopic density', 'We regard it as a continuous fluid and ignore its discrete molecular constitution.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.24 Definition of linear, surface and volume charge densities', 'In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq', '(1.10)', 'Suppose a continuous charge distribution in space has a charge density ρ', 'Choose any convenient origin O and let the position vector of any point in the charge distribution be r', 'The charge density ρ may vary from point to point, i.e., it is a function of r', 'Divide the charge distribution into small volume elements of size ∆V', 'The charge in a volume element ∆V is ρ∆V', 'Now, consider any general point P (inside or outside the distribution) with position vector R', 'Electric field due to the charge ρ∆V is given by Coulomb’s law: (1.26) where r′ is the distance between the charge element and P, and ′ is a unit vector in the direction from the charge element to P', 'By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements: (1.27) Note that ρ, r′, all can vary from point to point', 'In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity', 'In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous', '1.14 Gauss’s Law As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre', 'Divide the sphere into small area elements, as shown in 5', 'The flux through an area element ∆S is (1.28) where we have used Coulomb’s law for the electric field due to a single charge q', 'The unit vector is along the radius vector from the centre to the area element', 'Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S and have the same direction', 'Therefore,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* At the microscopic level, charge distribution is discontinuous, because they are discrete charges separated by intervening space where there is no charge.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.29) since the magnitude of a unit vector is 1', 'The total flux through the sphere is obtained by adding up flux through all the different area elements: Since each area element of the sphere is at the same distance r from the charge,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Now S, the total area of the sphere, equals 4πr2', 'Thus, (1.30) Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law', 'We state Gauss’s law without proof: Electric flux through a closed surface S']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= q/ε0 (1.31) q = total charge enclosed by S', 'The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface', 'We can see that explicitly in the simple situation of 6', 'Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E', 'The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface', 'Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0', 'Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E', 'Therefore, φ1 = –E S1, φ2 = +E S2 S1 = S2 = S where S is the area of circular cross-section', 'Thus, the total flux is zero, as expected by Gauss’s law', 'Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero', 'The great significance of Gauss’s law Eq', '(1.31), is that it is true in general, and not only for the simple cases we have considered above', 'Let us note some important points regarding this law: Gauss’s law is true for any closed surface, no matter what its shape or size', 'The term q on the right side of Gauss’s law, Eq', '(1.31), includes the sum of all charges enclosed by the surface', 'The charges may be located anywhere inside the surface', 'In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq', '(1.31)] is due to all the charges, both inside and outside S', 'The term q on the right side of Gauss’s law, however, represents only the total charge inside S']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface', 'You may choose any Gaussian surface and apply Gauss’s law', 'However, take care not to let the Gaussian surface pass through any discrete charge', 'This is because electric field due to a system of discrete charges is not well defined at the location of any charge', '(As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution', '(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry', 'This is facilitated by the choice of a suitable Gaussian surface', '(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law', 'Any violation of Gauss’s law will indicate departure from the inverse square law', 'Example 1.11 The electric field components in 7 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2', 'Calculate (a) the flux through the cube, and (b) the charge within the cube', 'Assume that a = 0.1 m.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.27 Solution (a) Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2', 'Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones', 'Now the magnitude of the electric field at the left face is EL = αx1/2 = αa1/2 (x = a at the left face)', 'The magnitude of electric field at the right face is ER = α x1/2 = α (2a)1/2 (x = 2a at the right face)', 'The corresponding fluxes are φL= EL.∆S = =EL ∆S cosθ = –EL ∆S, since θ = 180° = –ELa2 φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0° = ERa2 Net flux through the cube = φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2] = αa5/2 = 800 (0.1)5/2 = 1.05 N m2 C–1 (b) We can use Gauss’s law to find the total charge q inside the cube', 'We have φ = q/ε0 or q = φε0', 'Therefore, q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C. Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x', 'It is given that E = 200 N/C for x > 0 and E = –200 N/C for x < 0', 'A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm', '(a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder? Solution (a) We can see from the figure that on the left face E and ∆S are parallel', 'Therefore, the outward flux is φL= E.∆S = – 200 = + 200 ∆S, since = – ∆S = + 200 × π (0.05)2 = + 1.57 N m2 C–1 On the right face, E and ∆S are parallel and therefore φR = E.∆S = + 1.57 N m2 C–1', '(b) For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0', 'Therefore, the flux out of the side of the cylinder is zero', '(c) Net outward flux through the cylinder φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.28 (d) The net charge within the cylinder can be found by using Gauss’s law which gives q = ε0φ = 3.14 × 8.854 × 10–12 C = 2.78 × 10–11 C 1.15 Applications of Gauss’s Law The electric field due to a general charge distribution is, as seen above, given by Eq', '(1.27)', 'In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space', 'For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law', 'This is best understood by some examples', '1.15.1 Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ', 'The wire is obviously an axis of symmetry', 'Suppose we take the radial vector from O to P and rotate it around the wire', 'The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire', 'This implies that the electric field must have the same magnitude at these points', 'The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0)', 'This is clear from 9', 'Consider a pair of line elements P1 and P2 of the wire, as shown', 'The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel)', 'This is true for any such pair and hence the total field at any point P is radial', 'Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire', 'In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r', 'To calculate the field, imagine a cylindrical Gaussian surface, as shown in the 9(b)', 'Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero', 'At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r', 'The surface area of the curved part is 2πrl, where l is the length of the cylinder', 'Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E × 2πrl The surface includes charge equal to λ l', 'Gauss’s law then gives E × 2πrl = λl/ε0 i.e., E = Vectorially, E at any point is given by (1.32) where is the radial unit vector in the plane normal to the wire passing through the point', 'E is directed outward if λ is positive and inward if λ is negative.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A , the scalar A is an algebraic number', 'It can be negative or positive', 'The direction of A will be the same as that of the unit vector if A > 0 and opposite to if A < 0', 'When we want to restrict to non-negative values, we use the symbol and call it the modulus of A', 'Thus,', 'Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire', 'Further, the assumption that the wire is infinitely long is crucial', 'Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface', 'However, Eq', '(1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored', '15.2 Field due to a uniformly charged infinite plane sheet Let σ be the uniform surface charge density of an infinite plane sheet', 'We take the x-axis normal to the given plane', 'By symmetry, the electric field will not depend on y and z coordinates and its direction at every point must be parallel to the x-direction', 'We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional area A, as shown', '(A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux', 'The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction', 'Therefore, flux E.∆S through both the surfaces are equal and add up', 'Therefore the net flux through the Gaussian surface is 2 EA', 'The charge enclosed by the closed surface is σA', 'Therefore by Gauss’s law, 2 EA = σA/ε0 or, E = σ/2ε0 Vectorically, (1.33) where is a unit vector normal to the plane and going away from it', 'E is directed away from the plate if σ is positive and toward the plate if σ is negative', 'Note that the above application of the Gauss’ law has brought out an additional fact: E is independent of x also', 'For a finite large planar sheet, Eq', '(1.33) is approximately true in the middle regions of the planar sheet, away from the ends', '15.3 Field due to a uniformly charged thin spherical shell Let σ be the uniform surface charge density of a thin spherical shell of radius R', 'The situation has obvious spherical symmetry', 'The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector)', 'Field outside the shell: Consider a point P outside the shell with radius vector r', 'To calculate E at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P', 'All points on this sphere are equivalent relative to the given charged configuration', '(That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point', 'Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S', 'Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2', 'The charge enclosed is σ × 4 π R2', 'By Gauss’s law E × 4 π r2 = Or, where q = 4 π R2 σ is the total charge on the spherical shell', 'Vectorially, (1.34) The electric field is directed outward if q > 0 and inward if q < 0', 'This, however, is exactly the field produced by a charge q placed at the centre O', 'Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Field inside the shell: In 1(b), the point P is inside the shell', 'The Gaussian surface is again a sphere through P centred at O', 'The flux through the Gaussian surface, calculated as before, is E × 4 π r2', 'However, in this case, the Gaussian surface encloses no charge', 'Gauss’s law then gives E × 4 π r2 = 0 i.e., E = 0 (r < R ) (1.35) that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*', 'This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law', 'The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law', 'Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R', 'The atom as a whole is neutral', 'For this model, what is the electric field at a distance r from the nucleus?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.32 Solution The charge distribution for this model of the atom is as shown in 2', 'The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral', 'This immediately gives us the negative charge density ρ, since we must have or To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law', 'Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r', 'Its direction is along (or opposite to) the radius vector r from the origin to the point P', 'The obvious Gaussian surface is a spherical surface centred at the nucleus', 'We consider two situations, namely, r < R and r > R', 'r < R : The electric flux φ enclosed by the spherical surface is φ = E (r) × 4 π r2 where E (r) is the magnitude of the electric field at r', 'This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface', 'The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r, i.e., Substituting for the charge density ρ obtained earlier, we have Gauss’s law then gives, The electric field is directed radially outward', 'r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral', 'Thus, from Gauss’s law, E (r) × 4 π r2 = 0 or E (r) = 0; r > R At r = R, both cases give the same result: E = 0', 'Summary 1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter', '2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract', 'By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative', 'Conductors allow movement of electric charge through them, insulators do not', 'In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile', '4. Electric charge has three basic properties: quantisation, additivity and conservation', 'Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, ...', 'Proton and electron have charges +e, –e, respectively', 'For macroscopic charges for which n is a very large number, quantisation of charge can be ignored', 'Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system', 'Conservation of electric charges means that the total charge of an isolated system remains unchanged with time', 'This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge', '5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21 separating them', 'Mathematically, F21 = force on q2 due to where is a unit vector in the direction from q1 to q2 and k = is the constant of proportionality', 'In SI units, the unit of charge is coulomb', 'The experimental value of the constant ε0 is ε0 = 8.854 × 10–12 C2 N–1 m–2 The approximate value of k is k = 9 × 109 N m2 C–2 6. The ratio of electric force and gravitational force between a proton and an electron is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Compare this with a uniform mass shell discussed in Section 8.5 of Class XI Textbook of Physics.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s)', 'For an assembly of charges q1, q2, q3, ..., the force on any charge, say q1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on', 'For each pair, the force is given by the Coulomb’s law for two charges stated earlier', '8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge', 'Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative', 'Like Coulomb force, electric field also satisfies superposition principle', '9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point', 'The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak', 'In regions of constant electric field, the field lines are uniformly spaced parallel straight lines']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['On symmetry operations In Physics, we often encounter systems with various symmetries', 'Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation', 'Consider, for example an infinite uniform sheet of charge (surface charge density σ) along the y-z plane', 'This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle', 'As the system is unchanged under such symmetry operation, so must its properties be', 'In particular, in this example, the electric field E must be unchanged', 'Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0)', 'Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same', 'By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction', 'Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate', 'It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space', 'The direction, however, is opposite of each other on either side ofthe sheet', 'Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['10. Some of the important properties of field lines are: Field lines are continuous curves without any breaks', 'Two field lines cannot cross each other', 'Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops', '11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a', 'Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q', '12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Dipole electric field on the axis at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2 dependence of electric field due to a point charge', '13. In a uniform electric field E, a dipole experiences a torque given by = p × E but experiences no net force', '. The flux ∆φ of electric field E through a small area element ∆S is given by ∆φ = E.∆S The vector area element ∆S is ∆S = ∆S where ∆S is the magnitude of the area element and is normal to the area element, which can be considered planar for sufficiently small ∆S', 'For an area element of a closed surface, is taken to be the direction of outward normal, by convention', '15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S', 'The law is especially useful in determining electric field E, when the source distribution has simple symmetry: Thin infinitely long straight wire of uniform linear charge density λ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where r is the perpendicular distance of the point from the wire and is the radial unit vector in the plane normal to the wire passing through the point', 'Infinite thin plane sheet of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where is a unit vector normal to the plane, outward on either side', 'Thin spherical shell of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['E = 0 (r < R) where r is the distance of the point from the centre of the shell and R the radius of the shell', 'q is the total charge of the shell: q = 4πR2σ', 'The electric field outside the shell is as though the total charge is concentrated at the centre', 'The same result is true for a solid sphere of uniform volume charge density', 'The field is zero at all points inside the shell', 'Points to Ponder 1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus', 'Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together', 'The range of distance where this force is effective is, however, very small ~10-14 m', 'This is precisely the size of the nucleus', 'Also the electrons are not allowed to sit on top of the protons, i.e', 'inside the nucleus, due to the laws of quantum mechanics', 'This gives the atoms their structure as they exist in nature', '2. Coulomb force and gravitational force follow the same inverse-square law', 'But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces', 'This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature', '3. The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law', 'In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s)', 'In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2 C–2', '4. The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces', 'Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects', 'The additive property of charge is not an ‘obvious’ property', 'It is related to the fact that electric charge has no direction associated with it; charge is a scalar', '6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion', 'This is not always true for every scalar', 'For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion', '7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6', 'Conservation refers to invariance in time in a given frame of reference', 'A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision)', 'On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system)', '8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass', '9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors', 'It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges', '10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges', 'For continuous volume charge distribution, it is defined at any point in the distribution', 'For a surface charge distribution, electric field is discontinuous across the surface', '11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge', 'An electric dipole is the simplest example of this fact', 'Exercises']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Physical quantity Symbol Dimensions Unit Remarks Vector area element ∆S [L2] m2 ∆S = ∆S Electric field E [MLT–3A–1] V m–1 Electric flux φ [ML3 T–3A–1] V m ∆φ = E.∆S Dipole moment p [LTA] C m Vector directed from negative to positive charge Charge density: linear λ [L–1 TA] C m–1 Charge/length surface σ [L–2 TA] C m–2 Charge/area volume ρ [L–3 TA] C m–3 Charge/volume']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.1 What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air? 1.2 The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N', '(a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? 1.3 Check that the ratio ke2/G memp is dimensionless', 'Look up a Table of Physical Constants and determine the value of this ratio', 'What does the ratio signify? 1.4 (a) Explain the meaning of the statement ‘electric charge of a body is quantised’', '(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? 1.5 When a glass rod is rubbed with a silk cloth, charges appear on both', 'A similar phenomenon is observed with many other pairs of bodies', 'Explain how this observation is consistent with the law of conservation of charge', '6 Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm', 'What is the force on a charge of 1 µC placed at the centre of the square? 1.7 (a) An electrostatic field line is a continuous curve', 'That is, a field line cannot have sudden breaks', 'Why not? (b) Explain why two field lines never cross each other at any point? 8 Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum', '(a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge? 1.9 A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively', 'What are the total charge and electric dipole moment of the system? 1.10 An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1', 'Calculate the magnitude of the torque acting on the dipole', '1.11 A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C', '(a) Estimate the number of electrons transferred (from which to which?) (b) Is there a transfer of mass from wool to polythene? 1.12 (a) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm', 'What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation', '(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? 1.13 Suppose the spheres A and B in Exercise 1.12 have identical sizes', 'A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both', 'What is the new force of repulsion between A and B? 1.14 Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field', 'Give the signs of the three charges', 'Which particle has the highest charge to mass ratio?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.33 1.15 Consider a uniform electric field E = 3 × 103 î N/C', '(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis? 1.16 What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes? 1.17 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C', '(a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? 1.18 A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in 4', 'What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.34 1.19 A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge', 'What is the net electric flux through the surface? 1.20 A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge', '(a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 1.21 A conducting sphere of radius 10 cm has an unknown charge', 'If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere? 1.22 A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2', '(a) Find the charge on the sphere', '(b) What is the total electric flux leaving the surface of the sphere? 1.23 An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm', 'Calculate the linear charge density', '1.24 Two large, thin metal plates are parallel and close to each other', 'On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2', 'What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates? Additional Exercises 1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment)', 'The density of the oil is 1.26 g cm–3', 'Estimate the radius of the drop', '(g = 9.81 m s–2; e = 1.60 × 10–19 C)', '1.26 Which among the curves shown in 5 cannot possibly represent electrostatic field lines? Figure 1.35 1.27 In a certain region of space, electric field is along the z-direction throughout', 'The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre', 'What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ? 1.28 (a) A conductor A with a cavity as shown in 6(a) is given a charge Q', 'Show that the entire charge must appear on the outer surface of the conductor', '(b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A', 'Show that the total charge on the outside surface of A is Q + q', '(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment', 'Suggest a possible way', 'Figure 1.36 1.29 A hollow charged conductor has a tiny hole cut into its surface', 'Show that the electric field in the hole is (σ/2ε0), where is the unit vector in the outward normal direction, and σ is the surface charge density near the hole', '30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law', '[Hint: Use Coulomb’s law directly and evaluate the necessary integral.] 1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks', 'A proton and a neutron consist of three quarks each', 'Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter', '(Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron', '32 (a) Consider an arbitrary electrostatic field configuration', 'A small test charge is placed at a null point (i.e., where E = 0) of the configuration', 'Show that the equilibrium of the test charge is necessarily unstable', '(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart', '33 A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in 3)', 'The length of plate is L and an uniform electric field E is maintained between the plates', 'Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2)', 'Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics', '1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1', 'If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Interactive animation on simple electrostatic experiments: http://demoweb.physics.ucla.edu/content/100-simple-electrostatic-experiments']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Interactive animation on charging a two-sphere system by induction: http://www.physicsclassroom.com/mmedia/estatics/itsn.cfm']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.3']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Charles Augustin de Coulomb (1736 –1806)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.4']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Interactive animation on Coulomb’s law: http://webphysics.davidson.edu/physlet_resources/bu_semester2/menu_semester2.html']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.5']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.5']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.6']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.6']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.7']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.7']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* An alternate unit V/m will be introduced in the next chapter.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.8']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.8']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.9']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.9']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Solid angle is a measure of a cone', 'Consider the intersection of the given cone with a sphere of radius R', 'The solid angle ∆Ω of the cone is defined to be equal to ∆S/R2, where ∆S is the area on the sphere cut out by the cone.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* It will not be proper to say that the number of field lines is equal to E∆S', 'The number of field lines is after all, a matter of how many field lines we choose to draw', 'What is physically significant is the relative number of field lines crossing a given area at different points.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.11']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.11']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.12']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Contents']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Chapter-1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Landmarks']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Cover']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Chapter One']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['ELECTRIC CHARGES AND FIELDS']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.1 Introduction']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['All of us have the experience of seeing a spark or hearing a crackle when we take off our synthetic clothes or sweater, particularly in dry weather', 'This is almost inevitable with ladies garments like a polyester saree', 'Have you ever tried to find any explanation for this phenomenon? Another common example of electric discharge is the lightning that we see in the sky during thunderstorms', 'We also experience a sensation of an electric shock either while opening the door of a car or holding the iron bar of a bus after sliding from our seat', 'The reason for these experiences is discharge of electric charges through our body, which were accumulated due to rubbing of insulating surfaces', 'You might have also heard that this is due to generation of static electricity', 'This is precisely the topic we are going to discuss in this and the next chapter', 'Static means anything that does not move or change with time', 'Electrostatics deals with the study of forces, fields and potentials arising from static charges']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.2 Electric Charge Historically the credit of discovery of the fact that amber rubbed with wool or silk cloth attracts light objects goes to Thales of Miletus, Greece, around 600 BC', 'The name electricity is coined from the Greek word elektron meaning amber', 'Many such pairs of materials were known which on rubbing could attract light objects like straw, pith balls and bits of papers', 'You can perform the following activity at home to experience such an effect', 'Cut out long thin strips of white paper and lightly iron them', 'Take them near a TV screen or computer monitor', 'You will see that the strips get attracted to the screen', 'In fact they remain stuck to the screen for a while']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.1 Rods and pith balls: like charges repel and unlike charges attract each other.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['It was observed that if two glass rods rubbed with wool or silk cloth are brought close to each other, they repel each other', 'The two strands of wool or two pieces of silk cloth, with which the rods were rubbed, also repel each other', 'However, the glass rod and wool attracted each other', 'Similarly, two plastic rods rubbed with cat’s fur repelled each other but attracted the fur', 'On the other hand, the plastic rod attracts the glass rod and repel the silk or wool with which the glass rod is rubbed', 'The glass rod repels the fur', 'If a plastic rod rubbed with fur is made to touch two small pith balls (now-a-days we can use polystyrene balls) suspended by silk or nylon thread, then the balls repel each other and are also repelled by the rod', 'A similar effect is found if the pith balls are touched with a glass rod rubbed with silk', 'A dramatic observation is that a pith ball touched with glass rod attracts another pith ball touched with plastic rod', 'These seemingly simple facts were established from years of efforts and careful experiments and their analyses', 'It was concluded, after many careful studies by different scientists, that there were only two kinds of an entity which is called the electric charge', 'We say that the bodies like glass or plastic rods, silk, fur and pith balls are electrified', 'They acquire an electric charge on rubbing', 'The experiments on pith balls suggested that there are two kinds of electrification and we find that like charges repel and unlike charges attract each other', 'The experiments also demonstrated that the charges are transferred from the rods to the pith balls on contact', 'It is said that the pith balls are electrified or are charged by contact', 'The property which differentiates the two kinds of charges is called the polarity of charge', 'When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge', 'This is true for any pair of objects that are rubbed to be electrified', 'Now if the electrified glass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other', 'They also do not attract or repel other light objects as they did on being electrified', 'Thus, the charges acquired after rubbing are lost when the charged bodies are brought in contact', 'What can you conclude from these observations? It just tells us that unlike charges acquired by the objects neutralise or nullify each other’s effect', 'Therefore, the charges were named as positive and negative by the American scientist Benjamin Franklin', 'We know that when we add a positive number to a negative number of the same magnitude, the sum is zero', 'This might have been the philosophy in naming the charges as positive and negative', 'By convention, the charge on glass rod or cat’s fur is called positive and that on plastic rod or silk is termed negative', 'If an object possesses an electric charge, it is said to be electrified or charged', 'When it has no charge it is said to be electrically neutral', 'A simple apparatus to detect charge on a body is the gold-leaf electroscope', 'It consists of a vertical metal rod housed in a box, with two thin gold leaves attached to its bottom end', 'When a charged object touches the metal knob at the top of the rod, charge flows on to the leaves and they diverge', 'The degree of divergance is an indicator of the amount of charge']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Unification of electricity and magnetism In olden days, electricity and magnetism were treated as separate subjects', 'Electricity dealt with charges on glass rods, cat’s fur, batteries, lightning, etc., while magnetism described interactions of magnets, iron filings, compass needles, etc', 'In 1820 Danish scientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle', 'Ampere and Faraday supported this observation by saying that electric charges in motion produce magnetic fields and moving magnets generate electricity', 'The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Lorentz put forward a theory where they showed the interdependence of these two subjects', 'This field is called electromagnetism', 'Most of the phenomena occurring around us can be described under electromagnetism', 'Virtually every force that we can think of like friction, chemical force between atoms holding the matter together, and even the forces describing processes occurring in cells of living organisms, have its origin in electromagnetic force', 'Electromagnetic force is one of the fundamental forces of nature', 'Maxwell put forth four equations that play the same role in classical electromagnetism as Newton’s equations of motion and gravitation law play in mechanics', 'He also argued that light is electromagnetic in nature and its speed can be found by making purely electric and magnetic measurements', 'He claimed that the science of optics is intimately related to that of electricity and magnetism', 'The science of electricity and magnetism is the foundation for the modern technological civilisation', 'Electric power, telecommunication, radio and television, and a wide variety of the practical appliances used in daily life are based on the principles of this science', 'Although charged particles in motion exert both electric and magnetic forces, in the frame of reference where all the charges are at rest, the forces are purely electrical', 'You know that gravitational force is a long-range force', 'Its effect is felt even when the distance between the interacting particles is very large because the force decreases inversely as the square of the distance between the interacting bodies', 'We will learn in this chapter that electric force is also as pervasive and is in fact stronger than the gravitational force by several orders of magnitude (refer to Chapter 1 of Class XI Physics Textbook).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Students can make a simple electroscope as follows : Take a thin aluminium curtain rod with ball ends fitted for hanging the curtain', 'Cut out a piece of length about 20 cm with the ball at one end and flatten the cut end', 'Take a large bottle that can hold this rod and a cork which will fit in the opening of the bottle', 'Make a hole in the cork sufficient to hold the curtain rod snugly', 'Slide the rod through the hole in the cork with the cut end on the lower side and ball end projecting above the cork', 'Fold a small, thin aluminium foil (about 6 cm in length) in the middle and attach it to the flattened end of the rod by cellulose tape', 'This forms the leaves of your electroscope', 'Fit the cork in the bottle with about 5 cm of the ball end projecting above the cork', 'A paper scale may be put inside the bottle in advance to measure the separation of leaves', 'The separation is a rough measure of the amount of charge on the electroscope.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.2 Electroscopes: (a) The gold leaf electroscope, (b) Schematics of a simple electroscope.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['To understand how the electroscope works, use the white paper strips we used for seeing the attraction of charged bodies', 'Fold the strips into half so that you make a mark of fold', 'Open the strip and iron it lightly with the mountain fold up, as shown in', 'Hold the strip by pinching it at the fold', 'You would notice that the two halves move apart', 'This shows that the strip has acquired charge on ironing', 'When you fold it into half, both the halves have the same charge', 'Hence they repel each other', 'The same effect is seen in the leaf electroscope', 'On charging the curtain rod by touching the ball end with an electrified body, charge is transferred to the curtain rod and the attached aluminium foil', 'Both the halves of the foil get similar charge and therefore repel each other', 'The divergence in the leaves depends on the amount of charge on them', 'Let us first try to understand why material bodies acquire charge', 'You know that all matter is made up of atoms and/or molecules', 'Although normally the materials are electrically neutral, they do contain charges; but their charges are exactly balanced', 'Forces that hold the molecules together, forces that hold atoms together in a solid, the adhesive force of glue, forces associated with surface tension, all are basically electrical in nature, arising from the forces between charged particles', 'Thus the electric force is all pervasive and it encompasses almost each and every field associated with our life', 'It is therefore essential that we learn more about such a force.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.3 Paper strip experiment.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['To electrify a neutral body, we need to add or remove one kind of charge', 'When we say that a body is charged, we always refer to this excess charge or deficit of charge', 'In solids, some of the electrons, being less tightly bound in the atom, are the charges which are transferred from one body to the other', 'A body can thus be charged positively by losing some of its electrons', 'Similarly, a body can be charged negatively by gaining electrons', 'When we rub a glass rod with silk, some of the electrons from the rod are transferred to the silk cloth', 'Thus the rod gets positively charged and the silk gets negatively charged', 'No new charge is created in the process of rubbing', 'Also the number of electrons, that are transferred, is a very small fraction of the total number of electrons in the material body', 'Also only the less tightly bound electrons in a material body can be transferred from it to another by rubbing', 'Therefore, when a body is rubbed with another, the bodies get charged and that is why we have to stick to certain pairs of materials to notice charging on rubbing the bodies']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.3 Conductors and Insulators A metal rod held in hand and rubbed with wool will not show any sign of being charged', 'However, if a metal rod with a wooden or plastic handle is rubbed without touching its metal part, it shows signs of charging', 'Suppose we connect one end of a copper wire to a neutral pith ball and the other end to a negatively charged plastic rod', 'We will find that the pith ball acquires a negative charge', 'If a similar experiment is repeated with a nylon thread or a rubber band, no transfer of charge will take place from the plastic rod to the pith ball', 'Why does the transfer of charge not take place from the rod to the ball? Some substances readily allow passage of electricity through them, others do not', 'Those which allow electricity to pass through them easily are called conductors', 'They have electric charges (electrons) that are comparatively free to move inside the material', 'Metals, human and animal bodies and earth are conductors', 'Most of the non-metals like glass, porcelain, plastic, nylon, wood offer high resistance to the passage of electricity through them', 'They are called insulators', 'Most substances fall into one of the two classes stated above*', 'When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor', 'In contrast, if some charge is put on an insulator, it stays at the same place', 'You will learn why this happens in the next chapter', 'This property of the materials tells you why a nylon or plastic comb gets electrified on combing dry hair or on rubbing, but a metal article like spoon does not', 'The charges on metal leak through our body to the ground as both are conductors of electricity', 'When we bring a charged body in contact with the earth, all the excess charge on the body disappears by causing a momentary current to pass to the ground through the connecting conductor (such as our body)', 'This process of sharing the charges with the earth is called grounding or earthing', 'Earthing provides a safety measure for electrical circuits and appliances', 'A thick metal plate is buried deep into the earth and thick wires are drawn from this plate; these are used in buildings for the purpose of earthing near the mains supply', 'The electric wiring in our houses has three wires: live, neutral and earth', 'The first two carry electric current from the power station and the third is earthed by connecting it to the buried metal plate', 'Metallic bodies of the electric appliances such as electric iron, refrigerator, TV are connected to the earth wire', 'When any fault occurs or live wire touches the metallic body, the charge flows to the earth without damaging the appliance and without causing any injury to the humans; this would have otherwise been unavoidable since the human body is a conductor of electricity.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* There is a third category called semiconductors, which offer resistance to the movement of charges which is intermediate between the conductors and insulators.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.4 Charging by Induction When we touch a pith ball with an electrified plastic rod, some of the negative charges on the rod are transferred to the pith ball and it also gets charged', 'Thus the pith ball is charged by contact', 'It is then repelled by the plastic rod but is attracted by a glass rod which is oppositely charged', 'However, why a electrified rod attracts light objects, is a question we have still left unanswered', 'Let us try to understand what could be happening by performing the following experiment.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.4 Charging by induction.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Bring two metal spheres, A and B, supported on insulating stands, in contact as shown in (a)', 'Bring a positively charged rod near one of the spheres, say A, taking care that it does not touch the sphere', 'The free electrons in the spheres are attracted towards the rod', 'This leaves an excess of positive charge on the rear surface of sphere B', 'Both kinds of charges are bound in the metal spheres and cannot escape', 'They, therefore, reside on the surfaces, as shown in (b)', 'The left surface of sphere A, has an excess of negative charge and the right surface of sphere B, has an excess of positive charge', 'However, not all of the electrons in the spheres have accumulated on the left surface of A', 'As the negative charge starts building up at the left surface of A, other electrons are repelled by these', 'In a short time, equilibrium is reached under the action of force of attraction of the rod and the force of repulsion due to the accumulated charges', '(b) shows the equilibrium situation', 'The process is called induction of charge and happens almost instantly', 'The accumulated charges remain on the surface, as shown, till the glass rod is held near the sphere', 'If the rod is removed, the charges are not acted by any outside force and they redistribute to their original neutral state', 'Separate the spheres by a small distance while the glass rod is still held near sphere A, as shown in (c)', 'The two spheres are found to be oppositely charged and attract each other', '(iv) Remove the rod', 'The charges on spheres rearrange themselves as shown in (d)', 'Now, separate the spheres quite apart', 'The charges on them get uniformly distributed over them, as shown in (e)', 'In this process, the metal spheres will each be equal and oppositely charged', 'This is charging by induction', 'The positively charged glass rod does not lose any of its charge, contrary to the process of charging by contact', 'When electrified rods are brought near light objects, a similar effect takes place', 'The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object', '[This happens even when the light object is not a conductor', 'The mechanism for how this happens is explained later in Sections 1.10 and 2.10.] The centres of the two types of charges are slightly separated', 'We know that opposite charges attract while similar charges repel', 'However, the magnitude of force depends on the distance between the charges and in this case the force of attraction overweighs the force of repulsion', 'As a result the particles like bits of paper or pith balls, being light, are pulled towards the rods']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.1 How can you charge a metal sphere positively without touching it? Solution Figure 1.5(a) shows an uncharged metallic sphere on an insulating metal stand', 'Bring a negatively charged rod close to the metallic sphere, as shown in (b)', 'As the rod is brought close to the sphere, the free electrons in the sphere move away due to repulsion and start piling up at the farther end', 'The near end becomes positively charged due to deficit of electrons', 'This process of charge distribution stops when the net force on the free electrons inside the metal is zero', 'Connect the sphere to the ground by a conducting wire', 'The electrons will flow to the ground while the positive charges at the near end will remain held there due to the attractive force of the negative charges on the rod, as shown in (c)', 'Disconnect the sphere from the ground', 'The positive charge continues to be held at the near end', 'Remove the electrified rod', 'The positive charge will spread uniformly over the sphere as shown in (e)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.5 In this experiment, the metal sphere gets charged by the process of induction and the rod does not lose any of its charge.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Similar steps are involved in charging a metal sphere negatively by induction, by bringing a positively charged rod near it', 'In this case the electrons will flow from the ground to the sphere when the sphere is connected to the ground with a wire', 'Can you explain why?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5 Basic Properties of Electric Charge We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other', 'Here, we shall now describe some other properties of the electric charge', 'If the sizes of charged bodies are very small as compared to the distances between them, we treat them as point charges', 'All the charge content of the body is assumed to be concentrated at one point in space.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5.1 Additivity of charges We have not as yet given a quantitative definition of a charge; we shall follow it up in the next section', 'We shall tentatively assume that this can be done and proceed', 'If a system contains two point charges q1 and q2, the total charge of the system is obtained simply by adding algebraically q1 and q2 , i.e., charges add up like real numbers or they are scalars like the mass of a body', 'If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn', 'Charge has magnitude but no direction, similar to mass', 'However, there is one difference between mass and charge', 'Mass of a body is always positive whereas a charge can be either positive or negative', 'Proper signs have to be used while adding the charges in a system', 'For example, the total charge of a system containing five charges +1, +2, –3, +4 and –5, in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the same unit.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5.2 Charge is conserved We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed', 'A picture of particles of electric charge enables us to understand the idea of conservation of charge', 'When we rub two bodies, what one body gains in charge the other body loses', 'Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found that the total charge of the isolated system is always conserved', 'Conservation of charge has been established experimentally', 'It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed in a process', 'Sometimes nature creates charged particles: a neutron turns into a proton and an electron', 'The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5.3 Quantisation of charge Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted by e', 'Thus charge q on a body is always given by q = ne where n is any integer, positive or negative', 'This basic unit of charge is the charge that an electron or proton carries', 'By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as –e and that on a proton as +e', 'The fact that electric charge is always an integral multiple of e is termed as quantisation of charge', 'There are a large number of situations in physics where certain physical quantities are quantised', 'The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday', 'It was experimentally demonstrated by Millikan in 1912', 'In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C', 'A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter', 'In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2 of Class XI, Physics Textbook , Part I)', 'In this system, the value of the basic unit of charge is e = 1.602192 × 10–19 C Thus, there are about 6 × 1018 electrons in a charge of –1C', 'In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 µC (micro coulomb) = 10–6 C or 1 mC (milli coulomb) = 10–3 C', 'If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples of e', 'Thus, if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (–e) = (n2 – n1) e', 'Since n1 and n2 are integers, their difference is also an integer', 'Thus the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e', 'The step size e is, however, very small because at the macroscopic level, we deal with charges of a few µC', 'At this scale the fact that charge of a body can increase or decrease in units of e is not visible', 'In this respect, the grainy nature of the charge is lost and it appears to be continuous', 'This situation can be compared with the geometrical concepts of points and lines', 'A dotted line viewed from a distance appears continuous to us but is not continuous in reality', 'As many points very close to each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution', 'At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e', 'Since e = 1.6 × 10–19 C, a charge of magnitude, say 1 µC, contains something like 1013 times the electronic charge', 'At this scale, the fact that charge can increase or decrease only in units of e is not very different from saying that charge can take continuous values', 'Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored', 'However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored', 'It is the magnitude of scale involved that is very important.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.2 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 C on the other body? Solution In one second 109 electrons move out of the body', 'Therefore the charge given out in one second is 1.6 × 10–19 × 109 C = 1.6 × 10–10 C', 'The time required to accumulate a charge of 1 C can then be estimated to be 1 C ÷ (1.6 × 10–10 C/s) = 6.25 × 109 s = 6.25 × 109 ÷ (365 × 24 × 3600) years = 198 years', 'Thus to collect a charge of one coulomb, from a body from which 109 electrons move out every second, we will need approximately 200 years', 'One coulomb is, therefore, a very large unit for many practical purposes', 'It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material', 'A cubic piece of copper of side 1 cm contains about 2.5 × 1024 electrons', 'Example 1.3 How much positive and negative charge is there in a cup of water? Solution Let us assume that the mass of one cup of water is 250 g', 'The molecular mass of water is 18g', 'Thus, one mole (= 6.02 × 1023 molecules) of water is 18 g', 'Therefore the number of molecules in one cup of water is (250/18) × 6.02 × 1023.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Each molecule of water contains two hydrogen atoms and one oxygen atom, i.e., 10 electrons and 10 protons', 'Hence the total positive and total negative charge has the same magnitude', 'It is equal to (250/18) × 6.02 × 1023 × 10 × 1.6 × 10–19 C = 1.34 × 107 C.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.6 Coulomb’s Law Coulomb’s law is a quantitative statement about the force between two point charges', 'When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges', 'Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges', 'Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by (1.1) How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres', 'When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges', 'However, the charges on the spheres were unknown, to begin with', 'How then could he discover a relation like Eq', '(1.1)? Coulomb thought of the following simple way: Suppose the charge on a metallic sphere is q', 'If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres', 'By symmetry, the charge on each sphere will be q/2*', 'Repeating this process, we can get charges q/2, q/4, etc', 'Coulomb varied the distance for a fixed pair of charges and measured the force for different separations', 'He then varied the charges in pairs, keeping the distance fixed for each pair', 'Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation, Eq', '(1.1)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* A torsion balance is a sensitive device to measure force', 'It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Coulomb’s law, a simple mathematical statement, was initially experimentally arrived at in the manner described above', 'While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level (r ~ 10–10 m)', 'Coulomb discovered his law without knowing the explicit magnitude of the charge', 'In fact, it is the other way round: Coulomb’s law can now be employed to furnish a definition for a unit of charge', 'In the relation, Eq', '(1.1), k is so far arbitrary', 'We can choose any positive value of k', 'The choice of k determines the size of the unit of charge', 'In SI units, the value of k is about 9 × 109', 'The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4', 'Putting this value of k in Eq', '(1.1), we see that for q1 = q2 = 1 C, r = 1 m F = 9 × 109 N']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Charles Augustin de Coulomb (1736 – 1806) Coulomb, a French physicist, began his career as a military engineer in the West Indies', 'In 1776, he returned to Paris and retired to a small estate to do his scientific research', 'He invented a torsion balance to measure the quantity of a force and used it for determination of forces of electric attraction or repulsion between small charged spheres', 'He thus arrived in 1785 at the inverse square law relation, now known as Coulomb’s law', 'The law had been anticipated by Priestley and also by Cavendish earlier, though Cavendish never published his results', 'Coulomb also found the inverse square law of force between unlike and like magnetic poles.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['That is, 1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N', 'One coulomb is evidently too big a unit to be used', 'In practice, in electrostatics, one uses smaller units like 1 mC or 1 µC', 'The constant k in Eq', '(1.1) is usually put as k = 1/4πε0 for later convenience, so that Coulomb’s law is written as (1.2) ε0 is called the permittivity of free space', 'The value of ε0 in SI units is = 8.854 × 10–12 C2 N–1m–2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Since force is a vector, it is better to write Coulomb’s law in the vector notation', 'Let the position vectors of charges q1 and q2 be r1 and r2 respectively [see Fig.1.6(a)]', 'We denote force on q1 due to q2 by F12 and force on q2 due to q1 by F21', 'The two point charges q1 and q2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21: r21 = r2 – r1 In the same way, the vector leading from 2 to 1 is denoted by r12: r12 = r1 – r2 = – r21 The magnitude of the vectors r21 and r12 is denoted by r21 and r12, respectively (r12 = r21)', 'The direction of a vector is specified by a unit vector along the vector', 'To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors: , Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as (1.3) Some remarks on Eq', '(1.3) are relevant:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.6 (a) Geometry and (b) Forces between charges']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['• Equation (1.3) is valid for any sign of q1 and q2 whether positive or negative', 'If q1 and q2 are of the same sign (either both positive or both negative), F21 is along 21, which denotes repulsion, as it should be for like charges', 'If q1 and q2 are of opposite signs, F21 is along –21(=12), which denotes attraction, as expected for unlike charges', 'Thus, we do not have to write separate equations for the cases of like and unlike charges', 'Equation (1.3) takes care of both cases correctly', '• The force F12 on charge q1 due to charge q2, is obtained from Eq', '(1.3), by simply interchanging 1 and 2, i.e., Thus, Coulomb’s law agrees with the Newton’s third law', '• Coulomb’s law [Eq', '(1.3)] gives the force between two charges q1 and q2 in vacuum', 'If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter', 'We shall consider electrostatics in matter in the next chapter.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.4 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges and masses respectively.(a) Compare the strength of these forces by determining the ratio of their magnitudes for an electron and a proton and for two protons', '(b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are1 Å (= 10-10 m) apart? (mp = 1.67 × 10–27 kg, me = 9.11 × 10–31 kg) Solution (a) The electric force between an electron and a proton at a distance r apart is:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where the negative sign indicates that the force is attractive', 'The corresponding gravitational force (always attractive) is:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where mp and me are the masses of a proton and an electron respectively.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['On similar lines, the ratio of the magnitudes of electric force to the gravitational force between two protons at a distance r apart is: 1.3 × 1036 However, it may be mentioned here that the signs of the two forces are different', 'For two protons, the gravitational force is attractive in nature and the Coulomb force is repulsive', 'The actual values of these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N, whereas, FG ~ 1.9 × 10–34 N', 'The (dimensionless) ratio of the two forces shows that electrical forces are enormously stronger than the gravitational forces', '(b) The electric force F exerted by a proton on an electron is same in magnitude to the force exerted by an electron on a proton; however, the masses of an electron and a proton are different', 'Thus, the magnitude of force is |F| == 8.987 × 109 Nm2/C2 × (1.6 ×10–19C)2 / (10–10m)2 = 2.3 × 10–8 N Using Newton’s second law of motion, F = ma, the acceleration that an electron will undergo is a = 2.3×10–8 N / 9.11 ×10–31 kg = 2.5 × 1022 m/s2 Comparing this with the value of acceleration due to gravity, we can conclude that the effect of gravitational field is negligible on the motion of electron and it undergoes very large accelerations under the action of Coulomb force due to a proton', 'The value for acceleration of the proton is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['2.3 × 10–8 N / 1.67 × 10–27 kg = 1.4 × 1019 m/s2 Example 1.5 A charged metallic sphere A is suspended by a nylon thread', 'Another charged metallic sphere B held by an insulating handle is brought close to A such that the distance between their centres is 10 cm, as shown in (a)', 'The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen)', 'Spheres A and B are touched by uncharged spheres C and D respectively, as shown in (b)', 'C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in (c).What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes', 'Ignore the sizes of A and B in comparison to the separation between their centres.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.7 Solution Let the original charge on sphere A be q and that on B be q′', 'At a distance r between their centres, the magnitude of the electrostatic force on each is given by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['neglecting the sizes of spheres A and B in comparison to r', 'When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2', 'Similarly, after D touches B, the redistributed charge on each is q′/2', 'Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus the electrostatic force on A, due to B, remains unaltered.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.7 Forces between Multiple Charges The mutual electric force between two charges is given by Coulomb’s law', 'How to calculate the force on a charge where there are not one but several charges around? Consider a system of nstationary charges q1, q2, q3, ..., qn in vacuum. What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question', 'Recall that forces of mechanical origin add according to the parallelogram law of addition', 'Is the same true for forces of electrostatic origin?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.8 A system of (a) three charges (b) multiple charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time', 'The individual forces are unaffected due to the presence of other charges', 'This is termed as theprinciple of superposition', 'To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in (a). The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges', 'Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq', '(1.3) even though other charges are present. Thus, F12']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['In the same way, the force on q1 due to q3, denoted by F13, is given by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which again is the Coulomb force on q1 due to q3, even though other charge q2 is present', 'Thus the total force F1 on q1 due to the two charges q2 and q3 is given as (1.4) The above calculation of force can be generalised to a system of charges more than three, as shown in (b)', 'The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn', 'The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n: i.e.,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.5) The vector sum is obtained as usual by the parallelogram law of addition of vectors', 'All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in ?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.9 Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC, AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l. By symmatry AO = BO = CO', 'Thus, Force F1 on Q due to charge q at A =along AO Force F2 on Q due to charge q at B =along BO Force F3 on Q due to charge q at C =along CO The resultant of forces F2 and F3 isalong OA, by the parallelogram law', 'Therefore, the total force on Q == 0, whereis the unit vector along OA.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['It is clear also by symmetry that the three forces will sum to zero', 'Suppose that the resultant force was non-zero but in some direction', 'Consider what would happen if the system was rotated through 60° about O', 'Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in 0', 'What is the force on each charge?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.10 Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in 0', 'By the parallelogram law, the total force F1 on the charge q at A is given by F1 = Fwhereis a unit vector along BC', 'The force of attraction or repulsion for each pair of charges has the same magnitude The total force F2 on charge q at B is thus F2 = F2, where2 is a unit vector along AC', 'Similarly the total force on charge –q at C is F3 =F, whereis the unit vector along the direction bisecting the ∠BCA', 'It is interesting to see that the sum of the forces on the three charges is zero, i.e., F1 + F2 + F3 = 0']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The result is not at all surprising', 'It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law', 'The proof is left to you as an exercise.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.8 Electric Field Let us consider a point charge Q placed in vacuum, at the origin O', 'If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law', 'We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P', 'In order to answer such questions, the early scientists introduced the concept offield', 'According to this, we say that the charge Q produces an electric field everywhere in the surrounding', 'When another charge q is brought at some point P, the field there acts on it and produces a force', 'The electric field produced by the charge Q at a point r is given as (1.6) wherer/r, is a unit vector from the origin to the point r', 'Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r', 'The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position', 'The effect of the charge has been incorporated in the existence of the electric field', 'We obtain the force F exerted by a charge Q on a charge q, as (1.7) Note that the charge q also exerts an equal and opposite force on the charge Q. The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa. If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q. Thus, F(r) = q E(r) (1.8) Equation (1.8) defines the SI unit of electric field as N/C*', 'Some important remarks may be made here: From Eq', '(1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it', 'Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point', 'The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge', 'Note that the source charge Q must remain at its original location', 'However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move', 'A way out of this difficulty is to make q negligibly small', 'The force F is then negligibly small but the ratio F/q is finite and defines the electric field: (1.9)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice', 'When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet', 'Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q', 'This is because F is proportional to q, so the ratio F/q does not depend on q', 'The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q. Thus, the electric field E due to Q is also dependent on the space coordinate r. For different positions of the charge q all over the space, we get different values of electric field E', 'The field exists at every point in three-dimensional space', 'For a positive charge, the electric field will be directed radially outwards from the charge', 'On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards', '(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r', 'Thus at equal distances from the charge Q, the magnitude of its electric field E is same. The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.8.1 Electric field due to a system of charges Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O', 'Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn', 'We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r', 'Electric field E1 at r due to q1 at r1 is given by E1 = whereis a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P', 'In the same manner, electric field E2 at r due to q2 at r2 is E2 =']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['whereis a unit vector in the direction from q2 to P and r2P is the distance between q2 and P', 'Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn', 'By the superposition principle, the electric field E at r due to the system of charges is (as shown in 2)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['E(r) = E1 (r) + E2 (r) + … + En(r) = E(r)(1.10) E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.8.2 Physical significance of electric field You may wonder why the notion of electric field has been introduced here at all', 'After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq', '(1.5)]', 'Why then introduce this intermediate quantity called the electric field? For electrostatics, the concept of electric field is convenient, but not really necessary', 'Electric field is an elegant way of characterising the electrical environment of a system of charges', 'Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system)', 'Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field', 'The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point', 'Electric field is a vector field, since force is a vector quantity.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena', 'Suppose we consider the force between two distant charges q1, q2 in accelerated motion', 'Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light', 'Thus, the effect of any motion of q1 on q2 cannot arise instantaneously', 'There will be some time delay between the effect (force on q2) and the cause (motion of q1)', 'It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful. The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2', 'The notion of field elegantly accounts for the time delay', 'Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs', 'They have an independent dynamics of their own, i.e., they evolve according to laws of their own', 'They can also transport energy', 'Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy', 'The concept of field was first introduced by Faraday and is now among the central concepts in physics.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1', 'The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance', 'Compute the time of fall in each case', 'Contrast the situation with that of ‘free fall under gravity’.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Solution In 3(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field', 'The acceleration of the electron is ae = eE/me where me is the mass of the electron', 'Starting from rest, the time required by the electron to fall through a distance h is given by For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m, te = 2.9 × 10–9s In 3 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE', 'The acceleration of the proton is ap = eE/mp where mp is the mass of the proton; mp = 1.67 × 10–27 kg', 'The time of fall for the proton is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus, the heavier particle (proton) takes a greater time to fall through the same distance', 'This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body', 'Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall', 'To see if this is justified, let us calculate the acceleration of the proton in the given electric field:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity', 'The acceleration of the electron is even greater', 'Thus, the effect of acceleration due to gravity can be ignored in this example', 'Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart', 'Calculate the electric fields at points A, B and C shown in 4.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.14 Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude', 'Hence the magnitude of the total electric field EA at A is EA = E1A + E2A = 7.2 × 104 N C–1 EA is directed toward the right', 'The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude = 4 × 103 N C–1 The magnitude of the total electric field at B is EB = E1B – E2B = 3.2 × 104 N C–1 EB is directed towards the left', 'The magnitude of each electric field vector at point C, due to charge q1 and q2 is = 9 × 103 N C–1 The directions in which these two vectors point are indicated in 4', 'The resultant of these two vectors is = 9 × 103 N C–1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['EC points towards the right.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.9 Electric Field Lines We have studied electric field in the last section', 'It is a vector quantity and can be represented as we represent vectors', 'Let us try to represent E due to a point charge pictorially', 'Let the point charge be placed at the origin', 'Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point', 'Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward', 'Figure 1.15 shows such a picture', 'In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow', 'Connect the arrows pointing in one direction and the resulting figure represents a field line', 'We thus get many field lines, all pointing outwards from the point charge', 'Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No', 'Now the magnitude of the field is represented by the density of field lines. E is strong near the charge, so the density of field lines is more near the charge and the lines are closer', 'Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines', 'Another person may draw more lines', 'But the number of lines is not important', 'In fact, an infinite number of lines can be drawn in any region', 'It is the relative density of lines in different regions which is important', 'We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions', 'So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines', 'Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge', 'We started by saying that the field lines carry information about the direction of electric field at different points in space', 'Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points', 'The field lines crowd where the field is strong and are spaced apart where it is weak', 'Figure 1.16 shows a set of field lines', 'We can imagine two equal and small elements of area placed at points R and S normal to the field lines there', 'The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points', 'The picture shows that the field at R is stronger than at S.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.15 Field of a point charge', 'To understand the dependence of the field lines on the area, or rather the solid anglesubtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions', 'Recall how a (plane) angle is defined in two-dimensions', 'Let a small transverse line element ∆l be placed at a distance r from a point O', 'Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r', 'Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2', 'We know that in a given solid angle the number of radial field lines is the same', 'In 6, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is∆Ω at P1 and an element of area']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['∆Ω at P2, respectively', 'The number of lines (say n) cutting these area elements are the same', 'The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively', 'Since n and ∆Ω are common, the strength of the field clearly has a 1/r2dependence', 'The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations', 'Faraday called them lines of force', 'This term is somewhat misleading, especially in case of magnetic fields', 'The more appropriate term is field lines (electric or magnetic) that we have adopted in this book', 'Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges', 'An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point', 'An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve', 'A field line is a space curve, i.e., a curve in three dimensions.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 Field lines due to some simple charge configurations.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 shows the field lines around some simple charge configurations', 'As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane', 'The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward', 'The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges', 'The field lines follow some important general properties: Field lines start from positive charges and end at negative charges', 'If there is a single charge, they may start or end at infinity', 'In a charge-free region, electric field lines can be taken to be continuous curves without any breaks', 'Two field lines can never cross each other', '(If they did, the field at the point of intersection will not have a unique direction, which is absurd.) (iv) Electrostatic field lines do not form any closed loops', 'This follows from the conservative nature of electric field (Chapter 2).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.10 Electric Flux Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface', 'The rate of flow of liquid is given by the volume crossing the area per unit time v dSand represents the flux of liquid flowing across the plane', 'If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ', 'Therefore, the flux going out of the surface dSis v.dS. For the case of the electric field, we define an analogous quantity and call it electric flux', 'We should, however, note that there is no flow of a physically observable quantity unlike the case of liquid flow', 'In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point', 'This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S', 'Now suppose we tilt the area element by angle θ', 'Clearly, the number of field lines crossing the area element will be smaller', 'The projection of the area element normal to E is ∆S cosθ', 'Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ', 'When θ = 90°, field lines will be parallel to ∆Sand will not cross it at all .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.18 Dependence of flux on the inclination θ between E and n']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The orientation of area element and not merely its magnitude is important in many contexts', 'For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring', 'If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation', 'This shows that an area element should be treated as a vector', 'It has a magnitude and also a direction', 'How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane', 'Thus the direction of a planar area vector is along its normal', 'How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements', 'Each small area element may be treated as planar and a vector associated with it, as explained before', 'Notice one ambiguity here', 'The direction of an area element is along its normal', 'But a normal can point in two directions', 'Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context', 'For the case of a closed surface, this convention is very simple', 'The vector associated with every area element of a closed surface is taken to be in the direction of theoutward normal', 'This is the convention used in 9', 'Thus, the area element vector ∆S at a point on a closed surface equals ∆Sn where ∆S is the magnitude of the area element and n is a unit vector in the direction of outward normal at that point', 'We now come to the definition of electric flux', 'Electric flux ∆φ through an area element ∆S is defined by ∆φ = E.∆S = E ∆S cosθ (1.11) which, as seen before, is proportional to the number of field lines cutting the area element', 'The angle θ here is the angle between E and ∆S', 'For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element', 'Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element', 'The unit of electric flux is N C–1 m2', 'The basic definition of electric flux given by Eq', '(1.11) can be used, in principle, to calculate the total flux through any given surface', 'All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up', 'Thus, the total flux φ through a surface S is φ ~ Σ E.∆S (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element', 'This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq', '(1.12) is written as an integral.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.19 Convention for defining normal']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.11 Electric Dipole An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a', 'The line connecting the two charges defines a direction in space', 'By convention, the direction from –q to q is said to be the direction of the dipole', 'The mid-point of locations of –q and q is called the centre of the dipole', 'The total charge of the electric dipole is obviously zero', 'This does not mean that the field of the electric dipole is zero', 'Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out', 'However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out', 'The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q)', 'These qualitative ideas are borne out by the explicit calculation as follows:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.11.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle', 'The results are simple for the following two cases: when the point is on the dipole axis, and when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre', 'The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+qdue to the charge q, by the parallelogram law of vectors', 'For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in 0(a)', 'Then [1.13(a)] whereis the unit vector along the dipole axis (from –q to q)', 'Also [1.13(b)] The total field at P is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.14) For r >> a (r >> a) (1.15)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole. p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q', 'For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by [1.16(a)] [1.16(b)] and are equal', 'The directions of E+q and E–q are as shown in 0(b)', 'Clearly, the components normal to the dipole axis cancel away', 'The components along the dipole axis add up', 'The total electric field is opposite to', 'We have E = – (E +q + E –q) cosθ (1.17) At large distances (r >> a), this reduces to (1.18) From Eqs', '(1.15) and (1.18), it is clear that the dipole field at large distances does not involveq and a separately; it depends on the product qa', 'This suggests the definition of dipole moment', 'The dipole moment vector p of an electric dipole is defined by p = q × 2a(1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q', 'In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis (r >> a) (1.20)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['At a point on the equatorial plane (r >> a) (1.21) Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3', 'Further, the magnitude and the direction of the dipole field depends not only on the distance rbut also on the angle between the position vector r and the dipole moment p', 'We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite', 'Such a dipole is referred to as a point dipole', 'For a point dipole, Eqs', '(1.20) and (1.21) are exact, true for any r.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.11.2 Physical significance of dipoles']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['In most molecules, the centres of positive charges and of negative charges* lie at the same place', 'Therefore, their dipole moment is zero', 'CO2 and CH4 are of this type of molecules', 'However, they develop a dipole moment when an electric field is applied', 'But in some molecules, the centres of negative charges and of positive charges do not coincide', 'Therefore they have a permanent electric dipole moment, even in the absence of an electric field', 'Such molecules are called polar molecules', 'Water molecules, H2O, is an example of this type', 'Various materials give rise to interesting properties and important applications in the presence or absence of electric field.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10 Two charges ±10 µC are placed 5.0 mm apart', 'Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in 1(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in 1(b).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['figure 1.21 Solution (a) Field at P due to charge +10 µC = = 4.13 × 106 N C–1 along BP Field at P due to charge –10 µC']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP', 'In this example, the ratio OP/OB is quite large (= 60)', 'Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole', 'For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude (r/a >> 1) where p = 2a q is the magnitude of the dipole moment', 'The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q)', 'Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, E == 2.6 × 105 N C–1 along the dipole moment direction AB, which is close to the result obtained earlier', '(b) Field at Q due to charge + 10 µC at B = = 3.99 × 106 N C–1 along BQ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Field at Q due to charge –10 µC at A = = 3.99 × 106 N C–1 along QA.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA', 'Therefore, the resultant electric field at Q due to the two charges at A and B is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 2 ×along BA = 1.33 × 105 N C–1 along BA', 'As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole: (r/a >> 1)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 1.33 × 105 N C–1.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The direction of electric field in this case is opposite to the direction of the dipole moment vector', 'Again, the result agrees with that obtained before.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.12 Dipole in a Uniform External Field Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in 2', '(By permanent dipole, we mean that p exists irrespective of E; it has not been induced byE.) There is a force qE on q and a force –qE on –q', 'The net force on the dipole is zero, since E is uniform', 'However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole', 'When the net force is zero, the torque (couple) is independent of the origin', 'Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.22 Dipole in a uniform electric field', 'Magnitude of torque = q E × 2 a sinθ = 2 q a E sinθ Its direction is normal to the plane of the paper, coming out of it', 'The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it', 'Thus, τ = p × E (1.22) This torque will tend to align the dipole with the field E', 'When p is aligned with E, the torque is zero', 'What happens if the field is not uniform? In that case, the net force will evidently be non-zero', 'In addition there will, in general, be a torque on the system as before', 'The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E', 'In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform', 'Figure 1.23 is self-explanatory', 'It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field', 'When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field', 'In general, the force depends on the orientation of p with respect to E.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['This brings us to a common observation in frictional electricity', 'A comb run through dry hair attracts pieces of paper', 'The comb, as we know, acquires charge through friction', 'But the paper is not charged', 'What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field', 'Further, the electric field due to the comb is not uniform', 'In this situation, it is easily seen that the paper should move in the direction of the comb!']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.13 Continuous Charge Distribution We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn', 'One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus', 'For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions', 'For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents', 'It is more feasible to consider an area element ∆S on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element', 'We then define a surface charge density σ at the area element by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.23) We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density', 'The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*. σrepresents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically', 'The units for σ are C/m2', 'Similar considerations apply for a line charge distribution and a volume charge distribution', 'The linear charge density λ of a wire is defined by Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.24) where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element', 'The units for λ are C/m', 'The volume charge density (sometimes simply called charge density) is defined in a similar manner: (1.25) where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents', 'The units for ρ are C/m3', 'The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics', 'When we refer to the density of a liquid, we are referring to its macroscopic density', 'We regard it as a continuous fluid and ignore its discrete molecular constitution.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.24 Definition of linear, surface and volume charge densities. In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq', '(1.10)', 'Suppose a continuous charge distribution in space has a charge density ρ', 'Choose any convenient origin O and let the position vector of any point in the charge distribution be r', 'The charge density ρ may vary from point to point, i.e., it is a function of r', 'Divide the charge distribution into small volume elements of size ∆V', 'The charge in a volume element ∆V is ρ∆V', 'Now, consider any general point P (inside or outside the distribution) with position vector R', 'Electric field due to the charge ρ∆V is given by Coulomb’s law: (1.26) where r′ is the distance between the charge element and P, and′ is a unit vector in the direction from the charge element to P', 'By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements: (1.27) Note that ρ, r′,all can vary from point to point', 'In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity', 'In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.14 Gauss’s Law As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre', 'Divide the sphere into small area elements, as shown in 5.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The flux through an area element ∆S is (1.28) where we have used Coulomb’s law for the electric field due to a single charge q', 'The unit vectoris along the radius vector from the centre to the area element', 'Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S andhave the same direction', 'Therefore, (1.29) since the magnitude of a unit vector is 1', 'The total flux through the sphere is obtained by adding up flux through all the different area elements:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Since each area element of the sphere is at the same distance r from the charge,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Now S, the total area of the sphere, equals 4πr2', 'Thus, (1.30)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law', 'We state Gauss’s law without proof: Electric flux through a closed surface S']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= q/ε0 (1.31) q = total charge enclosed by S', 'The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface', 'We can see that explicitly in the simple situation of 6', 'Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E', 'The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface', 'Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0', 'Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E', 'Therefore, φ1 = –E S1, φ2 = +E S2 S1 = S2 = S where S is the area of circular cross-section', 'Thus, the total flux is zero, as expected by Gauss’s law', 'Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero', 'The great significance of Gauss’s law Eq', '(1.31), is that it is true in general, and not only for the simple cases we have considered above', 'Let us note some important points regarding this law: Gauss’s law is true for any closed surface, no matter what its shape or size', 'The term q on the right side of Gauss’s law, Eq', '(1.31), includes the sum of all charges enclosed by the surface', 'The charges may be located anywhere inside the surface', 'In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq', '(1.31)] is due to all the charges, both inside and outside S', 'The term q on the right side of Gauss’s law, however, represents only the total charge inside S', '(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface', 'You may choose any Gaussian surface and apply Gauss’s law', 'However, take care not to let the Gaussian surface pass through any discrete charge', 'This is because electric field due to a system of discrete charges is not well defined at the location of any charge', '(As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution', '(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface', '(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law', 'Any violation of Gauss’s law will indicate departure from the inverse square law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.11 The electric field components in 7 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2', 'Calculate (a) the flux through the cube, and (b) the charge within the cube', 'Assume that a = 0.1 m.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.27 Solution (a) Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2', 'Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones', 'Now the magnitude of the electric field at the left face is EL = αx1/2 = αa1/2 (x = a at the left face)', 'The magnitude of electric field at the right face is ER = α x1/2 = α (2a)1/2 (x = 2a at the right face)', 'The corresponding fluxes are φL= EL.∆S ==EL ∆S cosθ = –EL ∆S, since θ = 180° = –ELa2 φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0° = ERa2 Net flux through the cube = φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2] = αa5/2 = 800 (0.1)5/2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 1.05 N m2 C–1 (b) We can use Gauss’s law to find the total charge q inside the cube', 'We have φ = q/ε0 or q =φε0', 'Therefore,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C', 'Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x', 'It is given that E= 200N/C for x > 0 and E = –200N/C for x < 0', 'A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm', '(a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder? Solution (a) We can see from the figure that on the left face E and ∆S are parallel', 'Therefore, the outward flux is φL= E.∆S = – 200 = + 200 ∆S, since= – ∆S = + 200 × π (0.05)2 = + 1.57 N m2 C–1 On the right face, E and ∆S are parallel and therefore φR = E.∆S = + 1.57 N m2 C–1', '(b) For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0', 'Therefore, the flux out of the side of the cylinder is zero', '(c) Net outward flux through the cylinder φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.28 (d) The net charge within the cylinder can be found by using Gauss’s law which gives q = ε0φ = 3.14 × 8.854 × 10–12 C']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 2.78 × 10–11 C']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15 Applications of Gauss’s Law The electric field due to a general charge distribution is, as seen above, given by Eq', '(1.27)', 'In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space', 'For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law', 'This is best understood by some examples.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15.1 Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ', 'The wire is obviously an axis of symmetry', 'Suppose we take the radial vector from O to P and rotate it around the wire', 'The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire', 'This implies that the electric field must have the same magnitude at these points', 'The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0)', 'This is clear from 9.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density', 'Consider a pair of line elements P1 and P2 of the wire, as shown', 'The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel)', 'This is true for any such pair and hence the total field at any point P is radial', 'Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire', 'In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r', 'To calculate the field, imagine a cylindrical Gaussian surface, as shown in the 9(b).Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero', 'At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r', 'The surface area of the curved part is 2πrl, where l is the length of the cylinder', 'Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E × 2πrl The surface includes charge equal to λ l', 'Gauss’s law then gives E × 2πrl = λl/ε0 i.e., E = Vectorially, E at any point is given by (1.32) whereis the radial unit vector in the plane normal to the wire passing through the point. E is directed outward if λ is positive and inward if λ is negative.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A, the scalar A is an algebraic number', 'It can be negative or positive', 'The direction of A will be the same as that of the unit vectorif A > 0 and opposite toif A < 0', 'When we want to restrict to non-negative values, we use the symboland call it the modulus of A', 'Thus,', 'Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire', 'Further, the assumption that the wire is infinitely long is crucial', 'Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface', 'However, Eq', '(1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15.2 Field due to a uniformly charged infinite plane sheet Let σ be the uniform surface charge density of an infinite plane sheet', 'We take thex-axis normal to the given plane', 'By symmetry, the electric field will not depend on y and zcoordinates and its direction at every point must be parallel to the x-direction', 'We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional areaA, as shown', '(A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction', 'Therefore, flux E.∆S through both the surfaces are equal and add up', 'Therefore the net flux through the Gaussian surface is 2 EA', 'The charge enclosed by the closed surface is σA', 'Therefore by Gauss’s law, 2 EA = σA/ε0 or, E = σ/2ε0 Vectorically, (1.33) where n is a unit vector normal to the plane and going away from it', 'E is directed away from the plate if σ is positive and toward the plate if σ is negative', 'Note that the above application of the Gauss’ law has brought out an additional fact: E is independent ofx also', 'For a finite large planar sheet, Eq', '(1.33) is approximately true in the middle regions of the planar sheet, away from the ends.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15.3 Field due to a uniformly charged thin spherical shell Let σ be the uniform surface charge density of a thin spherical shell of radius R', 'The situation has obvious spherical symmetry', 'The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector)', 'Field outside the shell: Consider a point P outside the shell with radius vector r', 'To calculateE at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P', 'All points on this sphere are equivalent relative to the given charged configuration', '(That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point', 'Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S. Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2', 'The charge enclosed is σ × 4 π R2', 'By Gauss’s law E × 4 π r2 =']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Or, where q = 4 π R2 σ is the total charge on the spherical shell', 'Vectorially, (1.34)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R', 'The electric field is directed outward if q > 0 and inward if q < 0. This, however, is exactly the field produced by a charge q placed at the centre O', 'Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Field inside the shell: In 1(b), the point P is inside the shell', 'The Gaussian surface is again a sphere through P centred at O. The flux through the Gaussian surface, calculated as before, is E × 4 π r2', 'However, in this case, the Gaussian surface encloses no charge', 'Gauss’s law then gives E × 4 π r2 = 0 i.e., E = 0 (r < R ) (1.35) that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*', 'This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law', 'The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R', 'The atom as a whole is neutral', 'For this model, what is the electric field at a distance r from the nucleus?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.32 Solution The charge distribution for this model of the atom is as shown in 2', 'The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral', 'This immediately gives us the negative charge density ρ, since we must have or To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law', 'Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r', 'Its direction is along (or opposite to) the radius vector r from the origin to the point P', 'The obvious Gaussian surface is a spherical surface centred at the nucleus', 'We consider two situations, namely, r < R and r > R', 'r < R : The electric flux φ enclosed by the spherical surface is φ = E (r) × 4 π r2 where E (r) is the magnitude of the electric field at r', 'This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface', 'The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r, i.e., Substituting for the charge density ρ obtained earlier, we have Gauss’s law then gives,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The electric field is directed radially outward', 'r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral', 'Thus, from Gauss’s law, E (r) × 4 π r2 = 0 or E (r) = 0; r > R']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['At r = R, both cases give the same result: E = 0.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['On symmetry operations In Physics, we often encounter systems with various symmetries', 'Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation', 'Consider, for example an infinite uniform sheet of charge (surface charge densityσ) along the y-z plane', 'This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle', 'As the system is unchanged under such symmetry operation, so must its properties be', 'In particular, in this example, the electric field E must be unchanged', 'Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0)', 'Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same', 'By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction', 'Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate', 'It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space', 'The direction, however, is opposite of each other on either side ofthe sheet', 'Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Summary 1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter', '2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract', 'By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative', '3. Conductors allow movement of electric charge through them, insulators do not', 'In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile', '4. Electric charge has three basic properties: quantisation, additivity and conservation', 'Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, ...', 'Proton and electron have charges +e, –e, respectively', 'For macroscopic charges for which n is a very large number, quantisation of charge can be ignored', 'Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system', 'Conservation of electric charges means that the total charge of an isolated system remains unchanged with time', 'This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge', '5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21separating them', 'Mathematically, F21 = force on q2 due to whereis a unit vector in the direction from q1 to q2 and k =is the constant of proportionality', 'In SI units, the unit of charge is coulomb', 'The experimental value of the constant ε0 is ε0 = 8.854 × 10–12 C2 N–1 m–2 The approximate value of k is k = 9 × 109 N m2 C–2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['6. The ratio of electric force and gravitational force between a proton and an electron is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s)', 'For an assembly of charges q1, q2, q3, ..., the force on any charge, sayq1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on', 'For each pair, the force is given by the Coulomb’s law for two charges stated earlier', '8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge', 'Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative', 'Like Coulomb force, electric field also satisfies superposition principle', '9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point', 'The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak', 'In regions of constant electric field, the field lines are uniformly spaced parallel straight lines.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['10. Some of the important properties of field lines are: Field lines are continuous curves without any breaks', 'Two field lines cannot cross each other', 'Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops', '11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a', 'Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q', '12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Dipole electric field on the axis at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2dependence of electric field due to a point charge', '13. In a uniform electric field E, a dipole experiences a torquegiven by = p × E but experiences no net force', '14. The flux ∆φ of electric field E through a small area element ∆S is given by ∆φ = E.∆S The vector area element ∆S is ∆S = ∆S where ∆S is the magnitude of the area element andis normal to the area element, which can be considered planar for sufficiently small ∆S', 'For an area element of a closed surface,is taken to be the direction of outward normal, by convention', '15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S', 'The law is especially useful in determining electric field E, when the source distribution has simple symmetry: Thin infinitely long straight wire of uniform linear charge density λ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where r is the perpendicular distance of the point from the wire andis the radial unit vector in the plane normal to the wire passing through the point', 'Infinite thin plane sheet of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['whereis a unit vector normal to the plane, outward on either side', 'Thin spherical shell of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['E = 0 (r < R) where r is the distance of the point from the centre of the shell and R the radius of the shell. q is the total charge of the shell: q = 4πR2σ', 'The electric field outside the shell is as though the total charge is concentrated at the centre', 'The same result is true for a solid sphere of uniform volume charge density', 'The field is zero at all points inside the shell.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Points to Ponder 1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus', 'Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together', 'The range of distance where this force is effective is, however, very small ~10-14 m', 'This is precisely the size of the nucleus', 'Also the electrons are not allowed to sit on top of the protons, i.e', 'inside the nucleus, due to the laws of quantum mechanics', 'This gives the atoms their structure as they exist in nature', '2. Coulomb force and gravitational force follow the same inverse-square law', 'But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces', 'This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature', '3. The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law', 'In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s)', 'In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2C–2', '4. The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces', 'Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects', '5. The additive property of charge is not an ‘obvious’ property', 'It is related to the fact that electric charge has no direction associated with it; charge is a scalar', '6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion', 'This is not always true for every scalar', 'For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion', '7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6', 'Conservation refers to invariance in time in a given frame of reference', 'A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision)', 'On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system)', '8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass', '9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors', 'It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges', '10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges', 'For continuous volume charge distribution, it is defined at any point in the distribution', 'For a surface charge distribution, electric field is discontinuous across the surface.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge', 'An electric dipole is the simplest example of this fact.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Exercises 1.1 What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.2 The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N', '(a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? 1.3 Check that the ratio ke2/G memp is dimensionless', 'Look up a Table of Physical Constants and determine the value of this ratio', 'What does the ratio signify? 1.4 (a) Explain the meaning of the statement ‘electric charge of a body is quantised’', '(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? 1.5 When a glass rod is rubbed with a silk cloth, charges appear on both', 'A similar phenomenon is observed with many other pairs of bodies', 'Explain how this observation is consistent with the law of conservation of charge', '1.6 Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm', 'What is the force on a charge of 1 µC placed at the centre of the square? 1.7 (a) An electrostatic field line is a continuous curve', 'That is, a field line cannot have sudden breaks', 'Why not? (b) Explain why two field lines never cross each other at any point? 1.8 Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum', '(a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge? 1.9 A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively', 'What are the total charge and electric dipole moment of the system? 1.10 An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1', 'Calculate the magnitude of the torque acting on the dipole', '1.11 A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C', '(a) Estimate the number of electrons transferred (from which to which?) (b) Is there a transfer of mass from wool to polythene? 1.12 (a) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm', 'What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation', '(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? 1.13 Suppose the spheres A and B in Exercise 1.12 have identical sizes', 'A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both', 'What is the new force of repulsion between A and B? 1.14 Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field', 'Give the signs of the three charges', 'Which particle has the highest charge to mass ratio?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.33 1.15 Consider a uniform electric field E = 3 × 103 î N/C', '(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis? 1.16 What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes? 1.17 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C', '(a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? 1.18 A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in 4', 'What is the magnitude of the electric flux through the square? (Hint:Think of the square as one face of a cube with edge 10 cm.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.34 1.19 A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge', 'What is the net electric flux through the surface? 1.20 A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge', '(a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 1.21 A conducting sphere of radius 10 cm has an unknown charge', 'If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere? 1.22 A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2', '(a) Find the charge on the sphere', '(b) What is the total electric flux leaving the surface of the sphere? 1.23 An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm', 'Calculate the linear charge density.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.24 Two large, thin metal plates are parallel and close to each other', 'On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2', 'What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Additional Exercises 1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment)', 'The density of the oil is 1.26 g cm–3', 'Estimate the radius of the drop. (g = 9.81 m s–2; e = 1.60 × 10–19 C)', '1.26 Which among the curves shown in 5 cannot possibly represent electrostatic field lines?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.35 1.27 In a certain region of space, electric field is along the z-direction throughout', 'The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre', 'What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ? 1.28 (a) A conductor A with a cavity as shown in 6(a) is given a charge Q', 'Show that the entire charge must appear on the outer surface of the conductor', '(b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A', 'Show that the total charge on the outside surface of A is Q + q', '(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment', 'Suggest a possible way.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.36 1.29 A hollow charged conductor has a tiny hole cut into its surface', 'Show that the electric field in the hole is (σ/2ε0), whereis the unit vector in the outward normal direction, and σ is the surface charge density near the hole', '1.30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law', '[Hint: Use Coulomb’s law directly and evaluate the necessary integral.] 1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks', 'A proton and a neutron consist of three quarks each', 'Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter', '(Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron', '1.32 (a) Consider an arbitrary electrostatic field configuration', 'A small test charge is placed at a null point (i.e., where E = 0) of the configuration', 'Show that the equilibrium of the test charge is necessarily unstable', '(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart', '1.33 A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in 3)', 'The length of plate is Land an uniform electric field E is maintained between the plates', 'Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2)', 'Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1', 'If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Chapter 1 ELECTRIC CHARGES AND FIELDS']}]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(textDict)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "all_sentences = []\n",
+    "\n",
+    "for i in textDict:\n",
+    "    for j in i['text']:\n",
+    "        all_sentences.append(j)\n",
+    "with open('sentences-test.txt', 'a') as f:\n",
+    "    f.write('\\n'.join(all_sentences))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "for j in textDict:\n",
+    "    inputTextArray = j['text']\n",
+    "    inputTextTitle = j['title']\n",
+    "#     print(j)\n",
+    "    nFeatures = 10\n",
+    "\n",
+    "    data = [ [] for i in range(len(inputTextArray))]\n",
+    "\n",
+    "#     similarityFeatures = ahish.getCosSims(inputTextArray)\n",
+    "    similarityFeatures = [-1 for i in range(len(inputTextArray))]\n",
+    "#     simToSum = SimToSumm.getSimtoSum(inputTextArray, ' '.join(inputTextArray))\n",
+    "    simToTitle = SimToTitle.SimToTitle(inputTextTitle, inputTextArray)\n",
+    "\n",
+    "    maxLen = 0\n",
+    "\n",
+    "    for i in inputTextArray:\n",
+    "        curr = len(i.split())\n",
+    "        if curr > maxLen:\n",
+    "            maxLen = curr\n",
+    "\n",
+    "    for i in range(len(inputTextArray)):\n",
+    "        data[i] += [len(inputTextArray[i].split())/maxLen, i/len(inputTextArray)]\n",
+    "#         data[i] += [similarityFeatures[i]]\n",
+    "#         data[i] += [simToSum[i]]\n",
+    "        data[i] += [simToTitle[i]]\n",
+    "        curr = inputTextArray[i]\n",
+    "        data[i]+= [pranav.keyword_present(curr) , pranav.has_number(curr), pranav.number_of_superlatives(curr), pranav.abbrevs_per_length(curr), pranav.nouns_per_length(curr), pranav.pronouns_per_length(curr), discourse_connector.discourse_connector(curr)]\n",
+    "                \n",
+    "    with open(\"features-test.csv\", \"a\", newline=\"\") as f:\n",
+    "        writer = csv.writer(f)\n",
+    "        writer.writerows(data)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
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diff --git a/.ipynb_checkpoints/compiled-feature-generator-checkpoint.ipynb b/.ipynb_checkpoints/compiled-feature-generator-checkpoint.ipynb
index ce5e54a..b634bca 100644
--- a/.ipynb_checkpoints/compiled-feature-generator-checkpoint.ipynb
+++ b/.ipynb_checkpoints/compiled-feature-generator-checkpoint.ipynb
@@ -52,7 +52,7 @@
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-      "[{'title': 'Chapter Eight', 'text': ['Chapter Eight']}, {'title': 'Chapter Eight', 'text': ['Gravitation']}, {'title': 'Chapter Eight', 'text': ['8.1 Introduction']}, {'title': 'Chapter Eight', 'text': ['8.1 Introduction 8.2 Kepler’s laws 8.3 Universal law of gravitation 8.4 The gravitational constant 8.5 Acceleration due to gravity of the earth 8.6 Acceleration due to gravity below and above the surface of earth 8.7 Gravitational potential energy 8.8 Escape speed 8.9 Earth satellites 8.10 Energy of an orbiting satellite 8.11 Geostationary and polar satellites 8.12 Weightlessness Summary Points to ponder Exercises Additional exercises']}, {'title': 'Chapter Eight', 'text': ['8.1 Introduction Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth', 'Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena', 'Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration', 'It is said that he made a public demonstration of this fact', 'To find the truth, he certainly did experiments with bodies rolling down inclined planes and arrived at a value of the acceleration due to gravity which is close to the more accurate value obtained later', 'A seemingly unrelated phenomenon, observation of stars, planets and their motion has been the subject of attention in many countries since the earliest of times', 'Observations since early times recognised stars which appeared in the sky with positions unchanged year after year', 'The more interesting objects are the planets which seem to have regular motions against the background of stars', 'The earliest recorded model for planetary motions proposed by Ptolemy about 2000 years ago was a ‘geocentric’ model in which all celestial objects, stars, the sun and the planets, all revolved around the earth', 'The only motion that was thought to be possible for celestial objects was motion in a circle', 'Complicated schemes of motion were put forward by Ptolemy in order to describe the observed motion of the planets', 'The planets were described as moving in circles with the centre of the circles themselves moving in larger circles', 'Similar theories were also advanced by Indian astronomers some 400 years later', 'However a more elegant model in which the Sun was the centre around which the planets revolved – the ‘heliocentric’ model – was already mentioned by Aryabhatta (5th century A.D.) in his treatise', 'A thousand years later, a Polish monk named Nicolas Copernicus (1473-1543) proposed a definitive model in which the planets moved in circles around a fixed central sun', 'His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs', 'It was around the same time as Galileo, a nobleman called Tycho Brahe (1546-1601) hailing from Denmark, spent his entire lifetime recording observations of the planets with the naked eye', 'His compiled data were analysed later by his assistant Johannes Kepler (1571-1640)', 'He could extract from the data three elegant laws that now go by the name of Kepler’s laws', 'These laws were known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation']}, {'title': 'Chapter Eight', 'text': ['8.2 Kepler’s laws The three laws of Kepler can be stated as follows: 1', 'Law of orbits : All planets move in elliptical orbits with the Sun situated at one of the foci of the ellipse', 'This law was a deviation from the Copernican model which allowed only circular orbits', 'The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows.']}, {'title': 'Chapter Eight', 'text': ['* Refer to information given in the Box on Page 182']}, {'title': 'Chapter Eight', 'text': ['Select two points F1 and F2', 'Take a length of a string and fix its ends at F1 and F2 by pins', 'With the tip of a pencil stretch the string taut and then draw a curve by moving the pencil keeping the string taut throughout', ') The closed curve you get is called an ellipse', 'Clearly for any point T on the ellipse, the sum of the distances from F1 and F2 is a constant', 'F1, F2 are called the focii', 'Join the points F1 and F2 and extend the line to intersect the ellipse at points P and A as shown in (b)', 'The midpoint of the line PA is the centre of the ellipse O and the length PO = AO is called the semi-major axis of the ellipse', 'For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle', '2', 'Law of areas : The line that joins any planet to the sun sweeps equal areas in equal intervals of time', 'This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer.']}, {'title': 'Chapter Eight', 'text': ['Table 8.1 Data from measurement of planetary motions given below confirm Kepler’s Law of Periods (a ≡ Semi-major axis in units of 1010 m', 'T ≡ Time period of revolution of the planet in years(y)', 'Q ≡ The quotient ( T2/a3 ) in units of 10 -34 y2 m-3.)']}, {'title': 'Chapter Eight', 'text': ['The law of areas can be understood as a consequence of conservation of angular momentum whch is valid for any central force', 'A central force is such that the force on the planet is along the vector joining the Sun and the planet', 'Let the Sun be at the origin and let the position and momentum of the planet be denoted by r and p respectively', 'Then the area swept out by the planet of mass m in time interval ∆t is ∆A given by ∆A = ½ (r × v∆t) (8.1) Hence ∆A /∆t =½ (r × p)/m, (since v = p/m) = L / (2 m) (8.2) where v is the velocity, L is the angular momentum equal to ( r × p)', 'For a central force, which is directed along r, L is a constant as the planet goes around', 'Hence, ∆A /∆t is a constant according to the last equation', 'This is the law of areas', 'Gravitation is a central force and hence the law of areas follows.']}, {'title': 'Chapter Eight', 'text': ['Johannes Kepler (1571–1630) was a scientist of German origin', 'He formulated the three laws of planetary motion based on the painstaking observations of Tycho Brahe and coworkers', 'Kepler himself was an assistant to Brahe and it took him sixteen long years to arrive at the three planetary laws', 'He is also known as the founder of geometrical optics, being the first to describe what happens to light after it enters a telescope.']}, {'title': 'Chapter Eight', 'text': ['* Refer to information given in the Box on Page 182']}, {'title': 'Chapter Eight', 'text': ['Example 8.1 Let the speed of the planet at the perihelion P in (a) be vP and the Sun-planet distance SP be rP', 'Relate {rP, vP} to the corresponding quantities at the aphelion {rA, vA}', 'Will the planet take equal times to traverse BAC and CPB ?']}, {'title': 'Chapter Eight', 'text': ['Answer The magnitude of the angular momentum at P is Lp = mp rp vp, since inspection tells us that rp and vp are mutually perpendicular', 'Similarly, LA = mp rA vA', 'From angular momentum conservation mp rp vp = mp rA vA or Since rA > rp, vp > vA', 'The area SBAC bounded by the ellipse and the radius vectors SB and SC is larger than SBPC in', 'From Kepler’s second law, equal areas are swept in equal times', 'Hence the planet will take a longer time to traverse BAC than CPB.']}, {'title': 'Chapter Eight', 'text': ['8.3 Universal law of gravitation Legend has it that observing an apple falling from a tree, Newton was inspired to arrive at an universal law of gravitation that led to an explanation of terrestrial gravitation as well as of Kepler’s laws', 'Newton’s reasoning was that the moon revolving in an orbit of radius Rm was subject to a centripetal acceleration due to earth’s gravity of magnitude (8.3)']}, {'title': 'Chapter Eight', 'text': ['where V is the speed of the moon related to the time period T by the relation', 'The time period T is about 27.3 days and Rm was already known then to be about 3.84 × 10\\xad8m', 'If we substitute these numbers in Eq', '(8.3), we get a value of am much smaller than the value of acceleration due to gravity g on the surface of the earth, arising also due to earth’s gravitational attraction']}, {'title': 'Chapter Eight', 'text': ['Central Forces']}, {'title': 'Chapter Eight', 'text': ['We know the time rate of change of the angular momentum of a single particle about the origin is']}, {'title': 'Chapter Eight', 'text': ['The angular momentum of the particle is conserved, if the torquedue to the force F on it vanishes', 'This happens either when F is zero or when F is along r', 'We are interested in forces which satisfy the latter condition', 'Central forces satisfy this condition', 'A ‘central’ force is always directed towards or away from a fixed point, i.e., along the position vector of the point of application of the force with respect to the fixed point', '(See Figure below.) Further, the magnitude of a central force F depends on r, the distance of the point of application of the force from the fixed point; F = F(r)', 'In the motion under a central force the angular momentum is always conserved', 'Two important results follow from this: (1) The motion of a particle under the central force is always confined to a plane', '(2) The position vector of the particle with respect to the centre of the force (i.e', 'the fixed point) has a constant areal velocity', 'In other words the position vector sweeps out equal areas in equal times as the particle moves under the influence of the central force', 'Try to prove both these results', 'You may need to know that the areal velocity is given by : dA/dt = ½ r v sin α', 'An immediate application of the above discussion can be made to the motion of a planet under the gravitational force of the sun', 'For convenience the sun may be taken to be so heavy that it is at rest', 'The gravitational force of the sun on the planet is directed towards the sun', 'This force also satisfies the requirement F = F(r), since F = G m1m2/r2 where m1and m2 are respectively the masses of the planet and the sun and G is the universal constant of gravitation', 'The two results (1) and (2) described above, therefore, apply to the motion of the planet', 'In fact, the result (2) is the well-known second law of Kepler.']}, {'title': 'Chapter Eight', 'text': ['Tr is the trejectory of the particle under the central force', 'At a position P, the force is directed along OP, O is the centre of the force taken as the origin', 'In time ∆t, the particle moves from P to P′, arc PP′ = ∆s = v ∆t', 'The tangent PQ at P to the trajectory gives the direction of the velocity at P', 'The area swept in ∆t is the area of sector POP′ ≈ (r sin α ) PP′/2 = (r v sin a) ∆t/2.)']}, {'title': 'Chapter Eight', 'text': ['This clearly shows that the force due to earth’s gravity decreases with distance', 'If one assumes that the gravitational force due to the earth decreases in proportion to the inverse square of the distance from the centre of the earth, we will have and we get (8.4) in agreement with a value of g ≅ 9.8 m s-2 and the value of am from Eq', '(8.3)', 'These observations led Newton to propose the following Universal Law of Gravitation : Every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them', 'The quotation is essentially from Newton’s famous treatise called ‘Mathematical Principles of Natural Philosophy’ (Principia for short)', 'Stated Mathematically, Newton’s gravitation law reads : The force F on a point mass m2 due to another point mass m1 has the magnitude (8.5)']}, {'title': 'Chapter Eight', 'text': ['Equation (8.5) can be expressed in vector form as']}, {'title': 'Chapter Eight', 'text': ['where G is the universal gravitational constant, is the unit vector from m1 to m2 and r = r2 – r1 as shown in .']}, {'title': 'Chapter Eight', 'text': ['The gravitational force is attractive, i.e., the force F is along – r', 'The force on point mass m1 due to m2 is of course – F by Newton’s third law', 'Thus, the gravitational force F12 on the body 1 due to 2 and F21 on the body 2 due to 1 are related as F12 = – F21.']}, {'title': 'Chapter Eight', 'text': ['Before we can apply Eq', '(8.5) to objects under consideration, we have to be careful since the law refers to point masses whereas we deal with extended objects which have finite size', 'If we have a collection of point masses, the force on any one of them is the vector sum of the gravitational forces exerted by the other point masses as shown in Fig 8.4.']}, {'title': 'Chapter Eight', 'text': ['Example 8.2 Three equal masses of m kg each are fixed at the vertices of an equilateral triangle ABC', '(a) What is the force acting on a mass 2m placed at the centroid G of the triangle? (b) What is the force if the mass at the vertex A is doubled ? Take AG = BG = CG = 1 m (see )']}, {'title': 'Chapter Eight', 'text': ['Answer (a) The angle between GC and the positive x-axis is 30° and so is the angle between GB and the negative x-axis', 'The individual forces in vector notation are']}, {'title': 'Chapter Eight', 'text': ['From the principle of superposition and the law of vector addition, the resultant gravitational force FR on (2m) is FR = FGA + FGB + FGC']}, {'title': 'Chapter Eight', 'text': ['Alternatively, one expects on the basis of symmetry that the resultant force ought to be zero', '(b) Now if the mass at vertex A is doubled then']}, {'title': 'Chapter Eight', 'text': ['For the gravitational force between an extended object (like the earth) and a point mass, Eq', '(8.5) is not directly applicable', 'Each point mass in the extended object will exert a force on the given point mass and these force will not all be in the same direction', 'We have to add up these forces vectorially for all the point masses in the extended object to get the total force', 'This is easily done using calculus', 'For two special cases, a simple law results when you do that : (1) The force of attraction between a hollow spherical shell of uniform density and a point mass situated outside is just as if the entire mass of the shell is concentrated at the centre of the shell', 'Qualitatively this can be understood as follows: Gravitational forces caused by the various regions of the shell have components along the line joining the point mass to the centre as well as along a direction prependicular to this line', 'The components prependicular to this line cancel out when summing over all regions of the shell leaving only a resultant force along the line joining the point to the centre', 'The magnitude of this force works out to be as stated above.']}, {'title': 'Chapter Eight', 'text': ['Newton’s Principia']}, {'title': 'Chapter Eight', 'text': ['Kepler had formulated his third law by 1619', 'The announcement of the underlying universal law of gravitation came about seventy years later with the publication in 1687 of Newton’s masterpiece Philosophiae Naturalis Principia Mathematica, often simply called thePrincipia.']}, {'title': 'Chapter Eight', 'text': ['Around 1685, Edmund Halley (after whom the famous Halley’s comet is named), came to visit Newton at Cambridge and asked him about the nature of the trajectory of a body moving under the influence of an inverse square law', 'Without hesitation Newton replied that it had to be an ellipse, and further that he had worked it out long ago around 1665 when he was forced to retire to his farm house from Cambridge on account of a plague outbreak', 'Unfortunately, Newton had lost his papers', 'Halley prevailed upon Newton to produce his work in book form and agreed to bear the cost of publication', 'Newton accomplished this feat in eighteen months of superhuman effort', 'The Principia is a singular scientific masterpiece and in the words of Lagrange it is “the greatest production of the human mind.” The Indian born astrophysicist and Nobel laureate S', 'Chandrasekhar spent ten years writing a treatise on the Principia', 'His book, Newton’s Principia for the Common Reader brings into sharp focus the beauty, clarity and breath taking economy of Newton’s methods.']}, {'title': 'Chapter Eight', 'text': ['(2) The force of attraction due to a hollow spherical shell of uniform density, on a point mass situated inside it is zero', 'Qualitatively, we can again understand this result', 'Various regions of the spherical shell attract the point mass inside it in various directions', 'These forces cancel each other completely.']}, {'title': 'Chapter Eight', 'text': ['8.4 The Gravitational Constant The value of the gravitational constant G entering the Universal law of gravitation can be determined experimentally and this was first done by English scientist Henry Cavendish in 1798', 'The apparatus used by him is schematically shown in figure.8.6']}, {'title': 'Chapter Eight', 'text': ['If L is the length of the bar AB , then the torque arising out of F is F multiplied by L', 'At equilibrium, this is equal to the restoring torque and hence (8.7)']}, {'title': 'Chapter Eight', 'text': ['Observation of θ thus enables one to calculate G from this equation', 'Since Cavendish’s experiment, the measurement of G has been refined and the currently accepted value is G = 6.67×10-11 N m2/kg2 (8.8)']}, {'title': 'Chapter Eight', 'text': ['8.5 Acceleration due to gravity of the earth The earth can be imagined to be a sphere made of a large number of concentric spherical shells with the smallest one at the centre and the largest one at its surface', 'A point outside the earth is obviously outside all the shells', 'Thus, all the shells exert a gravitational force at the point outside just as if their masses are concentrated at their common centre according to the result stated in section 8.3', 'The total mass of all the shells combined is just the mass of the earth', 'Hence, at a point outside the earth, the gravitational force is just as if its entire mass of the earth is concentrated at its centre', 'For a point inside the earth, the situation is different', 'This is illustrated in .']}, {'title': 'Chapter Eight', 'text': ['We assume that the entire earth is of uniform density and hence its mass is where ME is the mass of the earth RE is its radius and ρ is the density', 'On the other hand the mass of the sphere Mr of radius r is and hence (8.10)']}, {'title': 'Chapter Eight', 'text': ['If the mass m is situated on the surface of earth, then r = RE and the gravitational force on it is, from Eq', '(8.10) (8.11)']}, {'title': 'Chapter Eight', 'text': ['The acceleration experienced by the mass m, which is usually denoted by the symbol g is related to F by Newton’s 2nd law by relation F = mg', 'Thus (8.12)']}, {'title': 'Chapter Eight', 'text': ['Acceleration g is readily measurable', 'RE is a known quantity', 'The measurement of G by Cavendish’s experiment (or otherwise), combined with knowledge of g and RE enables one to estimate ME from Eq', '(8.12)', 'This is the reason why there is a popular statement regarding Cavendish : “Cavendish weighed the earth”']}, {'title': 'Chapter Eight', 'text': ['8.6 Acceleration due to gravity below and above the surface of earth Consider a point mass m at a height h above the surface of the earth as shown in (a)', 'The radius of the earth is denoted by RE', 'Since this point is outside the earth, its distance from the centre of the earth is (RE + h )', 'If F (h) denoted the magnitude of the force on the point mass m , we get from Eq', '(8.5) :']}, {'title': 'Chapter Eight', 'text': ['(8.13)']}, {'title': 'Chapter Eight', 'text': ['This is clearly less than the value of g on the surface of earth : For we can expand the RHS of Eq', '(8.14) : For , using binomial expression, . (8.15)']}, {'title': 'Chapter Eight', 'text': ['Equation (8.15) thus tells us that for small heights h above the value of g decreases by a factor Now, consider a point mass m at a depth d below the surface of the earth ), so that its distance from the centre of the earth is as shown in the figure', 'The earth can be thought of as being composed of a smaller sphere of radius (RE – d ) and a spherical shell of thickness d', 'The force on m due to the outer shell of thickness d is zero because the result quoted in the previous section', 'As far as the smaller sphere of radius ( RE – d ) is concerned, the point mass is outside it and hence according to the result quoted earlier, the force due to this smaller sphere is just as if the entire mass of the smaller sphere is concentrated at the centre', 'If Ms is the mass of the smaller sphere, then, Ms/ME = ( RE – d)3 / RE3 ( 8.16) Since mass of a sphere is proportional to be cube of its radius.']}, {'title': 'Chapter Eight', 'text': ['(b) g at a depth d', 'In this case only the smaller sphere of radius (RE–d) contributes to g', 'Thus the force on the point mass is F (d) = G Ms m / (RE – d ) 2 (8.17) Substituting for Ms from above , we get F (d) = G ME m ( RE – d ) / RE 3 (8.18) and hence the acceleration due to gravity at a depth d, g(d) = is (8.19)']}, {'title': 'Chapter Eight', 'text': ['Thus, as we go down below earth’s surface, the acceleration due gravity decreases by a factor The remarkable thing about acceleration due to earth’s gravity is that it is maximum on its surface decreasing whether you go up or down.']}, {'title': 'Chapter Eight', 'text': ['8.7 Gravitational potential energy We had discussed earlier the notion of potential energy as being the energy stored in the body at its given position', 'If the position of the particle changes on account of forces acting on it, then the change in its potential energy is just the amount of work done on the body by the force', 'As we had discussed earlier, forces for which the work done is independent of the path are the conservative forces', 'The force of gravity is a conservative force and we can calculate the potential energy of a body arising out of this force, called the gravitational potential energy', 'Consider points close to the surface of earth, at distances from the surface much smaller than the radius of the earth', 'In such cases, the force of gravity is practically a constant equal to mg, directed towards the centre of the earth', 'If we consider a point at a height h1 from the surface of the earth and another point vertically above it at a height h2 from the surface, the work done in lifting the particle of mass m from the first to the second position is denoted by W12 W12 = Force × displacement = mg (h2 – h1) (8.20) If we associate a potential energy W(h) at a point at a height h above the surface such that W(h) = mgh + Wo (8.21) (where Wo = constant) ; then it is clear that W12 = W(h2) – W(h1) (8.22) The work done in moving the particle is just the difference of potential energy between its final and initial positions.Observe that the constant Wo cancels out in Eq', '(8.22)', 'Setting h = 0 in the last equation, we get W ( h = 0 ) = Wo', 'h = 0 means points on the surface of the earth', 'Thus, Wo is the potential energy on the surface of the earth', 'If we consider points at arbitrary distance from the surface of the earth, the result just derived is not valid since the assumption that the gravitational force mg is a constant is no longer valid', 'However, from our discussion we know that a point outside the earth, the force of gravitation on a particle directed towards the centre of the earth is (8.23)']}, {'title': 'Chapter Eight', 'text': ['where ME = mass of earth, m = mass of the particle and r its distance from the centre of the earth', 'If we now calculate the work done in lifting a particle from r = r1 to r = r2 (r2 > r1) along a vertical path, we get instead of Eq', '(8.20) (8.24)']}, {'title': 'Chapter Eight', 'text': ['In place of Eq', '(8.21), we can thus associate a potential energy W(r) at a distance r, such that (8.25)']}, {'title': 'Chapter Eight', 'text': ['valid for r > R , so that once again W12 = W(r2) – W(r1)', 'Setting r = infinity in the last equation, we get W ( r = infinity ) = W1', 'Thus, W1 is the potential energy at infinity', 'One should note that only the difference of potential energy between two points has a definite meaning from Eqs', '(8.22) and (8.24)', 'One conventionally sets W1 equal to zero, so that the potential energy at a point is just the amount of work done in displacing the particle from infinity to that point', 'We have calculated the potential energy at a point of a particle due to gravitational forces on it due to the earth and it is proportional to the mass of the particle', 'The gravitational potential due to the gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point', 'From the earlier discussion, we learn that the gravitational potential energy associated with two particles of masses m1 and m2 separated by distance by a distance r is given by (if we choose V = 0 as r → ∞ ) It should be noted that an isolated system of particles will have the total potential energy that equals the sum of energies (given by the above equation) for all possible pairs of its constituent particles', 'This is an example of the application of the superposition principle.']}, {'title': 'Chapter Eight', 'text': ['Example 8.3 Find the potential energy of a system of four particles placed at the vertices of a square of side l', 'Also obtain the potential at the centre of the square.']}, {'title': 'Chapter Eight', 'text': ['Answer Consider four masses each of mass m at the corners of a square of side l; See', 'We have four mass pairs at distance l and two diagonal pairs at distance Hence,']}, {'title': 'Chapter Eight', 'text': ['8.8 Escape Speed If a stone is thrown by hand, we see it falls back to the earth', 'Of course using machines we can shoot an object with much greater speeds and with greater and greater initial speed, the object scales higher and higher heights', 'A natural query that arises in our mind is the following: ‘can we throw an object with such high initial speeds that it does not fall back to the earth?’ The principle of conservation of energy helps us to answer this question', 'Suppose the object did reach infinity and that its speed there was Vf', 'The energy of an object is the sum of potential and kinetic energy', 'As before W1 denotes that gravitational potential energy of the object at infinity', 'The total energy of the projectile at infinity then is (8.26)']}, {'title': 'Chapter Eight', 'text': ['If the object was thrown initially with a speed Vi from a point at a distance (h+RE) from the centre of the earth (RE = radius of the earth), its energy initially was (8.27)']}, {'title': 'Chapter Eight', 'text': ['By the principle of energy conservation Eqs', '(8.26) and (8.27) must be equal', 'Hence (8.28)']}, {'title': 'Chapter Eight', 'text': ['The R.H.S', 'is a positive quantity with a minimum value zero hence so must be the L.H.S', 'Thus, an object can reach infinity as long as Vi is such that']}, {'title': 'Chapter Eight', 'text': ['(8.29)']}, {'title': 'Chapter Eight', 'text': ['The minimum value of Vi corresponds to the case when the L.H.S', 'of Eq', '(8.29) equals zero', 'Thus, the minimum speed required for an object to reach infinity (i.e', 'escape from the earth) corresponds to (8.30)']}, {'title': 'Chapter Eight', 'text': ['If the object is thrown from the surface of the earth, h = 0, and we get (8.31)']}, {'title': 'Chapter Eight', 'text': ['Using the relation , we get (8.32)']}, {'title': 'Chapter Eight', 'text': ['Using the value of g and RE, numerically (Vi)min≈11.2 km/s', 'This is called the escape speed, sometimes loosely called the escape velocity', 'Equation (8.32) applies equally well to an object thrown from the surface of the moon with g replaced by the acceleration due to Moon’s gravity on its surface and rE replaced by the radius of the moon', 'Both are smaller than their values on earth and the escape speed for the moon turns out to be 2.3 km/s, about five times smaller', 'This is the reason that moon has no atmosphere', 'Gas molecules if formed on the surface of the moon having velocities larger than this will escape the gravitational pull of the moon.']}, {'title': 'Chapter Eight', 'text': ['Example 8.4 Two uniform solid spheres of equal radii R, but mass M and 4 M have a centre to centre separation 6 R, as shown in 0', 'The two spheres are held fixed', 'A projectile of mass m is projected from the surface of the sphere of mass M directly towards the centre of the second sphere', 'Obtain an expression for the minimum speed v of the projectile so that it reaches the surface of the second sphere.']}, {'title': 'Chapter Eight', 'text': ['Answer The projectile is acted upon by two mutually opposing gravitational forces of the two spheres', 'The neutral point N (see 0) is defined as the position where the two forces cancel each other exactly', 'If ON = r, we have (6R – r)2 = 4r2 6R – r = ±2r r = 2R or – 6R', 'The neutral point r = – 6R does not concern us in this example', 'Thus ON = r = 2R', 'It is sufficient to project the particle with a speed which would enable it to reach N', 'Thereafter, the greater gravitational pull of 4M would suffice', 'The mechanical energy at the surface of M is', 'At the neutral point N, the speed approaches zero', 'The mechanical energy at N is purely potential', 'From the principle of conservation of mechanical energy or A point to note is that the speed of the projectile is zero at N, but is nonzero when it strikes the heavier sphere 4 M', 'The calculation of this speed is left as an exercise to the students.']}, {'title': 'Chapter Eight', 'text': ['8.9 Earth Satellites Earth satellites are objects which revolve around the earth', 'Their motion is very similar to the motion of planets around the Sun and hence Kepler’s laws of planetary motion are equally applicable to them', 'In particular, their orbits around the earth are circular or elliptic', 'Moon is the only natural satellite of the earth with a near circular orbit with a time period of approximately 27.3 days which is also roughly equal to the rotational period of the moon about its own axis', 'Since, 1957, advances in technology have enabled many countries including India to launch artificial earth satellites for practical use in fields like telecommunication, geophysics and meteorology', 'We will consider a satellite in a circular orbit of a distance (RE + h) from the centre of the earth, where RE = radius of the earth', 'If m is the mass of the satellite and V its speed, the centripetal force required for this orbit is F(centripetal) = (8.33)']}, {'title': 'Chapter Eight', 'text': ['directed towards the centre', 'This centripetal force is provided by the gravitational force, which is F(gravitation) = (8.34)']}, {'title': 'Chapter Eight', 'text': ['where ME is the mass of the earth', 'Equating R.H.S of Eqs', '(8.33) and (8.34) and cancelling out m, we get (8.35)']}, {'title': 'Chapter Eight', 'text': ['Thus V decreases as h increases', 'From ,the speed V for h = 0 is (8.36)']}, {'title': 'Chapter Eight', 'text': ['where we have used the relation g =', 'In every orbit, the satellite traverses a distance 2π(RE + h) with speed V', 'Its time period T therefore is (8.37)']}, {'title': 'Chapter Eight', 'text': ['on substitution of value of V from Eq', '(8.35)', 'Squaring both sides of Eq', '(8.37), we get T 2 = k ( RE + h)3 (where k = 4 π2 / GME) (8.38) which is Kepler’s law of periods, as applied to motion of satellites around the earth', 'For a satellite very close to the surface of earth h can be neglected in comparison to RE in Eq', '(8.38)', 'Hence, for such satellites, T is To, where (8.39)']}, {'title': 'Chapter Eight', 'text': ['If we substitute the numerical values g ≅ 9.8 m s-2 and RE = 6400 km., we get']}, {'title': 'Chapter Eight', 'text': ['Which is approximately 85 minutes.']}, {'title': 'Chapter Eight', 'text': ['Example 8.5 The planet Mars has two moons, phobos and delmos', 'phobos has a period 7 hours, 39 minutes and an orbital radius of 9.4 ×103 km', 'Calculate the mass of mars', 'Assume that earth and mars move in circular orbits around the sun, with the martian orbit being 1.52 times the orbital radius of the earth', 'What is the length of the martian year in days ?']}, {'title': 'Chapter Eight', 'text': ['Answer We employ Eq', '(8.38) with the sun’s mass replaced by the martian mass Mm = 6.48 × 1023 kg', 'Once again Kepler’s third law comes to our aid,']}, {'title': 'Chapter Eight', 'text': ['where RMS is the mars -sun distance and RES is the earth-sun distance', '∴ TM = (1.52)3/2 × 365 = 684 days We note that the orbits of all planets except Mercury, Mars and Pluto* are very close to being circular', 'For example, the ratio of the semi-minor to semi-major axis for our Earth is, b/a = 0.99986.']}, {'title': 'Chapter Eight', 'text': ['Example 8.6 Weighing the Earth : You are given the following data: g = 9.81 ms–2, RE = 6.37×106 m, the distance to the moon R = 3.84×108 m and the time period of the moon’s revolution is 27.3 days', 'Obtain the mass of the Earth ME in two different ways.']}, {'title': 'Chapter Eight', 'text': ['Answer From Eq', '(8.12) we have']}, {'title': 'Chapter Eight', 'text': ['= 5.97× 1024 kg', 'The moon is a satellite of the Earth', 'From the derivation of Kepler’s third law [see Eq', '(8.38)] Both methods yield almost the same answer, the difference between them being less than 1%.']}, {'title': 'Chapter Eight', 'text': ['Example 8.7 Express the constant k of Eq', '(8.38) in days and kilometres', 'Given k = 10–13 s2 m–3', 'The moon is at a distance of 3.84 × 105 km from the earth', 'Obtain its time-period of revolution in days.']}, {'title': 'Chapter Eight', 'text': ['Answer Given k = 10–13 s2 m–3 = = 1.33 ×10–14 d2 km–3 Using Eq', '(8.38) and the given value of k, the time period of the moon is T2 = (1.33 × 10-14)(3.84 × 105)3 T = 27.3 d Note that Eq', '(8.38) also holds for elliptical orbits if we replace (RE+h) by the semi-major axis of the ellipse', 'The earth will then be at one of the foci of this ellipse']}, {'title': 'Chapter Eight', 'text': ['8.10 Energy of an orbiting Satellite Using Eq', '(8.35), the kinetic energy of the satellite in a circular orbit with speed v is']}, {'title': 'Chapter Eight', 'text': ['(8.40)']}, {'title': 'Chapter Eight', 'text': ['Considering gravitational potential energy at infinity to be zero, the potential energy at distance (RE+h) from the centre of the earth is (8.41)']}, {'title': 'Chapter Eight', 'text': ['The K.E is positive whereas the P.E is negative', 'However, in magnitude the K.E', 'is half the P.E, so that the total E is (8.42)']}, {'title': 'Chapter Eight', 'text': ['The total energy of an circularly orbiting satellite is thus negative, with the potential energy being negative but twice is magnitude of the positive kinetic energy', 'When the orbit of a satellite becomes elliptic, both the K.E', 'and P.E', 'vary from point to point', 'The total energy which remains constant is negative as in the circular orbit case', 'This is what we expect, since as we have discussed before if the total energy is positive or zero, the object escapes to infinity', 'Satellites are always at finite distance from the earth and hence their energies cannot be positive or zero']}, {'title': 'Chapter Eight', 'text': ['Example 8.8 A 400 kg satellite is in a circular orbit of radius 2RE about the Earth', 'How much energy is required to transfer it to a circular orbit of radius 4RE ? What are the changes in the kinetic and potential energies ?']}, {'title': 'Chapter Eight', 'text': ['Answer Initially, While finally The change in the total energy is ∆E = Ef – Ei The kinetic energy is reduced and it mimics ∆E, namely, ∆K = Kf – Ki = – 3.13 × 109 J', 'The change in potential energy is twice the change in the total energy, namely ∆V = Vf – Vi = – 6.25 × 109 J']}, {'title': 'Chapter Eight', 'text': ['8.11 Geostationary and Polar Satellites An interesting phenomenon arises if in we arrange the value of (RE+ h) such that T in Eq', '(8.37) becomes equal to 24 hours', 'If the circular orbit is in the equatorial plane of the earth, such a satellite, having the same period as the period of rotation of the earth about its own axis would appear stationery viewed from a point on earth', 'The (RE + h) for this purpose works out to be large as compared to RE : (8.43)']}, {'title': 'Chapter Eight', 'text': ['and for T = 24 hours, h works out to be 35800 km', 'which is much larger than RE', 'Satellites in a circular orbits around the earth in the equatorial plane with T = 24 hours are called Geostationery Satellites', 'Clearly, since the earth rotates with the same period, the satellite would appear fixed from any point on earth', 'It takes very powerful rockets to throw up a satellite to such large heights above the earth but this has been done in view of the several benefits of many practical applications.']}, {'title': 'Chapter Eight', 'text': ['India’s Leap into Space India started its space programme in 1962 when Indian National Committee for Space Research was set up by the Government of India which was superseded by the Indian Space Research Organisation (ISRO) in 1969', 'ISRO identified the role and importance of space technology in nation’s development and bringing space to the service of the common man', 'India launched its first low orbit satellite Aryabhata in 1975, for which the launch vehicle was provided by the erstwhile Soviet Union', 'ISRO started employing its indigenous launching vehicle in 1979 by sending Rohini series of satellites into space from its main launch site at Satish Dhawan Space Center, Sriharikota, Andhra Pradesh', 'The tremendous progress in India’s space programme has made ISRO one of the six largest space agencies in the world', 'ISRO develops and delivers application specific satellite products and tools for broadcasts, communication, weather forecasts, disaster management tools, Geographic Information System, cartography, navigation, telemedicine, dedicated distance education satellite etc', 'In order to achieve complete self-reliance in these applications, cost effective and reliable Polar Satellite Launch Vehicle (PSLV) was developed in early 1990s', 'PSLV has thus become a favoured carrier for satellites of various countries, promoting unprecedented international collaboration', 'In 2001, the Geosynchronous Satellite Launch Vehicle (GSLV) was developed for launching heavier and more demanding Geosynchronous communication satellites', 'Various research centers and autonomous institutions for remote sensing, astronomy and astrophysics, atmospheric sciences and space research are functioning under the aegis of the Department of Space, Government of India', 'Success of lunar (Chandrayaan) and inter planetary (Mangalyaan) missions along with other scientific projects has been landmark achievements of ISRO', 'Future endeavors of ISRO include human space flight projects, the development of heavy lift launchers, reusable launch vehicles, semi-cryogenic engines, single and two stage to orbit (SSTO and TSTO) vehicles, development and use of composite materials for space application etc', 'In 1984 Rakesh Sharma became the first Indian to go into outer space aboard in a USSR spaceship', '(www.isro.gov.in)']}, {'title': 'Chapter Eight', 'text': ['Thus radio waves broadcast from an antenna can be received at points far away where the direct wave fail to reach on account of the curvature of the earth', 'Waves used in television broadcast or other forms of communication have much higher frequencies and thus cannot be received beyond the line of sight', 'A Geostationery satellite, appearing fixed above the broadcasting station can however receive these signals and broadcast them back to a wide area on earth', 'The INSAT group of satellites sent up by India are one such group of Geostationary satellites widely used for telecommunications in India', 'Another class of satellites are called the Polar satellites', 'These are low altitude (h ≈ 500 to 800 km) satellites, but they go around the poles of the earth in a north-south direction whereas the earth rotates around its axis in an east-west direction', 'Since its time period is around 100 minutes it crosses any altitude many times a day', 'However, since its height h above the earth is about 500-800 km, a camera fixed on it can view only small strips of the earth in one orbit', 'Adjacent strips are viewed in the next orbit, so that in effect the whole earth can be viewed strip by strip during the entire day', 'These satellites can view polar and equatorial regions at close distances with good resolution', 'Information gathered from such satellites is extremely useful for remote sensing, meterology as well as for environmental studies of the earth.']}, {'title': 'Chapter Eight', 'text': ['8.12 Weightlessness Weight of an object is the force with which the earth attracts it', 'We are conscious of our own weight when we stand on a surface, since the surface exerts a force opposite to our weight to keep us at rest', 'The same principle holds good when we measure the weight of an object by a spring balance hung from a fixed point e.g', 'the ceiling', 'The object would fall down unless it is subject to a force opposite to gravity', 'This is exactly what the spring exerts on the object', 'This is because the spring is pulled down a little by the gravitational pull of the object and in turn the spring exerts a force on the object vertically upwards', 'Now, imagine that the top end of the balance is no longer held fixed to the top ceiling of the room', 'Both ends of the spring as well as the object move with identical acceleration g', 'The spring is not stretched and does not exert any upward force on the object which is moving down with acceleration g due to gravity', 'The reading recorded in the spring balance is zero since the spring is not stretched at all', 'If the object were a human being, he or she will not feel his weight since there is no upward force on him', 'Thus, when an object is in free fall, it is weightless and this phenomenon is usually called the phenomenon of weightlessness', 'In a satellite around the earth, every part and parcel of the satellite has an acceleration towards the centre of the earth which is exactly the value of earth’s acceleration due to gravity at that position', 'Thus in the satellite everything inside it is in a state of free fall', 'This is just as if we were falling towards the earth from a height', 'Thus, in a manned satellite, people inside experience no gravity', 'Gravity for us defines the vertical direction and thus for them there are no horizontal or vertical directions, all directions are the same', 'Pictures of astronauts floating in a satellite show this fact']}, {'title': 'Chapter Eight', 'text': ['Summary 1. Newton’s law of universal gravitation states that the gravitational force of attraction between any two particles of masses m1 and m2 separated by a distance r has the magnitude where G is the universal gravitational constant, which has the value 6.672 ×10–11 N m2 kg–2', 'If we have to find the resultant gravitational force acting on the particle m due to a number of masses M1, M2, ….Mn etc', 'we use the principle of superposition', 'Let F1, F2, ….Fn be the individual forces due to M1, M2, ….Mn, each given by the law of gravitation', 'From the principle of superposition each force acts independently and uninfluenced by the other bodies', 'The resultant force FR is then found by vector addition FR = F1 + F2 + ……+ Fn = where the symbol ‘Σ’ stands for summation', 'Kepler’s laws of planetary motion state that (a) All planets move in elliptical orbits with the Sun at one of the focal points (b) The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals', 'This follows from the fact that the force of gravitation on the planet is central and hence angular momentum is conserved', '(c) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical orbit of the planet The period T and radius R of the circular orbit of a planet about the Sun are related by where Ms is the mass of the Sun', 'Most planets have nearly circular orbits about the Sun', 'For elliptical orbits, the above equation is valid if R is replaced by the semi-major axis, a', '4. The acceleration due to gravity', '(a) at a height h above the earth’s surface for h << RE (b) at depth d below the earth’s surface is 5. The gravitational force is a conservative force, and therefore a potential energy function can be defined', 'The gravitational potential energy associated with two particles separated by a distance r is given by where V is taken to be zero at r → ∞', 'The total potential energy for a system of particles is the sum of energies for all pairs of particles, with each pair represented by a term of the form given by above equation', 'This prescription follows from the principle of superposition', '6. If an isolated system consists of a particle of mass m moving with a speed v in the vicinity of a massive body of mass M, the total mechanical energy of the particle is given by That is, the total mechanical energy is the sum of the kinetic and potential energies', 'The total energy is a constant of motion', 'If m moves in a circular orbit of radius a about M, where M >> m, the total energy of the system is with the choice of the arbitrary constant in the potential energy given in the point 5., above', 'The total energy is negative for any bound system, that is, one in which the orbit is closed, such as an elliptical orbit', 'The kinetic and potential energies are The escape speed from the surface of the earth is = and has a value of 11.2 km s–1', 'If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attracts the particle as though the mass of the sphere or shell were concentrated at the centre of the sphere', '10. If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero', 'If a particle is inside a homogeneous solid sphere, the force on the particle acts toward the centre of the sphere', 'This force is exerted by the spherical mass interior to the particle', '. A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at a approximate distance of 4.22 × 104 km from the earth’s centre']}, {'title': 'Chapter Eight', 'text': ['Points to Ponder 1. In considering motion of an object under the gravitational influence of another object the following quantities are conserved: (a) Angular momentum (b) Total mechanical energy Linear momentum is not conserved 2. Angular momentum conservation leads to Kepler’s second law', 'However, it is not special to the inverse square law of gravitation', 'It holds for any central force', '3. In Kepler’s third law (see Eq', '(8.1) and T2 = KS R3', 'The constant KS is the same for all planets in circular orbits', 'This applies to satellites orbiting the Earth [(Eq', '(8.38)]', '4. An astronaut experiences weightlessness in a space satellite', 'This is not because the gravitational force is small at that location in space', 'It is because both the astronaut and the satellite are in “free fall” towards the Earth', '5. The gravitational potential energy associated with two particles separated by a distance r is given by The constant can be given any value', 'The simplest choice is to take it to be zero', 'With this choice This choice implies that V → 0 as r → ∞', 'Choosing location of zero of the gravitational energy is the same as choosing the arbitrary constant in the potential energy', 'Note that the gravitational force is not altered by the choice of this constant', '6. The total mechanical energy of an object is the sum of its kinetic energy (which is always positive) and the potential energy', 'Relative to infinity (i.e', 'if we presume that the potential energy of the object at infinity is zero), the gravitational potential energy of an object is negative', 'The total energy of a satellite is negative', '7. The commonly encountered expression m g h for the potential energy is actually an approximation to the difference in the gravitational potential energy discussed in the point 6, above', '8. Although the gravitational force between two particles is central, the force between two finite rigid bodies is not necessarily along the line joining their centre of mass', 'For a spherically symmetric body however the force on a particle external to the body is as if the mass is concentrated at the centre and this force is therefore central', 'The gravitational force on a particle inside a spherical shell is zero', 'However, (unlike a metallic shell which shields electrical forces) the shell does not shield other bodies outside it from exerting gravitational forces on a particle inside', 'Gravitational shielding is not possible.']}, {'title': 'Chapter Eight', 'text': ['Exercises 8.1 Answer the following : (a) You can shield a charge from electrical forces by putting it inside a hollow conductor', 'Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means ? (b) An astronaut inside a small space ship orbiting around the earth cannot detect gravity', 'If the space station orbiting around the earth has a large size, can he hope to detect gravity ? (c) If you compare the gravitational force on the earth due to the sun to that due to the moon, you would find that the Sun’s pull is greater than the moon’s pull', '(you can check this yourself using the data available in the succeeding exercises)', 'However, the tidal effect of the moon’s pull is greater than the tidal effect of sun', 'Why ? 8.2 Choose the correct alternative : (a) Acceleration due to gravity increases/decreases with increasing altitude', '(b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density)', '(c) Acceleration due to gravity is independent of mass of the earth/mass of the body', '(d) The formula –G Mm(1/r2 – 1/r1) is more/less accurate than the formulamg(r2 – r1) for the difference of potential energy between two points r2 and r1 distance away from the centre of the earth', '8.3 Suppose there existed a planet that went around the sun twice as fast as the earth', 'What would be its orbital size as compared to that of the earth ? 8.4 Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m', 'Show that the mass of Jupiter is about one-thousandth that of the sun', '8.5 Let us assume that our galaxy consists of 2.5 × 1011 stars each of one solar mass', 'How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution? Take the diameter of the Milky Way to be 105 ly', '8.6 Choose the correct alternative: (a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy', '(b) The energy required to launch an orbiting satellite out of earth’s gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth’s influence', '8.7 Does the escape speed of a body from the earth depend on (a) the mass of the body, (b) the location from where it is projected, (c) the direction of projection, (d) the height of the location from where the body is launched? 8 A comet orbits the sun in a highly elliptical orbit', 'Does the comet have a constant (a) linear speed, (b) angular speed, (c) angular momentum, (d) kinetic energy, (e) potential energy, (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun', '8.9 Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem', '8.10 In the following two exercises, choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12) a, b,c, (iv) 0.']}, {'title': 'Chapter Eight', 'text': ['APPENDIX 8.1 : LIST OF INDIAN SATELLITES']}, {'title': 'Chapter Eight', 'text': ['India has so far also launched 239 foreign satellites of 28 countries from Satish Dhawan Space Center, Sriharikota, Andhra Pradesh: May 26, 1999 (02); Oct', '22, 2001 (02); Jan', '10, 2007 (02); Apr', '23, 2007 (01); Jan', '21, 2008 (01); Apr', '28,2008 (08); Sep', '23,2009 (06); July 12, 2010 (03); Jan', '12,2011 (01); Apr', '20, 2011 (01) Sep', '09, 2012 (02); Feb', '25, 2013 (06); June 30, 2014 (05); July 10, 2015 (05); Sep', '28, 2015 (06); Dec', '16, 2015 (06); June 22, 2016 (17); Sep', '26, 2016 (05); Feb', '15, 2017 (101) and thus setting a world record; June 23, 2017 (29)', 'Jan 12, 2018 (28); Sep', '16, 2018 (02)', 'Details can be seen at www.isro.gov.in', 'a Launched from Kapustin Yar Missile and Space Complex, Soviet Union (now Russia) b Launched from Satish Dhawan Space Centre, Sriharikota, Andhra Pradesh c Launched from Centre Spatial Guyanais, Kourou, French Guiana d Launched from Air Force Eastern Test Range, Florida e Launched from Baikonur Cosmodrome, Kazakhstan']}, {'title': 'Chapter Eight', 'text': ['Table of Contents']}, {'title': 'Chapter Eight', 'text': ['Chapter Eight']}, {'title': 'Chapter Eight', 'text': ['Gravitation']}, {'title': 'Chapter Eight', 'text': ['8.1 Introduction']}, {'title': 'Chapter Eight', 'text': ['8.2 Kepler’s laws']}, {'title': 'Chapter Eight', 'text': ['8.4 The Gravitational Constant']}, {'title': 'Chapter Eight', 'text': ['8.5 Acceleration due to gravity of the earth']}, {'title': 'Chapter Eight', 'text': ['8.6 Acceleration due to gravity below and above the surface of earth']}, {'title': 'Chapter Eight', 'text': ['8.7 Gravitational potential energy']}, {'title': 'Chapter Eight', 'text': ['8.8 Escape Speed']}, {'title': 'Chapter Eight', 'text': ['8.9 Earth Satellites']}, {'title': 'Chapter Eight', 'text': ['8.10 Energy of an orbiting Satellite']}, {'title': 'Chapter Eight', 'text': ['8.11 Geostationary and Polar Satellites']}, {'title': 'Chapter Eight', 'text': ['8.12 Weightlessness']}, {'title': 'Chapter Eight', 'text': ['Summary']}, {'title': 'Chapter Eight', 'text': ['Exercises']}, {'title': 'Chapter Eight', 'text': ['Landmarks']}, {'title': 'Chapter Eight', 'text': ['Cover']}, {'title': 'Chapter Eight', 'text': ['Chapter EightGRAVITATION']}, {'title': 'Chapter Eight', 'text': []}]\n"
+      "[{'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['From the last chapter, we recall that all living organisms are made of cells', 'In unicellular organisms, a single cell performs all basic functions', 'For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion', 'But in multi-cellular organisms there are millions of cells', 'Most of these cells are specialised to carry out specific functions', 'Each specialised function is taken up by a different group of cells', 'Since these cells carry out only a particular function, they do it very efficiently', 'In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on', 'In plants, vascular tissues conduct food and water from one part of the plant to other parts', 'So, multi-cellular organisms show division of labour', 'Cells specialising in one function are often grouped together in the body', 'This means that a particular function is carried out by a cluster of cells at a definite place in the body', 'This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function', 'Blood, phloem and muscle are all examples of tissues', 'Chapter 6']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Tissues A group of cells that are similar in structure and/or work together to achieve a particular function forms a tissue', '6.1 Are Plants and Animals Made of Same Types of Tissues? Let us compare their structure and functions', 'Do plants and animals have the same structure? Do they both perform similar functions? There are noticeable differences between the two', 'Plants are stationary or fixed – they don’t move', 'Since they have to be upright, they have a large quantity of supportive tissue', 'The supportive tissue generally has dead cells', 'Animals on the other hand move around in search of food, mates and shelter', 'They consume more energy as compared to plants', 'Most of the tissues they contain are living', 'Another difference between animals and plants is in the pattern of growth', 'The growth in plants is limited to certain regions, while this is not so in animals', 'There are some tissues in plants that divide throughout their life', 'These tissues are localised in certain regions', 'Based on the dividing capacity of the tissues, various plant tissues can be classified as growing or meristematic tissue and permanent tissue', 'Cell growth in animals is more uniform', 'So, there is no such demarcation of dividing and non-dividing regions in animals', 'The structural organisation of organs and organ systems is far more specialised and localised in complex animals than even in very complex plants', 'This fundamental difference reflects the different modes of life pursued by these two major groups of organisms, particularly in their different feeding methods', 'Also, they are differently adapted for a sedentary existence on one hand (plants) and active locomotion on the other (animals), contributing to this difference in organ system design', 'It is with reference to these complex animal and plant bodies that we will now talk about the concept of tissues in some detail.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Questions 1. What is a tissue? 2. What is the utility of tissues in multi-cellular organisms?']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2 Plant Tissues']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.1 Meristematic tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Jar 1 Jar 2 : Growth of roots in onion bulbs']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Activity 6.1 • Take two glass jars and fill them with water', '• Now, take two onion bulbs and place one on each jar, as shown in', '• Observe the growth of roots in both the bulbs for a few days', '• Measure the length of roots on day 1, 2 and 3', '• On day 4, cut the root tips of the onion bulb in jar 2 by about 1 cm', 'After this, observe the growth of roots in both the jars and measure their lengths each day for five more days and record the observations in tables, like the table below: Length Day 1 Day 2 Day 3 Day 4 Day 5 Jar 1 Jar 2']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['• From the above observations, answer the following questions: 1. Which of the two onions has longer roots? Why? 2. Do the roots continue growing even after we have removed their tips? 3. Why would the tips stop growing in jar 2 after we cut them?']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['The growth of plants occurs only in certain specific regions', 'This is because the dividing tissue, also known as meristematic tissue, is located only at these points', 'Depending on the region where they are present, meristematic tissues are classified as apical, lateral and intercalary', 'New cells produced by meristem are initially like those of meristem itself, but as they grow and mature, their characteristics slowly change and they become differentiated as components of other tissues.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Apical meristem is present at the growing tips of stems and roots and increases the length of the stem and the root', 'The girth of the stem or root increases due to lateral meristem (cambium)', 'Intercalary meristem seen in some plants is located near the node', 'Cells of meristematic tissue are very active, they have dense cytoplasm, thin cellulose walls and prominent nuclei', 'They lack vacuoles', 'Can we think why they would lack vacuoles? (You might want to refer to the functions of vacuoles in the chapter on cells.)']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.2 Permanent tissue What happens to the cells formed by meristematic tissue? They take up a specific role and lose the ability to divide', 'As a result, they form a permanent tissue', 'This process of taking up a permanent shape, size, and a function is called differentiation', 'Differentiation leads to the development of various types of permanent tissues.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Activity 6.2 • Take a plant stem and with the help of your teacher cut into very thin slices or sections', '• Now, stain the slices with safranin', 'Place one neatly cut section on a slide, and put a drop of glycerine', '• Cover with a cover-slip and observe under a microscope', 'Observe the various types of cells and their arrangement', 'Compare it with', '• Now, answer the following on the basis of your observation: 1. Are all cells similar in structure? 2. How many types of cells can be seen? Can we think of reasons why there would be so many types of cells? • We can also try to cut sections of plant roots', 'We can even try cutting sections of root and stem of different plants.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.2 Simple permanent tissue A few layers of cells beneath the epidermis are generally simple permanent tissue', 'Parenchyma is the most common simple permanent tissue', 'It consists of relatively unspecialised cells with thin cell walls', 'They are living cells', 'They are usually loosely arranged, thus large spaces between cells (intercellular spaces) are found in this tissue', 'This tissue generally stores food', 'In some situations, it contains chlorophyll and performs photosynthesis, and then it is called chlorenchyma', 'In aquatic plants, large air cavities are present in parenchyma to help them float', 'Such a parenchyma type is called aerenchyma', 'The flexibility in plants is due to another permanent tissue, collenchyma', 'It allows bending of various parts of a plant like tendrils and stems of climbers without breaking', 'It also provides mechanical support', 'We can find this tissue in leaf stalks below the epidermis', 'The cells of this tissue are living, elongated and irregularly thickened at the corners', 'There is very little intercellular space .']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Activity 6.3 • Take a freshly plucked leaf of Rhoeo', '• Stretch and break it by applying pressure', '• While breaking it, keep it stretched gently so that some peel or skin projects out from the cut', '• Remove this peel and put it in a petri dish filled with water', '• Add a few drops of safranin', '• Wait for a couple of minutes and then transfer it onto a slide', 'Gently place a cover slip over it', '• Observe under microscope.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['What you observe is the outermost layer of cells, called epidermis', 'The epidermis is usually made of a single layer of cells', 'In some plants living in very dry habitats, the epidermis may be thicker since protection against water loss is critical', 'The entire surface of a plant has an outer covering epidermis', 'It protects all the parts of the plant', 'Epidermal cells on the aerial parts of the plant often secrete a waxy, water-resistant layer on their outer surface', 'This aids in protection against loss of water, mechanical injury and invasion by parasitic fungi', 'Since it has a protective role to play, cells of epidermal tissue form a continuous layer without intercellular spaces', 'Most epidermal cells are relatively flat', 'Often their outer and side walls are thicker than the inner wall.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['(a) (b) : Guard cells and epidermal cells: (a) lateral view, (b) surface view']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['We can observe small pores here and there in the epidermis of the leaf', 'These pores are called stomata', 'Stomata are enclosed by two kidney-shaped cells called guard cells', 'They are necessary for exchanging gases with the atmosphere', 'Transpiration (loss of water in the form ofwater vapour) also takes place through stomata', 'Recall which gas is required for photosynthesis', 'Find out the role of transpiration in plants', 'Epidermal cells of the roots, whose function is water absorption, commonly bear long hair-like parts that greatly increase the total absorptive surface area', 'In some plants like desert plants, epidermis has a thick waxy coating of cutin (chemical substance with waterproof quality) on its outer surface', 'Can we think of a reason for this? Is the outer layer of a branch of a tree different from the outer layer of a young stem? As plants grow older, the outer protective tissue undergoes certain changes', 'A strip of secondary meristem located in the cortex forms layers of cells which constitute the cork', 'Cells of cork are dead and compactly arranged without intercellular spaces', 'They also have a substance called suberin in their walls that makes them impervious to gases and water.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.2 Complex permanent tissue The different types of tissues we have discussed until now are all made of one type of cells, which look like each other', 'Such tissues are called simple permanent tissue', 'Yet another type of permanent tissue is complex tissue', 'Complex tissues are made of more than one type of cells', 'All these cells coordinate to perform a common function', 'Xylem and phloem are examples of such complex tissues', 'They are both conducting tissues and constitute a vascular bundle', 'Vascular tissue is a distinctive feature of the complex plants, one that has made possible their survival in the terrestrial environment', 'In showing a section of stem, can you see different types of cells in the vascular bundle? Xylem consists of tracheids, vessels, xylem parenchyma and xylem fibres', 'Tracheids and vessels have thick walls, and many are dead cells when mature', 'Tracheids and vessels are tubular structures', 'This allows them to transport water and minerals vertically', 'The parenchyma stores food', 'Xylem fibres are mainly supportive in function', 'Phloem is made up of five types of cells: sieve cells, sieve tubes, companion cells, phloem fibres and the phloem parenchyma', 'Sieve tubes are tubular cells with perforated walls', 'Phloem transports food from leaves to other parts of the plant', 'Except phloem fibres, other phloem cells are living cells.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Questions']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['1. Name types of simple tissues', '2. Where is apical meristem found? 3. Which tissue makes up the husk of coconut? 4. What are the constituents of phloem?']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3 Animal Tissues']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['When we breathe we can actually feel the movement of our chest', 'How do these body parts move? For this we have specialised cells called muscle cells', 'The contraction and relaxation of these cells result in movement.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['During breathing we inhale oxygen', 'Where does this oxygen go? It is absorbed in the lungs and then is transported to all the body cells through blood', 'Why would cells need oxygen? The functions of mitochondria we studied earlier provide a clue to this question', 'Blood flows and carries various substances from one part of the body to the other', 'For example, it carries oxygen and food to all cells', 'It also collects wastes from all parts of the body and carries them to the liver and kidney for disposal', 'Blood and muscles are both examples of tissues found in our body', 'On the basis of the functions they perform we can think of different types of animal tissues, such as epithelial tissue, connective tissue, muscular tissue and nervous tissue', 'Blood is a type of connective tissue, and muscle forms muscular tissue.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.1 Epithelial tissue The covering or protective tissues in the animal body are epithelial tissues', 'Epithelium covers most organs and cavities within the body', 'It also forms a barrier to keep different body systems separate', 'The skin, the lining of the mouth, the lining of blood vessels, lung alveoli and kidney tubules are all made of epithelial tissue', 'Epithelial tissue cells are tightly packed and form a continuous sheet', 'They have only a small amount of cementing material between them and almost no intercellular spaces', 'Obviously, anything entering or leaving the body must cross at least one layer of epithelium', 'As a result, the permeability of the cells of various epithelia play an important role in regulating the exchange of materials between the body and the external environment and also between different parts of the body', 'Regardless of the type, all epithelium is usually separated from the underlying tissue by an extracellular fibrous basement membrane.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Different epithelia show differing structures that correlate with their unique functions', 'For example, in cells lining blood vessels or lung alveoli, where transportation of substances occurs through a selectively permeable surface, there is a simple flat kind of epithelium', 'This is called the simple squamous epithelium (squama means scale of skin)', 'Simple squamous epithelial cells are extremely thin and flat and form a delicate lining', 'The oesophagus and the lining of the mouth are also covered with squamous epithelium', 'The skin, which protects the body, is also made of squamous epithelium', 'Skin epithelial cells are arranged in many layers to prevent wear and tear', 'Since they are arranged in a pattern of layers, the epithelium is called stratified squamous epithelium', 'Where absorption and secretion occur, as in the inner lining of the intestine, tall epithelial cells are present', 'This columnar (meaning ‘pillar-like’) epithelium facilitates movement across the epithelial barrier', 'In the respiratory tract, the columnar epithelial tissue also has cilia, which are hair-like projections on the outer surfaces of epithelial cells', 'These cilia can move, and their movement pushes the mucus forward to clear it', 'This type of epithelium is thus ciliated columnar epithelium', 'Cuboidal epithelium (with cube-shaped cells) forms the lining of kidney tubules and ducts of salivary glands, where it provides mechanical support', 'Epithelial cells often acquire additional specialisation as gland cells, which can secrete substances at the epithelial surface', 'Sometimes a portion of the epithelial tissue folds inward, and a multicellular gland is formed', 'This is glandular epithelium.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.2 Connective tissue Blood is a type of connective tissue', 'Why would it be called ‘connective’ tissue? A clue is provided in the introduction of this chapter! Now, let us look at this type of tissue in some more detail', 'The cells of connective tissue are loosely spaced and embedded in an intercellular matrix', 'The matrix may be jelly like, fluid, dense or rigid', 'The nature of matrix differs in concordance with the function of the particular connective tissue.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Activity 6.4 Take a drop of blood on a slide and observe different cells present in it under a microscope.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Blood has a fluid (liquid) matrix called plasma, in which red blood corpuscles (RBCs), white blood corpuscles (WBCs) and platelets are suspended', 'The plasma contains proteins, salts and hormones', 'Blood flows and transports gases, digested food, hormones and waste materials to different parts of the body', 'Bone is another example of a connective tissue', 'It forms the framework that supports the body', 'It also anchors the muscles and supports the main organs of the body', 'It is a strong and nonflexible tissue (what would be the advantage of these properties for bone functions?)', 'Bone cells are embedded in a hard matrix that is composed of calcium and phosphorus compounds', 'Two bones can be connected to each other by another type of connective tissue called the ligament', 'This tissue is very elastic', 'It has considerable strength', 'Ligaments contain very little matrix and connect bones with bones', 'Tendons connect muscles to bones and are another type of connective tissue', 'Tendons are fibrous tissue with great strength but limited flexibility', 'Another type of connective tissue, cartilage, has widely spaced cells', 'The solid matrix is composed of proteins and sugars', 'Cartilage smoothens bone surfaces at joints and is also present in the nose, ear, trachea and larynx', 'We can fold the cartilage of the ears, but we cannot bend the bones in our arms', 'Think of how the two tissues are different! Areolar connective tissue is found between the skin and muscles, around blood vessels and nerves and in the bone marrow', 'It fills the space inside the organs, supports internal organs and helps in repair of tissues', 'Where are fats stored in our body? Fat-storing adipose tissue is found below the skin and between internal organs', 'The cells of this tissue are filled with fat globules', 'Storage of fats also lets it act as an insulator.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Where are fats stored in our body? Fatstoring adipose tissue is found below the skin and between internal organs', 'The cells of this tissue are filled with fat globules', 'Storage of fats also lets it act as an insulator.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.3 Muscular tissue Muscular tissue consists of elongated cells, also called muscle fibres', 'This tissue is responsible for movement in our body', 'Muscles contain special proteins called contractile proteins, which contract and relax to cause movement', 'We can move some muscles by conscious will', 'Muscles present in our limbs move when we want them to, and stop when we so decide', 'Such muscles are called voluntary muscles', 'These muscles are also called skeletal muscles as they are mostly attached to bones and help in body movement', 'Under the microscope, these muscles show alternate light and dark bands or striations when stained appropriately', 'As a result, they are also called striated muscles', 'The cells of this tissue are long, cylindrical, unbranched and multinucleate (having many nuclei)', 'The movement of food in the alimentary canal or the contraction and relaxation of blood vessels are involuntary movements', 'We cannot really start them or stop them simply by wanting to do so! Smooth muscles or involuntary muscles control such movements', 'They are also found in the iris of the eye, in ureters and in the bronchi of the lungs', 'The cells are long with pointed ends (spindle-shaped) and uninucleate (having a single nucleus)', 'They are also called unstriated muscles – why would they be called that? Muscular tissue consists of elongated cells, also called muscle fibres', 'This tissue is responsible for movement in our body. Muscles contain special proteins called contractile proteins, which contract and relax to cause movement']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['We can move some muscles by conscious will', 'Muscles present in our limbs move when we want them to, and stop when we so decide', 'Such muscles are called voluntary muscles', 'These muscles are also called skeletal muscles as they are mostly attached to bones and help in body movement', 'Under the microscope, these muscles show alternate light and dark bands or striations when stained appropriately', 'As a result, they are also called striated muscles', 'The cells of this tissue are long, cylindrical, unbranched and multinucleate (having many nuclei). The movement of food in the alimentary canal or the contraction and relaxation of blood vessels are involuntary movements', 'We cannot really start them or stop them simply by wanting to do so! Smooth muscles or involuntary muscles control such movements', 'They are also found in the iris of the eye, in ureters and in the bronchi of the lungs', 'The cells are long with pointed ends (spindle-shaped) and uninucleate (having a single nucleus)', 'They are also called unstriated muscles – why would they be called that? The muscles of the heart show rhythmic contraction and relaxation throughout life', 'These involuntary muscles are called cardiac muscles', 'Heart muscle cells are cylindrical, branched and uninucleate']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Activity 6.5 Compare the structures of different types of muscular tissues', 'Note down their shape, number of nuclei and position of nuclei within the cell in the Table 6.1.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.4 Nervous tissue All cells possess the ability to respond to stimuli', 'However, cells of the nervous tissue are highly specialised for being stimulated and then transmitting the stimulus very rapidly from one place to another within the body', 'The brain, spinal cord and nerves are all composed of the nervous tissue', 'The cells of this tissue are called nerve cells or neurons', 'A neuron consists of a cell body with a nucleus and cytoplasm, from which long thin hair-like parts arise', 'Usually each neuron has a single long part (process), called the axon, and many short, branched parts (processes) called dendrites', 'An individual nerve cell may be up to a metre long', 'Many nerve fibres bound together by connective tissue make up a nerve.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['The signal that passes along the nerve fibre is called a nerve impulse', 'Nerve impulses allow us to move our muscles when we want to', 'The functional combination of nerve and muscle tissue is fundamental to most animals', 'This combination enables animals to move rapidly in response to stimuli.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Questions 1. Name the tissue responsible for movement in our body', '2. What does a neuron look like? 3. Give three features of cardiac muscles', '4. What are the functions of areolar tissue?']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['What you have learnt • Tissue is a group of cells similar in structure and function', '• Plant tissues are of two main types – meristematic and permanent', '• Meristematic tissue is the dividing tissue present in the growing regions of the plant', '• Permanent tissues are derived from meristematic tissue once they lose the ability to divide', 'They are classified as simple and complex tissues', '• Parenchyma, collenchyma and sclerenchyma are three types of simple tissues', 'Xylem and phloem are types of complex tissues', '• Animal tissues can be epithelial, connective, muscular and nervous tissue', '• Depending on shape and function, epithelial tissue is classified as squamous, cuboidal, columnar, ciliated and glandular', '• The different types of connective tissues in our body include areolar tissue, adipose tissue, bone, tendon, ligament, cartilage and blood', '• Striated, unstriated and cardiac are three types of muscle tissues', '• Nervous tissue is made of neurons that receive and conduct impulses.']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Exercises 1. Define the term “tissue”', '2. How many types of elements together make up the xylem tissue? Name them', '3. How are simple tissues different from complex tissues in plants? 4. Differentiate between parenchyma, collenchyma and sclerenchyma on the basis of their cell wall', '5. What are the functions of the stomata? 6. Diagrammatically show the difference between the three types of muscle fibres', '7. What is the specific function of the cardiac muscle? 8. Differentiate between striated, unstriated and cardiac muscles on the basis of their structure and site/location in the body', '9. Draw a labelled diagram of a neuron', '10. Name the following', '(a) Tissue that forms the inner lining of our mouth', '(b) Tissue that connects muscle to bone in humans', '(c) Tissue that transports food in plants', '(d) Tissue that stores fat in our body', '(e) Connective tissue with a fluid matrix', '(f) Tissue present in the brain', '11. Identify the type of tissue in the following: skin, bark of tree, bone, lining of kidney tubule, vascular bundle', '12. Name the regions in which parenchyma tissue is present', '13. What is the role of epidermis in plants?']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['14. How does the cork act as a protective tissue? 15. Complete the following chart:']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Table of Contents']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Chapter 6']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Tissues']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Questions']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2 Plant Tissues']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.1 Meristematic tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.2 Permanent tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.2 Simple permanent tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.2.2 Complex permanent tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Questions']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3 Animal Tissues']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.1 Epithelial tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.2 Connective tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.3 Muscular tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['6.3.4 Nervous tissue']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Questions']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['What you have learnt']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Exercises']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Landmarks']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Cover']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': ['Chapter 6Tissues']}, {'title': 'From the last chapter, we recall that all living organisms are made of cells. In unicellular organisms, a single cell performs all basic functions. For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion. But in multi-cellular organisms there are millions of cells. Most of these cells are specialised to carry out specific functions. Each specialised function is taken up by a different group of cells. Since these cells carry out only a particular function, they do it very efficiently. In human beings, muscle cells contract and relax to cause movement, nerve cells carry messages, blood flows to transport oxygen, food, hormones and waste material and so on. In plants, vascular tissues conduct food and water from one part of the plant to other parts. So, multi-cellular organisms show division of labour. Cells specialising in one function are often grouped together in the body. This means that a particular function is carried out by a cluster of cells at a definite place in the body. This cluster of cells, called a tissue, is arranged and designed so as to give the highest possible efficiency of function. Blood, phloem and muscle are all examples of tissues. Chapter 6', 'text': []}]\n"
      ]
     }
    ],
diff --git a/.ipynb_checkpoints/model-checkpoint.ipynb b/.ipynb_checkpoints/model-checkpoint.ipynb
index 6ec376d..646a6d7 100644
--- a/.ipynb_checkpoints/model-checkpoint.ipynb
+++ b/.ipynb_checkpoints/model-checkpoint.ipynb
@@ -117,145 +117,145 @@
       "WARNING:tensorflow:From /home/pranav_kirsur/.pyenv/versions/3.6.8/envs/megathon2019/lib/python3.6/site-packages/keras/backend/tensorflow_backend.py:422: The name tf.global_variables is deprecated. Please use tf.compat.v1.global_variables instead.\n",
       "\n",
       "Epoch 1/300\n",
-      "1100/1100 [==============================] - 0s 165us/step - loss: 1.3875 - mean_absolute_error: 0.4967\n",
+      "1100/1100 [==============================] - 0s 154us/step - loss: 1.3119 - mean_absolute_error: 0.4619\n",
       "Epoch 2/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.3781 - mean_absolute_error: 0.4943\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.3004 - mean_absolute_error: 0.4579\n",
       "Epoch 3/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3686 - mean_absolute_error: 0.4919\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.2886 - mean_absolute_error: 0.4537\n",
       "Epoch 4/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3590 - mean_absolute_error: 0.4894\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.2767 - mean_absolute_error: 0.4493\n",
       "Epoch 5/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.3498 - mean_absolute_error: 0.4870\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.2646 - mean_absolute_error: 0.4448\n",
       "Epoch 6/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.3406 - mean_absolute_error: 0.4847\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2522 - mean_absolute_error: 0.4401\n",
       "Epoch 7/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 1.3314 - mean_absolute_error: 0.4823\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.2398 - mean_absolute_error: 0.4352\n",
       "Epoch 8/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.3225 - mean_absolute_error: 0.4799\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 1.2272 - mean_absolute_error: 0.4301\n",
       "Epoch 9/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3133 - mean_absolute_error: 0.4775\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2143 - mean_absolute_error: 0.4248\n",
       "Epoch 10/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3040 - mean_absolute_error: 0.4749\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.2013 - mean_absolute_error: 0.4193\n",
       "Epoch 11/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2945 - mean_absolute_error: 0.4723\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1881 - mean_absolute_error: 0.4137\n",
       "Epoch 12/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.2850 - mean_absolute_error: 0.4695\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 1.1749 - mean_absolute_error: 0.4079\n",
       "Epoch 13/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2753 - mean_absolute_error: 0.4667\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1616 - mean_absolute_error: 0.4019\n",
       "Epoch 14/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2655 - mean_absolute_error: 0.4637\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1482 - mean_absolute_error: 0.3958\n",
       "Epoch 15/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2557 - mean_absolute_error: 0.4607\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1348 - mean_absolute_error: 0.3895\n",
       "Epoch 16/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.2458 - mean_absolute_error: 0.4576\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1213 - mean_absolute_error: 0.3831\n",
       "Epoch 17/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.2359 - mean_absolute_error: 0.4545\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1078 - mean_absolute_error: 0.3766\n",
       "Epoch 18/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.2259 - mean_absolute_error: 0.4512\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0942 - mean_absolute_error: 0.3699\n",
       "Epoch 19/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2157 - mean_absolute_error: 0.4478\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0807 - mean_absolute_error: 0.3631\n",
       "Epoch 20/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2055 - mean_absolute_error: 0.4444\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0672 - mean_absolute_error: 0.3562\n",
       "Epoch 21/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1953 - mean_absolute_error: 0.4408\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0536 - mean_absolute_error: 0.3492\n",
       "Epoch 22/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1849 - mean_absolute_error: 0.4372\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0401 - mean_absolute_error: 0.3420\n",
       "Epoch 23/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1746 - mean_absolute_error: 0.4334\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0267 - mean_absolute_error: 0.3348\n",
       "Epoch 24/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.1642 - mean_absolute_error: 0.4296\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0132 - mean_absolute_error: 0.3274\n",
       "Epoch 25/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1537 - mean_absolute_error: 0.4257\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9998 - mean_absolute_error: 0.3200\n",
       "Epoch 26/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 1.1431 - mean_absolute_error: 0.4217\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9865 - mean_absolute_error: 0.3124\n",
       "Epoch 27/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1325 - mean_absolute_error: 0.4176\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9733 - mean_absolute_error: 0.3048\n",
       "Epoch 28/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1218 - mean_absolute_error: 0.4134\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9602 - mean_absolute_error: 0.2971\n",
       "Epoch 29/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1110 - mean_absolute_error: 0.4090\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9472 - mean_absolute_error: 0.2893\n",
       "Epoch 30/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1001 - mean_absolute_error: 0.4046\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9344 - mean_absolute_error: 0.2816\n",
       "Epoch 31/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0892 - mean_absolute_error: 0.4000\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9217 - mean_absolute_error: 0.2738\n",
       "Epoch 32/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.0780 - mean_absolute_error: 0.3952\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.9093 - mean_absolute_error: 0.2659\n",
       "Epoch 33/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0669 - mean_absolute_error: 0.3904\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.8970 - mean_absolute_error: 0.2581\n",
       "Epoch 34/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0557 - mean_absolute_error: 0.3854\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.8849 - mean_absolute_error: 0.2504\n",
       "Epoch 35/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0444 - mean_absolute_error: 0.3803\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8731 - mean_absolute_error: 0.2426\n",
       "Epoch 36/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0330 - mean_absolute_error: 0.3750\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8616 - mean_absolute_error: 0.2350\n",
       "Epoch 37/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0216 - mean_absolute_error: 0.3697\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8503 - mean_absolute_error: 0.2274\n",
       "Epoch 38/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0100 - mean_absolute_error: 0.3641\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8394 - mean_absolute_error: 0.2199\n",
       "Epoch 39/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.9985 - mean_absolute_error: 0.3585\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8288 - mean_absolute_error: 0.2126\n",
       "Epoch 40/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.9868 - mean_absolute_error: 0.3527\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.8185 - mean_absolute_error: 0.2054\n",
       "Epoch 41/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.9750 - mean_absolute_error: 0.3467\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.8086 - mean_absolute_error: 0.1983\n",
       "Epoch 42/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.9633 - mean_absolute_error: 0.3406\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7990 - mean_absolute_error: 0.1915\n",
       "Epoch 43/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.9516 - mean_absolute_error: 0.3344\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7898 - mean_absolute_error: 0.1849\n",
       "Epoch 44/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.9398 - mean_absolute_error: 0.3281\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7810 - mean_absolute_error: 0.1784\n",
       "Epoch 45/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.9280 - mean_absolute_error: 0.3217\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7725 - mean_absolute_error: 0.1723\n",
       "Epoch 46/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9162 - mean_absolute_error: 0.3151\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7645 - mean_absolute_error: 0.1663\n",
       "Epoch 47/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.9044 - mean_absolute_error: 0.3084\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.7568 - mean_absolute_error: 0.1606\n",
       "Epoch 48/300\n",
-      "1100/1100 [==============================] - 0s 8us/step - loss: 0.8926 - mean_absolute_error: 0.3017\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7495 - mean_absolute_error: 0.1552\n",
       "Epoch 49/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.8809 - mean_absolute_error: 0.2948\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7425 - mean_absolute_error: 0.1501\n",
       "Epoch 50/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.8693 - mean_absolute_error: 0.2879\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7358 - mean_absolute_error: 0.1452\n",
       "Epoch 51/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8577 - mean_absolute_error: 0.2809\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.7295 - mean_absolute_error: 0.1406\n",
       "Epoch 52/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8463 - mean_absolute_error: 0.2738\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7235 - mean_absolute_error: 0.1363\n",
       "Epoch 53/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8349 - mean_absolute_error: 0.2667\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7178 - mean_absolute_error: 0.1322\n",
       "Epoch 54/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.8238 - mean_absolute_error: 0.2596\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7124 - mean_absolute_error: 0.1284\n",
       "Epoch 55/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8127 - mean_absolute_error: 0.2524\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.7072 - mean_absolute_error: 0.1248\n",
       "Epoch 56/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.8019 - mean_absolute_error: 0.2454\n",
+      "1100/1100 [==============================] - 0s 7us/step - loss: 0.7022 - mean_absolute_error: 0.1215\n",
       "Epoch 57/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7913 - mean_absolute_error: 0.2383\n",
+      "1100/1100 [==============================] - 0s 8us/step - loss: 0.6975 - mean_absolute_error: 0.1184\n",
       "Epoch 58/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.7809 - mean_absolute_error: 0.2313\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6930 - mean_absolute_error: 0.1156\n",
       "Epoch 59/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7707 - mean_absolute_error: 0.2243\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6886 - mean_absolute_error: 0.1130\n",
       "Epoch 60/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7609 - mean_absolute_error: 0.2175\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6844 - mean_absolute_error: 0.1106\n",
       "Epoch 61/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7512 - mean_absolute_error: 0.2107\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6803 - mean_absolute_error: 0.1084\n",
       "Epoch 62/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7419 - mean_absolute_error: 0.2041\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6764 - mean_absolute_error: 0.1063\n",
       "Epoch 63/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7329 - mean_absolute_error: 0.1976\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6725 - mean_absolute_error: 0.1045\n",
       "Epoch 64/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7241 - mean_absolute_error: 0.1912\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6688 - mean_absolute_error: 0.1028\n",
       "Epoch 65/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7157 - mean_absolute_error: 0.1851\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6651 - mean_absolute_error: 0.1013\n",
       "Epoch 66/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7076 - mean_absolute_error: 0.1791\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6615 - mean_absolute_error: 0.0999\n",
       "Epoch 67/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6999 - mean_absolute_error: 0.1733\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6580 - mean_absolute_error: 0.0987\n",
       "Epoch 68/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6924 - mean_absolute_error: 0.1677\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6545 - mean_absolute_error: 0.0975\n",
       "Epoch 69/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6853 - mean_absolute_error: 0.1623\n",
+      "1100/1100 [==============================] - 0s 8us/step - loss: 0.6511 - mean_absolute_error: 0.0966\n",
       "Epoch 70/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6784 - mean_absolute_error: 0.1572\n"
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6477 - mean_absolute_error: 0.0957\n"
      ]
     },
     {
@@ -263,147 +263,147 @@
      "output_type": "stream",
      "text": [
       "Epoch 71/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6719 - mean_absolute_error: 0.1522\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6443 - mean_absolute_error: 0.0949\n",
       "Epoch 72/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6657 - mean_absolute_error: 0.1475\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6410 - mean_absolute_error: 0.0942\n",
       "Epoch 73/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6598 - mean_absolute_error: 0.1430\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6377 - mean_absolute_error: 0.0936\n",
       "Epoch 74/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6541 - mean_absolute_error: 0.1388\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6345 - mean_absolute_error: 0.0931\n",
       "Epoch 75/300\n",
-      "1100/1100 [==============================] - 0s 9us/step - loss: 0.6487 - mean_absolute_error: 0.1347\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6312 - mean_absolute_error: 0.0927\n",
       "Epoch 76/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6436 - mean_absolute_error: 0.1309\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6280 - mean_absolute_error: 0.0923\n",
       "Epoch 77/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6387 - mean_absolute_error: 0.1274\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6248 - mean_absolute_error: 0.0920\n",
       "Epoch 78/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6340 - mean_absolute_error: 0.1240\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6217 - mean_absolute_error: 0.0918\n",
       "Epoch 79/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6295 - mean_absolute_error: 0.1209\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6185 - mean_absolute_error: 0.0916\n",
       "Epoch 80/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6252 - mean_absolute_error: 0.1179\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6154 - mean_absolute_error: 0.0915\n",
       "Epoch 81/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6211 - mean_absolute_error: 0.1152\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6123 - mean_absolute_error: 0.0914\n",
       "Epoch 82/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6172 - mean_absolute_error: 0.1127\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6092 - mean_absolute_error: 0.0913\n",
       "Epoch 83/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6134 - mean_absolute_error: 0.1103\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6062 - mean_absolute_error: 0.0913\n",
       "Epoch 84/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6097 - mean_absolute_error: 0.1081\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6031 - mean_absolute_error: 0.0913\n",
       "Epoch 85/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6061 - mean_absolute_error: 0.1061\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6001 - mean_absolute_error: 0.0913\n",
       "Epoch 86/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6027 - mean_absolute_error: 0.1042\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5971 - mean_absolute_error: 0.0914\n",
       "Epoch 87/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5993 - mean_absolute_error: 0.1025\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5941 - mean_absolute_error: 0.0915\n",
       "Epoch 88/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5960 - mean_absolute_error: 0.1010\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5912 - mean_absolute_error: 0.0916\n",
       "Epoch 89/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5928 - mean_absolute_error: 0.0996\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5883 - mean_absolute_error: 0.0917\n",
       "Epoch 90/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5897 - mean_absolute_error: 0.0983\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5854 - mean_absolute_error: 0.0919\n",
       "Epoch 91/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5866 - mean_absolute_error: 0.0971\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5825 - mean_absolute_error: 0.0920\n",
       "Epoch 92/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5836 - mean_absolute_error: 0.0960\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5796 - mean_absolute_error: 0.0922\n",
       "Epoch 93/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5806 - mean_absolute_error: 0.0950\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5768 - mean_absolute_error: 0.0923\n",
       "Epoch 94/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5776 - mean_absolute_error: 0.0942\n",
+      "1100/1100 [==============================] - 0s 7us/step - loss: 0.5740 - mean_absolute_error: 0.0925\n",
       "Epoch 95/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5747 - mean_absolute_error: 0.0934\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5712 - mean_absolute_error: 0.0926\n",
       "Epoch 96/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5719 - mean_absolute_error: 0.0927\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5684 - mean_absolute_error: 0.0928\n",
       "Epoch 97/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5690 - mean_absolute_error: 0.0921\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5657 - mean_absolute_error: 0.0930\n",
       "Epoch 98/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5662 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5630 - mean_absolute_error: 0.0931\n",
       "Epoch 99/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5634 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5603 - mean_absolute_error: 0.0933\n",
       "Epoch 100/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5606 - mean_absolute_error: 0.0908\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5576 - mean_absolute_error: 0.0934\n",
       "Epoch 101/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5578 - mean_absolute_error: 0.0904\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5549 - mean_absolute_error: 0.0936\n",
       "Epoch 102/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5551 - mean_absolute_error: 0.0901\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5523 - mean_absolute_error: 0.0937\n",
       "Epoch 103/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5523 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5497 - mean_absolute_error: 0.0938\n",
       "Epoch 104/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5496 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5471 - mean_absolute_error: 0.0940\n",
       "Epoch 105/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5469 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5445 - mean_absolute_error: 0.0941\n",
       "Epoch 106/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5442 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5420 - mean_absolute_error: 0.0942\n",
       "Epoch 107/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5416 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5394 - mean_absolute_error: 0.0943\n",
       "Epoch 108/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5389 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5369 - mean_absolute_error: 0.0943\n",
       "Epoch 109/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5363 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.5344 - mean_absolute_error: 0.0944\n",
       "Epoch 110/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5337 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5319 - mean_absolute_error: 0.0945\n",
       "Epoch 111/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5311 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5294 - mean_absolute_error: 0.0945\n",
       "Epoch 112/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5285 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5270 - mean_absolute_error: 0.0945\n",
       "Epoch 113/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5260 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5246 - mean_absolute_error: 0.0946\n",
       "Epoch 114/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5235 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5222 - mean_absolute_error: 0.0946\n",
       "Epoch 115/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5210 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5198 - mean_absolute_error: 0.0946\n",
       "Epoch 116/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5185 - mean_absolute_error: 0.0898\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5174 - mean_absolute_error: 0.0946\n",
       "Epoch 117/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5160 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5150 - mean_absolute_error: 0.0946\n",
       "Epoch 118/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5136 - mean_absolute_error: 0.0901\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5127 - mean_absolute_error: 0.0945\n",
       "Epoch 119/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5111 - mean_absolute_error: 0.0902\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5103 - mean_absolute_error: 0.0945\n",
       "Epoch 120/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5087 - mean_absolute_error: 0.0903\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5080 - mean_absolute_error: 0.0945\n",
       "Epoch 121/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5064 - mean_absolute_error: 0.0904\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5057 - mean_absolute_error: 0.0944\n",
       "Epoch 122/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5040 - mean_absolute_error: 0.0905\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5035 - mean_absolute_error: 0.0944\n",
       "Epoch 123/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5016 - mean_absolute_error: 0.0906\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5012 - mean_absolute_error: 0.0943\n",
       "Epoch 124/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4993 - mean_absolute_error: 0.0908\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4990 - mean_absolute_error: 0.0943\n",
       "Epoch 125/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4970 - mean_absolute_error: 0.0909\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4967 - mean_absolute_error: 0.0942\n",
       "Epoch 126/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4947 - mean_absolute_error: 0.0910\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4945 - mean_absolute_error: 0.0941\n",
       "Epoch 127/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4924 - mean_absolute_error: 0.0910\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4924 - mean_absolute_error: 0.0940\n",
       "Epoch 128/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4902 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4902 - mean_absolute_error: 0.0940\n",
       "Epoch 129/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4879 - mean_absolute_error: 0.0912\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4880 - mean_absolute_error: 0.0939\n",
       "Epoch 130/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4857 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4859 - mean_absolute_error: 0.0938\n",
       "Epoch 131/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4835 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4838 - mean_absolute_error: 0.0937\n",
       "Epoch 132/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4813 - mean_absolute_error: 0.0914\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4816 - mean_absolute_error: 0.0936\n",
       "Epoch 133/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4791 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4795 - mean_absolute_error: 0.0936\n",
       "Epoch 134/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.4770 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4775 - mean_absolute_error: 0.0935\n",
       "Epoch 135/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4749 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4754 - mean_absolute_error: 0.0934\n",
       "Epoch 136/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4727 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4734 - mean_absolute_error: 0.0933\n",
       "Epoch 137/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4706 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4713 - mean_absolute_error: 0.0932\n",
       "Epoch 138/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4686 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4693 - mean_absolute_error: 0.0931\n",
       "Epoch 139/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4665 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4673 - mean_absolute_error: 0.0931\n",
       "Epoch 140/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4644 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4653 - mean_absolute_error: 0.0930\n",
       "Epoch 141/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4624 - mean_absolute_error: 0.0916\n"
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4633 - mean_absolute_error: 0.0929\n"
      ]
     },
     {
@@ -411,147 +411,147 @@
      "output_type": "stream",
      "text": [
       "Epoch 142/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4604 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4614 - mean_absolute_error: 0.0928\n",
       "Epoch 143/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4584 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4594 - mean_absolute_error: 0.0927\n",
       "Epoch 144/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4564 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4575 - mean_absolute_error: 0.0927\n",
       "Epoch 145/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4544 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4556 - mean_absolute_error: 0.0926\n",
       "Epoch 146/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4524 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4537 - mean_absolute_error: 0.0925\n",
       "Epoch 147/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4505 - mean_absolute_error: 0.0914\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4518 - mean_absolute_error: 0.0925\n",
       "Epoch 148/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4485 - mean_absolute_error: 0.0914\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4499 - mean_absolute_error: 0.0924\n",
       "Epoch 149/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4466 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4480 - mean_absolute_error: 0.0923\n",
       "Epoch 150/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4447 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4462 - mean_absolute_error: 0.0923\n",
       "Epoch 151/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4428 - mean_absolute_error: 0.0912\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4443 - mean_absolute_error: 0.0922\n",
       "Epoch 152/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4409 - mean_absolute_error: 0.0912\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4425 - mean_absolute_error: 0.0921\n",
       "Epoch 153/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4391 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4407 - mean_absolute_error: 0.0921\n",
       "Epoch 154/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4372 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4389 - mean_absolute_error: 0.0920\n",
       "Epoch 155/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4354 - mean_absolute_error: 0.0910\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4371 - mean_absolute_error: 0.0919\n",
       "Epoch 156/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4336 - mean_absolute_error: 0.0909\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4354 - mean_absolute_error: 0.0919\n",
       "Epoch 157/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4318 - mean_absolute_error: 0.0909\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4336 - mean_absolute_error: 0.0918\n",
       "Epoch 158/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4300 - mean_absolute_error: 0.0908\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4318 - mean_absolute_error: 0.0918\n",
       "Epoch 159/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4282 - mean_absolute_error: 0.0907\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4301 - mean_absolute_error: 0.0917\n",
       "Epoch 160/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4264 - mean_absolute_error: 0.0907\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4284 - mean_absolute_error: 0.0917\n",
       "Epoch 161/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4247 - mean_absolute_error: 0.0906\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4267 - mean_absolute_error: 0.0916\n",
       "Epoch 162/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4229 - mean_absolute_error: 0.0905\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4250 - mean_absolute_error: 0.0916\n",
       "Epoch 163/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4212 - mean_absolute_error: 0.0905\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4233 - mean_absolute_error: 0.0915\n",
       "Epoch 164/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4195 - mean_absolute_error: 0.0904\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.4216 - mean_absolute_error: 0.0915\n",
       "Epoch 165/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4178 - mean_absolute_error: 0.0903\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4200 - mean_absolute_error: 0.0914\n",
       "Epoch 166/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4161 - mean_absolute_error: 0.0903\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4183 - mean_absolute_error: 0.0914\n",
       "Epoch 167/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4144 - mean_absolute_error: 0.0902\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4167 - mean_absolute_error: 0.0913\n",
       "Epoch 168/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4128 - mean_absolute_error: 0.0902\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4151 - mean_absolute_error: 0.0913\n",
       "Epoch 169/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4111 - mean_absolute_error: 0.0901\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4135 - mean_absolute_error: 0.0913\n",
       "Epoch 170/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4095 - mean_absolute_error: 0.0900\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4119 - mean_absolute_error: 0.0912\n",
       "Epoch 171/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4079 - mean_absolute_error: 0.0900\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4103 - mean_absolute_error: 0.0912\n",
       "Epoch 172/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4063 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4087 - mean_absolute_error: 0.0911\n",
       "Epoch 173/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4047 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4071 - mean_absolute_error: 0.0911\n",
       "Epoch 174/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4031 - mean_absolute_error: 0.0898\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4056 - mean_absolute_error: 0.0910\n",
       "Epoch 175/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4015 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4040 - mean_absolute_error: 0.0910\n",
       "Epoch 176/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3999 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4025 - mean_absolute_error: 0.0910\n",
       "Epoch 177/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3984 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4010 - mean_absolute_error: 0.0909\n",
       "Epoch 178/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3968 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3995 - mean_absolute_error: 0.0909\n",
       "Epoch 179/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3953 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3980 - mean_absolute_error: 0.0909\n",
       "Epoch 180/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3938 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.3965 - mean_absolute_error: 0.0908\n",
       "Epoch 181/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3923 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3950 - mean_absolute_error: 0.0908\n",
       "Epoch 182/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3908 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.3936 - mean_absolute_error: 0.0907\n",
       "Epoch 183/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3893 - mean_absolute_error: 0.0893\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3921 - mean_absolute_error: 0.0907\n",
       "Epoch 184/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3878 - mean_absolute_error: 0.0893\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3907 - mean_absolute_error: 0.0907\n",
       "Epoch 185/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3864 - mean_absolute_error: 0.0893\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3892 - mean_absolute_error: 0.0906\n",
       "Epoch 186/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3849 - mean_absolute_error: 0.0892\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3878 - mean_absolute_error: 0.0906\n",
       "Epoch 187/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3835 - mean_absolute_error: 0.0892\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3864 - mean_absolute_error: 0.0906\n",
       "Epoch 188/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3821 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3850 - mean_absolute_error: 0.0905\n",
       "Epoch 189/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3807 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.3836 - mean_absolute_error: 0.0905\n",
       "Epoch 190/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3793 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3823 - mean_absolute_error: 0.0905\n",
       "Epoch 191/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3779 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3809 - mean_absolute_error: 0.0904\n",
       "Epoch 192/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3765 - mean_absolute_error: 0.0890\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3795 - mean_absolute_error: 0.0904\n",
       "Epoch 193/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3751 - mean_absolute_error: 0.0890\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3782 - mean_absolute_error: 0.0904\n",
       "Epoch 194/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3737 - mean_absolute_error: 0.0890\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3768 - mean_absolute_error: 0.0903\n",
       "Epoch 195/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3724 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3755 - mean_absolute_error: 0.0903\n",
       "Epoch 196/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3711 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3742 - mean_absolute_error: 0.0903\n",
       "Epoch 197/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3697 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3729 - mean_absolute_error: 0.0902\n",
       "Epoch 198/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3684 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3716 - mean_absolute_error: 0.0902\n",
       "Epoch 199/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3671 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3703 - mean_absolute_error: 0.0902\n",
       "Epoch 200/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3658 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3690 - mean_absolute_error: 0.0901\n",
       "Epoch 201/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3645 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3678 - mean_absolute_error: 0.0901\n",
       "Epoch 202/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3632 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3665 - mean_absolute_error: 0.0901\n",
       "Epoch 203/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3620 - mean_absolute_error: 0.0887\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3653 - mean_absolute_error: 0.0900\n",
       "Epoch 204/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3607 - mean_absolute_error: 0.0887\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.3640 - mean_absolute_error: 0.0900\n",
       "Epoch 205/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3594 - mean_absolute_error: 0.0887\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3628 - mean_absolute_error: 0.0900\n",
       "Epoch 206/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3582 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3616 - mean_absolute_error: 0.0900\n",
       "Epoch 207/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3570 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3604 - mean_absolute_error: 0.0899\n",
       "Epoch 208/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3557 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3592 - mean_absolute_error: 0.0899\n",
       "Epoch 209/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3545 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3580 - mean_absolute_error: 0.0899\n",
       "Epoch 210/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3533 - mean_absolute_error: 0.0885\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3568 - mean_absolute_error: 0.0898\n",
       "Epoch 211/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3521 - mean_absolute_error: 0.0885\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3556 - mean_absolute_error: 0.0898\n",
       "Epoch 212/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3509 - mean_absolute_error: 0.0885\n"
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3544 - mean_absolute_error: 0.0898\n"
      ]
     },
     {
@@ -559,147 +559,147 @@
      "output_type": "stream",
      "text": [
       "Epoch 213/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3498 - mean_absolute_error: 0.0885\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3533 - mean_absolute_error: 0.0897\n",
       "Epoch 214/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3486 - mean_absolute_error: 0.0884\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3521 - mean_absolute_error: 0.0897\n",
       "Epoch 215/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3474 - mean_absolute_error: 0.0884\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3510 - mean_absolute_error: 0.0897\n",
       "Epoch 216/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3463 - mean_absolute_error: 0.0884\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3499 - mean_absolute_error: 0.0897\n",
       "Epoch 217/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3451 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3487 - mean_absolute_error: 0.0896\n",
       "Epoch 218/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3440 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3476 - mean_absolute_error: 0.0896\n",
       "Epoch 219/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3429 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3465 - mean_absolute_error: 0.0896\n",
       "Epoch 220/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3418 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3454 - mean_absolute_error: 0.0895\n",
       "Epoch 221/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3407 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3443 - mean_absolute_error: 0.0895\n",
       "Epoch 222/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3396 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3432 - mean_absolute_error: 0.0895\n",
       "Epoch 223/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3385 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3422 - mean_absolute_error: 0.0895\n",
       "Epoch 224/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3374 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3411 - mean_absolute_error: 0.0894\n",
       "Epoch 225/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3363 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3401 - mean_absolute_error: 0.0894\n",
       "Epoch 226/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3353 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3390 - mean_absolute_error: 0.0894\n",
       "Epoch 227/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3342 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3380 - mean_absolute_error: 0.0894\n",
       "Epoch 228/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3332 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3369 - mean_absolute_error: 0.0893\n",
       "Epoch 229/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3321 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3359 - mean_absolute_error: 0.0893\n",
       "Epoch 230/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3311 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3349 - mean_absolute_error: 0.0893\n",
       "Epoch 231/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3301 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3339 - mean_absolute_error: 0.0892\n",
       "Epoch 232/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3290 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3329 - mean_absolute_error: 0.0892\n",
       "Epoch 233/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3280 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3319 - mean_absolute_error: 0.0892\n",
       "Epoch 234/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3270 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3309 - mean_absolute_error: 0.0892\n",
       "Epoch 235/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3260 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3299 - mean_absolute_error: 0.0892\n",
       "Epoch 236/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3251 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3290 - mean_absolute_error: 0.0891\n",
       "Epoch 237/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3241 - mean_absolute_error: 0.0878\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3280 - mean_absolute_error: 0.0891\n",
       "Epoch 238/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3231 - mean_absolute_error: 0.0878\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3270 - mean_absolute_error: 0.0891\n",
       "Epoch 239/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3221 - mean_absolute_error: 0.0878\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3261 - mean_absolute_error: 0.0891\n",
       "Epoch 240/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3212 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3251 - mean_absolute_error: 0.0890\n",
       "Epoch 241/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3202 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3242 - mean_absolute_error: 0.0890\n",
       "Epoch 242/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3193 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3233 - mean_absolute_error: 0.0890\n",
       "Epoch 243/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3184 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3224 - mean_absolute_error: 0.0890\n",
       "Epoch 244/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3174 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3214 - mean_absolute_error: 0.0889\n",
       "Epoch 245/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3165 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3205 - mean_absolute_error: 0.0889\n",
       "Epoch 246/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3156 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3196 - mean_absolute_error: 0.0889\n",
       "Epoch 247/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3147 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3187 - mean_absolute_error: 0.0889\n",
       "Epoch 248/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3138 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3179 - mean_absolute_error: 0.0889\n",
       "Epoch 249/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3129 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3170 - mean_absolute_error: 0.0888\n",
       "Epoch 250/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3120 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3161 - mean_absolute_error: 0.0888\n",
       "Epoch 251/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3111 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3152 - mean_absolute_error: 0.0888\n",
       "Epoch 252/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3103 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3144 - mean_absolute_error: 0.0888\n",
       "Epoch 253/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3094 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3135 - mean_absolute_error: 0.0887\n",
       "Epoch 254/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3085 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3127 - mean_absolute_error: 0.0887\n",
       "Epoch 255/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3077 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3118 - mean_absolute_error: 0.0887\n",
       "Epoch 256/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3069 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3110 - mean_absolute_error: 0.0887\n",
       "Epoch 257/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3060 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3102 - mean_absolute_error: 0.0886\n",
       "Epoch 258/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3052 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3094 - mean_absolute_error: 0.0886\n",
       "Epoch 259/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3044 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3085 - mean_absolute_error: 0.0886\n",
       "Epoch 260/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3035 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3077 - mean_absolute_error: 0.0886\n",
       "Epoch 261/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3027 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3069 - mean_absolute_error: 0.0886\n",
       "Epoch 262/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3019 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3061 - mean_absolute_error: 0.0885\n",
       "Epoch 263/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3011 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3053 - mean_absolute_error: 0.0885\n",
       "Epoch 264/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3003 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3045 - mean_absolute_error: 0.0885\n",
       "Epoch 265/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2995 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3038 - mean_absolute_error: 0.0885\n",
       "Epoch 266/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2988 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3030 - mean_absolute_error: 0.0884\n",
       "Epoch 267/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2980 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3022 - mean_absolute_error: 0.0884\n",
       "Epoch 268/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2972 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3015 - mean_absolute_error: 0.0884\n",
       "Epoch 269/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2964 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3007 - mean_absolute_error: 0.0884\n",
       "Epoch 270/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2957 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3000 - mean_absolute_error: 0.0883\n",
       "Epoch 271/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2949 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2992 - mean_absolute_error: 0.0883\n",
       "Epoch 272/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2942 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2985 - mean_absolute_error: 0.0883\n",
       "Epoch 273/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2934 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2977 - mean_absolute_error: 0.0883\n",
       "Epoch 274/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2927 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2970 - mean_absolute_error: 0.0882\n",
       "Epoch 275/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2920 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2963 - mean_absolute_error: 0.0882\n",
       "Epoch 276/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2913 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2956 - mean_absolute_error: 0.0882\n",
       "Epoch 277/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2905 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2948 - mean_absolute_error: 0.0882\n",
       "Epoch 278/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2898 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2941 - mean_absolute_error: 0.0881\n",
       "Epoch 279/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2891 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2934 - mean_absolute_error: 0.0881\n",
       "Epoch 280/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2884 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2927 - mean_absolute_error: 0.0881\n",
       "Epoch 281/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2877 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2920 - mean_absolute_error: 0.0881\n",
       "Epoch 282/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2870 - mean_absolute_error: 0.0868\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2914 - mean_absolute_error: 0.0881\n",
       "Epoch 283/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2863 - mean_absolute_error: 0.0868\n"
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2907 - mean_absolute_error: 0.0880\n"
      ]
     },
     {
@@ -707,45 +707,45 @@
      "output_type": "stream",
      "text": [
       "Epoch 284/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2856 - mean_absolute_error: 0.0868\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2900 - mean_absolute_error: 0.0880\n",
       "Epoch 285/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2850 - mean_absolute_error: 0.0867\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2893 - mean_absolute_error: 0.0880\n",
       "Epoch 286/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2843 - mean_absolute_error: 0.0867\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2887 - mean_absolute_error: 0.0880\n",
       "Epoch 287/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2836 - mean_absolute_error: 0.0866\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2880 - mean_absolute_error: 0.0879\n",
       "Epoch 288/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2830 - mean_absolute_error: 0.0866\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2873 - mean_absolute_error: 0.0879\n",
       "Epoch 289/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2823 - mean_absolute_error: 0.0866\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2867 - mean_absolute_error: 0.0879\n",
       "Epoch 290/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2816 - mean_absolute_error: 0.0865\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2860 - mean_absolute_error: 0.0879\n",
       "Epoch 291/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2810 - mean_absolute_error: 0.0865\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2854 - mean_absolute_error: 0.0879\n",
       "Epoch 292/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2803 - mean_absolute_error: 0.0865\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2848 - mean_absolute_error: 0.0878\n",
       "Epoch 293/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2797 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2841 - mean_absolute_error: 0.0878\n",
       "Epoch 294/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2791 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2835 - mean_absolute_error: 0.0878\n",
       "Epoch 295/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2784 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2829 - mean_absolute_error: 0.0878\n",
       "Epoch 296/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2778 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2823 - mean_absolute_error: 0.0878\n",
       "Epoch 297/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2772 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2817 - mean_absolute_error: 0.0877\n",
       "Epoch 298/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2766 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2810 - mean_absolute_error: 0.0877\n",
       "Epoch 299/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2760 - mean_absolute_error: 0.0863\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2804 - mean_absolute_error: 0.0877\n",
       "Epoch 300/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2754 - mean_absolute_error: 0.0863\n"
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2798 - mean_absolute_error: 0.0877\n"
      ]
     },
     {
      "data": {
       "text/plain": [
-       "<keras.callbacks.callbacks.History at 0x7f1f34672518>"
+       "<keras.callbacks.callbacks.History at 0x7f9e41c6ffd0>"
       ]
      },
      "execution_count": 6,
@@ -766,8 +766,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "1100/1100 [==============================] - 0s 50us/step\n",
-      "Mean Absolute Error Percentage: 0.006589039818931172\n"
+      "1100/1100 [==============================] - 0s 51us/step\n",
+      "Mean Absolute Error Percentage: 0.006692540895847874\n"
      ]
     }
    ],
@@ -794,13 +794,216 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[0.0627239 ]\n",
-      " [0.07016328]\n",
-      " [0.08498865]\n",
-      " ...\n",
-      " [0.03661832]\n",
-      " [0.03941888]\n",
-      " [0.03665248]]\n"
+      "[[0.05987275]\n",
+      " [0.07047856]\n",
+      " [0.08760342]\n",
+      " [0.04979923]\n",
+      " [0.05166233]\n",
+      " [0.04934117]\n",
+      " [0.07812735]\n",
+      " [0.05524576]\n",
+      " [0.04840016]\n",
+      " [0.0965617 ]\n",
+      " [0.10365805]\n",
+      " [0.08732992]\n",
+      " [0.10197255]\n",
+      " [0.07169345]\n",
+      " [0.08316922]\n",
+      " [0.08781105]\n",
+      " [0.06089982]\n",
+      " [0.05451554]\n",
+      " [0.04658008]\n",
+      " [0.06053025]\n",
+      " [0.07305136]\n",
+      " [0.06277847]\n",
+      " [0.03665629]\n",
+      " [0.098389  ]\n",
+      " [0.0868187 ]\n",
+      " [0.09067354]\n",
+      " [0.10324144]\n",
+      " [0.0848611 ]\n",
+      " [0.09628972]\n",
+      " [0.06930184]\n",
+      " [0.04741266]\n",
+      " [0.0641183 ]\n",
+      " [0.07043156]\n",
+      " [0.08346471]\n",
+      " [0.06404978]\n",
+      " [0.0701538 ]\n",
+      " [0.04513595]\n",
+      " [0.05924329]\n",
+      " [0.06592053]\n",
+      " [0.06626862]\n",
+      " [0.03470093]\n",
+      " [0.04479468]\n",
+      " [0.02683493]\n",
+      " [0.02569604]\n",
+      " [0.07547492]\n",
+      " [0.06036592]\n",
+      " [0.06471756]\n",
+      " [0.10437256]\n",
+      " [0.07415253]\n",
+      " [0.07223606]\n",
+      " [0.08091393]\n",
+      " [0.08793288]\n",
+      " [0.06360474]\n",
+      " [0.0709784 ]\n",
+      " [0.04671657]\n",
+      " [0.05989459]\n",
+      " [0.04279709]\n",
+      " [0.08662194]\n",
+      " [0.08666131]\n",
+      " [0.06101134]\n",
+      " [0.08401617]\n",
+      " [0.07692432]\n",
+      " [0.06763563]\n",
+      " [0.0451411 ]\n",
+      " [0.06105676]\n",
+      " [0.08719212]\n",
+      " [0.05750605]\n",
+      " [0.0865581 ]\n",
+      " [0.08127308]\n",
+      " [0.0760203 ]\n",
+      " [0.08066425]\n",
+      " [0.07768643]\n",
+      " [0.07854018]\n",
+      " [0.06418386]\n",
+      " [0.06303421]\n",
+      " [0.05167019]\n",
+      " [0.0704079 ]\n",
+      " [0.06521657]\n",
+      " [0.04750559]\n",
+      " [0.05712697]\n",
+      " [0.06134903]\n",
+      " [0.07936773]\n",
+      " [0.04882196]\n",
+      " [0.05709308]\n",
+      " [0.08182999]\n",
+      " [0.06822082]\n",
+      " [0.05949762]\n",
+      " [0.03348103]\n",
+      " [0.05904639]\n",
+      " [0.08303881]\n",
+      " [0.07341632]\n",
+      " [0.09511766]\n",
+      " [0.09009409]\n",
+      " [0.08403727]\n",
+      " [0.07363299]\n",
+      " [0.07215634]\n",
+      " [0.07517365]\n",
+      " [0.10830417]\n",
+      " [0.0831027 ]\n",
+      " [0.06821123]\n",
+      " [0.07043773]\n",
+      " [0.06828451]\n",
+      " [0.0673196 ]\n",
+      " [0.07388246]\n",
+      " [0.05345121]\n",
+      " [0.06180677]\n",
+      " [0.04424304]\n",
+      " [0.05934063]\n",
+      " [0.05631447]\n",
+      " [0.07077795]\n",
+      " [0.06338841]\n",
+      " [0.0643796 ]\n",
+      " [0.07952979]\n",
+      " [0.06385201]\n",
+      " [0.03907123]\n",
+      " [0.0926834 ]\n",
+      " [0.07305819]\n",
+      " [0.09337723]\n",
+      " [0.07430211]\n",
+      " [0.08875009]\n",
+      " [0.06230545]\n",
+      " [0.06216493]\n",
+      " [0.08424464]\n",
+      " [0.06766856]\n",
+      " [0.05280164]\n",
+      " [0.05075416]\n",
+      " [0.05293822]\n",
+      " [0.0629158 ]\n",
+      " [0.04674768]\n",
+      " [0.04449037]\n",
+      " [0.10647008]\n",
+      " [0.09053552]\n",
+      " [0.09864247]\n",
+      " [0.07641321]\n",
+      " [0.07405573]\n",
+      " [0.09431157]\n",
+      " [0.05055869]\n",
+      " [0.06393123]\n",
+      " [0.06474605]\n",
+      " [0.07344761]\n",
+      " [0.05588779]\n",
+      " [0.07900479]\n",
+      " [0.06947374]\n",
+      " [0.0356732 ]\n",
+      " [0.02500269]\n",
+      " [0.04629824]\n",
+      " [0.06867462]\n",
+      " [0.08204171]\n",
+      " [0.07958868]\n",
+      " [0.05840129]\n",
+      " [0.05575734]\n",
+      " [0.07071397]\n",
+      " [0.06452814]\n",
+      " [0.06130123]\n",
+      " [0.06218821]\n",
+      " [0.05575269]\n",
+      " [0.0510855 ]\n",
+      " [0.03636277]\n",
+      " [0.02983409]\n",
+      " [0.03601119]\n",
+      " [0.07719433]\n",
+      " [0.07829916]\n",
+      " [0.06654462]\n",
+      " [0.07042193]\n",
+      " [0.09790522]\n",
+      " [0.06097481]\n",
+      " [0.07226038]\n",
+      " [0.0603691 ]\n",
+      " [0.05554828]\n",
+      " [0.04249924]\n",
+      " [0.05688986]\n",
+      " [0.0637235 ]\n",
+      " [0.0360972 ]\n",
+      " [0.04375321]\n",
+      " [0.03207108]\n",
+      " [0.03577149]\n",
+      " [0.02954644]\n",
+      " [0.04560512]\n",
+      " [0.04873624]\n",
+      " [0.08011109]\n",
+      " [0.07926044]\n",
+      " [0.07943588]\n",
+      " [0.06822565]\n",
+      " [0.057161  ]\n",
+      " [0.06316799]\n",
+      " [0.02389175]\n",
+      " [0.03852141]\n",
+      " [0.0356386 ]\n",
+      " [0.03183255]\n",
+      " [0.04871386]\n",
+      " [0.03614715]\n",
+      " [0.0496394 ]\n",
+      " [0.03470093]\n",
+      " [0.02884576]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.0327442 ]\n",
+      " [0.0327442 ]\n",
+      " [0.03470093]\n",
+      " [0.02569604]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.03470093]\n",
+      " [0.06807911]\n",
+      " [0.03470093]\n",
+      " [0.0341661 ]\n",
+      " [0.03470089]\n",
+      " [0.03035369]]\n"
      ]
     }
    ],
@@ -830,25 +1033,29 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 16,
    "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "54\n"
-     ]
-    }
-   ],
+   "outputs": [],
+   "source": [
+    "test_X = loadtxt(\"features-test.csv\", delimiter=',')\n",
+    "T_X = test_X[:, 0:nFeatures]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "predictions = model.predict(T_X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [],
    "source": [
-    "count = 0\n",
-    "for i in range(len(predictions)):\n",
-    "    if predictions[i][0] > 0.065 and CV_Y[i][0] == 1:\n",
-    "        count+=1\n",
-    "#     elif predictions[i][0] < 0.04 and CV_Y[i][0] == 0:\n",
-    "#         count+=1\n",
-    "print(count)\n",
     "a = [0 for i in range(len(predictions))]\n",
     "for i in range(len(predictions)):\n",
     "    a[i] = predictions[i][0] > 0.065\n"
@@ -856,7 +1063,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 99,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -937,131 +1144,95 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 101,
-   "metadata": {
-    "collapsed": true
-   },
+   "execution_count": 22,
+   "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[('Classification', 'NN')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('Periodicity', 'NN')]\n",
-      "[('Properties', 'NNS')]\n",
-      "[('Unit', 'NN')]\n",
-      "[('concept', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('accordance', 'NN')]\n",
-      "[('properties', 'NNS')]\n",
-      "[('development', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('remarkable', 'JJ'), ('demonstration', 'NN')]\n",
-      "[('fact', 'NN')]\n",
-      "[('chemical', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('random', 'NN'), ('cluster', 'NN')]\n",
-      "[('entities', 'NNS')]\n",
-      "[('trends', 'NNS')]\n",
-      "[('families', 'NNS')]\n",
-      "[('Seaborg', 'NN')]\n",
-      "[('Unit', 'NN')]\n",
-      "[('historical', 'JJ'), ('development', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('today', 'NN')]\n",
-      "[('Modern', 'JJ')]\n",
+      "[('•', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Law', 'NN')]\n",
-      "[('periodic', 'JJ'), ('classification', 'NN')]\n",
-      "[('logical', 'JJ'), ('consequence', 'NN')]\n",
+      "[('•', 'NN')]\n",
+      "[('significance', 'NN')]\n",
+      "[('atomic', 'JJ'), ('number', 'NN')]\n",
       "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('3.1', 'CD')]\n",
-      "[('WE', 'NN')]\n",
-      "[('NEED', 'NN')]\n",
-      "[('TO', 'NN')]\n",
-      "[('CLASSIFY', 'NN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
+      "[('basis', 'NN')]\n",
+      "[('periodic', 'JJ'), ('classification', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('basic', 'JJ'), ('units', 'NNS')]\n",
-      "[('types', 'NNS')]\n",
-      "[('matter', 'NN')]\n",
-      "[('1800', 'CD')]\n",
-      "[('31', 'CD')]\n",
+      "[('Z', 'NN')]\n",
+      "[('100', 'CD')]\n",
+      "[('IUPAC', 'NN')]\n",
+      "[('nomenclature', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('1865', 'CD')]\n",
-      "[('number', 'NN')]\n",
-      "[('identified', 'VBN'), ('elements', 'NNS')]\n",
-      "[('63', 'CD')]\n",
-      "[('present', 'JJ'), ('114', 'CD')]\n",
+      "[('s', 'NN')]\n",
+      "[('p', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('f', 'NN')]\n",
+      "[('blocks', 'NNS')]\n",
+      "[('main', 'JJ'), ('characteristics', 'NNS')]\n",
+      "[('•', 'NN')]\n",
+      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
+      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
-      "[('known', 'VBN'), ('chemical', 'JJ'), ('facts', 'NNS')]\n",
+      "[('•', 'NN')]\n",
+      "[('reactivity', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('new', 'JJ'), ('ones', 'NNS')]\n",
-      "[('further', 'RB'), ('study', 'NN')]\n",
-      "[('3.2', 'CD')]\n",
-      "[('GENESIS', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('CLASSIFICATION', 'NN')]\n",
-      "[('Classification', 'NN')]\n",
+      "[('occurrence', 'NN')]\n",
+      "[('nature', 'NN')]\n",
+      "[('relationship', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('metallic', 'JJ'), ('character', 'NN')]\n",
+      "[('scientific', 'JJ'), ('vocabulary', 'NN')]\n",
+      "[('ideas', 'NNS')]\n",
+      "[('certain', 'JJ'), ('important', 'JJ'), ('properties', 'NNS')]\n",
+      "[('atoms', 'NNS')]\n",
+      "[('atomic/', 'NN'), ('ionic', 'JJ'), ('radii', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('valence', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('groups', 'NNS')]\n",
-      "[('development', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Law', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
-      "[('consequences', 'NNS')]\n",
-      "[('knowledge', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('scientists', 'NNS')]\n",
-      "[('observations', 'NNS')]\n",
-      "[('experiments', 'NNS')]\n",
-      "[('case', 'NN')]\n",
-      "[('middle', 'JJ'), ('element', 'NN')]\n",
-      "[('Triads', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('weight', 'NN')]\n",
-      "[('way', 'NN')]\n",
-      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
-      "[('other', 'JJ'), ('two', 'CD')]\n",
-      "[('Table', 'JJ'), ('3.1', 'CD')]\n",
+      "[('important', 'JJ'), ('concept', 'NN')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('practice', 'NN')]\n",
+      "[('everyday', 'JJ'), ('support', 'NN')]\n",
+      "[('students', 'NNS')]\n",
+      "[('new', 'JJ'), ('avenues', 'NNS')]\n",
+      "[('research', 'NN')]\n",
+      "[('professionals', 'NNS')]\n",
+      "[('succinct', 'JJ'), ('organization', 'NN')]\n",
+      "[('whole', 'JJ')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('Glenn', 'NNP')]\n",
+      "[('T', 'NN')]\n",
+      "[('Efforts', 'NNS')]\n",
+      "[('new', 'JJ'), ('elements', 'NNS')]\n",
+      "[('large', 'JJ'), ('number', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('innumerable', 'JJ'), ('compounds', 'NNS')]\n",
       "[('properties', 'NNS')]\n",
       "[('middle', 'JJ'), ('element', 'NN')]\n",
       "[('other', 'JJ'), ('two', 'CD')]\n",
       "[('members', 'NNS')]\n",
-      "[('Dobereiner', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN')]\n",
-      "[('relationship', 'NN')]\n",
-      "[('Law', 'NN')]\n",
-      "[('Triads', 'NNS')]\n",
-      "[('few', 'JJ'), ('elements', 'NNS')]\n",
-      "[('coincidence', 'NN')]\n",
-      "[('attempt', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('French', 'JJ'), ('geologist', 'NN')]\n",
-      "[('A.E.B', 'NN')]\n",
-      "[('Chancourtois', 'NN')]\n",
-      "[('1862', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('order', 'NN')]\n",
-      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
-      "[('cylindrical', 'JJ'), ('table', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('periodic', 'JJ'), ('recurrence', 'NN')]\n",
-      "[('properties', 'NNS')]\n",
       "[('much', 'JJ'), ('attention', 'NN')]\n",
-      "[('relationship', 'NN')]\n",
-      "[('eighth', 'JJ'), ('note', 'NN')]\n",
-      "[('octaves', 'NNS')]\n",
-      "[('music', 'NN')]\n",
+      "[('English', 'JJ')]\n",
+      "[('chemist', 'NN')]\n",
+      "[('John', 'NNP')]\n",
+      "[('Alexander', 'NN')]\n",
+      "[('Newlands', 'NNS')]\n",
+      "[('1865', 'CD')]\n",
+      "[('Law', 'NN')]\n",
+      "[('Octaves', 'NNS')]\n",
       "[('Newlands', 'NNS')]\n",
       "[('s', 'NN'), ('Law', 'NN')]\n",
       "[('Octaves', 'NNS')]\n",
@@ -1079,6 +1250,25 @@
       "[('Dobereiner', 'NN')]\n",
       "[('’', 'NN')]\n",
       "[('Triads', 'NNS')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Law', 'NN')]\n",
+      "[('today', 'NN')]\n",
+      "[('development', 'NN')]\n",
+      "[('Russian', 'JJ'), ('chemist', 'NN')]\n",
+      "[('Dmitri', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('German', 'JJ'), ('chemist', 'NN')]\n",
+      "[('Lothar', 'NN')]\n",
+      "[('Meyer', 'NN')]\n",
+      "[('1830-1895', 'JJ')]\n",
+      "[('chemists', 'NNS')]\n",
+      "[('1869', 'CD')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('order', 'NN')]\n",
+      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
+      "[('similarities', 'NNS')]\n",
+      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
+      "[('regular', 'JJ'), ('intervals', 'NNS')]\n",
       "[('Lothar', 'NN')]\n",
       "[('Meyer', 'NN')]\n",
       "[('physical', 'JJ'), ('properties', 'NNS')]\n",
@@ -1093,11 +1283,50 @@
       "[('change', 'NN')]\n",
       "[('length', 'NN')]\n",
       "[('pattern', 'NN')]\n",
+      "[('1868', 'CD')]\n",
+      "[('Lothar', 'NN')]\n",
+      "[('Meyer', 'NN')]\n",
+      "[('table', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('Modern', 'JJ')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('Dmitri', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('scientist', 'NN')]\n",
+      "[('development', 'NN')]\n",
+      "[('Modern', 'JJ')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('Table', 'JJ'), ('3.2', 'CD')]\n",
+      "[('Newlands', 'NNS')]\n",
+      "[('’', 'NN'), ('Octaves', 'NNS')]\n",
+      "[('Dobereiner', 'NN')]\n",
+      "[('study', 'NN')]\n",
+      "[('periodic', 'NN'), ('relationship', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Law', 'NN')]\n",
+      "[('first', 'JJ'), ('time', 'NN')]\n",
       "[('properties', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
       "[('periodic', 'JJ'), ('function', 'NN')]\n",
       "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
       "[('Mendeleev', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('horizontal', 'NN'), ('rows', 'NNS')]\n",
+      "[('vertical', 'JJ'), ('columns', 'NN')]\n",
+      "[('table', 'NN')]\n",
+      "[('order', 'NN')]\n",
+      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
+      "[('way', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('similar', 'JJ'), ('properties', 'NNS')]\n",
+      "[('same', 'JJ'), ('vertical', 'JJ'), ('column', 'NN')]\n",
+      "[('group', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
       "[('’', 'NN')]\n",
       "[('s', 'NN')]\n",
       "[('system', 'NN')]\n",
@@ -1122,11 +1351,6 @@
       "[('classification', 'NN')]\n",
       "[('order', 'NN')]\n",
       "[('atomic', 'JJ'), ('weight', 'NN')]\n",
-      "[('order', 'NN')]\n",
-      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('measurements', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('similar', 'JJ'), ('properties', 'NNS')]\n",
       "[('example', 'NN')]\n",
       "[('iodine', 'NN')]\n",
       "[('lower', 'JJR'), ('atomic', 'NN')]\n",
@@ -1141,31 +1365,30 @@
       "[('bromine', 'NN')]\n",
       "[('similarities', 'NNS')]\n",
       "[('properties', 'NNS')]\n",
-      "[('example', 'NN')]\n",
+      "[('gap', 'NN')]\n",
+      "[('aluminium', 'NN')]\n",
+      "[('gap', 'NN')]\n",
+      "[('silicon', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('Eka-Aluminium', 'NN')]\n",
+      "[('Eka-Silicon', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('existence', 'NN')]\n",
       "[('gallium', 'NN')]\n",
       "[('germanium', 'NN')]\n",
-      "[('time', 'NN')]\n",
-      "[('Mendeleev', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
+      "[('general', 'JJ'), ('physical', 'NN'), ('properties', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
       "[('properties', 'NNS')]\n",
       "[('Mendeleev', 'NN')]\n",
       "[('elements', 'NNS')]\n",
       "[('Table', 'JJ'), ('3.3', 'CD')]\n",
-      "[('Table', 'JJ'), ('3.3', 'CD')]\n",
-      "[('Mendeleev', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'JJ'), ('Predictions', 'NNS')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('Eka-aluminium', 'NN')]\n",
-      "[('Gallium', 'NN')]\n",
-      "[('Eka-silicon', 'NN')]\n",
-      "[('Germanium', 'NN')]\n",
+      "[('boldness', 'NN')]\n",
       "[('Mendeleev', 'NN')]\n",
       "[('’', 'NN')]\n",
+      "[('quantitative', 'JJ'), ('predictions', 'NNS')]\n",
+      "[('eventual', 'JJ'), ('success', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
-      "[('1905', 'CD')]\n",
       "[('PERIODIC', 'NN')]\n",
       "[('SYSTEM', 'NN')]\n",
       "[('THE', 'DT')]\n",
@@ -1182,17 +1405,14 @@
       "[('straight', 'JJ'), ('line', 'NN')]\n",
       "[('plot', 'NN')]\n",
       "[('vs', 'JJ'), ('atomic', 'JJ'), ('mass', 'NN')]\n",
+      "[('atomic', 'JJ'), ('number', 'NN')]\n",
+      "[('fundamental', 'JJ'), ('property', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('atomic', 'JJ'), ('mass', 'NN')]\n",
       "[('Mendeleev', 'NN')]\n",
       "[('’', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Law', 'NN')]\n",
-      "[('Modern', 'JJ')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Law', 'NN')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('periodic', 'JJ'), ('functions', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Law', 'NN')]\n",
       "[('important', 'JJ'), ('analogies', 'NNS')]\n",
@@ -1206,108 +1426,45 @@
       "[('blende', 'NN')]\n",
       "[('ore', 'NN')]\n",
       "[('uranium', 'NN')]\n",
-      "[('interest', 'NN')]\n",
-      "[('Inorganic', 'JJ')]\n",
-      "[('Chemistry', 'NN')]\n",
-      "[('creation', 'NN')]\n",
-      "[('short-lived', 'JJ'), ('elements', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('nuclear', 'JJ'), ('charge', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('protons', 'NNS')]\n",
-      "[('number', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('neutral', 'JJ'), ('atom', 'NN')]\n",
-      "[('significance', 'NN')]\n",
-      "[('quantum', 'NN')]\n",
-      "[('numbers', 'NNS')]\n",
-      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
-      "[('periodicity', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('modern', 'JJ'), ('version', 'NN')]\n",
-      "[('so-called', 'JJ'), ('“', 'NN')]\n",
-      "[('long', 'RB'), ('form', 'NN')]\n",
-      "[('”', 'NN')]\n",
+      "[('Numerous', 'JJ'), ('forms', 'NNS')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('period', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('corresponds', 'NNS')]\n",
-      "[('highest', 'JJS'), ('principal', 'JJ'), ('quantum', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('period', 'NN')]\n",
-      "[('seventh', 'JJ'), ('period', 'NN')]\n",
-      "[('sixth', 'JJ'), ('period', 'NN')]\n",
-      "[('theoretical', 'JJ'), ('maximum', 'NN')]\n",
-      "[('basis', 'NN')]\n",
-      "[('quantum', 'NN'), ('numbers', 'NNS')]\n",
-      "[('32', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
+      "[('time', 'NN')]\n",
+      "[('time', 'NN')]\n",
+      "[('recommendation', 'NN')]\n",
+      "[('International', 'NNP')]\n",
+      "[('Union', 'NNP')]\n",
+      "[('Pure', 'NN')]\n",
+      "[('Applied', 'NNP')]\n",
+      "[('Chemistry', 'NN')]\n",
+      "[('IUPAC', 'NN')]\n",
+      "[('groups', 'NNS')]\n",
+      "[('1', 'CD')]\n",
+      "[('18', 'CD')]\n",
+      "[('older', 'JJR'), ('notation', 'NN')]\n",
+      "[('groups', 'NNS')]\n",
+      "[('IA', 'NN')]\n",
+      "[('…', 'NN')]\n",
+      "[('VIIA', 'NN')]\n",
+      "[('VIII', 'NN')]\n",
+      "[('IB', 'NN')]\n",
+      "[('…', 'NN')]\n",
+      "[('VIIB', 'NN')]\n",
+      "[('0', 'CD')]\n",
       "[('synthesis', 'NN')]\n",
       "[('characterisation', 'NN')]\n",
       "[('sophisticated', 'VBN'), ('costly', 'JJ'), ('equipment', 'NN')]\n",
       "[('laboratory', 'NN')]\n",
-      "[('Such', 'JJ'), ('work', 'NN')]\n",
-      "[('competitive', 'JJ'), ('spirit', 'NN')]\n",
-      "[('laboratories', 'NNS')]\n",
-      "[('world', 'NN')]\n",
-      "[('Scientists', 'NNS')]\n",
-      "[('reliable', 'JJ'), ('data', 'NNS')]\n",
-      "[('new', 'JJ'), ('element', 'NN')]\n",
-      "[('times', 'NNS')]\n",
-      "[('discovery', 'NN')]\n",
-      "[('example', 'NN')]\n",
-      "[('American', 'JJ')]\n",
-      "[('Soviet', 'JJ'), ('scientists', 'NNS')]\n",
-      "[('credit', 'NN')]\n",
-      "[('element', 'NN')]\n",
-      "[('104', 'CD')]\n",
-      "[('Americans', 'NNS')]\n",
-      "[('Rutherfordium', 'NN')]\n",
-      "[('whereas', 'NNS'), ('Soviets', 'NNS')]\n",
-      "[('Kurchatovium', 'NN')]\n",
-      "[('such', 'JJ'), ('problems', 'NNS')]\n",
-      "[('IUPAC', 'NN')]\n",
-      "[('recommendation', 'NN')]\n",
-      "[('new', 'JJ'), ('element', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN'), ('discovery', 'NN')]\n",
-      "[('name', 'NN')]\n",
-      "[('systematic', 'JJ'), ('nomenclature', 'NN')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('element', 'NN')]\n",
-      "[('numerical', 'JJ'), ('roots', 'NNS')]\n",
-      "[('0', 'CD')]\n",
-      "[('numbers', 'NNS')]\n",
-      "[('Table', 'JJ'), ('3.4', 'CD')]\n",
-      "[('roots', 'NNS')]\n",
-      "[('order', 'NN')]\n",
-      "[('digits', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('“', 'NN'), ('ium', 'NN')]\n",
-      "[('”', 'NN')]\n",
-      "[('end', 'NN')]\n",
-      "[('IUPAC', 'NN')]\n",
-      "[('names', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('Z', 'NN')]\n",
-      "[('100', 'CD')]\n",
-      "[('Table', 'JJ'), ('3.5', 'CD')]\n",
-      "[('*', 'NN'), ('Glenn', 'NNP')]\n",
-      "[('T', 'NN')]\n",
-      "[('1951', 'CD')]\n",
-      "[('Seaborg', 'NN')]\n",
-      "[('Nobel', 'NN')]\n",
-      "[('Prize', 'VB')]\n",
-      "[('chemistry', 'NN')]\n",
-      "[('work', 'NN')]\n",
       "[('Element', 'NN')]\n",
       "[('106', 'CD')]\n",
       "[('Seaborgium', 'NN')]\n",
       "[('Sg', 'NN')]\n",
       "[('honour', 'NN')]\n",
+      "[('Table', 'JJ'), ('3.4', 'CD')]\n",
+      "[('Notation', 'NN')]\n",
+      "[('IUPAC', 'NN')]\n",
+      "[('Nomenclature', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
       "[('Table', 'JJ'), ('3.5', 'CD')]\n",
       "[('Nomenclature', 'NN')]\n",
       "[('Elements', 'NNS')]\n",
@@ -1315,6 +1472,22 @@
       "[('Number', 'NNP')]\n",
       "[('Above', 'IN')]\n",
       "[('100', 'CD')]\n",
+      "[('new', 'JJ'), ('element', 'NN')]\n",
+      "[('temporary', 'JJ'), ('name', 'NN')]\n",
+      "[('symbol', 'NN')]\n",
+      "[('consisting', 'VBG')]\n",
+      "[('three', 'CD')]\n",
+      "[('letters', 'NNS')]\n",
+      "[('permanent', 'JJ'), ('name', 'NN')]\n",
+      "[('country', 'NN')]\n",
+      "[('state', 'NN')]\n",
+      "[('country', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('tribute', 'NN')]\n",
+      "[('notable', 'JJ'), ('scientist', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
+      "[('118', 'CD')]\n",
       "[('Problem', 'NN')]\n",
       "[('3.1', 'CD')]\n",
       "[('IUPAC', 'NN')]\n",
@@ -1331,26 +1504,6 @@
       "[('0', 'CD')]\n",
       "[('bi', 'NN')]\n",
       "[('nil', 'NN')]\n",
-      "[('3.5', 'CD')]\n",
-      "[('ELECTRONIC', 'NN')]\n",
-      "[('CONFIGURATIONS', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
-      "[('THE', 'DT')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('TABLE', 'NN')]\n",
-      "[('unit', 'NN')]\n",
-      "[('electron', 'NN')]\n",
-      "[('atom', 'NN')]\n",
-      "[('set', 'NN')]\n",
-      "[('four', 'CD')]\n",
-      "[('quantum', 'NN')]\n",
-      "[('numbers', 'NNS')]\n",
-      "[('principal', 'JJ'), ('quantum', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('main', 'JJ'), ('energy', 'NN')]\n",
-      "[('level', 'NN')]\n",
-      "[('shell', 'NN')]\n",
       "[('filling', 'VBG')]\n",
       "[('electrons', 'NNS')]\n",
       "[('different', 'JJ'), ('subshells', 'NNS')]\n",
@@ -1360,13 +1513,19 @@
       "[('d', 'NN')]\n",
       "[('f', 'NN')]\n",
       "[('atom', 'NN')]\n",
-      "[('section', 'NN')]\n",
-      "[('direct', 'JJ'), ('connection', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('long', 'RB'), ('form', 'NN')]\n",
+      "[('distribution', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('orbitals', 'NNS')]\n",
+      "[('atom', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('s', 'NN')]\n",
+      "[('location', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
+      "[('quantum', 'NN'), ('numbers', 'NNS')]\n",
+      "[('last', 'JJ'), ('orbital', 'JJ')]\n",
       "[('Electronic', 'JJ'), ('Configurations', 'NNS')]\n",
       "[('Periods', 'NNS')]\n",
       "[('period', 'NN')]\n",
@@ -1375,23 +1534,6 @@
       "[('outermost', 'NN')]\n",
       "[('valence', 'NN')]\n",
       "[('shell', 'NN')]\n",
-      "[('other', 'JJ'), ('words', 'NNS')]\n",
-      "[('successive', 'JJ'), ('period', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('filling', 'VBG')]\n",
-      "[('next', 'JJ'), ('higher', 'JJR'), ('principal', 'JJ'), ('energy', 'NN')]\n",
-      "[('level', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('1', 'CD')]\n",
-      "[('=', '$'), ('2', 'CD')]\n",
-      "[('number', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('period', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('atomic', 'JJ'), ('orbitals', 'NNS')]\n",
-      "[('energy', 'NN')]\n",
-      "[('level', 'NN')]\n",
       "[('first', 'JJ'), ('period', 'NN')]\n",
       "[('n', 'NNS'), ('=', 'VBP')]\n",
       "[('1', 'CD')]\n",
@@ -1407,6 +1549,12 @@
       "[('first', 'RB'), ('shell', 'NN')]\n",
       "[('K', 'NNP')]\n",
       "[('next', 'JJ'), ('element', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('four', 'CD')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('1s22s2', 'CD')]\n",
+      "[('next', 'JJ'), ('element', 'NN')]\n",
       "[('boron', 'NN')]\n",
       "[('2p', 'CD')]\n",
       "[('orbitals', 'NNS')]\n",
@@ -1418,13 +1566,41 @@
       "[('8', 'CD')]\n",
       "[('elements', 'NNS')]\n",
       "[('second', 'JJ'), ('period', 'NN')]\n",
+      "[('third', 'JJ'), ('period', 'NN')]\n",
+      "[('n', 'NNS'), ('=', 'VBP')]\n",
+      "[('3', 'CD')]\n",
+      "[('sodium', 'NN')]\n",
+      "[('added', 'VBD'), ('electron', 'NN')]\n",
+      "[('3s', 'CD')]\n",
+      "[('orbital', 'NN')]\n",
       "[('fourth', 'JJ'), ('period', 'NN')]\n",
-      "[('krypton', 'NN')]\n",
+      "[('n', 'NNS'), ('=', 'VBP')]\n",
+      "[('4', 'CD')]\n",
+      "[('starts', 'NNS')]\n",
+      "[('potassium', 'NN')]\n",
+      "[('added', 'VBD'), ('electrons', 'NNS')]\n",
+      "[('4s', 'CD')]\n",
+      "[('orbital', 'NN')]\n",
       "[('4p', 'CD')]\n",
+      "[('orbital', 'NN')]\n",
+      "[('3d', 'CD')]\n",
       "[('orbitals', 'NNS')]\n",
-      "[('18', 'CD')]\n",
+      "[('called', 'VBN'), ('3d', 'CD')]\n",
+      "[('transition', 'NN')]\n",
+      "[('series', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('fourth', 'JJ'), ('period', 'NN')]\n",
+      "[('scandium', 'NN')]\n",
+      "[('Z', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('21', 'CD')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('3d14s2', 'CD')]\n",
+      "[('3d', 'CD')]\n",
+      "[('orbitals', 'NNS')]\n",
+      "[('zinc', 'NN')]\n",
+      "[('Z=30', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('3d104s2', 'CD')]\n",
       "[('fifth', 'JJ'), ('period', 'NN')]\n",
       "[('n', 'NNS'), ('=', 'VBP')]\n",
       "[('5', 'CD')]\n",
@@ -1438,263 +1614,117 @@
       "[('=', 'NN')]\n",
       "[('39', 'CD')]\n",
       "[('period', 'NN')]\n",
-      "[('xenon', 'NN')]\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "[('xenon', 'NN')]\n",
       "[('5p', 'CD')]\n",
       "[('orbitals', 'NNS')]\n",
-      "[('sixth', 'JJ'), ('period', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('6', 'CD')]\n",
-      "[('32', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('successive', 'JJ'), ('electrons', 'NNS')]\n",
-      "[('6s', 'CD')]\n",
-      "[('4f', 'CD')]\n",
-      "[('5d', 'CD')]\n",
-      "[('6p', 'CD')]\n",
-      "[('orbitals', 'NNS')]\n",
       "[('order', 'NN')]\n",
-      "[('—', 'NN')]\n",
-      "[('4f', 'CD')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('cerium', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('58', 'CD')]\n",
-      "[('lutetium', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('71', 'CD')]\n",
-      "[('4f-inner', 'JJ'), ('transition', 'NN')]\n",
-      "[('series', 'NN')]\n",
-      "[('lanthanoid', 'NN')]\n",
-      "[('series', 'NN')]\n",
-      "[('seventh', 'JJ'), ('period', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('7', 'CD')]\n",
-      "[('sixth', 'JJ'), ('period', 'NN')]\n",
-      "[('successive', 'NN')]\n",
-      "[('7s', 'CD')]\n",
-      "[('5f', 'CD')]\n",
-      "[('6d', 'CD')]\n",
-      "[('7p', 'CD')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('man-made', 'JJ'), ('radioactive', 'JJ'), ('elements', 'NNS')]\n",
-      "[('Problem', 'NN')]\n",
-      "[('3.2', 'CD')]\n",
-      "[('How', 'WRB')]\n",
-      "[('presence', 'NN')]\n",
-      "[('18', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('5th', 'CD'), ('period', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('n', 'NN')]\n",
-      "[('5', 'CD')]\n",
-      "[('l', 'NN')]\n",
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-      "[('2', 'CD')]\n",
-      "[('3', 'CD')]\n",
-      "[('total', 'JJ'), ('number', 'NN')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('9', 'CD')]\n",
+      "[('energy', 'NN')]\n",
+      "[('available', 'JJ'), ('orbitals', 'NNS')]\n",
+      "[('4d', 'CD')]\n",
+      "[('5s', 'CD')]\n",
+      "[('5p', 'CD')]\n",
+      "[('increases', 'NNS')]\n",
+      "[('5s', 'CD')]\n",
+      "[('<', '$'), ('4d', 'CD')]\n",
+      "[('5p', 'CD')]\n",
       "[('maximum', 'JJ'), ('number', 'NN')]\n",
       "[('electrons', 'NNS')]\n",
       "[('18', 'CD')]\n",
       "[('18', 'CD')]\n",
       "[('elements', 'NNS')]\n",
       "[('5th', 'CD'), ('period', 'NN')]\n",
-      "[('3.6', 'CD')]\n",
-      "[('ELECTRONIC', 'NN')]\n",
-      "[('CONFIGURATIONS', 'NN')]\n",
-      "[('AND', 'CC')]\n",
-      "[('TYPES', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
-      "[('f-', 'NN'), ('BLOCKS', 'NNP')]\n",
-      "[('aufbau', 'NN')]\n",
-      "[('principle', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('theoretical', 'JJ'), ('foundation', 'NN')]\n",
-      "[('periodic', 'JJ'), ('classification', 'NN')]\n",
+      "[('example', 'NN')]\n",
+      "[('Group', 'NNP')]\n",
+      "[('1', 'CD')]\n",
       "[('elements', 'NNS')]\n",
-      "[('vertical', 'JJ'), ('column', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('family', 'NN')]\n",
-      "[('exhibit', 'NN')]\n",
-      "[('similar', 'JJ'), ('chemical', 'NN')]\n",
-      "[('behaviour', 'NN')]\n",
-      "[('similarity', 'NN')]\n",
+      "[('alkali', 'JJ'), ('metals', 'NNS')]\n",
+      "[('valence', 'NN')]\n",
+      "[('shell', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('same', 'JJ'), ('number', 'NN')]\n",
-      "[('same', 'JJ'), ('distribution', 'NN')]\n",
+      "[('four', 'CD')]\n",
+      "[('blocks', 'NNS')]\n",
+      "[('f-block', 'NN'), ('depending', 'VBG')]\n",
+      "[('type', 'NN')]\n",
+      "[('atomic', 'JJ'), ('orbitals', 'NNS')]\n",
       "[('electrons', 'NNS')]\n",
-      "[('outermost', 'NN')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('two', 'CD')]\n",
-      "[('exceptions', 'NNS')]\n",
-      "[('categorisation', 'NN')]\n",
-      "[('other', 'JJ'), ('exception', 'NN')]\n",
+      "[('other', 'JJ'), ('exception', 'NN')]\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
       "[('hydrogen', 'NN')]\n",
       "[('3.6.1', 'CD')]\n",
       "[('s-Block', 'NN')]\n",
       "[('Elements', 'NNS')]\n",
+      "[('3.6.3', 'CD')]\n",
+      "[('d-Block', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('Transition', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
       "[('Group', 'NNP')]\n",
-      "[('1', 'CD')]\n",
-      "[('alkali', 'JJ'), ('metals', 'NNS')]\n",
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-      "[('2', 'CD')]\n",
-      "[('alkaline', 'NN'), ('earth', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
-      "[('ns1', 'NN')]\n",
-      "[('ns2', 'NNS'), ('outermost', 'VBD')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('belong', 'NN')]\n",
-      "[('s-Block', 'JJ'), ('Elements', 'NNS')]\n",
-      "[('reactive', 'JJ'), ('metals', 'NNS')]\n",
-      "[('low', 'JJ'), ('ionization', 'NN')]\n",
-      "[('enthalpies', 'NNS')]\n",
-      "[('outermost', 'NN')]\n",
-      "[('electron', 'NN')]\n",
-      "[('1+', 'CD')]\n",
-      "[('ion', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('alkali', 'JJ'), ('metals', 'NNS')]\n",
-      "[('2+', 'CD')]\n",
-      "[('ion', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('alkaline', 'JJ'), ('earth', 'NN'), ('metals', 'NNS')]\n",
-      "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('reactivity', 'NN')]\n",
-      "[('increase', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('compounds', 'NNS')]\n",
-      "[('s-block', 'JJ'), ('elements', 'NNS')]\n",
-      "[('exception', 'NN')]\n",
-      "[('lithium', 'NN')]\n",
-      "[('beryllium', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
+      "[('3', 'CD')]\n",
+      "[('12', 'CD')]\n",
+      "[('centre', 'NN')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
       "[('coloured', 'VBN'), ('ions', 'NNS')]\n",
       "[('exhibit', 'NN')]\n",
       "[('variable', 'JJ'), ('valence', 'NN')]\n",
       "[('oxidation', 'NN'), ('states', 'NNS')]\n",
       "[('paramagnetism', 'NN')]\n",
       "[('catalysts', 'NNS')]\n",
-      "[('Zn', 'NN')]\n",
-      "[('Cd', 'NN')]\n",
-      "[('Hg', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('d10ns2', 'NN')]\n",
-      "[('properties', 'NNS')]\n",
-      "[('transition', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('way', 'NN')]\n",
-      "[('transition', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
-      "[('bridge', 'NN')]\n",
-      "[('active', 'JJ'), ('metals', 'NNS')]\n",
-      "[('s-block', 'JJ'), ('elements', 'NNS')]\n",
-      "[('active', 'JJ'), ('elements', 'NNS')]\n",
-      "[('Groups', 'NNS')]\n",
-      "[('13', 'CD')]\n",
-      "[('14', 'CD')]\n",
-      "[('familiar', 'JJ'), ('name', 'NN')]\n",
-      "[('“', 'NN')]\n",
-      "[('Transition', 'NN')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('”', 'NN')]\n",
-      "[('Problem', 'NN')]\n",
-      "[('3.4', 'CD')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('position', 'NN')]\n",
-      "[('periodic', 'NN')]\n",
-      "[('table', 'NN')]\n",
-      "[('following', 'VBG'), ('elements', 'NNS')]\n",
-      "[('order', 'NN')]\n",
-      "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('Si', 'NN')]\n",
-      "[('Be', 'VB')]\n",
-      "[('Mg', 'NN')]\n",
-      "[('Na', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('order', 'NN')]\n",
-      "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('Si', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('Be', 'VB')]\n",
-      "[('<', 'NN')]\n",
-      "[('Mg', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('Na', 'NN')]\n",
-      "[('3.7', 'CD')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('TRENDS', 'NNS')]\n",
-      "[('IN', 'NN')]\n",
-      "[('PROPERTIES', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
-      "[('many', 'JJ'), ('observable', 'JJ'), ('patterns', 'NNS')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('early', 'JJ'), ('actinoids', 'NNS')]\n",
+      "[('corresponding', 'VBG'), ('lanthanoids', 'NNS')]\n",
+      "[('large', 'JJ'), ('number', 'NN')]\n",
+      "[('oxidation', 'NN')]\n",
+      "[('states', 'NNS')]\n",
+      "[('actinoid', 'JJ'), ('elements', 'NNS')]\n",
+      "[('3.6.5', 'CD')]\n",
+      "[('Metals', 'NNS')]\n",
+      "[('Non-metals', 'NNS')]\n",
+      "[('Metalloids', 'NNS')]\n",
+      "[('addition', 'NN')]\n",
+      "[('classification', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('group', 'NN')]\n",
-      "[('period', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('example', 'NN')]\n",
-      "[('period', 'NN')]\n",
-      "[('chemical', 'NN'), ('reactivity', 'NN')]\n",
-      "[('Group', 'NNP')]\n",
-      "[('1', 'CD')]\n",
-      "[('metals', 'NNS')]\n",
+      "[('f-blocks', 'NNS')]\n",
+      "[('broad', 'JJ'), ('classification', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('middle', 'NN')]\n",
-      "[('table', 'NN')]\n",
-      "[('maximum', 'NN')]\n",
-      "[('Group', 'NNP')]\n",
-      "[('17', 'CD')]\n",
-      "[('non-metals', 'NNS')]\n",
-      "[('section', 'NN')]\n",
+      "[('properties', 'NNS')]\n",
       "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
+      "[('respect', 'NN')]\n",
+      "[('ionic', 'JJ'), ('radii', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('Atomic', 'NNP')]\n",
+      "[('Radius', 'NN')]\n",
+      "[('size', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('lot', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('ball', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('cloud', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('sharp', 'JJ'), ('boundary', 'JJ')]\n",
+      "[('determination', 'NN')]\n",
+      "[('atomic', 'JJ'), ('size', 'NN')]\n",
+      "[('trends', 'NNS')]\n",
       "[('terms', 'NNS')]\n",
-      "[('number', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('energy', 'NN')]\n",
-      "[('levels', 'NNS')]\n",
-      "[('3.7.1', 'CD')]\n",
-      "[('Trends', 'NNS')]\n",
-      "[('Physical', 'JJ'), ('Properties', 'NNS')]\n",
-      "[('numerous', 'JJ'), ('physical', 'JJ'), ('properties', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('points', 'NNS')]\n",
-      "[('heats', 'NNS')]\n",
-      "[('fusion', 'NN')]\n",
-      "[('vaporization', 'NN')]\n",
+      "[('nuclear', 'JJ'), ('charge', 'NN')]\n",
       "[('energy', 'NN')]\n",
-      "[('atomization', 'NN')]\n",
-      "[('periodic', 'JJ'), ('variations', 'NNS')]\n",
-      "[('other', 'JJ'), ('words', 'NNS')]\n",
-      "[('practical', 'JJ'), ('way', 'NN')]\n",
-      "[('size', 'NN')]\n",
-      "[('individual', 'JJ'), ('atom', 'NN')]\n",
-      "[('estimate', 'NN')]\n",
+      "[('level', 'NN')]\n",
       "[('atomic', 'JJ'), ('size', 'NN')]\n",
-      "[('distance', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('combined', 'VBN'), ('state', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('second', 'JJ'), ('period', 'NN')]\n",
       "[('family', 'NN')]\n",
       "[('vertical', 'JJ'), ('column', 'NN')]\n",
       "[('periodic', 'NNS'), ('table', 'NN')]\n",
@@ -1702,68 +1732,106 @@
       "[('increases', 'NNS')]\n",
       "[('atomic', 'JJ'), ('number', 'NN')]\n",
       "[('b', 'NN')]\n",
-      "[('size', 'NN')]\n",
-      "[('anion', 'NN')]\n",
-      "[('parent', 'NN')]\n",
-      "[('atom', 'NN')]\n",
-      "[('addition', 'NN')]\n",
-      "[('one', 'CD')]\n",
-      "[('more', 'RBR'), ('electrons', 'NNS')]\n",
-      "[('increased', 'VBN'), ('repulsion', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('decrease', 'NN')]\n",
-      "[('effective', 'JJ'), ('nuclear', 'JJ'), ('charge', 'NN')]\n",
-      "[('example', 'NN')]\n",
-      "[('ionic', 'JJ'), ('radius', 'NN')]\n",
-      "[('fluoride', 'JJ'), ('ion', 'NN')]\n",
-      "[('F–', 'NN')]\n",
-      "[('136', 'CD')]\n",
-      "[('pm', 'NN')]\n",
-      "[('atomic', 'JJ'), ('radius', 'NN')]\n",
-      "[('fluorine', 'NN')]\n",
-      "[('64', 'CD')]\n",
-      "[('pm', 'NN')]\n",
-      "[('*', 'CD'), ('Two', 'CD')]\n",
-      "[('more', 'JJR'), ('species', 'NNS')]\n",
-      "[('same', 'JJ'), ('number', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('same', 'JJ'), ('number', 'NN')]\n",
-      "[('valence', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('same', 'JJ'), ('structure', 'NN')]\n",
-      "[('nature', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('first', 'JJ'), ('ionization', 'NN')]\n",
-      "[('enthalpies', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
-      "[('60', 'CD')]\n",
-      "[('situation', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('alkali', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
-      "[('ns-electron', 'NN')]\n",
-      "[('noble', 'JJ'), ('gas', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('fact', 'NN')]\n",
+      "[('radii', 'NN')]\n",
+      "[('noble', 'JJ'), ('gases', 'NNS')]\n",
+      "[('covalent', 'NN')]\n",
+      "[('radii', 'NN')]\n",
+      "[('van', 'NN')]\n",
+      "[('der', 'NN')]\n",
+      "[('Waals', 'NNS')]\n",
+      "[('radii', 'NN')]\n",
+      "[('other', 'JJ'), ('elements', 'NNS')]\n",
       "[('case', 'NN')]\n",
-      "[('increase', 'NN')]\n",
+      "[('net', 'JJ'), ('repulsion', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
       "[('nuclear', 'JJ'), ('charge', 'NN')]\n",
-      "[('removal', 'NN')]\n",
-      "[('outermost', 'NN')]\n",
+      "[('ion', 'NN')]\n",
+      "[('size', 'NN')]\n",
+      "[('largest', 'JJS'), ('species', 'NNS')]\n",
+      "[('Mg', 'NN')]\n",
+      "[('smallest', 'JJS'), ('one', 'CD')]\n",
+      "[('Al3+', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('Ionization', 'NN')]\n",
+      "[('Enthalpy', 'NN')]\n",
+      "[('A', 'DT')]\n",
+      "[('quantitative', 'JJ'), ('measure', 'NN')]\n",
+      "[('tendency', 'NN')]\n",
+      "[('element', 'NN')]\n",
       "[('electron', 'NN')]\n",
-      "[('less', 'RBR'), ('energy', 'NN')]\n",
-      "[('group', 'NN')]\n",
+      "[('Ionization', 'NN')]\n",
+      "[('Enthalpy', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('isolated', 'VBN'), ('gaseous', 'JJ'), ('atom', 'NN')]\n",
+      "[('X', 'NN')]\n",
+      "[('ground', 'NN')]\n",
+      "[('state', 'NN')]\n",
+      "[('X', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('→', 'NN')]\n",
+      "[('X+', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('+', 'NN')]\n",
+      "[('e–', 'NN')]\n",
+      "[('3.1', 'CD')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('units', 'NNS')]\n",
+      "[('kJ', 'NN')]\n",
+      "[('mol–1', 'NN')]\n",
+      "[('X+', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('→', 'NN')]\n",
+      "[('X2+', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('+', 'NN')]\n",
+      "[('e–', 'NN')]\n",
+      "[('3.2', 'CD')]\n",
+      "[('Energy', 'NNP')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('atom', 'NN')]\n",
+      "[('hence', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpies', 'NNS')]\n",
+      "[('first', 'JJ'), ('ionization', 'NN')]\n",
+      "[('enthalpies', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
+      "[('60', 'CD')]\n",
+      "[('periodicity', 'NN')]\n",
+      "[('graph', 'NN')]\n",
+      "[('addition', 'NN')]\n",
+      "[('two', 'CD')]\n",
       "[('first', 'JJ'), ('ionization', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('boron', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('5', 'CD')]\n",
-      "[('beryllium', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('greater', 'JJR'), ('nuclear', 'JJ'), ('charge', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('decreases', 'NNS')]\n",
+      "[('group', 'NN')]\n",
+      "[('effective', 'JJ'), ('nuclear', 'JJ'), ('charge', 'NN')]\n",
+      "[('valence', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('actual', 'JJ'), ('charge', 'NN')]\n",
+      "[('nucleus', 'NN')]\n",
+      "[('“', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('valence', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('nucleus', 'NN')]\n",
+      "[('intervening', 'VBG')]\n",
+      "[('core', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('example', 'NN')]\n",
+      "[('2s', 'CD')]\n",
+      "[('electron', 'NN')]\n",
+      "[('lithium', 'NN')]\n",
+      "[('nucleus', 'NN')]\n",
+      "[('inner', 'NN'), ('core', 'NN')]\n",
+      "[('1s', 'CD')]\n",
+      "[('electrons', 'NNS')]\n",
       "[('same', 'JJ'), ('principal', 'JJ'), ('quantum', 'NN')]\n",
       "[('level', 'NN')]\n",
       "[('s-electron', 'NN')]\n",
@@ -1777,135 +1845,138 @@
       "[('ionization', 'NN')]\n",
       "[('boron', 'NN')]\n",
       "[('p-electron', 'NN')]\n",
-      "[('Predict', 'NN')]\n",
-      "[('first', 'RB'), ('∆i', 'NN')]\n",
-      "[('H', 'NN')]\n",
-      "[('value', 'NN')]\n",
-      "[('Al', 'NN')]\n",
-      "[('575', 'CD')]\n",
-      "[('760', 'CD')]\n",
-      "[('kJ', 'NN')]\n",
+      "[('2p-electron', 'JJ')]\n",
+      "[('boron', 'NN')]\n",
+      "[('removal', 'NN')]\n",
+      "[('2s-', 'JJ'), ('electron', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('boron', 'NN')]\n",
+      "[('smaller', 'JJR'), ('first', 'JJ'), ('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('Problem', 'NN')]\n",
+      "[('3.7', 'CD')]\n",
+      "[('Which', 'WDT')]\n",
+      "[('following', 'VBG')]\n",
+      "[('negative', 'JJ'), ('electron', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('P', 'NN')]\n",
+      "[('S', 'NN')]\n",
+      "[('Cl', 'NN')]\n",
+      "[('F', 'NN')]\n",
       "[('answer', 'NN')]\n",
-      "[('many', 'JJ'), ('elements', 'NNS')]\n",
-      "[('energy', 'NN')]\n",
+      "[('Solution', 'NN')]\n",
+      "[('Electron', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('period', 'NN')]\n",
       "[('electron', 'NN')]\n",
-      "[('atom', 'NN')]\n",
+      "[('2p-orbital', 'JJ'), ('leads', 'NNS')]\n",
+      "[('greater', 'JJR'), ('repulsion', 'NN')]\n",
       "[('electron', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('negative', 'JJ'), ('electron', 'NN')]\n",
       "[('gain', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('electron', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('least', 'RBS'), ('negative', 'JJ'), ('electron', 'NN')]\n",
       "[('gain', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('size', 'NN')]\n",
+      "[('e', 'NN')]\n",
+      "[('Electronegativity', 'NN')]\n",
+      "[('A', 'DT')]\n",
+      "[('qualitative', 'JJ'), ('measure', 'NN')]\n",
+      "[('ability', 'NN')]\n",
       "[('atom', 'NN')]\n",
-      "[('increases', 'NNS')]\n",
-      "[('added', 'VBD'), ('electron', 'NN')]\n",
-      "[('nucleus', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('3.7', 'CD')]\n",
+      "[('chemical', 'NN')]\n",
+      "[('compound', 'NN')]\n",
+      "[('shared', 'VBN'), ('electrons', 'NNS')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electron', 'NN')]\n",
       "[('gain', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('O', 'NN')]\n",
-      "[('F', 'NN')]\n",
-      "[('succeeding', 'VBG')]\n",
-      "[('element', 'NN')]\n",
+      "[('measureable', 'JJ'), ('quantity', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('scale', 'NN')]\n",
+      "[('energy', 'NN')]\n",
       "[('electron', 'NN')]\n",
-      "[('O', 'NN')]\n",
-      "[('F', 'NN')]\n",
-      "[('added', 'VBD'), ('electron', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('quantum', 'NNS'), ('level', 'NN')]\n",
-      "[('suffers', 'NNS')]\n",
-      "[('significant', 'JJ'), ('repulsion', 'NN')]\n",
-      "[('other', 'JJ'), ('electrons', 'NNS')]\n",
-      "[('level', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('3', 'CD')]\n",
-      "[('quantum', 'NN')]\n",
-      "[('level', 'NN')]\n",
-      "[('S', 'NN')]\n",
-      "[('Cl', 'NN')]\n",
-      "[('added', 'VBD'), ('electron', 'NN')]\n",
-      "[('larger', 'JJR'), ('region', 'NN')]\n",
-      "[('space', 'NN')]\n",
-      "[('electron-electron', 'JJ'), ('repulsion', 'NN')]\n",
-      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('figure', 'NN')]\n",
-      "[('3.7', 'CD')]\n",
-      "[('Periodicity', 'NN')]\n",
-      "[('Valence', 'NN')]\n",
-      "[('Oxidation', 'NN')]\n",
-      "[('States', 'NNS')]\n",
-      "[('valence', 'NN')]\n",
-      "[('characteristic', 'JJ'), ('property', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('terms', 'NNS')]\n",
-      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
-      "[('valence', 'NN')]\n",
-      "[('representative', 'JJ'), ('elements', 'NNS')]\n",
-      "[('number', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('outermost', 'NN')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('/', 'NN')]\n",
-      "[('eight', 'CD')]\n",
-      "[('minus', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('outermost', 'NN'), ('electrons', 'NNS')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('Silicon', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('14', 'CD')]\n",
-      "[('element', 'NN')]\n",
-      "[('valence', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('bromine', 'NN'), ('belongs', 'NNS')]\n",
-      "[('halogen', 'NN')]\n",
-      "[('family', 'NN')]\n",
-      "[('valence', 'NN')]\n",
-      "[('1', 'CD')]\n",
+      "[('atom', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('affinity', 'NN')]\n",
+      "[('convention', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('affinity', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('negative', 'JJ'), ('sign', 'NN')]\n",
+      "[('electron', 'NN'), ('affinity', 'NN')]\n",
+      "[('absolute', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('other', 'JJ'), ('temperature', 'NN')]\n",
+      "[('T', 'NN')]\n",
+      "[('heat', 'NN')]\n",
+      "[('capacities', 'NNS')]\n",
+      "[('reactants', 'NNS')]\n",
+      "[('products', 'NNS')]\n",
+      "[('account', 'NN')]\n",
+      "[('∆egH', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('–Ae', 'NN')]\n",
+      "[('5/2', 'CD')]\n",
+      "[('RT', 'NN')]\n",
+      "[('trend', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('Table', 'JJ'), ('3.8', 'CD')]\n",
+      "[('Electronegativity', 'NN')]\n",
+      "[('Values', 'NNS')]\n",
+      "[('Pauling', 'VBG')]\n",
+      "[('scale', 'NN')]\n",
+      "[('Periods', 'NNS')]\n",
+      "[('Table', 'JJ'), ('3.8', 'CD')]\n",
       "[('b', 'NN')]\n",
-      "[('Aluminium', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('13', 'CD')]\n",
-      "[('valence', 'NN')]\n",
-      "[('3', 'CD')]\n",
-      "[('sulphur', 'NN'), ('belongs', 'NNS')]\n",
-      "[('group', 'NN')]\n",
-      "[('16', 'CD')]\n",
+      "[('Electronegativity', 'NN')]\n",
+      "[('Values', 'NNS')]\n",
+      "[('Pauling', 'VBG')]\n",
+      "[('scale', 'NN')]\n",
+      "[('3.7.2', 'CD')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Trends', 'NNS')]\n",
+      "[('Chemical', 'NNP')]\n",
+      "[('Properties', 'NNS')]\n",
+      "[('Most', 'JJS')]\n",
+      "[('trends', 'NNS')]\n",
+      "[('chemical', 'NN')]\n",
+      "[('properties', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
+      "[('diagonal', 'JJ'), ('relationships', 'NNS')]\n",
+      "[('inert', 'NN'), ('pair', 'NN')]\n",
+      "[('effect', 'NN')]\n",
+      "[('effects', 'NNS')]\n",
+      "[('lanthanoid', 'JJ'), ('contraction', 'NN')]\n",
+      "[('etc', 'NN')]\n",
+      "[('term', 'NN')]\n",
+      "[('oxidation', 'NN')]\n",
+      "[('state', 'NN')]\n",
       "[('valence', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('Hence', 'NN')]\n",
-      "[('formula', 'NN')]\n",
-      "[('compound', 'NN')]\n",
-      "[('Al2S3', 'NN')]\n",
-      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('valence', 'NN')]\n",
+      "[('Problem', 'NN')]\n",
+      "[('3.8', 'CD')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('formulas', 'NNS')]\n",
+      "[('compounds', 'NNS')]\n",
+      "[('following', 'VBG'), ('pairs', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
-      "[('hydrides', 'NNS')]\n",
-      "[('oxides', 'NNS')]\n",
-      "[('Table', 'JJ'), ('3.9', 'CD')]\n",
+      "[('silicon', 'NN')]\n",
+      "[('bromine', 'NN')]\n",
       "[('b', 'NN')]\n",
-      "[('Anomalous', 'JJ')]\n",
-      "[('Properties', 'NNS')]\n",
-      "[('Second', 'JJ')]\n",
-      "[('Period', 'NN')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('first', 'RB'), ('element', 'NN')]\n",
-      "[('groups', 'NNS')]\n",
-      "[('1', 'CD')]\n",
-      "[('lithium', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('beryllium', 'NN')]\n",
-      "[('groups', 'NNS')]\n",
-      "[('boron', 'NN')]\n",
-      "[('differs', 'NNS')]\n",
-      "[('many', 'JJ'), ('respects', 'NNS')]\n",
-      "[('other', 'JJ'), ('members', 'NNS')]\n",
-      "[('respective', 'JJ'), ('group', 'NN')]\n",
+      "[('aluminium', 'NN')]\n",
+      "[('sulphur', 'NN')]\n",
       "[('example', 'NN')]\n",
       "[('lithium', 'NN')]\n",
       "[('other', 'JJ'), ('alkali', 'JJ'), ('metals', 'NNS')]\n",
@@ -1917,138 +1988,222 @@
       "[('other', 'JJ'), ('members', 'NNS')]\n",
       "[('groups', 'NNS')]\n",
       "[('ionic', 'JJ'), ('compounds', 'NNS')]\n",
-      "[('Problem', 'NN')]\n",
-      "[('3.9', 'CD')]\n",
-      "[('Are', 'NN')]\n",
-      "[('oxidation', 'NN')]\n",
-      "[('state', 'NN')]\n",
-      "[('covalency', 'NN')]\n",
-      "[('Al', 'NN')]\n",
-      "[('[', 'NN')]\n",
-      "[('AlCl', 'NN')]\n",
-      "[('H2O', 'NN')]\n",
-      "[('5', 'CD')]\n",
-      "[('2+', 'CD')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('No', 'DT')]\n",
-      "[('Oxides', 'NNS')]\n",
+      "[('fact', 'NN')]\n",
+      "[('behaviour', 'NN')]\n",
+      "[('lithium', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('second', 'JJ'), ('element', 'NN')]\n",
+      "[('group', 'NN')]\n",
+      "[('magnesium', 'NN')]\n",
+      "[('aluminium', 'NN')]\n",
+      "[('chemical', 'NN')]\n",
+      "[('physical', 'JJ'), ('properties', 'NNS')]\n",
+      "[('manifestation', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
       "[('elements', 'NNS')]\n",
+      "[('results', 'NNS')]\n",
+      "[('high', 'JJ'), ('chemical', 'NN'), ('reactivity', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('extremes', 'NNS')]\n",
       "[('centre', 'NN')]\n",
-      "[('Al2O3', 'NN')]\n",
-      "[('As2O3', 'NN')]\n",
-      "[('CO', 'NN')]\n",
-      "[('NO', 'DT')]\n",
-      "[('N2O', 'NN')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('Na2O', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('forms', 'NNS')]\n",
-      "[('strong', 'JJ'), ('base', 'NN')]\n",
-      "[('whereas', 'NNS')]\n",
-      "[('Cl2O7', 'NN')]\n",
-      "[('forms', 'NNS')]\n",
-      "[('strong', 'JJ'), ('acid', 'NN')]\n",
-      "[('Na2O', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('H2O', 'NN')]\n",
-      "[('2NaOH', 'CD')]\n",
-      "[('Cl2O7', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('H2O', 'NN')]\n",
-      "[('2HClO4', 'CD')]\n",
-      "[('Their', 'PRP$')]\n",
-      "[('basic', 'JJ')]\n",
-      "[('acidic', 'JJ'), ('nature', 'NN')]\n",
-      "[('litmus', 'JJ'), ('paper', 'NN')]\n",
+      "[('property', 'NN')]\n",
+      "[('reducing', 'VBG')]\n",
+      "[('behaviour', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
       "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('increases', 'NNS')]\n",
-      "[('group', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('left', 'NN'), ('decreases', 'NNS')]\n",
       "[('non-metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('decreases', 'NNS')]\n",
-      "[('Non-metals', 'NNS')]\n",
-      "[('top', 'NN')]\n",
-      "[('periodic', 'NNS'), ('table', 'NN')]\n",
-      "[('twenty', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('atomic', 'JJ'), ('radii', 'NN')]\n",
-      "[('decrease', 'NN')]\n",
+      "[('increases', 'NNS')]\n",
       "[('period', 'NN')]\n",
-      "[('increase', 'NN')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
+      "[('consequence', 'NN')]\n",
       "[('group', 'NN')]\n",
+      "[('1', 'CD')]\n",
+      "[('2', 'CD')]\n",
+      "[('metals', 'NNS')]\n",
+      "[('Four', 'CD')]\n",
+      "[('types', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('periodic', 'NNS'), ('table', 'NN')]\n",
+      "[('basis', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
+      "[('s-block', 'NN')]\n",
+      "[('p-block', 'NN')]\n",
+      "[('d-block', 'NN')]\n",
+      "[('f-block', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('Hydrogen', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('electron', 'NN')]\n",
+      "[('1s', 'CD')]\n",
+      "[('orbital', 'JJ'), ('occupies', 'NNS')]\n",
+      "[('unique', 'JJ'), ('position', 'NN')]\n",
+      "[('periodic', 'NN')]\n",
+      "[('table', 'NN')]\n",
+      "[('Oxides', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('left', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('right', 'NN')]\n",
+      "[('nature', 'NN')]\n",
       "[('Oxides', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
       "[('centre', 'NN')]\n",
+      "[('3.15', 'CD')]\n",
+      "[('Energy', 'NNP')]\n",
+      "[('electron', 'NN')]\n",
+      "[('ground', 'NN')]\n",
+      "[('state', 'NN')]\n",
+      "[('hydrogen', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('3.22', 'CD')]\n",
+      "[('basic', 'JJ'), ('difference', 'NN')]\n",
+      "[('terms', 'NNS')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('3.23', 'CD')]\n",
+      "[('statement', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('N', 'NN')]\n",
+      "[('Pauling', 'VBG')]\n",
+      "[('scale', 'NN')]\n",
+      "[('3.0', 'CD')]\n",
+      "[('nitrogen', 'NN')]\n",
+      "[('compounds', 'NNS')]\n",
+      "[('3.24', 'CD')]\n",
+      "[('Describe', 'NN')]\n",
+      "[('theory', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('gains', 'NNS')]\n",
+      "[('electron', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('3.25', 'CD')]\n",
+      "[('Would', 'MD')]\n",
+      "[('first', 'JJ'), ('ionization', 'NN')]\n",
+      "[('enthalpies', 'NNS')]\n",
+      "[('two', 'CD')]\n",
+      "[('isotopes', 'NNS')]\n",
+      "[('same', 'JJ'), ('element', 'NN')]\n",
+      "[('answer', 'NN')]\n",
       "[('b', 'NN')]\n",
       "[('element', 'NN')]\n",
       "[('two', 'CD')]\n",
       "[('electrons', 'NNS')]\n",
+      "[('above', 'IN'), ('elements', 'NNS')]\n",
+      "[('least', 'RBS'), ('reactive', 'JJ'), ('element', 'NN')]\n",
       "[('c', 'NNS')]\n",
-      "[('element', 'NN')]\n",
-      "[('two', 'CD')]\n",
+      "[('c', 'NNS')]\n",
+      "[('block', 'NN')]\n",
+      "[('number', 'NN')]\n",
+      "[('number', 'NN')]\n",
       "[('electrons', 'NNS')]\n",
-      "[('d', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('metal', 'NN')]\n",
-      "[('liquid', 'NN')]\n",
-      "[('gas', 'NN')]\n",
-      "[('room', 'NN')]\n",
-      "[('temperature', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('Contents', 'NNS')]\n",
-      "[('Unit', 'NN')]\n",
-      "[('3', 'CD')]\n",
-      "[('3.2', 'CD')]\n",
-      "[('GENESIS', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('CLASSIFICATION', 'NN')]\n",
-      "[('3.7.2', 'CD')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Trends', 'NNS')]\n",
-      "[('Chemical', 'NNP')]\n",
+      "[('subshell', 'NN')]\n",
+      "[('3.35', 'CD')]\n",
+      "[('Anything', 'NN')]\n",
+      "[('valence', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('Classification', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('Periodicity', 'NN')]\n",
       "[('Properties', 'NNS')]\n",
-      "[('3.7.3', 'CD')]\n",
-      "[('Periodic', 'NN')]\n",
+      "[('3.4', 'CD')]\n",
+      "[('NOMENCLATURE', 'NN')]\n",
+      "[('ELEMENTS', 'NNS')]\n",
+      "[('WITH', 'NN')]\n",
+      "[('ATOMIC', 'NN')]\n",
+      "[('NUMBERS', 'NNS')]\n",
+      "[('100', 'CD')]\n",
+      "[('3.6.1', 'CD')]\n",
+      "[('s-Block', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('3.6.2', 'CD')]\n",
+      "[('p-Block', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('3.7.1', 'CD')]\n",
       "[('Trends', 'NNS')]\n",
-      "[('Chemical', 'NNP')]\n",
-      "[('Reactivity', 'NN')]\n",
-      "[('thousand', 'NN')]\n",
+      "[('Physical', 'JJ')]\n",
+      "[('Properties', 'NNS')]\n",
+      "[('SUMMARY', 'NN')]\n",
+      "[('Exercises', 'NNS')]\n",
+      "[('8.1', 'CD')]\n",
+      "[('Introduction', 'NN')]\n",
+      "[('lives', 'NNS')]\n",
+      "[('tendency', 'NN')]\n",
+      "[('material', 'NN'), ('objects', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('falls', 'NNS')]\n",
+      "[('towards', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('downhill', 'NN')]\n",
+      "[('clouds', 'NNS')]\n",
+      "[('fall', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('many', 'JJ'), ('other', 'JJ'), ('such', 'JJ'), ('phenomena', 'NNS')]\n",
+      "[('Italian', 'JJ'), ('Physicist', 'NN')]\n",
+      "[('Galileo', 'NN')]\n",
+      "[('fact', 'NN')]\n",
+      "[('bodies', 'NNS')]\n",
+      "[('irrespective', 'JJ')]\n",
+      "[('masses', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('constant', 'JJ'), ('acceleration', 'NN')]\n",
+      "[('public', 'JJ'), ('demonstration', 'NN')]\n",
+      "[('fact', 'NN')]\n",
+      "[('only', 'RB'), ('motion', 'NN')]\n",
+      "[('celestial', 'JJ'), ('objects', 'NNS')]\n",
+      "[('motion', 'NN')]\n",
+      "[('circle', 'NN')]\n",
+      "[('Similar', 'JJ'), ('theories', 'NNS')]\n",
+      "[('Indian', 'JJ'), ('astronomers', 'NNS')]\n",
+      "[('400', 'CD')]\n",
       "[('years', 'NNS')]\n",
-      "[('Polish', 'JJ'), ('monk', 'NN')]\n",
-      "[('Nicolas', 'NNS')]\n",
-      "[('Copernicus', 'NN')]\n",
-      "[('definitive', 'JJ'), ('model', 'NN')]\n",
-      "[('planets', 'NNS')]\n",
-      "[('circles', 'NNS')]\n",
-      "[('fixed', 'JJ'), ('central', 'JJ'), ('sun', 'NN')]\n"
+      "[('data', 'NNS')]\n",
+      "[('assistant', 'NN')]\n",
+      "[('Johannes', 'NNS')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('1571-1640', 'JJ')]\n",
+      "[('length', 'NN')]\n",
+      "[('string', 'NN')]\n",
+      "[('ends', 'NNS')]\n",
+      "[('F1', 'NN')]\n",
+      "[('F2', 'NN')]\n",
+      "[('pins', 'NNS')]\n",
+      "[('circle', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('focii', 'NN')]\n",
+      "[('merge', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('semi-major', 'JJ'), ('axis', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('circle', 'NN')]\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[('theory', 'NN')]\n",
-      "[('church', 'NN')]\n",
-      "[('notable', 'JJ'), ('amongst', 'NN')]\n",
-      "[('supporters', 'NNS')]\n",
-      "[('Galileo', 'NN')]\n",
-      "[('prosecution', 'NN')]\n",
-      "[('state', 'NN')]\n",
-      "[('beliefs', 'NNS')]\n",
-      "[('ellipse', 'NN')]\n",
-      "[('circle', 'NN')]\n",
-      "[('special', 'JJ'), ('case', 'NN')]\n",
-      "[('closed', 'VBD'), ('curve', 'NN')]\n",
-      "[('points', 'NNS')]\n",
-      "[('F1', 'NN')]\n",
-      "[('F2', 'NN')]\n",
-      "[('line', 'NN')]\n",
-      "[('ellipse', 'NN')]\n",
-      "[('points', 'NNS')]\n",
-      "[('P', 'NN')]\n",
-      "[('A', 'DT')]\n",
-      "[('b', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('observations', 'NNS')]\n",
+      "[('sun', 'NN')]\n",
+      "[('Table', 'JJ'), ('8.1', 'CD')]\n",
+      "[('Data', 'NNS')]\n",
+      "[('measurement', 'NN')]\n",
+      "[('planetary', 'JJ'), ('motions', 'NNS')]\n",
+      "[('confirm', 'NN')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('Law', 'NN')]\n",
+      "[('Periods', 'NNS')]\n",
+      "[('≡', 'NN')]\n",
+      "[('Semi-major', 'JJ'), ('axis', 'NN')]\n",
+      "[('units', 'NNS')]\n",
+      "[('1010', 'CD')]\n",
+      "[('m', 'NN')]\n",
       "[('T', 'NN')]\n",
       "[('≡', 'JJ'), ('Time', 'NNP')]\n",
       "[('period', 'NN')]\n",
@@ -2056,96 +2211,108 @@
       "[('planet', 'NN')]\n",
       "[('years', 'NNS')]\n",
       "[('y', 'NN')]\n",
+      "[('central', 'JJ'), ('force', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('planet', 'NN')]\n",
+      "[('vector', 'NN')]\n",
       "[('Sun', 'NNP')]\n",
-      "[('origin', 'NN')]\n",
-      "[('position', 'NN')]\n",
-      "[('momentum', 'NN')]\n",
       "[('planet', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('p', 'NN')]\n",
       "[('law', 'NN')]\n",
       "[('areas', 'NNS')]\n",
-      "[('8.1', 'CD')]\n",
-      "[('speed', 'NN')]\n",
+      "[('three', 'CD')]\n",
+      "[('laws', 'NNS')]\n",
+      "[('planetary', 'JJ'), ('motion', 'NN')]\n",
+      "[('painstaking', 'VBG'), ('observations', 'NNS')]\n",
+      "[('Tycho', 'NN')]\n",
+      "[('Brahe', 'NN')]\n",
+      "[('coworkers', 'NNS')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('assistant', 'NN')]\n",
+      "[('Brahe', 'NN')]\n",
+      "[('years', 'NNS')]\n",
+      "[('three', 'CD')]\n",
+      "[('planetary', 'JJ'), ('laws', 'NNS')]\n",
       "[('planet', 'NN')]\n",
-      "[('perihelion', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('Sun-planet', 'NN')]\n",
-      "[('distance', 'NN')]\n",
-      "[('SP', 'NN')]\n",
-      "[('LA', 'NNP')]\n",
-      "[('=', 'NN')]\n",
-      "[('mp', 'NN')]\n",
-      "[('rA', 'NN')]\n",
-      "[('vA', 'NN')]\n",
-      "[('two', 'CD')]\n",
-      "[('results', 'NNS')]\n",
+      "[('longer', 'JJR'), ('time', 'NN')]\n",
+      "[('BAC', 'NN')]\n",
+      "[('CPB', 'NN')]\n",
+      "[('time', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('T', 'NN')]\n",
+      "[('27.3', 'CD')]\n",
+      "[('days', 'NNS')]\n",
+      "[('Rm', 'NN')]\n",
+      "[('3.84', 'CD')]\n",
+      "[('×', 'NN')]\n",
+      "[('10\\xad8m', 'CD')]\n",
+      "[('angular', 'JJ'), ('momentum', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('torquedue', 'JJ')]\n",
+      "[('force', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('vanishes', 'NNS')]\n",
+      "[('Central', 'JJ'), ('forces', 'NNS')]\n",
+      "[('condition', 'NN')]\n",
+      "[('Two', 'CD')]\n",
+      "[('important', 'JJ'), ('results', 'NNS')]\n",
       "[('1', 'CD')]\n",
-      "[('2', 'CD')]\n",
       "[('motion', 'NN')]\n",
-      "[('planet', 'NN')]\n",
-      "[('fact', 'NN')]\n",
-      "[('result', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('well-known', 'JJ'), ('second', 'JJ'), ('law', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('Tr', 'NN')]\n",
-      "[('trejectory', 'NN')]\n",
       "[('particle', 'NN')]\n",
       "[('central', 'JJ'), ('force', 'NN')]\n",
-      "[('position', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('force', 'NN')]\n",
-      "[('OP', 'NN')]\n",
-      "[('O', 'NN')]\n",
-      "[('centre', 'NN')]\n",
-      "[('force', 'NN')]\n",
-      "[('origin', 'NN')]\n",
-      "[('area', 'NN')]\n",
-      "[('∆t', 'NN')]\n",
-      "[('area', 'NN')]\n",
-      "[('sector', 'NN')]\n",
-      "[('POP′', 'NN')]\n",
-      "[('≈', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('sin', 'NN')]\n",
-      "[('α', 'NN')]\n",
-      "[('PP′/2', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('v', 'NN')]\n",
-      "[('∆t/2', 'NN')]\n",
-      "[('8.3', 'CD')]\n",
+      "[('plane', 'NN')]\n",
+      "[('Mathematically', 'RB')]\n",
+      "[('Newton', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('gravitation', 'NN')]\n",
+      "[('law', 'NN')]\n",
       "[('force', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('m2', 'NN')]\n",
+      "[('point', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('2m', 'CD')]\n",
-      "[('centroid', 'NN'), ('G', 'NNP')]\n",
-      "[('triangle', 'NN')]\n",
+      "[('m1', 'NN')]\n",
+      "[('magnitude', 'NN')]\n",
+      "[('8.5', 'CD')]\n",
+      "[('one', 'CD')]\n",
+      "[('basis', 'NN')]\n",
+      "[('symmetry', 'NN')]\n",
+      "[('resultant', 'JJ'), ('force', 'NN')]\n",
+      "[('zero', 'NN')]\n",
       "[('b', 'NN')]\n",
-      "[('force', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('vertex', 'NN')]\n",
-      "[('A', 'DT')]\n",
-      "[('Take', 'VB')]\n",
-      "[('AG', 'NNP')]\n",
-      "[('=', 'NN')]\n",
-      "[('BG', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('CG', 'NN')]\n",
+      "[('calculus', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('special', 'JJ'), ('cases', 'NNS')]\n",
+      "[('simple', 'NN'), ('law', 'NN')]\n",
+      "[('results', 'NNS')]\n",
       "[('1', 'CD')]\n",
-      "[('m', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('attraction', 'NN')]\n",
+      "[('hollow', 'JJ'), ('spherical', 'NN'), ('shell', 'NN')]\n",
+      "[('uniform', 'JJ'), ('density', 'NN')]\n",
       "[('point', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('extended', 'VBN'), ('object', 'NN')]\n",
-      "[('force', 'NN')]\n",
+      "[('entire', 'JJ'), ('mass', 'NN')]\n",
+      "[('shell', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('shell', 'NN')]\n",
+      "[('Gravitational', 'JJ'), ('forces', 'NNS')]\n",
+      "[('various', 'JJ'), ('regions', 'NNS')]\n",
+      "[('shell', 'NN')]\n",
+      "[('components', 'NNS')]\n",
+      "[('line', 'NN')]\n",
       "[('point', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('force', 'NN')]\n",
-      "[('same', 'JJ'), ('direction', 'NN')]\n",
-      "[('calculus', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('third', 'JJ'), ('law', 'NN')]\n",
-      "[('1619', 'CD')]\n",
+      "[('centre', 'NN')]\n",
+      "[('direction', 'NN')]\n",
+      "[('prependicular', 'NN')]\n",
+      "[('line', 'NN')]\n",
+      "[('forces', 'NNS')]\n",
+      "[('point', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('shells', 'NNS')]\n",
       "[('total', 'JJ'), ('mass', 'NN')]\n",
       "[('shells', 'NNS')]\n",
       "[('mass', 'NN')]\n",
@@ -2158,31 +2325,77 @@
       "[('RE', 'NN')]\n",
       "[('gravitational', 'JJ'), ('force', 'NN')]\n",
       "[('Eq', 'NN')]\n",
-      "[('8.10', 'CD')]\n",
-      "[('8.11', 'CD')]\n",
-      "[('8.6', 'CD')]\n",
-      "[('Acceleration', 'NN')]\n",
-      "[('gravity', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('earth', 'NN')]\n",
-      "[('point', 'NN')]\n",
-      "[('mass', 'NN')]\n",
-      "[('m', 'NN')]\n",
-      "[('height', 'NN')]\n",
-      "[('h', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('earth', 'NN')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
       "[('radius', 'NN')]\n",
-      "[('earth', 'NN')]\n",
       "[('RE', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
       "[('point', 'NN')]\n",
-      "[('earth', 'NN')]\n",
-      "[('distance', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('result', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
+      "[('entire', 'JJ'), ('mass', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
       "[('centre', 'NN')]\n",
-      "[('earth', 'NN')]\n",
+      "[('Ms', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
+      "[('Ms/ME', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('RE', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('3', 'CD')]\n",
+      "[('/', 'NN'), ('RE3', 'NNP')]\n",
+      "[('8.16', 'CD')]\n",
+      "[('mass', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('cube', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('depth', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('case', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('RE–d', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('G', 'NN')]\n",
+      "[('Ms', 'NN')]\n",
+      "[('m', 'NN')]\n",
+      "[('/', 'NN')]\n",
+      "[('RE', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('8.17', 'CD')]\n",
+      "[('Ms', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('G', 'NN')]\n",
+      "[('ME', 'NN')]\n",
+      "[('m', 'NN')]\n",
       "[('RE', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('h', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('3', 'CD')]\n",
+      "[('8.18', 'CD')]\n",
+      "[('acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('depth', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('8.19', 'CD')]\n",
       "[('earth', 'NN'), ('’', 'NNS')]\n",
       "[('s', 'NN')]\n",
       "[('surface', 'NN')]\n",
@@ -2195,14 +2408,6 @@
       "[('’', 'NN')]\n",
       "[('s', 'NN'), ('gravity', 'NN')]\n",
       "[('surface', 'NN')]\n",
-      "[('8.7', 'CD')]\n",
-      "[('Gravitational', 'JJ')]\n",
-      "[('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('notion', 'NN')]\n",
-      "[('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('energy', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('position', 'NN')]\n",
       "[('position', 'NN')]\n",
       "[('particle', 'NN')]\n",
       "[('changes', 'NNS')]\n",
@@ -2214,131 +2419,279 @@
       "[('work', 'NN')]\n",
       "[('body', 'NN')]\n",
       "[('force', 'NN')]\n",
-      "[('course', 'NN')]\n",
-      "[('machines', 'NNS')]\n",
-      "[('object', 'NN')]\n",
-      "[('much', 'JJ'), ('greater', 'JJR'), ('speeds', 'NNS')]\n",
-      "[('greater', 'JJR'), ('initial', 'JJ'), ('speed', 'NN')]\n",
-      "[('object', 'NN')]\n",
-      "[('higher', 'JJR'), ('heights', 'NNS')]\n",
-      "[('reason', 'NN')]\n",
-      "[('moon', 'NN')]\n",
-      "[('atmosphere', 'RB')]\n",
-      "[('greater', 'JJR'), ('gravitational', 'JJ'), ('pull', 'NN')]\n",
-      "[('4M', 'CD')]\n",
-      "[('mechanical', 'JJ'), ('energy', 'NN')]\n",
+      "[('forces', 'NNS')]\n",
+      "[('work', 'NN')]\n",
+      "[('path', 'NN')]\n",
+      "[('conservative', 'JJ'), ('forces', 'NNS')]\n",
+      "[('force', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('conservative', 'JJ'), ('force', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('points', 'NNS')]\n",
       "[('surface', 'NN')]\n",
-      "[('M', 'NN')]\n",
-      "[('neutral', 'JJ'), ('point', 'NN')]\n",
-      "[('N', 'NN')]\n",
-      "[('speed', 'NN')]\n",
-      "[('zero', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('distances', 'NNS')]\n",
+      "[('surface', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('height', 'NN'), ('h1', 'NN')]\n",
+      "[('surface', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('height', 'NN'), ('h2', 'NN')]\n",
+      "[('surface', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('m', 'NN')]\n",
+      "[('second', 'JJ'), ('position', 'NN')]\n",
+      "[('W12', 'NN')]\n",
+      "[('W12', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('Force', 'NN')]\n",
+      "[('×', 'NN')]\n",
+      "[('displacement', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('mg', 'NN')]\n",
+      "[('h2', 'NN'), ('–', 'NN')]\n",
+      "[('h1', 'NN')]\n",
+      "[('8.20', 'CD')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('height', 'NN')]\n",
+      "[('h', 'NN')]\n",
+      "[('surface', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h', 'NN')]\n",
+      "[('mgh', 'NN'), ('+', 'NN')]\n",
+      "[('Wo', 'NN')]\n",
+      "[('8.21', 'CD')]\n",
+      "[('Wo', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('constant', 'JJ')]\n",
+      "[('W12', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h2', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h1', 'NN')]\n",
+      "[('8.22', 'CD')]\n",
+      "[('work', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('difference', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('initial', 'JJ'), ('positions.Observe', 'NN')]\n",
+      "[('constant', 'JJ'), ('Wo', 'NNP')]\n",
+      "[('cancels', 'NNS')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('r', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('r1', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('r2', 'NN')]\n",
+      "[('r2', 'NN'), ('>', 'NN')]\n",
+      "[('r1', 'NN')]\n",
+      "[('vertical', 'JJ'), ('path', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.20', 'CD')]\n",
+      "[('8.24', 'CD')]\n",
+      "[('8.22', 'CD')]\n",
+      "[('8.24', 'CD')]\n",
+      "[('example', 'NN')]\n",
+      "[('application', 'NN')]\n",
+      "[('superposition', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('Answer', 'NN')]\n",
+      "[('Consider', 'VB')]\n",
+      "[('four', 'CD')]\n",
+      "[('mass', 'NN')]\n",
+      "[('m', 'NN')]\n",
+      "[('corners', 'NNS')]\n",
+      "[('square', 'NN')]\n",
+      "[('side', 'NN')]\n",
+      "[('l', 'NN')]\n",
+      "[('8.8', 'CD')]\n",
+      "[('Escape', 'NN')]\n",
+      "[('Speed', 'NN')]\n",
+      "[('stone', 'NN')]\n",
+      "[('hand', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('natural', 'JJ'), ('query', 'NN')]\n",
+      "[('mind', 'NN')]\n",
+      "[('‘', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('such', 'JJ'), ('high', 'JJ'), ('initial', 'JJ'), ('speeds', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
       "[('principle', 'NN')]\n",
       "[('conservation', 'NN')]\n",
-      "[('mechanical', 'JJ'), ('energy', 'NN')]\n",
-      "[('point', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('question', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('conservation', 'NN')]\n",
+      "[('Eqs', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('Vi', 'NN')]\n",
+      "[('8.29', 'CD')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('Moon', 'NNP')]\n",
+      "[('only', 'RB'), ('natural', 'JJ'), ('satellite', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('near', 'IN'), ('circular', 'JJ'), ('orbit', 'NN')]\n",
+      "[('time', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('27.3', 'CD')]\n",
+      "[('days', 'NNS')]\n",
+      "[('rotational', 'JJ'), ('period', 'NN')]\n",
+      "[('moon', 'NN')]\n",
+      "[('own', 'JJ'), ('axis', 'NN')]\n",
+      "[('R.H.S', 'NN')]\n",
+      "[('Eqs', 'NN')]\n",
       "[('speed', 'NN')]\n",
-      "[('projectile', 'NN')]\n",
-      "[('zero', 'NN')]\n",
-      "[('N', 'NN')]\n",
-      "[('heavier', 'JJR'), ('sphere', 'RB'), ('4', 'CD')]\n",
-      "[('M', 'NN')]\n",
-      "[('8.35', 'CD')]\n",
-      "[('8.37', 'CD')]\n",
-      "[('T', 'CD'), ('2', 'CD')]\n",
-      "[('=', 'NN'), ('k', 'NN')]\n",
-      "[('RE', 'NN')]\n",
-      "[('+', 'NN')]\n",
+      "[('V', 'NN')]\n",
       "[('h', 'NN')]\n",
-      "[('3', 'CD')]\n",
-      "[('k', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('π2', 'NN')]\n",
-      "[('/', 'NN')]\n",
-      "[('GME', 'NN')]\n",
-      "[('8.38', 'CD')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('’', 'NN')]\n",
-      "[('law', 'NN')]\n",
-      "[('periods', 'NNS')]\n",
-      "[('motion', 'NN')]\n",
-      "[('satellites', 'NNS')]\n",
+      "[('=', '$'), ('0', 'CD')]\n",
+      "[('8.36', 'CD')]\n",
+      "[('relation', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('sides', 'NNS')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('surface', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('numerical', 'JJ'), ('values', 'NNS')]\n",
-      "[('9.8', 'CD')]\n",
+      "[('h', 'NN')]\n",
+      "[('comparison', 'NN')]\n",
       "[('RE', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('6400', 'CD')]\n",
-      "[('km', 'NN'), ('.', '.')]\n",
-      "[('85', 'CD')]\n",
-      "[('minutes', 'NNS')]\n",
-      "[('8.5', 'CD')]\n",
-      "[('planet', 'NN')]\n",
-      "[('Mars', 'NNS')]\n",
-      "[('two', 'CD')]\n",
-      "[('moons', 'NNS')]\n",
-      "[('phobos', 'NNS')]\n",
-      "[('delmos', 'NN')]\n",
-      "[('phobos', 'NNS')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[('5.97×', 'CD')]\n",
+      "[('1024', 'CD')]\n",
+      "[('kg', 'NN')]\n",
+      "[('moon', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('Earth', 'NN')]\n",
+      "[('8.7', 'CD')]\n",
+      "[('Express', 'NN')]\n",
+      "[('constant', 'JJ'), ('k', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[('value', 'NN')]\n",
+      "[('k', 'NN')]\n",
+      "[('time', 'NN')]\n",
       "[('period', 'NN')]\n",
-      "[('7', 'CD')]\n",
-      "[('hours', 'NNS')]\n",
-      "[('39', 'CD')]\n",
-      "[('minutes', 'NNS')]\n",
-      "[('orbital', 'JJ'), ('radius', 'NN')]\n",
-      "[('9.4', 'CD')]\n",
-      "[('×103', 'NN')]\n",
-      "[('km', 'NN')]\n",
-      "[('length', 'NN')]\n",
-      "[('martian', 'JJ'), ('year', 'NN')]\n",
-      "[('days', 'NNS')]\n",
-      "[('∴', 'NN'), ('TM', 'NNP')]\n",
+      "[('moon', 'NN')]\n",
+      "[('T2', 'NN')]\n",
       "[('=', 'NN')]\n",
-      "[('1.52', 'CD')]\n",
-      "[('3/2', 'CD')]\n",
-      "[('365', 'CD')]\n",
-      "[('=', '$'), ('684', 'CD')]\n",
-      "[('days', 'NNS')]\n",
-      "[('orbits', 'NNS')]\n",
-      "[('planets', 'NNS')]\n",
-      "[('Mercury', 'NN')]\n",
-      "[('Mars', 'NNS')]\n",
-      "[('Pluto*', 'NN')]\n",
-      "[('derivation', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('’', 'NN')]\n",
-      "[('third', 'JJ'), ('law', 'NN')]\n",
-      "[('[', 'NN')]\n",
+      "[('1.33', 'CD')]\n",
+      "[('×', 'NN')]\n",
+      "[('10-14', 'JJ')]\n",
+      "[('3.84', 'CD')]\n",
+      "[('105', 'CD')]\n",
+      "[('3', 'CD')]\n",
+      "[('T', 'NN')]\n",
+      "[('27.3', 'CD')]\n",
+      "[('d', 'NN'), ('Note', 'NN')]\n",
       "[('Eq', 'NN')]\n",
-      "[('moon', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[('elliptical', 'JJ'), ('orbits', 'NNS')]\n",
+      "[('RE+h', 'NN')]\n",
+      "[('semi-major', 'JJ'), ('axis', 'NN')]\n",
+      "[('ellipse', 'NN')]\n",
+      "[('8.40', 'CD')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
       "[('distance', 'NN')]\n",
-      "[('3.84', 'CD')]\n",
-      "[('×', '$'), ('105', 'CD')]\n",
-      "[('km', 'NN')]\n",
+      "[('RE+h', 'NN')]\n",
+      "[('centre', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('8.40', 'CD')]\n",
-      "[('magnitude', 'NN')]\n",
+      "[('8.41', 'CD')]\n",
+      "[('K.E', 'NNP')]\n",
+      "[('P.E', 'NN')]\n",
+      "[('orbit', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
       "[('K.E', 'NNP')]\n",
-      "[('much', 'JJ'), ('energy', 'NN')]\n",
-      "[('circular', 'JJ'), ('orbit', 'NN')]\n",
-      "[('radius', 'NN')]\n",
-      "[('4RE', 'CD')]\n",
-      "[('changes', 'NNS')]\n",
-      "[('potential', 'JJ'), ('energies', 'NNS')]\n",
-      "[('8.11', 'CD')]\n",
-      "[('Geostationary', 'JJ')]\n",
-      "[('Polar', 'NN')]\n",
       "[('Satellites', 'NNS')]\n",
-      "[('interesting', 'VBG'), ('phenomenon', 'NN')]\n",
-      "[('arises', 'NNS')]\n",
-      "[('value', 'NN')]\n",
-      "[('RE+', 'NN')]\n",
+      "[('finite', 'JJ'), ('distance', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('energies', 'NNS')]\n",
+      "[('zero', 'NN')]\n",
+      "[('8.37', 'CD')]\n",
+      "[('24', 'CD')]\n",
+      "[('hours', 'NNS')]\n",
+      "[('T', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('24', 'CD')]\n",
+      "[('hours', 'NNS')]\n",
       "[('h', 'NN')]\n",
+      "[('35800', 'CD')]\n",
+      "[('km', 'NN')]\n",
+      "[('Satellites', 'NNS')]\n",
+      "[('circular', 'JJ'), ('orbits', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('equatorial', 'JJ'), ('plane', 'NN')]\n",
       "[('T', 'NN')]\n",
-      "[('Eq', 'NN')]\n",
-      "[('8.37', 'CD')]\n",
+      "[('=', 'NN')]\n",
       "[('24', 'CD')]\n",
       "[('hours', 'NNS')]\n",
+      "[('Geostationery', 'NN')]\n",
+      "[('Satellites', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('same', 'JJ'), ('period', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('powerful', 'JJ'), ('rockets', 'NNS')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('such', 'JJ'), ('large', 'JJ'), ('heights', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('view', 'NN')]\n",
+      "[('several', 'JJ'), ('benefits', 'NNS')]\n",
+      "[('many', 'JJ'), ('practical', 'JJ'), ('applications', 'NNS')]\n",
+      "[('India', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('s', 'NN')]\n",
+      "[('Leap', 'VB')]\n",
+      "[('Space', 'NN')]\n",
+      "[('India', 'NNP')]\n",
+      "[('space', 'NN')]\n",
+      "[('programme', 'NN')]\n",
+      "[('1962', 'CD')]\n",
+      "[('Indian', 'JJ'), ('National', 'NNP')]\n",
+      "[('Committee', 'NNP')]\n",
+      "[('Space', 'NN')]\n",
+      "[('Research', 'NN')]\n",
+      "[('Government', 'NN')]\n",
+      "[('India', 'NNP')]\n",
+      "[('Indian', 'JJ'), ('Space', 'NNP')]\n",
+      "[('Research', 'NN')]\n",
+      "[('Organisation', 'NN')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('1969', 'CD')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('role', 'NN')]\n",
+      "[('importance', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('technology', 'NN')]\n",
+      "[('nation', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('s', 'NN')]\n",
+      "[('development', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('service', 'NN')]\n",
+      "[('common', 'JJ'), ('man', 'NN')]\n",
       "[('India', 'NNP')]\n",
       "[('first', 'RB'), ('low', 'JJ'), ('orbit', 'NN')]\n",
       "[('satellite', 'NN')]\n",
@@ -2349,26 +2702,58 @@
       "[('erstwhile', 'NN')]\n",
       "[('Soviet', 'JJ')]\n",
       "[('Union', 'NNP')]\n",
-      "[('Various', 'JJ'), ('research', 'NN')]\n",
-      "[('centers', 'NNS')]\n",
-      "[('autonomous', 'JJ'), ('institutions', 'NNS')]\n",
-      "[('remote', 'NN'), ('sensing', 'VBG')]\n",
-      "[('astronomy', 'NN')]\n",
-      "[('astrophysics', 'NNS')]\n",
-      "[('atmospheric', 'JJ'), ('sciences', 'NNS')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('indigenous', 'JJ'), ('launching', 'VBG')]\n",
+      "[('vehicle', 'NN')]\n",
+      "[('1979', 'CD')]\n",
+      "[('Rohini', 'NN')]\n",
+      "[('series', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
       "[('space', 'NN')]\n",
-      "[('research', 'NN')]\n",
-      "[('aegis', 'NN')]\n",
-      "[('Department', 'NNP')]\n",
+      "[('main', 'JJ'), ('launch', 'NN'), ('site', 'NN')]\n",
+      "[('Satish', 'JJ')]\n",
+      "[('Dhawan', 'NNP')]\n",
       "[('Space', 'NN')]\n",
-      "[('Government', 'NN')]\n",
+      "[('Center', 'NNP')]\n",
+      "[('Sriharikota', 'NN')]\n",
+      "[('Andhra', 'NN')]\n",
+      "[('Pradesh', 'NN')]\n",
+      "[('tremendous', 'JJ'), ('progress', 'NN')]\n",
       "[('India', 'NNP')]\n",
-      "[('Geostationery', 'NN')]\n",
-      "[('satellite', 'NN')]\n",
-      "[('station', 'NN')]\n",
-      "[('signals', 'NNS')]\n",
-      "[('wide', 'JJ'), ('area', 'NN')]\n",
-      "[('earth', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('space', 'NN')]\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[('programme', 'NN')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('six', 'CD')]\n",
+      "[('largest', 'JJS'), ('space', 'NN')]\n",
+      "[('agencies', 'NNS')]\n",
+      "[('world', 'NN')]\n",
+      "[('order', 'NN')]\n",
+      "[('complete', 'JJ'), ('self-reliance', 'NN')]\n",
+      "[('applications', 'NNS')]\n",
+      "[('cost', 'NN')]\n",
+      "[('reliable', 'JJ'), ('Polar', 'NN')]\n",
+      "[('Satellite', 'NN')]\n",
+      "[('Launch', 'NN')]\n",
+      "[('Vehicle', 'NN')]\n",
+      "[('PSLV', 'NN')]\n",
+      "[('1990s', 'CD')]\n",
+      "[('PSLV', 'NN')]\n",
+      "[('favoured', 'VBN'), ('carrier', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
+      "[('various', 'JJ'), ('countries', 'NNS')]\n",
+      "[('unprecedented', 'JJ'), ('international', 'JJ'), ('collaboration', 'NN')]\n",
+      "[('class', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
+      "[('Polar', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
       "[('low', 'JJ'), ('altitude', 'NN')]\n",
       "[('500', 'CD')]\n",
       "[('800', 'CD')]\n",
@@ -2380,117 +2765,122 @@
       "[('earth', 'NN')]\n",
       "[('axis', 'NN')]\n",
       "[('east-west', 'JJ'), ('direction', 'NN')]\n",
-      "[('Information', 'NN')]\n",
-      "[('such', 'JJ'), ('satellites', 'NNS')]\n",
-      "[('remote', 'NN'), ('sensing', 'VBG')]\n",
-      "[('meterology', 'NN')]\n",
-      "[('environmental', 'JJ'), ('studies', 'NNS')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('Weightlessness', 'NN')]\n",
+      "[('Weight', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('force', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('spring', 'NN')]\n",
-      "[('gravitational', 'JJ'), ('pull', 'NN')]\n",
+      "[('same', 'JJ'), ('principle', 'NN')]\n",
+      "[('weight', 'NN')]\n",
       "[('object', 'NN')]\n",
-      "[('turn', 'NN')]\n",
       "[('spring', 'NN')]\n",
-      "[('force', 'NN')]\n",
+      "[('balance', 'NN')]\n",
+      "[('hung', 'NN')]\n",
+      "[('fixed', 'VBN'), ('point', 'NN')]\n",
+      "[('e.g', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('exerts', 'NNS')]\n",
       "[('object', 'NN')]\n",
-      "[('upwards', 'NNS')]\n",
       "[('imagine', 'NN')]\n",
       "[('top', 'JJ'), ('end', 'NN')]\n",
       "[('balance', 'NN')]\n",
       "[('top', 'JJ'), ('ceiling', 'NN')]\n",
       "[('room', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('upward', 'RB'), ('force', 'NN')]\n",
       "[('object', 'NN')]\n",
-      "[('free', 'JJ'), ('fall', 'NN')]\n",
-      "[('phenomenon', 'NN')]\n",
-      "[('phenomenon', 'NN')]\n",
-      "[('weightlessness', 'NN')]\n",
-      "[('resultant', 'JJ'), ('force', 'NN')]\n",
-      "[('FR', 'NN')]\n",
-      "[('vector', 'NN')]\n",
-      "[('addition', 'NN')]\n",
-      "[('FR', 'NN')]\n",
-      "[('=', 'NN')]\n",
+      "[('acceleration', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('reading', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('balance', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('height', 'NN')]\n",
+      "[('Summary', 'JJ'), ('1', 'CD')]\n",
+      "[('Newton', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('universal', 'JJ'), ('gravitation', 'NN')]\n",
+      "[('states', 'NNS')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('attraction', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('particles', 'NNS')]\n",
+      "[('masses', 'NNS')]\n",
+      "[('m1', 'NN')]\n",
+      "[('m2', 'NN')]\n",
+      "[('distance', 'NN')]\n",
+      "[('r', 'NN')]\n",
+      "[('magnitude', 'NN')]\n",
+      "[('G', 'NN')]\n",
+      "[('universal', 'JJ'), ('gravitational', 'JJ'), ('constant', 'NN')]\n",
+      "[('value', 'NN')]\n",
+      "[('6.672', 'CD')]\n",
+      "[('×10–11', 'NN')]\n",
+      "[('N', 'NN')]\n",
+      "[('m2', 'NN')]\n",
+      "[('kg–2', 'NN')]\n",
       "[('F1', 'NN')]\n",
-      "[('+', 'NN')]\n",
       "[('F2', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('……+', 'NN')]\n",
-      "[('Fn', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('symbol', 'NN')]\n",
-      "[('‘', 'NN')]\n",
-      "[('Σ', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('summation', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('’', 'NN')]\n",
-      "[('laws', 'NNS')]\n",
-      "[('planetary', 'JJ'), ('motion', 'NN')]\n",
-      "[('state', 'NN')]\n",
-      "[('planets', 'NNS')]\n",
-      "[('elliptical', 'JJ'), ('orbits', 'NNS')]\n",
-      "[('Sun', 'NNP')]\n",
-      "[('one', 'CD')]\n",
-      "[('focal', 'JJ'), ('points', 'NNS')]\n",
-      "[('b', 'NN')]\n",
-      "[('radius', 'NN')]\n",
-      "[('vector', 'NN')]\n",
-      "[('drawn', 'NN')]\n",
+      "[('individual', 'JJ'), ('forces', 'NNS')]\n",
+      "[('M1', 'NN')]\n",
+      "[('M2', 'NN')]\n",
+      "[('….Mn', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('gravitation', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('superposition', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('other', 'JJ'), ('bodies', 'NNS')]\n",
+      "[('Most', 'JJS'), ('planets', 'NNS')]\n",
+      "[('circular', 'JJ'), ('orbits', 'NNS')]\n",
       "[('Sun', 'NNP')]\n",
-      "[('planet', 'NN')]\n",
-      "[('equal', 'JJ'), ('areas', 'NNS')]\n",
-      "[('equal', 'JJ'), ('time', 'NN')]\n",
-      "[('intervals', 'NNS')]\n",
       "[('elliptical', 'JJ'), ('orbits', 'NNS')]\n",
       "[('above', 'IN'), ('equation', 'NN')]\n",
       "[('R', 'NN')]\n",
       "[('semi-major', 'JJ'), ('axis', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('acceleration', 'NN')]\n",
-      "[('gravity', 'NN')]\n",
-      "[('height', 'NN')]\n",
-      "[('h', 'NN')]\n",
-      "[('earth', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN')]\n",
+      "[('total', 'JJ'), ('energy', 'NN')]\n",
+      "[('bound', 'NN')]\n",
+      "[('system', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('orbit', 'NN')]\n",
+      "[('elliptical', 'JJ'), ('orbit', 'NN')]\n",
+      "[('potential', 'JJ'), ('energies', 'NNS')]\n",
+      "[('escape', 'NN')]\n",
+      "[('speed', 'NN')]\n",
       "[('surface', 'NN')]\n",
-      "[('h', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('RE', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('depth', 'NN')]\n",
-      "[('d', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('5', 'CD')]\n",
+      "[('value', 'NN')]\n",
+      "[('11.2', 'CD')]\n",
+      "[('km', 'NN')]\n",
+      "[('s–1', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('uniform', 'JJ'), ('spherical', 'NN'), ('shell', 'NN')]\n",
+      "[('solid', 'NNS'), ('sphere', 'RB')]\n",
+      "[('symmetric', 'JJ'), ('internal', 'JJ'), ('mass', 'NN')]\n",
+      "[('distribution', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('shell', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('10', 'CD')]\n",
+      "[('particle', 'NN')]\n",
+      "[('uniform', 'JJ'), ('spherical', 'NN'), ('shell', 'NN')]\n",
       "[('gravitational', 'JJ'), ('force', 'NN')]\n",
-      "[('conservative', 'JJ'), ('force', 'NN')]\n",
-      "[('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('function', 'NN')]\n",
-      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
-      "[('two', 'CD')]\n",
-      "[('particles', 'NNS')]\n",
-      "[('distance', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('V', 'NN')]\n",
+      "[('particle', 'NN')]\n",
       "[('zero', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('→', 'NN')]\n",
-      "[('∞', 'NN')]\n",
-      "[('total', 'JJ'), ('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('system', 'NN')]\n",
-      "[('particles', 'NNS')]\n",
-      "[('sum', 'NN')]\n",
-      "[('energies', 'NNS')]\n",
-      "[('pairs', 'NNS')]\n",
-      "[('particles', 'NNS')]\n",
-      "[('pair', 'NN')]\n",
-      "[('term', 'NN')]\n",
-      "[('form', 'NN')]\n",
-      "[('equation', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('homogeneous', 'JJ'), ('solid', 'NNS')]\n",
+      "[('force', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
       "[('geosynchronous', 'JJ'), ('communication', 'NN')]\n",
       "[('satellite', 'NN')]\n",
       "[('moves', 'NNS')]\n",
@@ -2504,18 +2894,106 @@
       "[('’', 'NN')]\n",
       "[('s', 'NN')]\n",
       "[('centre', 'NN')]\n",
+      "[('Points', 'NNS')]\n",
+      "[('1', 'CD')]\n",
+      "[('motion', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('influence', 'NN')]\n",
+      "[('following', 'VBG'), ('quantities', 'NNS')]\n",
+      "[('Angular', 'JJ')]\n",
+      "[('momentum', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('Total', 'JJ')]\n",
+      "[('mechanical', 'JJ'), ('energy', 'NN')]\n",
+      "[('Linear', 'JJ')]\n",
+      "[('momentum', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('Angular', 'JJ'), ('momentum', 'NN')]\n",
+      "[('conservation', 'NN')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('second', 'JJ'), ('law', 'NN')]\n",
+      "[('inverse', 'NN')]\n",
+      "[('square', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('gravitation', 'NN')]\n",
+      "[('central', 'JJ'), ('force', 'NN')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('third', 'JJ'), ('law', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
       "[('8.1', 'CD')]\n",
       "[('T2', 'NN')]\n",
       "[('=', 'NN')]\n",
       "[('KS', 'NNP')]\n",
       "[('R3', 'NN')]\n",
-      "[('Exercises', 'NNS')]\n",
-      "[('8.1', 'CD')]\n",
-      "[('Answer', 'NN')]\n",
-      "[('following', 'VBG')]\n",
-      "[('charge', 'NN')]\n",
-      "[('electrical', 'JJ'), ('forces', 'NNS')]\n",
-      "[('hollow', 'NN'), ('conductor', 'NN')]\n",
+      "[('constant', 'JJ'), ('KS', 'NNP')]\n",
+      "[('planets', 'NNS')]\n",
+      "[('circular', 'JJ'), ('orbits', 'NNS')]\n",
+      "[('satellites', 'NNS')]\n",
+      "[('Earth', 'NN')]\n",
+      "[('[', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[(']', 'NN')]\n",
+      "[('4', 'CD')]\n",
+      "[('astronaut', 'NN'), ('experiences', 'NNS')]\n",
+      "[('weightlessness', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('location', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('astronaut', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('“', 'NN')]\n",
+      "[('free', 'JJ'), ('fall', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('Earth', 'NN')]\n",
+      "[('5', 'CD')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('particles', 'NNS')]\n",
+      "[('distance', 'NN')]\n",
+      "[('r', 'NN')]\n",
+      "[('constant', 'JJ')]\n",
+      "[('value', 'NN')]\n",
+      "[('choice', 'NN')]\n",
+      "[('choice', 'NN')]\n",
+      "[('V', 'NN')]\n",
+      "[('0', 'CD')]\n",
+      "[('r', 'NN')]\n",
+      "[('→', 'NN')]\n",
+      "[('∞', 'NN')]\n",
+      "[('Note', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('choice', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('i.e', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('total', 'JJ'), ('energy', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('8', 'CD')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('particles', 'NNS')]\n",
+      "[('force', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('finite', 'JJ'), ('rigid', 'JJ'), ('bodies', 'NNS')]\n",
+      "[('line', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('spherical', 'JJ'), ('shell', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('Gravitational', 'JJ')]\n",
+      "[('shielding', 'VBG')]\n",
       "[('8.2', 'CD')]\n",
       "[('Choose', 'VB')]\n",
       "[('correct', 'JJ'), ('alternative', 'NN')]\n",
@@ -2523,13 +3001,58 @@
       "[('gravity', 'NN')]\n",
       "[('increases/decreases', 'NNS')]\n",
       "[('altitude', 'NN')]\n",
-      "[('8.5', 'CD')]\n",
-      "[('Let', 'VB')]\n",
-      "[('galaxy', 'NN')]\n",
-      "[('2.5', 'CD')]\n",
-      "[('×', 'NN'), ('1011', 'CD')]\n",
-      "[('one', 'CD')]\n",
-      "[('solar', 'JJ'), ('mass', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('Acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('increases/decreases', 'NNS')]\n",
+      "[('depth', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('uniform', 'JJ'), ('density', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('Acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('earth/mass', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('formula', 'NN')]\n",
+      "[('–G', 'NN')]\n",
+      "[('Mm', 'NN')]\n",
+      "[('1/r2', 'CD')]\n",
+      "[('1/r1', 'CD')]\n",
+      "[('more/less', 'NN'), ('accurate', 'NN')]\n",
+      "[('formulamg', 'NN')]\n",
+      "[('r2', 'NN'), ('–', 'NN')]\n",
+      "[('r1', 'NN')]\n",
+      "[('difference', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('points', 'NNS')]\n",
+      "[('r1', 'NN'), ('distance', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('8.7', 'CD')]\n",
+      "[('escape', 'NN')]\n",
+      "[('speed', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('location', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('direction', 'NN')]\n",
+      "[('projection', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('height', 'NN')]\n",
+      "[('location', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('8', 'CD')]\n",
+      "[('A', 'DT')]\n",
+      "[('comet', 'NN')]\n",
+      "[('sun', 'NN')]\n",
+      "[('elliptical', 'JJ'), ('orbit', 'NN')]\n",
       "[('8.9', 'CD')]\n",
       "[('Which', 'WDT')]\n",
       "[('following', 'VBG'), ('symptoms', 'NNS')]\n",
@@ -2544,6 +3067,29 @@
       "[('d', 'NN')]\n",
       "[('orientational', 'JJ')]\n",
       "[('problem', 'NN')]\n",
+      "[('8.10', 'CD')]\n",
+      "[('following', 'VBG'), ('two', 'CD')]\n",
+      "[('exercises', 'NNS')]\n",
+      "[('correct', 'NN')]\n",
+      "[('answer', 'NN')]\n",
+      "[('ones', 'NNS')]\n",
+      "[('gravitational', 'JJ'), ('intensity', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('hemispherical', 'JJ'), ('shell', 'NN')]\n",
+      "[('uniform', 'JJ'), ('mass', 'NN')]\n",
+      "[('density', 'NN')]\n",
+      "[('direction', 'NN')]\n",
+      "[('arrow', 'NN')]\n",
+      "[('Fig', 'NN')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('b', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('iv', 'NN')]\n",
+      "[('0', 'CD')]\n",
+      "[('8.1', 'CD')]\n",
+      "[('LIST', 'NN')]\n",
+      "[('INDIAN', 'NN')]\n",
+      "[('SATELLITES', 'NNS')]\n",
       "[('India', 'NNP')]\n",
       "[('239', 'CD')]\n",
       "[('foreign', 'JJ'), ('satellites', 'NNS')]\n",
@@ -2559,51 +3105,137 @@
       "[('26', 'CD')]\n",
       "[('1999', 'CD')]\n",
       "[('02', 'CD')]\n",
-      "[('15', 'CD')]\n",
-      "[('2017', 'CD')]\n",
-      "[('101', 'CD')]\n",
-      "[('world', 'NN')]\n",
-      "[('record', 'NN')]\n",
-      "[('June', 'NNP')]\n",
       "[('23', 'CD')]\n",
-      "[('2017', 'CD')]\n",
-      "[('29', 'CD')]\n",
+      "[('2007', 'CD')]\n",
+      "[('01', 'CD')]\n",
+      "[('Jan', 'NN')]\n",
+      "[('20', 'CD')]\n",
+      "[('2011', 'CD')]\n",
+      "[('01', 'CD')]\n",
+      "[('Sep', 'NN')]\n",
+      "[('09', 'CD')]\n",
+      "[('2012', 'CD')]\n",
+      "[('02', 'CD')]\n",
+      "[('Feb', 'NN')]\n",
+      "[('25', 'CD')]\n",
+      "[('2013', 'CD')]\n",
+      "[('06', 'CD')]\n",
+      "[('June', 'NNP')]\n",
+      "[('30', 'CD')]\n",
+      "[('2014', 'CD')]\n",
+      "[('05', 'CD')]\n",
+      "[('July', 'NNP')]\n",
+      "[('10', 'CD')]\n",
+      "[('2015', 'CD')]\n",
+      "[('05', 'CD')]\n",
+      "[('Sep', 'NN')]\n",
+      "[('26', 'CD')]\n",
+      "[('2016', 'CD')]\n",
+      "[('05', 'CD')]\n",
+      "[('Feb', 'NN')]\n",
+      "[('Jan', 'NN')]\n",
+      "[('12', 'CD')]\n",
+      "[('2018', 'CD')]\n",
+      "[('28', 'CD')]\n",
+      "[('Sep', 'NN')]\n",
       "[('Table', 'NN')]\n",
       "[('Contents', 'NNS')]\n",
+      "[('Chapter', 'NN')]\n",
+      "[('Eight', 'CD')]\n",
+      "[('Gravitation', 'NN')]\n",
+      "[('8.1', 'CD')]\n",
+      "[('Introduction', 'NN')]\n",
+      "[('8.2', 'CD')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('laws', 'NNS')]\n",
+      "[('8.4', 'CD')]\n",
+      "[('Gravitational', 'JJ')]\n",
+      "[('Constant', 'JJ')]\n",
+      "[('8.5', 'CD')]\n",
+      "[('Acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('earth', 'NN')]\n",
       "[('8.6', 'CD')]\n",
       "[('Acceleration', 'NN')]\n",
       "[('gravity', 'NN')]\n",
       "[('surface', 'NN')]\n",
       "[('earth', 'NN')]\n",
+      "[('8.7', 'CD')]\n",
+      "[('Gravitational', 'JJ')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('8.8', 'CD')]\n",
+      "[('Escape', 'NN')]\n",
+      "[('Speed', 'NN')]\n",
       "[('8.9', 'CD')]\n",
       "[('Earth', 'NN')]\n",
       "[('Satellites', 'NNS')]\n",
+      "[('8.10', 'CD')]\n",
+      "[('Energy', 'NNP')]\n",
+      "[('Satellite', 'NN')]\n",
       "[('8.11', 'CD')]\n",
       "[('Geostationary', 'JJ')]\n",
       "[('Polar', 'NN')]\n",
       "[('Satellites', 'NNS')]\n",
-      "[('example', 'NN')]\n",
-      "[('Amoeba', 'NN')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('Weightlessness', 'NN')]\n",
+      "[('Cover', 'NN')]\n",
+      "[('last', 'JJ'), ('chapter', 'NN')]\n",
+      "[('living', 'NN')]\n",
+      "[('organisms', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('unicellular', 'JJ'), ('organisms', 'NNS')]\n",
       "[('single', 'JJ'), ('cell', 'NN')]\n",
-      "[('movement', 'NN')]\n",
-      "[('intake', 'NN')]\n",
-      "[('food', 'NN')]\n",
-      "[('gaseous', 'JJ'), ('exchange', 'NN')]\n",
-      "[('excretion', 'NN')]\n",
-      "[('specialised', 'VBN'), ('function', 'NN')]\n",
-      "[('different', 'JJ'), ('group', 'NN')]\n",
+      "[('performs', 'NNS')]\n",
+      "[('basic', 'JJ'), ('functions', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
-      "[('plants', 'NNS')]\n",
-      "[('vascular', 'JJ'), ('tissues', 'NNS')]\n",
-      "[('food', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('one', 'CD')]\n",
-      "[('part', 'NN')]\n",
-      "[('plant', 'NN')]\n",
-      "[('other', 'JJ'), ('parts', 'NNS')]\n",
-      "[('6.2', 'CD')]\n",
-      "[('Plant', 'NN')]\n",
+      "[('specific', 'JJ'), ('functions', 'NNS')]\n",
+      "[('particular', 'JJ'), ('function', 'NN')]\n",
+      "[('cluster', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('definite', 'JJ'), ('place', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('group', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('structure', 'NN')]\n",
+      "[('and/or', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('particular', 'JJ'), ('function', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('6.1', 'CD')]\n",
+      "[('Are', 'NN'), ('Plants', 'NNS')]\n",
+      "[('Animals', 'NNS')]\n",
+      "[('Made', 'VBN')]\n",
+      "[('Same', 'NN')]\n",
+      "[('Types', 'NNS')]\n",
       "[('Tissues', 'NNS')]\n",
+      "[('structure', 'NN')]\n",
+      "[('functions', 'NNS')]\n",
+      "[('plants', 'NNS')]\n",
+      "[('animals', 'NNS')]\n",
+      "[('same', 'JJ'), ('structure', 'NN')]\n",
+      "[('similar', 'JJ'), ('functions', 'NNS')]\n",
+      "[('noticeable', 'JJ'), ('differences', 'NNS')]\n",
+      "[('two', 'CD')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('certain', 'JJ'), ('regions', 'NNS')]\n",
+      "[('capacity', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('various', 'JJ'), ('plant', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
+      "[('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Cell', 'NN')]\n",
+      "[('growth', 'NN')]\n",
+      "[('animals', 'NNS')]\n",
+      "[('fundamental', 'JJ'), ('difference', 'NN')]\n",
+      "[('different', 'JJ'), ('modes', 'NNS')]\n",
+      "[('life', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('major', 'JJ'), ('groups', 'NNS')]\n",
+      "[('organisms', 'NNS')]\n",
+      "[('different', 'JJ'), ('feeding', 'NN')]\n",
+      "[('methods', 'NNS')]\n",
       "[('Jar', 'NN')]\n",
       "[('1', 'CD')]\n",
       "[('Jar', 'NN')]\n",
@@ -2612,23 +3244,125 @@
       "[('roots', 'NNS')]\n",
       "[('onion', 'NN')]\n",
       "[('bulbs', 'NN')]\n",
-      "[('girth', 'NN')]\n"
+      "[('two', 'CD')]\n",
+      "[('onion', 'NN')]\n",
+      "[('bulbs', 'NN')]\n",
+      "[('place', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('jar', 'NN')]\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[('stem', 'NN')]\n",
-      "[('root', 'NN')]\n",
-      "[('increases', 'NNS')]\n",
-      "[('lateral', 'JJ'), ('meristem', 'NN')]\n",
-      "[('cambium', 'NN')]\n",
-      "[('Activity', 'NN')]\n",
-      "[('6.3', 'CD')]\n",
       "[('•', 'NN')]\n",
-      "[('leaf', 'NN')]\n",
-      "[('Rhoeo', 'NN')]\n",
+      "[('Observe', 'NN')]\n",
+      "[('growth', 'NN')]\n",
+      "[('roots', 'NNS')]\n",
+      "[('bulbs', 'NN')]\n",
+      "[('few', 'JJ'), ('days', 'NNS')]\n",
+      "[('growth', 'NN')]\n",
+      "[('roots', 'NNS')]\n",
+      "[('jars', 'NNS')]\n",
+      "[('lengths', 'NNS')]\n",
+      "[('day', 'NN')]\n",
+      "[('five', 'CD')]\n",
+      "[('more', 'RBR'), ('days', 'NNS')]\n",
+      "[('record', 'NN')]\n",
+      "[('observations', 'NNS')]\n",
+      "[('tables', 'NNS')]\n",
+      "[('table', 'NN')]\n",
+      "[('Length', 'NN')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('1', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('2', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('3', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('4', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('5', 'CD')]\n",
+      "[('Jar', 'NN')]\n",
+      "[('1', 'CD')]\n",
+      "[('Jar', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('growth', 'NN')]\n",
+      "[('plants', 'NNS')]\n",
+      "[('certain', 'JJ'), ('specific', 'JJ'), ('regions', 'NNS')]\n",
+      "[('dividing', 'VBG')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
+      "[('points', 'NNS')]\n",
+      "[('region', 'NN')]\n",
+      "[('meristematic', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('New', 'NNP')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('meristem', 'NN')]\n",
+      "[('meristem', 'NN')]\n",
+      "[('characteristics', 'NNS')]\n",
+      "[('components', 'NNS')]\n",
+      "[('other', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('6.2.2', 'CD')]\n",
+      "[('Permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
+      "[('specific', 'JJ'), ('role', 'NN')]\n",
+      "[('ability', 'NN')]\n",
+      "[('process', 'NN')]\n",
+      "[('permanent', 'JJ'), ('shape', 'NN')]\n",
+      "[('size', 'NN')]\n",
+      "[('function', 'NN')]\n",
+      "[('differentiation', 'NN')]\n",
+      "[('Place', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('section', 'NN')]\n",
+      "[('slide', 'NN')]\n",
+      "[('drop', 'NN')]\n",
+      "[('glycerine', 'NN')]\n",
+      "[('sections', 'NNS')]\n",
+      "[('root', 'NN')]\n",
+      "[('stem', 'NN')]\n",
+      "[('different', 'JJ'), ('plants', 'NNS')]\n",
+      "[('6.2.2', 'CD')]\n",
+      "[('Simple', 'JJ'), ('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('A', 'DT')]\n",
+      "[('few', 'JJ'), ('layers', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('simple', 'JJ'), ('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('unspecialised', 'JJ'), ('cells', 'NNS')]\n",
+      "[('thin', 'NN'), ('cell', 'NN')]\n",
+      "[('walls', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('large', 'JJ'), ('spaces', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('intercellular', 'JJ'), ('spaces', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('stores', 'NNS')]\n",
+      "[('food', 'NN')]\n",
+      "[('situations', 'NNS')]\n",
+      "[('performs', 'NNS'), ('photosynthesis', 'NN')]\n",
+      "[('chlorenchyma', 'NN')]\n",
+      "[('parenchyma', 'NN')]\n",
+      "[('type', 'NN')]\n",
+      "[('aerenchyma', 'NN')]\n",
+      "[('bending', 'NN')]\n",
+      "[('various', 'JJ'), ('parts', 'NNS')]\n",
+      "[('plant', 'NN')]\n",
+      "[('tendrils', 'NNS')]\n",
+      "[('stems', 'NNS')]\n",
+      "[('climbers', 'NNS')]\n",
+      "[('mechanical', 'JJ'), ('support', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('leaf', 'NN'), ('stalks', 'NNS')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('corners', 'NNS')]\n",
+      "[('little', 'JJ'), ('intercellular', 'JJ'), ('space', 'NN')]\n",
       "[('•', 'NNS'), ('Stretch', 'VBP')]\n",
       "[('pressure', 'NN')]\n",
       "[('•', 'NN')]\n",
@@ -2636,71 +3370,185 @@
       "[('skin', 'NN')]\n",
       "[('projects', 'NNS')]\n",
       "[('cut', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('Guard', 'NN')]\n",
-      "[('cells', 'NNS')]\n",
-      "[('epidermal', 'JJ'), ('cells', 'NNS')]\n",
-      "[('lateral', 'JJ'), ('view', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('view', 'NN')]\n",
-      "[('pores', 'NNS')]\n",
-      "[('stomata', 'NNS')]\n",
-      "[('Transpiration', 'NN')]\n",
-      "[('loss', 'NN')]\n",
+      "[('Remove', 'VB')]\n",
+      "[('peel', 'NN')]\n",
+      "[('petri', 'NNS'), ('dish', 'NN')]\n",
       "[('water', 'NN')]\n",
-      "[('form', 'NN')]\n",
-      "[('vapour', 'NN')]\n",
+      "[('•', 'NN')]\n",
+      "[('Add', 'VB')]\n",
+      "[('few', 'JJ'), ('drops', 'NNS')]\n",
+      "[('safranin', 'NN')]\n",
+      "[('•', 'NNS'), ('Wait', 'VBP')]\n",
+      "[('couple', 'NN')]\n",
+      "[('minutes', 'NNS')]\n",
+      "[('slide', 'NN')]\n",
       "[('place', 'NN')]\n",
-      "[('stomata', 'NNS')]\n",
+      "[('cover', 'NN')]\n",
+      "[('slip', 'NN')]\n",
+      "[('•', 'NNS'), ('Observe', 'VBP')]\n",
+      "[('microscope', 'NN')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('single', 'JJ'), ('layer', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('entire', 'JJ'), ('surface', 'NN')]\n",
+      "[('plant', 'NN')]\n",
+      "[('outer', 'NN')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('gases', 'NNS')]\n",
+      "[('atmosphere', 'RB')]\n",
       "[('role', 'NN')]\n",
       "[('transpiration', 'NN')]\n",
       "[('plants', 'NNS')]\n",
-      "[('Epidermal', 'JJ')]\n",
+      "[('reason', 'NN')]\n",
+      "[('outer', 'NN'), ('layer', 'NN')]\n",
+      "[('branch', 'NN')]\n",
+      "[('outer', 'NN')]\n",
+      "[('layer', 'NN')]\n",
+      "[('young', 'JJ'), ('stem', 'NN')]\n",
+      "[('plants', 'NNS')]\n",
+      "[('outer', 'NN')]\n",
+      "[('protective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('undergoes', 'NNS')]\n",
+      "[('certain', 'JJ'), ('changes', 'NNS')]\n",
+      "[('strip', 'NN')]\n",
+      "[('secondary', 'JJ'), ('meristem', 'NN')]\n",
+      "[('cortex', 'NN')]\n",
+      "[('layers', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
-      "[('roots', 'NNS')]\n",
-      "[('function', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('absorption', 'NN')]\n",
-      "[('hair-like', 'NN'), ('parts', 'NNS')]\n",
-      "[('total', 'JJ'), ('absorptive', 'JJ'), ('surface', 'NN')]\n",
-      "[('area', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('minerals', 'NNS')]\n",
+      "[('cork', 'NN')]\n",
+      "[('Cells', 'NNS')]\n",
+      "[('cork', 'NN')]\n",
+      "[('intercellular', 'JJ'), ('spaces', 'NNS')]\n",
+      "[('Such', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('simple', 'JJ'), ('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Complex', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('one', 'CD')]\n",
+      "[('type', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('common', 'JJ'), ('function', 'NN')]\n",
+      "[('Xylem', 'NN')]\n",
+      "[('phloem', 'NN')]\n",
+      "[('examples', 'NNS')]\n",
+      "[('such', 'JJ'), ('complex', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('vascular', 'JJ'), ('bundle', 'NN')]\n",
+      "[('Vascular', 'JJ'), ('tissue', 'NN')]\n",
+      "[('distinctive', 'JJ'), ('feature', 'NN')]\n",
+      "[('complex', 'JJ'), ('plants', 'NNS')]\n",
+      "[('one', 'CD')]\n",
+      "[('survival', 'NN')]\n",
+      "[('terrestrial', 'JJ'), ('environment', 'NN')]\n",
+      "[('section', 'NN')]\n",
+      "[('stem', 'NN')]\n",
+      "[('different', 'JJ'), ('types', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('vascular', 'NN')]\n",
+      "[('bundle', 'NN')]\n",
+      "[('Xylem', 'NN')]\n",
+      "[('tracheids', 'NNS')]\n",
+      "[('vessels', 'NNS')]\n",
+      "[('xylem', 'NN')]\n",
       "[('parenchyma', 'NN')]\n",
-      "[('stores', 'NNS')]\n",
-      "[('food', 'NN')]\n",
-      "[('Sieve', 'NN')]\n",
-      "[('tubes', 'NNS')]\n",
-      "[('tubular', 'JJ'), ('cells', 'NNS')]\n",
-      "[('perforated', 'VBN'), ('walls', 'NNS')]\n",
-      "[('Phloem', 'NN')]\n",
-      "[('transports', 'NNS')]\n",
+      "[('xylem', 'NN')]\n",
+      "[('fibres', 'NNS')]\n",
+      "[('Questions', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('oxygen', 'NN')]\n",
+      "[('functions', 'NNS')]\n",
+      "[('mitochondria', 'NNS')]\n",
+      "[('clue', 'NN')]\n",
+      "[('question', 'NN')]\n",
+      "[('example', 'NN')]\n",
+      "[('oxygen', 'NN')]\n",
       "[('food', 'NN')]\n",
-      "[('leaves', 'NNS')]\n",
-      "[('other', 'JJ'), ('parts', 'NNS')]\n",
-      "[('plant', 'NN')]\n",
-      "[('1', 'CD')]\n",
-      "[('Name', 'NN')]\n",
-      "[('types', 'NNS')]\n",
-      "[('simple', 'NN'), ('tissues', 'NNS')]\n",
-      "[('movement', 'NN')]\n",
-      "[('chest', 'NN')]\n",
-      "[('small', 'JJ'), ('amount', 'NN')]\n",
-      "[('material', 'NN')]\n",
-      "[('intercellular', 'JJ'), ('spaces', 'NNS')]\n",
-      "[('anything', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('6.3.1', 'CD')]\n",
+      "[('Epithelial', 'JJ'), ('tissue', 'NN')]\n",
+      "[('covering', 'VBG')]\n",
+      "[('protective', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('animal', 'NN')]\n",
       "[('body', 'NN')]\n",
-      "[('least', 'JJS'), ('one', 'CD')]\n",
-      "[('layer', 'NN')]\n",
+      "[('epithelial', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('Epithelium', 'NN')]\n",
+      "[('organs', 'NNS')]\n",
+      "[('cavities', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('result', 'NN')]\n",
+      "[('permeability', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('various', 'JJ'), ('epithelia', 'NNS')]\n",
+      "[('important', 'JJ'), ('role', 'NN')]\n",
+      "[('exchange', 'NN')]\n",
+      "[('materials', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('external', 'JJ'), ('environment', 'NN')]\n",
+      "[('different', 'JJ'), ('parts', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('oesophagus', 'NN')]\n",
+      "[('lining', 'VBG')]\n",
+      "[('mouth', 'NN')]\n",
+      "[('squamous', 'JJ'), ('epithelium', 'NN')]\n",
+      "[('skin', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('squamous', 'JJ'), ('epithelium', 'NN')]\n",
+      "[('columnar', 'NN')]\n",
+      "[('pillar-like', 'JJ'), ('’', 'NN')]\n",
       "[('epithelium', 'NN')]\n",
-      "[('Different', 'NN')]\n",
-      "[('show', 'NN')]\n",
-      "[('structures', 'NNS')]\n",
-      "[('unique', 'JJ'), ('functions', 'NNS')]\n",
+      "[('movement', 'NN')]\n",
+      "[('epithelial', 'JJ'), ('barrier', 'NN')]\n",
+      "[('glandular', 'JJ'), ('epithelium', 'NN')]\n",
+      "[('framework', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('Bone', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('hard', 'JJ'), ('matrix', 'NN')]\n",
+      "[('calcium', 'NN')]\n",
+      "[('phosphorus', 'NN')]\n",
+      "[('compounds', 'NNS')]\n",
+      "[('Two', 'CD')]\n",
+      "[('bones', 'NNS')]\n",
+      "[('type', 'NN')]\n",
+      "[('connective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('ligament', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('considerable', 'JJ'), ('strength', 'NN')]\n",
+      "[('Tendons', 'NNS')]\n",
+      "[('muscles', 'NNS')]\n",
+      "[('bones', 'NNS')]\n",
+      "[('type', 'NN')]\n",
+      "[('connective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Tendons', 'NNS')]\n",
+      "[('fibrous', 'JJ'), ('tissue', 'NN')]\n",
+      "[('great', 'JJ'), ('strength', 'NN')]\n",
+      "[('limited', 'JJ'), ('flexibility', 'NN')]\n",
+      "[('type', 'NN')]\n",
+      "[('connective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('cartilage', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('solid', 'JJ'), ('matrix', 'NN')]\n",
+      "[('proteins', 'NNS')]\n",
+      "[('sugars', 'NNS')]\n",
+      "[('space', 'NN')]\n",
+      "[('organs', 'NNS')]\n",
+      "[('internal', 'JJ'), ('organs', 'NNS')]\n",
+      "[('repair', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('fats', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('Fat-storing', 'VBG'), ('adipose', 'JJ'), ('tissue', 'NN')]\n",
+      "[('skin', 'NN')]\n",
+      "[('internal', 'JJ'), ('organs', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('fat', 'JJ'), ('globules', 'NNS')]\n",
+      "[('Storage', 'NN')]\n",
+      "[('fats', 'NNS')]\n",
+      "[('insulator', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('fat', 'JJ'), ('globules', 'NNS')]\n",
       "[('muscles', 'NNS')]\n",
-      "[('Muscles', 'NNS')]\n",
-      "[('limbs', 'NNS')]\n",
       "[('Such', 'JJ'), ('muscles', 'NNS')]\n",
       "[('voluntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('muscles', 'NNS')]\n",
@@ -2709,17 +3557,20 @@
       "[('help', 'NN')]\n",
       "[('body', 'NN')]\n",
       "[('movement', 'NN')]\n",
-      "[('microscope', 'NN')]\n",
-      "[('muscles', 'NNS')]\n",
-      "[('alternate', 'NN'), ('light', 'NN')]\n",
-      "[('dark', 'JJ'), ('bands', 'NNS')]\n",
-      "[('striations', 'NNS')]\n",
       "[('result', 'NN')]\n",
       "[('muscles', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
       "[('tissue', 'NN')]\n",
       "[('multinucleate', 'NN')]\n",
       "[('many', 'JJ'), ('nuclei', 'NNS')]\n",
+      "[('movement', 'NN')]\n",
+      "[('food', 'NN')]\n",
+      "[('alimentary', 'JJ'), ('canal', 'NN')]\n",
+      "[('contraction', 'NN')]\n",
+      "[('relaxation', 'NN')]\n",
+      "[('blood', 'NN')]\n",
+      "[('vessels', 'NNS')]\n",
+      "[('involuntary', 'JJ'), ('movements', 'NNS')]\n",
       "[('Smooth', 'NNP')]\n",
       "[('involuntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('control', 'NN')]\n",
@@ -2732,73 +3583,22 @@
       "[('cells', 'NNS')]\n",
       "[('pointed', 'VBN'), ('ends', 'NNS')]\n",
       "[('single', 'JJ'), ('nucleus', 'NN')]\n",
-      "[('unstriated', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('Muscular', 'JJ'), ('tissue', 'NN')]\n",
-      "[('elongated', 'VBN'), ('cells', 'NNS')]\n",
-      "[('muscle', 'NN')]\n",
-      "[('fibres', 'NNS')]\n",
-      "[('tissue', 'NN')]\n",
-      "[('movement', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('Muscles', 'NNS')]\n",
-      "[('special', 'JJ'), ('proteins', 'NNS')]\n",
-      "[('contractile', 'NN'), ('proteins', 'NNS')]\n",
-      "[('contract', 'NN')]\n",
-      "[('relax', 'NN')]\n",
-      "[('movement', 'NN')]\n",
       "[('muscles', 'NNS')]\n",
+      "[('Muscles', 'NNS')]\n",
+      "[('limbs', 'NNS')]\n",
       "[('Such', 'JJ'), ('muscles', 'NNS')]\n",
       "[('voluntary', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('microscope', 'NN')]\n",
-      "[('muscles', 'NNS')]\n",
-      "[('alternate', 'NN'), ('light', 'NN')]\n",
-      "[('dark', 'JJ'), ('bands', 'NNS')]\n",
-      "[('striations', 'NNS')]\n",
       "[('result', 'NN')]\n",
       "[('muscles', 'NNS')]\n",
-      "[('cells', 'NNS')]\n",
-      "[('tissue', 'NN')]\n",
-      "[('multinucleate', 'NN')]\n",
-      "[('many', 'JJ'), ('nuclei', 'NNS')]\n",
-      "[('movement', 'NN')]\n",
-      "[('food', 'NN')]\n",
-      "[('alimentary', 'JJ'), ('canal', 'NN')]\n",
-      "[('contraction', 'NN')]\n",
-      "[('relaxation', 'NN')]\n",
-      "[('blood', 'NN')]\n",
-      "[('vessels', 'NNS')]\n",
-      "[('involuntary', 'JJ'), ('movements', 'NNS')]\n",
       "[('Smooth', 'NNP')]\n",
       "[('involuntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('control', 'NN')]\n",
       "[('such', 'JJ'), ('movements', 'NNS')]\n",
-      "[('iris', 'NN')]\n",
-      "[('eye', 'NN')]\n",
-      "[('ureters', 'NNS')]\n",
-      "[('bronchi', 'NN')]\n",
-      "[('lungs', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
       "[('pointed', 'VBN'), ('ends', 'NNS')]\n",
       "[('single', 'JJ'), ('nucleus', 'NN')]\n",
-      "[('unstriated', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('muscles', 'NNS')]\n",
-      "[('heart', 'NN')]\n",
-      "[('show', 'NN')]\n",
-      "[('rhythmic', 'JJ'), ('contraction', 'NN')]\n",
-      "[('relaxation', 'NN')]\n",
-      "[('life', 'NN')]\n",
       "[('involuntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('cardiac', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('Heart', 'NN')]\n",
-      "[('muscle', 'NN')]\n",
-      "[('cells', 'NNS')]\n",
-      "[('Activity', 'NN')]\n",
-      "[('6.5', 'CD')]\n",
-      "[('Compare', 'NN')]\n",
-      "[('structures', 'NNS')]\n",
-      "[('different', 'JJ'), ('types', 'NNS')]\n",
-      "[('muscular', 'NN')]\n",
-      "[('tissues', 'NNS')]\n",
       "[('shape', 'NN')]\n",
       "[('number', 'NN')]\n",
       "[('nuclei', 'NN')]\n",
@@ -2808,54 +3608,39 @@
       "[('Table', 'NN')]\n",
       "[('6.1', 'CD')]\n",
       "[('cells', 'NNS')]\n",
-      "[('nervous', 'JJ'), ('tissue', 'NN')]\n",
-      "[('one', 'CD')]\n",
-      "[('place', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('brain', 'NN')]\n",
-      "[('spinal', 'JJ'), ('cord', 'NN')]\n",
-      "[('nerves', 'NNS')]\n",
-      "[('nervous', 'JJ'), ('tissue', 'NN')]\n",
-      "[('Questions', 'NNS')]\n",
-      "[('1', 'CD')]\n",
       "[('tissue', 'NN')]\n",
-      "[('movement', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('neuron', 'NNS'), ('look', 'VBP')]\n",
-      "[('3', 'CD')]\n",
-      "[('three', 'CD')]\n",
-      "[('features', 'NNS')]\n",
-      "[('cardiac', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('•', 'JJ'), ('Permanent', 'NNP')]\n",
+      "[('nerve', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('neurons', 'NNS')]\n",
+      "[('4', 'CD')]\n",
+      "[('functions', 'NNS')]\n",
+      "[('areolar', 'JJ'), ('tissue', 'NN')]\n",
+      "[('•', 'NN')]\n",
+      "[('Plant', 'NN')]\n",
       "[('tissues', 'NNS')]\n",
-      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
-      "[('ability', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('main', 'JJ'), ('types', 'NNS')]\n",
+      "[('Xylem', 'NN')]\n",
+      "[('phloem', 'NN')]\n",
+      "[('types', 'NNS')]\n",
       "[('complex', 'JJ'), ('tissues', 'NNS')]\n",
       "[('•', 'NN')]\n",
-      "[('Parenchyma', 'NN')]\n",
-      "[('collenchyma', 'NN')]\n",
-      "[('sclerenchyma', 'NN')]\n",
+      "[('Striated', 'VBN')]\n",
       "[('three', 'CD')]\n",
       "[('types', 'NNS')]\n",
-      "[('simple', 'NN'), ('tissues', 'NNS')]\n",
-      "[('shape', 'NN')]\n",
-      "[('function', 'NN')]\n",
-      "[('epithelial', 'JJ'), ('tissue', 'NN')]\n",
-      "[('columnar', 'NN')]\n",
-      "[('different', 'JJ'), ('types', 'NNS')]\n",
-      "[('connective', 'JJ'), ('tissues', 'NNS')]\n",
-      "[('body', 'NN')]\n",
-      "[('areolar', 'JJ'), ('tissue', 'NN')]\n",
-      "[('adipose', 'JJ'), ('tissue', 'NN')]\n",
-      "[('bone', 'NN')]\n",
-      "[('tendon', 'NN')]\n",
-      "[('ligament', 'NN')]\n",
-      "[('cartilage', 'NN')]\n",
-      "[('blood', 'NN')]\n",
-      "[('Nervous', 'JJ'), ('tissue', 'NN')]\n",
-      "[('neurons', 'NNS')]\n",
-      "[('impulses', 'NNS')]\n",
+      "[('muscle', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('Exercises', 'NNS')]\n",
+      "[('1', 'CD')]\n",
+      "[('term', 'NN')]\n",
+      "[('“', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('many', 'JJ'), ('types', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('xylem', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
       "[('simple', 'NN'), ('tissues', 'NNS')]\n",
       "[('complex', 'JJ'), ('tissues', 'NNS')]\n",
       "[('plants', 'NNS')]\n",
@@ -2877,48 +3662,44 @@
       "[('types', 'NNS')]\n",
       "[('muscle', 'NN')]\n",
       "[('fibres', 'NNS')]\n",
-      "[('10', 'CD')]\n",
-      "[('following', 'VBG')]\n",
       "[('Tissue', 'NN')]\n",
       "[('inner', 'NN'), ('lining', 'NN')]\n",
       "[('mouth', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('Tissue', 'NN')]\n",
-      "[('muscle', 'NN')]\n",
-      "[('humans', 'NNS')]\n",
-      "[('c', 'NNS')]\n",
-      "[('Tissue', 'NN')]\n",
-      "[('food', 'NN')]\n",
-      "[('plants', 'NNS')]\n",
-      "[('d', 'NN')]\n",
+      "[('e', 'NN')]\n",
+      "[('Connective', 'JJ')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('fluid', 'NN')]\n",
+      "[('matrix', 'NN')]\n",
+      "[('f', 'NN')]\n",
       "[('Tissue', 'NN')]\n",
-      "[('stores', 'NNS')]\n",
-      "[('body', 'NN')]\n",
-      "[('12', 'CD')]\n",
-      "[('regions', 'NNS')]\n",
-      "[('parenchyma', 'NN'), ('tissue', 'NN')]\n",
-      "[('13', 'CD')]\n",
-      "[('role', 'NN')]\n",
-      "[('epidermis', 'NN')]\n",
-      "[('plants', 'NNS')]\n",
+      "[('present', 'NN')]\n",
+      "[('brain', 'NN')]\n",
+      "[('11', 'CD')]\n",
+      "[('type', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('following', 'VBG')]\n",
+      "[('skin', 'NN')]\n",
+      "[('bark', 'NN')]\n",
+      "[('tree', 'NN')]\n",
+      "[('bone', 'NN')]\n",
+      "[('lining', 'VBG')]\n",
+      "[('kidney', 'NN')]\n",
+      "[('tubule', 'NN')]\n",
+      "[('vascular', 'JJ'), ('bundle', 'NN')]\n",
       "[('14', 'CD')]\n",
       "[('cork', 'NN')]\n",
       "[('act', 'NN')]\n",
       "[('protective', 'JJ'), ('tissue', 'NN')]\n",
       "[('15', 'CD')]\n",
       "[('following', 'VBG'), ('chart', 'NN')]\n",
-      "[('Chapter', 'NN')]\n",
-      "[('6', 'CD')]\n",
-      "[('6.2', 'CD')]\n",
-      "[('Plant', 'NN')]\n",
-      "[('Tissues', 'NNS')]\n",
-      "[('6.3', 'CD')]\n",
-      "[('Animal', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('Contents', 'NNS')]\n",
       "[('Tissues', 'NNS')]\n",
-      "[('6.3.2', 'CD')]\n",
-      "[('Connective', 'JJ'), ('tissue', 'NN')]\n",
-      "[('6.3.4', 'CD')]\n",
-      "[('Nervous', 'JJ'), ('tissue', 'NN')]\n",
+      "[('6.2.2', 'CD')]\n",
+      "[('Permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('6.3.1', 'CD')]\n",
+      "[('Epithelial', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Landmarks', 'NNS')]\n",
       "[('Cover', 'NN')]\n",
       "[('Chapter', 'NN')]\n",
       "[('6Tissues', 'NNS')]\n"
@@ -2943,7 +3724,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 103,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -2956,14 +3737,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 104,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "124\n"
+      "39\n"
      ]
     }
    ],
diff --git a/11_biology_1.txt b/11_biology_1.txt
new file mode 100644
index 0000000..26ab0f7
--- /dev/null
+++ b/11_biology_1.txt
@@ -0,0 +1,282 @@
+ 
+
+Unit 1
+
+
+
+DIVERSITY IN THE LIVING WORLD
+
+
+
+Chapter 1
+The Living World
+ 
+Chapter 2
+Biological Classification
+ 
+Chapter 3
+Plant Kingdom
+ 
+Chapter 4
+Animal Kingdom
+
+Biology is the science of life forms and living processes. The living world comprises an amazing diversity of living organisms. Early man could easily perceive the difference between inanimate matter and living organisms. Early man deified some of the inanimate matter (wind, sea, fire etc.) and some among the animals and plants. A common feature of all such forms of inanimate and animate objects was the sense of awe or fear that they evoked. The description of living organisms including human beings began much later in human history. Societies which indulged in anthropocentric view of biology could register limited progress in biological knowledge. Systematic and monumental description of life forms brought in, out of necessity, detailed systems of identification, nomenclature and classification. The biggest spin off of such studies was the recognition of the sharing of similarities among living organisms both horizontally and vertically. That all present day living organisms are related to each other and also to all organisms that ever lived on this earth, was a revelation which humbled man and led to cultural movements for conservation of biodiversity. In the following chapters of this unit, you will get a description, including classification, of animals and plants from a taxonomist’s perspective.
+
+
+
+
+
+
+
+Ernst Mayr
+(1904 – 2004)
+Born on 5 July 1904, in Kempten, Germany, Ernst Mayr, the Harvard University evolutionary biologist who has been called ‘The Darwin of the 20th century’, was one of the 100 greatest scientists of all time. Mayr joined Harvard’s Faculty of Arts and Sciences in 1953 and retired in 1975, assuming the title Alexander Agassiz Professor of Zoology Emeritus. Throughout his nearly 80-year career, his research spanned ornithology, taxonomy, zoogeography, evolution, systematics, and the history and philosophy of biology. He almost single-handedly made the origin of species diversity the central question of evolutionary biology that it is today. He also pioneered the currently accepted definition of a biological species. Mayr was awarded the three prizes widely regarded as the triple crown of biology: the Balzan Prize in 1983, the International Prize for Biology in 1994, and the Crafoord Prize in 1999. Mayr died at the age of 100 in the year 2004.
+
+
+
+
+
+
+
+
+
+
+Chapter 1
+
+
+
+The Living World
+
+
+
+1.1 What is ‘Living’?
+1.2 Diversity in the Living World
+1.3 Taxonomic Categories
+1.4 Taxonomical Aids
+
+
+How wonderful is the living world ! The wide range of living types is amazing. The extraordinary habitats in which we find living organisms, be it cold mountains, deciduous forests, oceans, fresh water lakes, deserts or hot springs, leave us speechless. The beauty of a galloping horse, of the migrating birds, the valley of flowers or the attacking shark evokes awe and a deep sense of wonder. The ecological conflict and cooperation among members of a population and among populations of a community or even the molecular traffic inside a cell make us deeply reflect on – what indeed is life? This question has two implicit questions within it. The first is a technical one and seeks answer to what living is as opposed to the non-living, and the second is a philosophical one, and seeks answer to what the purpose of life is. As scientists, we shall not attempt answering the second question. We will try to reflect on – what is living?
+
+1.1 What is ‘Living’?
+When we try to define ‘living’, we conventionally look for distinctive characteristics exhibited by living organisms. Growth, reproduction, ability to sense environment and mount a suitable response come to our mind immediately as unique features of living organisms. One can add a few more features like metabolism, ability to self-replicate, self-organise, interact and emergence to this list. Let us try to understand each of these.
+All living organisms grow. Increase in mass and increase in number of individuals are twin characteristics of growth. A multicellular organism grows by cell division. In plants, this growth by cell division occurs continuously throughout their life span. In animals, this growth is seen only up to a certain age. However, cell division occurs in certain tissues to replace lost cells. Unicellular organisms grow by cell division. One can easily observe this in in vitro cultures by simply counting the number of cells under the microscope. In majority of higher animals and plants, growth and reproduction are mutually exclusive events. One must remember that increase in body mass is considered as growth. Non-living objects also grow if we take increase in body mass as a criterion for growth. Mountains, boulders and sand mounds do grow. However, this kind of growth exhibited by non-living objects is by accumulation of material on the surface. In living organisms, growth is from inside. Growth, therefore, cannot be taken as a defining property of living organisms. Conditions under which it can be observed in all living organisms have to be explained and then we understand that it is a characteristic of living systems. A dead organism does not grow.
+Reproduction, likewise, is a characteristic of living organisms. In multicellular organisms, reproduction refers to the production of progeny possessing features more or less similar to those of parents. Invariably and implicitly we refer to sexual reproduction. Organisms reproduce by asexual means also. Fungi multiply and spread easily due to the millions of asexual spores they produce. In lower organisms like yeast and hydra, we observe budding. In Planaria (flat worms), we observe true regeneration, i.e., a fragmented organism regenerates the lost part of its body and becomes, a new organism. The fungi, the filamentous algae, the protonema of mosses, all easily multiply by fragmentation. When it comes to unicellular organisms like bacteria, unicellular algae or Amoeba, reproduction is synonymous with growth, i.e., increase in number of cells. We have already defined growth as equivalent to increase in cell number or mass. Hence, we notice that in single-celled organisms, we are not very clear about the usage of these two terms – growth and reproduction. Further, there are many organisms which do not reproduce (mules, sterile worker bees, infertile human couples, etc). Hence, reproduction also cannot be an all-inclusive defining characteristic of living organisms. Of course, no non-living object is capable of reproducing or replicating by itself.
+Another characteristic of life is metabolism. All living organisms are made of chemicals. These chemicals, small and big, belonging to various classes, sizes, functions, etc., are constantly being made and changed into some other biomolecules. These conversions are chemical reactions or metabolic reactions. There are thousands of metabolic reactions occurring simultaneously inside all living organisms, be they unicellular or multicellular. All plants, animals, fungi and microbes exhibit metabolism. The sum total of all the chemical reactions occurring in our body is metabolism. No non­-living object exhibits metabolism. Metabolic reactions can be demonstrated outside the body in cell-free systems. An isolated metabolic reaction(s) outside the body of an organism, performed in a test tube is neither living nor non-living. Hence, while metabolism is a defining feature of all living organisms without exception, isolated metabolic reactions in vitro are not living things but surely living reactions. Hence, cellular organisation of the body is the defining feature of life forms.
+Perhaps, the most obvious and technically complicated feature of all living organisms is this ability to sense their surroundings or environment and respond to these environmental stimuli which could be physical, chemical or biological. We sense our environment through our sense organs. Plants respond to external factors like light, water, temperature, other organisms, pollutants, etc. All organisms, from the prokaryotes to the most complex eukaryotes can sense and respond to environmental cues. Photoperiod affects reproduction in seasonal breeders, both plants and animals. All organisms handle chemicals entering their bodies. All organisms therefore, are ‘aware’ of their surroundings. Human being is the only organism who is aware of himself, i.e., has self-consciousness. Consciousness therefore, becomes the defining property of living organisms.
+When it comes to human beings, it is all the more difficult to define the living state. We observe patients lying in coma in hospitals virtually supported by machines which replace heart and lungs. The patient is otherwise brain-dead. The patient has no self-­consciousness. Are such patients who never come back to normal life, living or non-living?
+In higher classes, you will come to know that all living phenomena are due to underlying interactions. Properties of tissues are not present in the constituent cells but arise as a result of interactions among the constituent cells. Similarly, properties of cellular organelles are not present in the molecular constituents of the organelle but arise as a result of interactions among the molecular components comprising the organelle. These interactions result in emergent properties at a higher level of organisation. This phenomenon is true in the hierarchy of organisational complexity at all levels. Therefore, we can say that living organisms are self-replicating, evolving and self-regulating interactive systems capable of responding to external stimuli. Biology is the story of life on earth. Biology is the story of evolution of living organisms on earth. All living organisms – present, past and future, are linked to one another by the sharing of the common genetic material, but to varying degrees.
+
+1.2 Diversity in the Living World
+If you look around you will see a large variety of living organisms, be it potted plants, insects, birds, your pets or other animals and plants. There are also several organisms that you cannot see with your naked eye but they are all around you. If you were to increase the area that you make observations in, the range and variety of organisms that you see would increase. Obviously, if you were to visit a dense forest, you would probably see a much greater number and kinds of living organisms in it. Each different kind of plant, animal or organism that you see, represents a species. The number of species that are known and described range between 1.7-1.8 million. This refers to biodiversity or the number and types of organisms present on earth. We should remember here that as we explore new areas, and even old ones, new organisms are continuously being identified.
+As stated earlier, there are millions of plants and animals in the world; we know the plants and animals in our own area by their local names. These local names would vary from place to place, even within a country. Probably you would recognise the confusion that would be created if we did not find ways and means to talk to each other, to refer to organisms we are talking about.
+Hence, there is a need to standardise the naming of living organisms such that a particular organism is known by the same name all over the world. This process is called nomenclature. Obviously, nomenclature or naming is only possible when the organism is described correctly and we know to what organism the name is attached to. This is identification.
+In order to facilitate the study, number of scientists have established procedures to assign a scientific name to each known organism. This is acceptable to biologists all over the world. For plants, scientific names are based on agreed principles and criteria, which are provided in International Code for Botanical Nomenclature (ICBN). You may ask, how are animals named? Animal taxonomists have evolved International Code of Zoological Nomenclature (ICZN). The scientific names ensure that each organism has only one name. Description of any organism should enable the people (in any part of the world) to arrive at the same name. They also ensure that such a name has not been used for any other known organism.
+Biologists follow universally accepted principles to provide scientific names to known organisms. Each name has two components – the Generic name and the specific epithet. This system of providing a name with two components is called Binomial nomenclature. This naming system given by Carolus Linnaeus is being practised by biologists all over the world. This naming system using a two word format was found convenient. Let us take the example of mango to understand the way of providing scientific names better. The scientific name of mango is written as Mangifera indica. Let us see how it is a binomial name. In this name Mangifera represents the genus while indica, is a particular species, or a specific epithet. Other universal rules of nomenclature are as follows:
+1. Biological names are generally in Latin and written in italics. They are Latinised or derived from Latin irrespective of their origin.
+2. The first word in a biological name represents the genus while the second component denotes the specific epithet.
+3. Both the words in a biological name, when handwritten, are separately underlined, or printed in italics to indicate their Latin origin.
+4. The first word denoting the genus starts with a capital letter while the specific epithet starts with a small letter. It can be illustrated with the example of Mangifera indica.
+Name of the author appears after the specific epithet, i.e., at the end of the biological name and is written in an abbreviated form, e.g., Mangifera indica Linn. It indicates that this species was first described by Linnaeus.
+Since it is nearly impossible to study all the living organisms, it is necessary to devise some means to make this possible. This process is classification. Classification is the process by which anything is grouped into convenient categories based on some easily observable characters. For example, we easily recognise groups such as plants or animals or dogs, cats or insects. The moment we use any of these terms, we associate certain characters with the organism in that group. What image do you see when you think of a dog ? Obviously, each one of us will see ‘dogs’ and not ‘cats’. Now, if we were to think of ‘Alsatians’ we know what we are talking about. Similarly, suppose we were to say ‘mammals’, you would, of course, think of animals with external ears and body hair. Likewise, in plants, if we try to talk of ‘Wheat’, the picture in each of our minds will be of wheat plants, not of rice or any other plant. Hence, all these - ‘Dogs’, ‘Cats’, ‘Mammals’, ‘Wheat’, ‘Rice’, ‘Plants’, ‘Animals’, etc., are convenient categories we use to study organisms. The scientific term for these categories is taxa. Here you must recognise that taxa can indicate categories at very different levels. ‘Plants’ – also form a taxa. ‘Wheat’ is also a taxa. Similarly, ‘animals’, ‘mammals’, ‘dogs’ are all taxa – but you know that a dog is a mammal and mammals are animals. Therefore, ‘animals’, ‘mammals’ and ‘dogs’ represent taxa at different levels.
+Hence, based on characteristics, all living organisms can be classified into different taxa. This process of classification is taxonomy. External and internal structure, along with the structure of cell, development process and ecological information of organisms are essential and form the basis of modern taxonomic studies.
+Hence, characterisation, identification, classification and nomenclature are the processes that are basic to taxonomy.
+Taxonomy is not something new. Human beings have always been interested in knowing more and more about the various kinds of organisms, particularly with reference to their own use. In early days, human beings needed to find sources for their basic needs of food, clothing and shelter. Hence, the earliest classifications were based on the ‘uses’ of various organisms.
+Human beings were, since long, not only interested in knowing more about different kinds of organisms and their diversities, but also the relationships among them. This branch of study was referred to as systematics. The word systematics is derived from the Latin word ‘systema’ which means systematic arrangement of organisms. Linnaeus used Systema Naturae as the title of his publication. The scope of systematics was later enlarged to include identification, nomenclature and classification. Systematics takes into account evolutionary relationships between organisms.
+
+1.3 Taxonomic Categories
+Classification is not a single step process but involves hierarchy of steps in which each step represents a rank or category. Since the category is a part of overall taxonomic arrangement, it is called the taxonomic category and all categories together constitute the taxonomic hierarchy. Each category, referred to as a unit of classification, in fact, represents a rank and is commonly termed as taxon (pl.: taxa).
+Taxonomic categories and hierarchy can be illustrated by an example. Insects represent a group of organisms sharing common features like three pairs of jointed legs. It means insects are recognisable concrete objects which can be classified, and thus were given a rank or category. Can you name other such groups of organisms? Remember, groups represent category. Category further denotes rank. Each rank or taxon, in fact, represents a unit of classification. These taxonomic groups/categories are distinct biological entities and not merely morphological aggregates.
+Taxonomical studies of all known organisms have led to the development of common categories such as kingdom, phylum or division (for plants), class, order, family, genus and species. All organisms, including those in the plant and animal kingdoms have species as the lowest category. Now the question you may ask is, how to place an organism in various categories? The basic requirement is the knowledge of characters of an individual or group of organisms. This helps in identifying similarities and dissimilarities among the individuals of the same kind of organisms as well as of other kinds of organisms.
+
+1.3.1 Species
+Taxonomic studies consider a group of individual organisms with fundamental similarities as a species. One should be able to distinguish one species from the other closely related species based on the distinct morphological differences. Let us consider Mangifera indica, Solanum tuberosum (potato) and Panthera leo (lion). All the three names, indica, tuberosum and leo, represent the specific epithets, while the first words Mangifera, Solanum and Panthera are genera and represents another higher level of taxon or category. Each genus may have one or more than one specific epithets representing different organisms, but having morphological similarities. For example, Panthera has another specific epithet called tigris and Solanum includes species like nigrum and melongena. Human beings belong to the species sapiens which is grouped in the genus Homo. The scientific name thus, for human being, is written as Homo sapiens.
+
+1.3.2 Genus
+Genus comprises a group of related species which has more characters in common in comparison to species of other genera. We can say that genera are aggregates of closely related species. For example, potato and brinjal are two different species but both belong to the genus Solanum. Lion (Panthera leo), leopard (P. pardus) and tiger (P. tigris) with several common features, are all species of the genus Panthera. This genus differs from another genus Felis which includes cats.
+
+1.3.3 Family
+The next category, Family, has a group of related genera with still less number of similarities as compared to genus and species. Families are characterised on the basis of both vegetative and reproductive features of plant species. Among plants for example, three different genera Solanum, Petunia and Datura are placed in the family Solanaceae. Among animals for example, genus Panthera, comprising lion, tiger, leopard is put along with genus, Felis (cats) in the family Felidae. Similarly, if you observe the features of a cat and a dog, you will find some similarities and some differences as well. They are separated into two different families – Felidae and Canidae, respectively.
+
+1.3.4 Order
+You have seen earlier that categories like species, genus and families are based on a number of similar characters. Generally, order and other higher taxonomic categories are identified based on the aggregates of characters. Order being a higher category, is the assemblage of families which exhibit a few similar characters. The similar characters are less in number as compared to different genera included in a family. Plant families like Convolvulaceae, Solanaceae are included in the order Polymoniales mainly based on the floral characters. The animal order, Carnivora, includes families like Felidae and Canidae.
+
+
+Figure 1.1 Taxonomic categories showing hierarchial arrangement in ascending order
+
+
+1.3.5 Class
+This category includes related orders. For example, order Primata comprising monkey, gorilla and gibbon is placed in class Mammalia along with order Carnivora that includes animals like tiger, cat and dog. Class Mammalia has other orders also.
+
+1.3.6 Phylum
+Classes comprising animals like fishes, amphibians, reptiles, birds along with mammals constitute the next higher category called Phylum. All these, based on the common features like presence of notochord and dorsal hollow neural system, are included in phylum Chordata. In case of plants, classes with a few similar characters are assigned to a higher category called Division.
+
+
+
+1.3.7 Kingdom
+All animals belonging to various phyla are assigned to the highest category called Kingdom Animalia in the classification system of animals. The Kingdom Plantae, on the other hand, is distinct, and comprises all plants from various divisions. Henceforth, we will refer to these two groups as animal and plant kingdoms.
+The taxonomic categories from species to kingdom have been shown in ascending order starting with species in Figure 1.1. These are broad categories. However, taxonomists have also developed sub-categories in this hierarchy to facilitate more sound and scientific placement of various taxa.
+Look at the hierarchy in Figure 1.1. Can you recall the basis of arrangement? Say, for example, as we go higher from species to kingdom, the number of common characteristics goes on decreasing. Lower the taxa, more are the characteristics that the members within the taxon share. Higher the category, greater is the difficulty of determining the relationship to other taxa at the same level. Hence, the problem of classification becomes more complex.
+
+
+
+
+
+
+
+
+
+
+        Table 1.1 indicates the taxonomic categories to which some common organisms like housefly, man, mango and wheat belong.
+
+
+
+
+
+TABLE 1.1  Organisms with their Taxonomic Categories
+
+
+
+
+
+
+
+
+
+1.4 Taxonomical Aids
+
+Taxonomic studies of various species of plants, animals and other organisms are useful in agriculture, forestry, industry and in general in knowing our bio-resources and their diversity. These studies would require correct classification and identification of organisms. Identification of organisms requires intensive laboratory and field studies. The collection of actual specimens of plant and animal species is essential and is the prime source of taxonomic studies. These are also fundamental to studies and essential for training in systematics. It is used for classification of an organism, and the information gathered is also stored along with the specimens. In some cases the specimen is preserved for future studies.
+Biologists have established certain procedures and techniques to store and preserve the information as well as the specimens. Some of these are explained to help you understand the usage of these aids.
+
+1.4.1 Herbarium
+Herbarium is a store house of collected plant specimens that are dried, pressed and preserved on sheets. Further, these sheets are arranged according to a universally accepted system of classification. These specimens, along with their descriptions on herbarium sheets, become a store house or repository for future use (Figure 1.2). The herbarium sheets also carry a label providing information about date and place of collection, English, local and botanical names, family, collector’s name, etc. Herbaria also serve as quick referral systems in taxonomical studies.
+
+
+Figure 1.2 Herbarium showing stored specimens
+
+
+1.4.2 Botanical Gardens
+These specialised gardens have collections of living plants for reference. Plant species in these gardens are grown for identification purposes and each plant is labelled indicating its botanical/scientific name and its family. The famous botanical gardens are at Kew (England), Indian Botanical Garden, Howrah (India) and at National Botanical Research Institute, Lucknow (India).
+
+1.4.3 Museum
+Biological museums are generally set up in educational institutes such as schools and colleges. Museums have collections of preserved plant and animal specimens for study and reference. Specimens are preserved in the containers or jars in preservative solutions. Plant and animal specimens may also be preserved as dry specimens. Insects are preserved in insect boxes after collecting, killing and pinning. Larger animals like birds and mammals are usually stuffed and preserved. Museums often have collections of skeletons of animals too.
+
+
+
+
+
+
+1.4.4 Zoological Parks
+These are the places where wild animals are kept in protected environments under human care and which enable us to learn about their food habits and behaviour. All animals in a zoo are provided, as far as possible, the conditions similar to their natural habitats. Children love visiting these parks, commonly called Zoos (Figure 1.3).
+
+ 
+    
+
+
+
+
+Figure 1.3 Pictures showing animals in different zoological parks of India
+
+
+1.4.5 Key
+Key is another taxonomical aid used for identification of plants and animals based on the similarities and dissimilarities. The keys are based on the contrasting characters generally in a pair called couplet. It represents the choice made between two opposite options. This results in acceptance of only one and rejection of the other. Each statement in the key is called a lead. Separate taxonomic keys are required for each taxonomic category such as family, genus and species for identification purposes. Keys are generally analytical in nature.
+Flora, manuals, monographs and catalogues are some other means of recording descriptions. They also help in correct identification. Flora contains the actual account of habitat and distribution of plants of a given area. These provide the index to the plant species found in a particular area. Manuals are useful in providing information for identification of names of species found in an area. Monographs contain information on any one taxon.
+
+Summary
+The living world is rich in variety. Millions of plants and animals have been identified and described but a large number still remains unknown. The very range of organisms in terms of size, colour, habitat, physiological and morphological features make us seek the defining characteristics of living organisms. In order to facilitate the study of kinds and diversity of organisms, biologists have evolved certain rules and principles for identification, nomenclature and classification of organisms. The branch of knowledge dealing with these aspects is referred to as taxonomy. The taxonomic studies of various species of plants and animals are useful in agriculture, forestry, industry and in general for knowing our bio-resources and their diversity. The basics of taxonomy like identification, naming and classification of organisms are universally evolved under international codes. Based on the resemblances and distinct differences, each organism is identified and assigned a correct scientific/biological name comprising two words as per the binomial system of nomenclature. An organism represents/occupies a place or position in the system of classification. There are many categories/ranks and are generally referred to as taxonomic categories or taxa. All the categories constitute a taxonomic hierarchy.
+Taxonomists have developed a variety of taxonomic aids to facilitate identification, naming and classification of organisms. These studies are carried out from the actual specimens which are collected from the field and preserved as referrals in the form of herbaria, museums and in botanical gardens and zoological parks. It requires special techniques for collection and preservation of specimens in herbaria and museums. Live specimens, on the other hand, of plants and animals, are found in botanical gardens or in zoological parks. Taxonomists also prepare and disseminate information through manuals and monographs for further taxonomic studies. Taxonomic keys are tools that help in identification based on characteristics.
+
+Exercises
+1. Why are living organisms classified?
+2. Why are the classification systems changing every now and then?
+3. What different criteria would you choose to classify people that you meet often?
+4. What do we learn from identification of individuals and populations?
+5. Given below is the scientific name of Mango. Identify the correctly written name.
+Mangifera Indica
+Mangifera indica
+6. Define a taxon. Give some examples of taxa at different hierarchical levels.
+7. Can you identify the correct sequence of taxonomical categories?
+(a) Species  Order  Phylum  Kingdom
+(b) Genus  Species  Order  Kingdom
+(c) Species  Genus  Order  Phylum
+8. Try to collect all the currently accepted meanings for the word ‘species’. Discuss with your teacher the meaning of species in case of higher plants and animals on one hand, and bacteria on the other hand.
+9. Define and understand the following terms:
+(i) Phylum   (ii) Class   (iii) Family  (iv) Order  (v) Genus
+10. How is a key helpful in the identification and classification of an organism?
+11. Illustrate the taxonomical hierarchy with suitable examples of a plant and an animal.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+The Living World
+
+
+
+
+
+
+
+Table of Contents
+
+Unit 1
+
+DIVERSITY IN THE LIVING WORLD
+
+Chapter 1
+Chapter 2
+Chapter 3
+Chapter 4
+Chapter 1
+
+The Living World
+
+1.1 What is ‘Living’?
+1.2 Diversity in the Living World
+1.3 Taxonomic Categories
+1.4 Taxonomical Aids
+1.1 What is ‘Living’?
+1.2 Diversity in the Living World
+1.3 Taxonomic Categories
+1.3.1 Species
+1.3.2 Genus
+1.3.3 Family
+1.3.4 Order
+1.3.5 Class
+1.3.6 Phylum
+1.3.7 Kingdom
+1.4 Taxonomical Aids
+1.4.1 Herbarium
+1.4.2 Botanical Gardens
+1.4.3 Museum
+1.4.4 Zoological Parks
+1.4.5 Key
+
+
+Summary
+Exercises
+
+
+
+
+
+Landmarks
+
+Table of Contents
+
diff --git a/12_chemistry_12.txt b/12_chemistry_12.txt
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+ 
+Unit 12
+Aldehydes, Ketones and Carboxylic Acids
+
+
+
+
+
+
+Objectives
+
+After studying this Unit, you will be able to
+• write the common and IUPAC names of aldehydes, ketones and carboxylic acids;
+• write the structures of the compounds containing functional groups namely carbonyl and carboxyl groups;
+• describe the important methods of preparation and reactions of these classes of compounds;
+• correlate physical properties and chemical reactions of aldehydes, ketones and carboxylic acids, with their structures;
+• explain the mechanism of a few selected reactions of aldehydes and ketones;
+• understand various factors affecting the acidity of carboxylic acids and their reactions;
+• describe the uses of aldehydes, ketones and carboxylic acids.
+
+
+
+
+
+
+
+
+Carbonyl compounds are of utmost importance to organic chemistry. They are constituents of fabrics, flavourings, plastics and drugs.
+
+
+
+
+In the previous Unit, you have studied organic compounds with functional groups containing carbon-oxygen single bond. In this Unit, we will study about the organic compounds containing carbon-oxygen double bond (>C=O) called carbonyl group, which is one of the most important functional groups in organic chemistry.
+In aldehydes, the carbonyl group is bonded to a carbon and hydrogen while in the ketones, it is bonded to two carbon atoms. The carbonyl compounds in which carbon of carbonyl group is bonded to carbon or hydrogen and oxygen of hydroxyl moiety (-OH) are known as carboxylic acids, while in compounds where carbon is attached to carbon or hydrogen and nitrogen of -NH2 moiety or to halogens are called amides and acyl halides respectively. Esters and anhydrides are derivatives of carboxylic acids. The general formulas of these classes of compounds are given below:
+
+
+
+
+ 
+
+
+
+
+
+Aldehydes, ketones and carboxylic acids are widespread in plants and animal kingdom. They play an important role in biochemical processes of life. They add fragrance and flavour to nature, for example, vanillin (from vanilla beans), salicylaldehyde (from meadow sweet) and cinnamaldehyde (from cinnamon) have very pleasant fragrances.
+
+
+
+
+
+
+
+
+
+
+They are used in many food products and pharmaceuticals to add flavours. Some of these families are manufactured for use as solvents (i.e., acetone) and for preparing materials like adhesives, paints, resins, perfumes, plastics, fabrics, etc.
+
+
+12.1 Nomenclature and Structure of Carbonyl Group
+
+
+
+
+12.1.1 Nomenclature
+
+
+I. Aldehydes and ketones
+Aldehydes and ketones are the simplest and most important carbonyl compounds.
+There are two systems of nomenclature of aldehydes and ketones.
+(a) Common names
+Aldehydes and ketones are often called by their common names instead of IUPAC names. The common names of most aldehydes are derived from the common names of the corresponding carboxylic acids [Section 12.6.1] by replacing the ending –ic of acid with aldehyde. At the same time, the names reflect the Latin or Greek term for the original source of the acid or aldehyde. The location of the substituent in the carbon chain is indicated by Greek letters α, β, γ, δ, etc. The α-carbon being the one directly linked to the aldehyde group, β-carbon the next, and so on. For example
+
+
+
+The common names of ketones are derived by naming two alkyl or aryl groups bonded to the carbonyl group. The locations of substituents are indicated by Greek letters, α α′, β β′ and so on beginning with the carbon atoms next to the carbonyl group, indicated as αα′. Some ketones have historical common names, the simplest dimethyl ketone is called acetone. Alkyl phenyl ketones are usually named by adding the name of acyl group as prefix to the word phenone. For example
+
+
+
+(b) IUPAC names
+The IUPAC names of open chain aliphatic aldehydes and ketones are derived from the names of the corresponding alkanes by replacing the ending –e with –al and –one respectively. In case of aldehydes the longest carbon chain is numbered starting from the carbon of the aldehyde group while in case of ketones the numbering begins from the end nearer to the carbonyl group. The substituents are prefixed in alphabetical order along with numerals indicating their positions in the carbon chain. The same applies to cyclic ketones, where the carbonyl carbon is numbered one. When the aldehyde group is attached to a ring, the suffix carbaldehyde is added after the full name of the cycloalkane. The numbering of the ring carbon atoms start from the carbon atom attached to the aldehyde group. The name of the simplest aromatic aldehyde carrying the aldehyde group on a benzene ring is benzenecarbaldehyde. However, the common name benzaldehyde is also accepted by IUPAC. Other aromatic aldehydes are hence named as substituted benzaldehydes.
+
+
+
+
+
+
+
+
+
+
+The common and IUPAC names of some aldehydes and ketones are given in Table 12.1.
+
+
+Table 12.1: Common and IUPAC Names of Some Aldehydes and Ketones
+
+
+
+
+
+
+12.1.2 Structure of the Carbonyl Group
+
+
+
+
+
+
+The carbonyl carbon atom is sp2-hybridised and forms three sigma (σ) bonds. The fourth valence electron of carbon remains in its p-orbital and forms a π-bond with oxygen by overlap with p-orbital of an oxygen. In addition, the oxygen atom also has two non bonding electron pairs. Thus, the carbonyl carbon and the three atoms attached to it lie in the same plane and the π-electron cloud is above and below this plane. The bond angles are approximately 120° as expected of a trigonal coplanar structure (Figure 12.1).
+
+
+
+
+
+
+
+
+
+
+Fig.12.1 Orbital diagram for the formation of carbonyl group
+
+
+
+
+
+
+
+The carbon-oxygen double bond is polarised due to higher electronegativity of oxygen relative to carbon. Hence, the carbonyl carbon is an electrophilic (Lewis acid), and carbonyl oxygen, a nucleophilic (Lewis base) centre. Carbonyl compounds have substantial dipole moments and are polar than ethers. The high polarity of the carbonyl group is explained on the basis of resonance involving a neutral (A) and a dipolar (B) structures as shown.
+
+
+
+
+
+
+
+
+
+
+
+Intext Questions
+12.1 Write the structures of the following compounds.
+(i) α-Methoxypropionaldehyde (ii) 3-Hydroxybutanal
+(iii) 2-Hydroxycyclopentane carbaldehyde (iv) 4-Oxopentanal
+(v) Di-sec. butyl ketone (vi) 4-Fluoroacetophenone
+
+
+
+
+
+
+
+
+
+
+
+12.2 Preparation of Aldehydes and Ketones
+Some important methods for the preparation of aldehydes and ketones are as follows:
+
+
+
+12.2.1 Preparation of Aldehydes and Ketones
+1. By oxidation of alcohols
+
+Aldehydes and ketones are generally prepared by oxidation of primary and secondary alcohols, respectively (Unit 11, Class XII).
+
+
+
+2. By dehydrogenation of alcohols
+This method is suitable for volatile alcohols and is of industrial application. In this method alcohol vapours are passed over heavy metal catalysts (Ag or Cu). Primary and secondary alcohols give aldehydes and ketones, respectively (Unit 11, Class XII).
+3. From hydrocarbons
+(i) By ozonolysis of alkenes: As we know, ozonolysis of alkenes followed by reaction with zinc dust and water gives aldehydes, ketones or a mixture of both depending on the substitution pattern of the alkene (Unit 13, Class XI).
+(ii) By hydration of alkynes: Addition of water to ethyne in the presence of H2SO4 and HgSO4 gives acetaldehyde. All other alkynes give ketones in this reaction (Unit 13, Class XI).
+
+12.2.2 Preparation of Aldehydes
+1. From acyl chloride (acid chloride)
+Acyl chloride (acid chloride) is hydrogenated over catalyst, palladium on barium sulphate. This reaction is called Rosenmund reduction.
+
+2. From nitriles and esters
+Nitriles are reduced to corresponding imine with stannous chloride in the presence of hydrochloric acid, which on hydrolysis give corresponding aldehyde.
+
+This reaction is called Stephen reaction.
+Alternatively, nitriles are selectively reduced by diisobutylaluminium hydride, (DIBAL-H) to imines followed by hydrolysis to aldehydes:
+
+Similarly, esters are also reduced to aldehydes with DIBAL-H.
+
+3. From hydrocarbons
+
+
+        Aromatic aldehydes (benzaldehyde and its derivatives) are prepared from aromatic hydrocarbons by the following methods:
+
+
+(i) By oxidation of methylbenzene
+Strong oxidising agents oxidise toluene and its derivatives to benzoic acids. However, it is possible to stop the oxidation at the aldehyde stage with suitable reagents that convert the methyl group to an intermediate that is difficult to oxidise further. The following methods are used for this purpose.
+(a) Use of chromyl chloride (CrO2Cl2): Chromyl chloride oxidises methyl group to a chromium complex, which on hydrolysis gives corresponding benzaldehyde.
+
+This reaction is called Etard reaction.
+(b) Use of chromic oxide (CrO3): Toluene or substituted toluene is converted to benzylidene diacetate on treating with chromic oxide in acetic anhydride. The benzylidene diacetate can be hydrolysed to corresponding benzaldehyde with aqueous acid.
+
+
+
+
+
+
+(ii) By side chain chlorination followed by hydrolysis
+Side chain chlorination of toluene gives benzal chloride, which on hydrolysis gives benzaldehyde. This is a commercial method of manufacture of benzaldehyde.
+
+(iii) By Gatterman – Koch reaction
+
+
+        When benzene or its derivative is treated with carbon monoxide and hydrogen chloride in the presence of anhydrous aluminium chloride or cuprous chloride, it gives benzaldehyde or substituted benzaldehyde.
+
+
+
+
+
+This reaction is known as Gatterman-Koch reaction.
+12.2.3 Preparation of Ketones
+1. From acyl chlorides
+
+
+        Treatment of acyl chlorides with dialkylcadmium, prepared by the reaction of cadmium chloride with Grignard reagent, gives ketones.
+
+
+
+2. From nitriles
+
+
+        Treating a nitrile with Grignard reagent followed by hydrolysis yields a ketone.
+
+
+
+
+
+
+
+
+3. From benzene or substituted benzenes
+When benzene or substituted benzene is treated with acid chloride in the presence of anhydrous aluminium chloride, it affords the corresponding ketone. This reaction is known as Friedel-Crafts acylation reaction.
+
+
+
+
+
+
+
+Example 12.1
+Give names of the reagents to bring about the following transformations:
+(i) Hexan-1-ol to hexanal (ii) Cyclohexanol to cyclohexanone
+(iii) p-Fluorotoluene to  p-fluorobenzaldehyde (iv) Ethanenitrile to ethanal
+
+(v) Allyl alcohol to propenal (vi) But-2-ene to ethanal
+Solution
+
+(i) C5H5NH+CrO3Cl-(PCC) (ii) Anhydrous CrO3
+(iii) CrO3 in the presence of acetic anhydride/1. CrO2Cl2 2. HOH 
+(iv) (Diisobutyl)aluminiumhydride (DIBAL-H)
+(v) PCC  (vi) O3/H2O-Zn dust
+
+
+
+
+Intext Question
+
+
+12.2 Write the structures of products of the following reactions;
+(i)  
+(ii)
+
+
+
+(iii) 
+
+
+
+(iv) 
+
+
+
+
+
+
+
+
+
+
+12.3 Physical Properties
+The physical properties of aldehydes and ketones are described as follows.
+Methanal is a gas at room temperature. Ethanal is a volatile liquid. Other aldehydes and ketones are liquid or solid at room temperature. The boiling points of aldehydes and ketones are higher than hydrocarbons and ethers of comparable molecular masses. It is due to weak molecular association in aldehydes and ketones arising out of the dipole-dipole interactions. Also, their boiling points are lower than those of alcohols of similar molecular masses due to absence of intermolecular hydrogen bonding. The following compounds of molecular masses 58 and 60 are ranked in order of increasing boiling points.
+
+
+
+
+
+
+The lower members of aldehydes and ketones such as methanal, ethanal and propanone are miscible with water in all proportions, because they form hydrogen bond with water.
+
+
+
+
+However, the solubility of aldehydes and ketones decreases rapidly on increasing the length of alkyl chain. All aldehydes and ketones are fairly soluble in organic solvents like benzene, ether, methanol, chloroform, etc. The lower aldehydes have sharp pungent odours. As the size of the molecule increases, the odour becomes less pungent and more fragrant. In fact, many naturally occurring aldehydes and ketones are used in the blending of perfumes and flavouring agents.
+
+
+
+
+
+Example 12.2
+
+
+            Arrange the following compounds in the increasing order of their boiling points:
+
+
+
+CH3CH2CH2CHO, CH3CH2CH2CH2OH, H5C2-O-C2H5, CH3CH2CH2CH3
+Solution
+The molecular masses of these compounds are in the range of 72 to 74. Since only butan-1-ol molecules are associated due to extensive intermolecular hydrogen bonding, therefore, the boiling point of butan-1-ol would be the highest. Butanal is more polar than ethoxyethane. Therefore, the intermolecular dipole-dipole attraction is stronger in the former. n-Pentane molecules have only weak van der Waals forces. Hence increasing order of boiling points of the given compounds is as follows:
+CH3CH2CH2CH3 < H5C2-O-C2H5 < CH3CH2CH2CHO < CH3CH2CH2CH2OH
+
+
+
+
+
+Intext Question
+12.3 Arrange the following compounds in increasing order of their boiling points.
+CH3CHO, CH3CH2OH, CH3OCH3, CH3CH2CH3
+
+
+12.4 Chemical Reactions
+
+
+
+Since aldehydes and ketones both possess the carbonyl functional group, they undergo similar chemical reactions.
+1. Nucleophilic addition reactions
+Contrary to electrophilic addition reactions observed in alkenes (refer Unit 13, Class XI), the aldehydes and ketones undergo nucleophilic addition reactions.
+
+
+
+
+Fig.12.2: Nucleophilic attack on carbonyl carbon
+
+
+
+
+(i) Mechanism of nucleophilic addition reactions
+A nucleophile attacks the electrophilic carbon atom of the polar carbonyl group from a direction approximately perpendicular to the plane of sp2 hybridised orbitals of carbonyl carbon (Fig. 12.2). The hybridisation of carbon changes from sp2 to sp3 in this process, and a tetrahedral alkoxide intermediate is produced. This intermediate captures a proton from the reaction medium to give the electrically neutral product. The net result is addition of Nu– and H+ across the carbon oxygen double bond as shown in Fig. 12.2.
+
+(ii) Reactivity
+Aldehydes are generally more reactive than ketones in nucleophilic addition reactions due to steric and electronic reasons. Sterically, the presence of two relatively large substituents in ketones hinders the approach of nucleophile to carbonyl carbon than in aldehydes having only one such substituent. Electronically, aldehydes are more reactive than ketones because two alkyl groups reduce the electrophilicity of the carbonyl carbon more effectively than in former.
+
+
+
+
+Example 12.3
+
+
+
+
+
+
+          Would you expect benzaldehyde to be more reactive or less reactive in nucleophilic addition reactions than propanal? Explain your answer.
+
+
+Solution
+The carbon atom of the carbonyl group of benzaldehyde is less electrophilic than carbon atom of the carbonyl group present in propanal. The polarity of the carbonyl group is reduced in benzaldehyde due to resonance as shown below and hence it is less reactive than propanal.
+
+
+
+
+
+
+(iii) Some important examples of nucleophilic addition and nucleophilic addition-elimination reactions:
+(a) Addition of hydrogen cyanide (HCN): Aldehydes and ketones react with hydrogen cyanide (HCN) to yield cyanohydrins. This reaction occurs very slowly with pure HCN. Therefore, it is catalysed by a base and the generated cyanide ion (CN-) being a stronger nucleophile readily adds to carbonyl compounds to yield corresponding cyanohydrin.
+Cyanohydrins are useful synthetic intermediates.
+(b) Addition of sodium hydrogensulphite: Sodium hydrogensulphite adds to aldehydes and ketones to form the addition products.
+
+
+
+The position of the equilibrium lies largely to the right hand side for most aldehydes and to the left for most ketones due to steric reasons. The hydrogensulphite addition compound is water soluble and can be converted back to the original carbonyl compound by treating it with dilute mineral acid or alkali. Therefore, these are useful for separation and purification of aldehydes.
+
+
+
+
+
+
+
+(c) Addition of Grignard reagents: (refer Unit 11, Class XII).
+(d) Addition of alcohols: Aldehydes react with one equivalent of monohydric alcohol in the presence of dry hydrogen chloride to yield alkoxyalcohol intermediate, known as hemiacetals, which further react with one more molecule of alcohol to give a gem-dialkoxy compound known as acetal as shown in the reaction.
+
+
+
+Ketones react with ethylene glycol under similar conditions to form cyclic products known as ethylene glycol ketals.
+Dry hydrogen chloride protonates the oxygen of the carbonyl compounds and therefore, increases the electrophilicity of the carbonyl carbon facilitating the nucleophilic attack of ethylene glycol. Acetals and ketals are hydrolysed with aqueous mineral acids to yield corresponding aldehydes and ketones respectively.
+(e) Addition of ammonia and its derivatives: Nucleophiles, such as ammonia and its derivatives H2N-Z add to the carbonyl group of aldehydes and ketones. The reaction is reversible and catalysed by acid. The equilibrium favours the product formation due to rapid dehydration of the intermediate to form >C=N-Z.
+
+
+
+
+
+
+Z = Alkyl, aryl, OH, NH2, C6H5NH, NHCONH2, etc.
+
+
+Table 12.2: Some N-Substituted Derivatives of Aldehydes and Ketones (>C=N-Z)
+
+
+
+
+
+* 2,4-DNP-derivatives are yellow, orange or red solids, useful for characterisation of aldehydes and ketones.
+
+
+2. Reduction
+
+(i) Reduction to alcohols: Aldehydes and ketones are reduced to primary and secondary alcohols respectively by sodium borohydride (NaBH4) or lithium aluminium hydride (LiAlH4) as well as by catalytic hydrogenation (Unit 11, Class XII).
+(ii) Reduction to hydrocarbons: The carbonyl group of aldehydes and ketones is reduced to CH2 group on treatment with zinc-amalgam and concentrated hydrochloric acid [Clemmensen reduction] or with hydrazine followed by heating with sodium or potassium hydroxide in high boiling solvent such as ethylene glycol (Wolff-Kishner reduction).
+
+
+
+
+
+        Bernhard Tollens (1841-1918) was a Professor of Chemistry at the University of Gottingen, Germany.
+
+
+3. Oxidation
+Aldehydes differ from ketones in their oxidation reactions. Aldehydes are easily oxidised to carboxylic acids on treatment with common oxidising agents like nitric acid, potassium permanganate, potassium dichromate, etc. Even mild oxidising agents, mainly Tollens’ reagent and Fehlings’ reagent also oxidise aldehydes.
+
+Ketones are generally oxidised under vigorous conditions, i.e., strong oxidising agents and at elevated temperatures. Their oxidation involves carbon-carbon bond cleavage to afford a mixture of carboxylic acids having lesser number of carbon atoms than the parent ketone.
+
+
+
+
+
+
+
+
+
+The mild oxidising agents given below are used to distinguish aldehydes from ketones:
+(i) Tollens’ test: On warming an aldehyde with freshly prepared ammoniacal silver nitrate solution (Tollens’ reagent), a bright silver mirror is produced due to the formation of silver metal. The aldehydes are oxidised to corresponding carboxylate anion. The reaction occurs in alkaline medium.
+
+(ii) Fehling’s test: Fehling reagent comprises of two solutions, Fehling solution A and Fehling solution B. Fehling solution A is aqueous copper sulphate and Fehling solution B is alkaline sodium potassium tartarate (Rochelle salt). These two solutions are mixed in equal amounts before test. On heating an aldehyde with Fehling’s reagent, a reddish brown precipitate is obtained. Aldehydes are oxidised to corresponding carboxylate anion. Aromatic aldehydes do not respond to this test.
+
+(iii) Oxidation of methyl ketones by haloform reaction: Aldehydes and ketones having at least one methyl group linked to the carbonyl carbon atom (methyl ketones) are oxidised by sodium hypohalite to sodium salts of corresponding carboxylic acids having one carbon atom less than that of carbonyl compound. The methyl group is converted to haloform. This oxidation does not affect a carbon-carbon double bond, if present in the molecule.
+
+Iodoform reaction with sodium hypoiodite is also used for detection of CH3CO group or CH3CH(OH) group which produces CH3CO group on oxidation.
+
+
+Example 12.4
+
+An organic compound (A) with molecular formula C8H8O forms an orange-red precipitate with 2,4-DNP reagent and gives yellow precipitate on heating with iodine in the presence of sodium hydroxide. It neither reduces Tollens’ or Fehlings’ reagent, nor does it decolourise bromine water or Baeyer’s reagent. On drastic oxidation with chromic acid, it gives a carboxylic acid (B) having molecular formula C7H6O2. Identify the compounds (A) and (B) and explain the reactions involved.
+
+
+Solution
+(A) forms 2,4-DNP derivative. Therefore, it is an aldehyde or a ketone. Since it does not reduce Tollens’ or Fehling reagent, (A) must be a ketone. (A) responds to iodoform test. Therefore, it should be a methyl ketone. The molecular formula of (A) indicates high degree of unsaturation, yet it does not decolourise bromine water or Baeyer’s reagent. This indicates the presence of unsaturation due to an aromatic ring.
+Compound (B), being an oxidation product of a ketone should be a carboxylic acid. The molecular formula of (B) indicates that it should be benzoic acid and compound (A) should, therefore, be a monosubstituted aromatic methyl ketone. The molecular formula of (A) indicates that it should be phenyl methyl ketone (acetophenone). Reactions are as follows:
+
+
+
+
+
+
+
+
+
+4. Reactions due to a-hydrogen
+Acidity of α-hydrogens of aldehydes and ketones: The aldehydes and ketones undergo a number of reactions due to the acidic nature of α-hydrogen. The acidity of α-hydrogen atoms of carbonyl compounds is due to the strong electron withdrawing effect of the carbonyl group and resonance stabilisation of the conjugate base.
+
+
+(i) Aldol condensation: Aldehydes and ketones having at least one α-hydrogen undergo a reaction in the presence of dilute alkali as catalyst to form β-hydroxy aldehydes (aldol) or β-hydroxy ketones (ketol), respectively. This is known as Aldol reaction.
+
+
+
+
+
+
+
+
+
+The name aldol is derived from the names of the two functional groups, aldehyde and alcohol, present in the products. The aldol and ketol readily lose water to give α,β-unsaturated carbonyl compounds which are aldol condensation products and the reaction is called Aldol condensation. Though ketones give ketols (compounds containing a keto and alcohol groups), the general name aldol condensation still applies to the reactions of ketones due to their similarity with aldehydes.
+
+
+
+
+
+
+
+
+(ii) Cross aldol condensation: When aldol condensation is carried out between two different aldehydes and / or ketones, it is called cross aldol condensation. If both of them contain α-hydrogen atoms, it gives a mixture of four products. This is illustrated below by aldol reaction of a mixture of ethanal and propanal.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ketones can also be used as one component in the cross aldol reactions.
+
+5. Other reactions
+
+
+(i) Cannizzaro reaction: Aldehydes which do not have an α-hydrogen atom, undergo self oxidation and reduction (disproportionation) reaction on heating with concentrated alkali. In this reaction, one molecule of the aldehyde is reduced to alcohol while another is oxidised to carboxylic acid salt.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(ii) Electrophilic substitution reaction: Aromatic aldehydes and ketones undergo electrophilic substitution at the ring in which the carbonyl group acts as a deactivating and meta-directing group.
+
+
+
+Intext Questions
+12.4 Arrange the following compounds in increasing order of their reactivity in nucleophilic addition reactions.
+(i) Ethanal, Propanal, Propanone, Butanone.
+(ii) Benzaldehyde, p-Tolualdehyde, p-Nitrobenzaldehyde, Acetophenone.
+Hint: Consider steric effect and electronic effect.
+12.5 Predict the products of the following reactions:
+(i) 
+(ii) 
+(iii) 
+
+(iv) 
+
+
+12.5 Uses of Aldehydes and Ketones
+In chemical industry aldehydes and ketones are used as solvents, starting materials and reagents for the synthesis of other products. Formaldehyde is well known as formalin (40%) solution used to preserve biological specimens and to prepare bakelite (a phenol-formaldehyde resin), urea-formaldehyde glues and other polymeric products. Acetaldehyde is used primarily as a starting material in the manufacture of acetic acid, ethyl acetate, vinyl acetate, polymers and drugs. Benzaldehyde is used in perfumery and in dye industries. Acetone and ethyl methyl ketone are common industrial solvents. Many aldehydes and ketones, e.g., butyraldehyde, vanillin, acetophenone, camphor, etc. are well known for their odours and flavours.
+
+
+
+Carboxylic Acids
+
+
+
+Carbon compounds containing a carboxyl functional group, –COOH are called carboxylic acids. The carboxyl group, consists of a carbonyl group attached to a hydroxyl group, hence its name carboxyl. Carboxylic acids may be aliphatic (RCOOH) or aromatic (ArCOOH) depending on the group, alkyl or aryl, attached to carboxylic carbon. Large number of carboxylic acids are found in nature. Some higher members of aliphatic carboxylic acids (C12 – C18) known as fatty acids, occur in natural fats as esters of glycerol. Carboxylic acids serve as starting material for several other important organic compounds such as anhydrides, esters, acid chlorides, amides, etc.
+
+
+
+
+
+
+
+
+12.6 Nomenclature and Structure of Carboxyl Group
+
+
+12.6.1 Nomenclature
+
+
+
+Since carboxylic acids are amongst the earliest organic compounds to be isolated from nature, a large number of them are known by their common names. The common names end with the suffix –ic acid and have been derived from Latin or Greek names of their natural sources. For example, formic acid (HCOOH) was first obtained from red ants (Latin: formica means ant), acetic acid (CH3COOH) from vinegar (Latin: acetum, means vinegar), butyric acid (CH3CH2CH2COOH) from rancid butter (Latin: butyrum, means butter).
+
+
+
+In the IUPAC system, aliphatic carboxylic acids are named by replacing the ending –e in the name of the corresponding alkane with – oic acid. In numbering the carbon chain, the carboxylic carbon is numbered one. For naming compounds containing more than one carboxyl group, the alkyl chain leaving carboxyl groups is numbered and the number of carboxyl groups is indicated by adding the multiplicative prefix, dicarboxylic acid, tricarboxylic acid, etc. to the name of parent alkyl chain. The position of –COOH groups are indicated by the arabic numeral before the multiplicative prefix. Some of the carboxylic acids along with their common and IUPAC names are listed in Table 12.3.
+
+
+
+
+
+Table 12.3 Names and Structures of Some Carboxylic Acids
+
+
+
+
+
+
+
+12.6.2 Structure of Carboxyl Group
+
+
+
+
+
+In carboxylic acids, the bonds to the carboxyl carbon lie in one plane and are separated by about 120°. The carboxylic carbon is less electrophilic than carbonyl carbon because of the possible resonance structure shown below:
+
+
+
+Intext Question
+12.6 Give the IUPAC names of the following compounds:
+(i) Ph CH2CH2COOH (ii) (CH3)2C=CHCOOH
+(iii)  (iv) 
+
+
+
+12.7 Methods of Preparation of Carboxylic Acids
+
+
+Some important methods of preparation of carboxylic acids are as follows.
+1. From primary alcohols and aldehydes
+Primary alcohols are readily oxidised to carboxylic acids with common oxidising agents such as potassium permanganate (KMnO4) in neutral, acidic or alkaline media or by potassium dichromate (K2Cr2O7) and chromium trioxide (CrO3) in acidic media (Jones reagent).
+
+
+
+
+
+
+
+
+
+
+Carboxylic acids are also prepared from aldehydes by the use of mild oxidising agents (Section 12.4).
+
+
+
+
+2. From alkylbenzenes
+Aromatic carboxylic acids can be prepared by vigorous oxidation of alkyl benzenes with chromic acid or acidic or alkaline potassium permanganate. The entire side chain is oxidised to the carboxyl group irrespective of length of the side chain. Primary and secondary alkyl groups are oxidised in this manner while tertiary group is not affected. Suitably substituted alkenes are also oxidised to carboxylic acids with these oxidising reagents (refer Unit 13, Class XI).
+
+
+3. From nitriles and amides
+Nitriles are hydrolysed to amides and then to acids in the presence of H+ or as catalyst. Mild reaction conditions are used to stop the reaction at the amide stage.
+
+
+
+
+4. From Grignard reagents
+Grignard reagents react with carbon dioxide (dry ice) to form salts of carboxylic acids which in turn give corresponding carboxylic acids after acidification with mineral acid.
+
+
+
+
+
+
+
+
+
+
+As we know, the Grignard reagents and nitriles can be prepared from alkyl halides (refer Unit 10, Class XII). The above methods
+    (3 and 4) are useful for converting alkyl halides into corresponding carboxylic acids having one carbon atom more than that present in alkyl halides (ascending the series).
+
+
+
+
+
+
+
+
+5. From acyl halides and anhydrides
+Acid chlorides when hydrolysed with water give carboxylic acids or more readily hydrolysed with aqueous base to give carboxylate ions which on acidification provide corresponding carboxylic acids. Anhydrides on the other hand are hydrolysed to corresponding acid(s) with water.
+
+
+6. From esters
+
+
+
+          Acidic hydrolysis of esters gives directly carboxylic acids while basic hydrolysis gives carboxylates, which on acidification give corresponding carboxylic acids.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Example 12.5
+Write chemical reactions to affect the following transformations:
+(i) Butan-1-ol to butanoic acid
+(ii) Benzyl alcohol to phenylethanoic acid
+(iii) 3-Nitrobromobenzene to 3-nitrobenzoic acid
+(iv) 4-Methylacetophenone to benzene-1,4-dicarboxylic acid
+(v) Cyclohexene to hexane-1,6-dioic acid
+
+(vi) Butanal to butanoic acid.
+Solution
+(i)
+(ii)
+
+(iii)
+(iv)
+
+(v)
+
+        (vi)
+
+
+
+
+
+
+
+Intext Question
+12.7 Show how each of the following compounds can be converted to benzoic acid.
+(i) Ethylbenzene (ii) Acetophenone
+(iii) Bromobenzene (iv) Phenylethene (Styrene)
+
+
+
+12.8 Physical Properties
+Aliphatic carboxylic acids upto nine carbon atoms are colourless liquids at room temperature with unpleasant odours. The higher acids are wax like solids and are practically odourless due to their low volatility. Carboxylic acids are higher boiling liquids than aldehydes, ketones and even alcohols of comparable molecular masses. This is due to more extensive association of carboxylic acid molecules through intermolecular hydrogen bonding. The hydrogen bonds are not broken completely even in the vapour phase. In fact, most carboxylic acids exist as dimer in the vapour phase or in the aprotic solvents.
+
+In vapour state or in aprotic solvent
+
+
+Simple aliphatic carboxylic acids having upto four carbon atoms are miscible in water due to the formation of hydrogen bonds with water. The solubility decreases with increasing number of carbon atoms. Higher carboxylic acids are practically insoluble in water due to the increased hydrophobic interaction of hydrocarbon part. Benzoic acid, the simplest aromatic carboxylic acid is nearly insoluble in cold water. Carboxylic acids are also soluble in less polar organic solvents like benzene, ether, alcohol, chloroform, etc.
+
+
+
+
+
+
+
+Hydrogen bonding of RCOOH with H2O
+
+
+
+
+
+
+
+12.9 Chemical Reactions
+The reaction of carboxylic acids are classified as follows:
+
+
+
+
+
+12.9.1 Reactions Involving Cleavage of O–H Bond
+
+
+Acidity
+
+
+Reactions with metals and alkalies
+
+
+
+
+
+
+
+
+          The carboxylic acids like alcohols evolve hydrogen with electropositive metals and form salts with alkalies similar to phenols. However, unlike phenols they react with weaker bases such as carbonates and hydrogencarbonates to evolve carbon dioxide. This reaction is used to detect the presence of carboxyl group in an organic compound.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Carboxylic acids dissociate in water to give resonance stabilised carboxylate anions and hydronium ion.
+
+
+
+
+
+
+
+
+
+For the above reaction:
+
+
+
+
+
+where Keq, is equilibrium constant and Ka is the acid dissociation constant.
+For convenience, the strength of an acid is generally indicated by its pka value rather than its Ka value.
+pKa = – log Ka
+The pKa of hydrochloric acid is –7.0, where as pKa of trifluoroacetic acid (the strongest carboxylic acid), benzoic acid and acetic acid are 0.23, 4.19 and 4.76, respectively.
+Smaller the pKa, the stronger the acid ( the better it is as a proton donor). Strong acids have pKa values < 1, the acids with pKa values between 1 and 5 are considered to be moderately strong acids, weak acids have pKa values between 5 and 15, and extremely weak acids have pKa values >15.
+
+
+Carboxylic acids are weaker than mineral acids, but they are stronger acids than alcohols and many simple phenols (pKa is ~16 for ethanol and 10 for phenol). In fact, carboxylic acids are amongst the most acidic organic compounds you have studied so far. You already know why phenols are more acidic than alcohols. The higher acidity of carboxylic acids as compared to phenols can be understood similarly. The conjugate base of carboxylic acid, a carboxylate ion, is stabilised by two equivalent resonance structures in which the negative charge is at the more electronegative oxygen atom. The conjugate base of phenol, a phenoxide ion, has non-equivalent resonance structures in which the negative charge is at the less electronegative carbon atom. Therefore, resonance in phenoxide ion is not as important as it is in carboxylate ion. Further, the negative charge is delocalised over two electronegative oxygen atoms in carboxylate ion whereas it is less effectively delocalised over one oxygen atom and less electronegative carbon atoms in phenoxide ion (Unit 11, Class XII). Thus, the carboxylate ion is more stabilised than phenoxide ion, so carboxylic acids are more acidic than phenols.
+
+
+
+
+
+
+
+
+Effect of substituents on the acidity of carboxylic acids: Substituents may affect the stability of the conjugate base and thus, also affect the acidity of the carboxylic acids. Electron withdrawing groups increase the acidity of carboxylic acids by stabilising the conjugate base through delocalisation of the negative charge by inductive and/or resonance effects. Conversely, electron donating groups decrease the acidity by destabilising the conjugate base.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Electron withdrawing group (EWG) stabilises the carboxylate anion and strengthens the acid
+
+
+
+
+
+
+
+
+
+
+Electron donating group (EDG) destabilises the carboxylate anion and weakens the acid
+
+
+
+
+
+
+The effect of the following groups in increasing acidity order is
+Ph < I < Br < Cl < F < CN < NO2 < CF3
+Thus, the following acids are arranged in order of increasing acidity (based on pKa values):
+CF3COOH > CCl3COOH > CHCl2COOH > NO2CH2COOH > NC-CH2COOH >
+
+
+FCH2COOH > ClCH2COOH > BrCH2COOH > HCOOH > ClCH2CH2COOH >
+(continue)
+C6H5COOH > C6H5CH2COOH > CH3COOH > CH3CH2COOH
+(continue )
+Direct attachment of groups such as phenyl or vinyl to the carboxylic acid, increases the acidity of corresponding carboxylic acid, contrary to the decrease expected due to resonance effect shown below:
+
+This is because of greater electronegativity of sp2 hybridised carbon to which carboxyl carbon is attached. The presence of electron withdrawing group on the phenyl of aromatic carboxylic acid increases their acidity while electron donating groups decrease their acidity.
+
+
+12.9.2 Reactions Involving Cleavage of C–OH Bond
+1. Formation of anhydride
+Carboxylic acids on heating with mineral acids such as H2SO4 or with P2O5 give corresponding anhydride.
+
+
+
+
+
+
+
+
+2. Esterification
+Carboxylic acids are esterified with alcohols or phenols in the presence of a mineral acid such as concentrated H2SO4 or HCl gas as a catalyst.
+
+
+
+Mechanism of esterification of carboxylic acids: The esterification of carboxylic acids with alcohols is a kind of nucleophilic acyl substitution. Protonation of the carbonyl oxygen activates the carbonyl group towards nucleophilic addition of the alcohol. Proton transfer in the tetrahedral intermediate converts the hydroxyl group into –+OH2 group, which, being a better leaving group, is eliminated as neutral water molecule. The protonated ester so formed finally loses a proton to give the ester.
+
+
+
+
+
+
+
+3. Reactions with PCl5, PCl3 and SOCl2
+The hydroxyl group of carboxylic acids, behaves like that of alcohols and is easily replaced by chlorine atom on treating with PCl5, PCl3 or SOCl2. Thionyl chloride (SOCl2) is preferred because the other two products are gaseous and escape the reaction mixture making the purification of the products easier.
+
+
+4. Reaction with ammonia
+Carboxylic acids react with ammonia to give ammonium salt which on further heating at high temperature give amides. For example:
+
+
+
+
+
+12.9.3 Reactions Involving –COOH Group
+1. Reduction
+Carboxylic acids are reduced to primary alcohols by lithium aluminium hydride or better with diborane. Diborane does not easily reduce functional groups such as ester, nitro, halo, etc. Sodium borohydride does not reduce the carboxyl group.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+      
+
+2. Decarboxylation
+Carboxylic acids lose carbon dioxide to form hydrocarbons when their sodium salts are heated with sodalime (NaOH and CaO in the ratio of 3 : 1). The reaction is known as decarboxylation.
+
+
+
+
+
+
+Alkali metal salts of carboxylic acids also undergo decarboxylation on electrolysis of their aqueous solutions and form hydrocarbons having twice the number of carbon atoms present in the alkyl group of the acid. The reaction is known as Kolbe electrolysis (Unit 13, Class XI).
+
+
+
+
+
+12.9.4. Substitution Reactions in the Hydrocarbon Part
+1. Halogenation
+
+
+
+Carboxylic acids having an α-hydrogen are halogenated at the α-position on treatment with chlorine or bromine in the presence of small amount of red phosphorus to give α-halocarboxylic acids. The reaction is known as Hell-Volhard-Zelinsky reaction.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+2. Ring substitution
+
+
+
+
+
+
+
+
+Aromatic carboxylic acids undergo electrophilic substitution reactions in which the carboxyl group acts as a deactivating and meta-directing group. They however, do not undergo Friedel-Crafts reaction (because the carboxyl group is deactivating and the catalyst aluminium chloride (Lewis acid) gets bonded to the carboxyl group).
+
+
+
+
+
+
+
+
+
+
+
+
+Intext Question
+12.8 Which acid of each pair shown here would you expect to be stronger?
+(i) CH3CO2H or CH2FCO2H 
+(ii) CH2FCO2H or CH2ClCO2H
+(iii) CH2FCH2CH2CO2H or CH3CHFCH2CO2H
+
+
+(iv)
+
+
+
+
+12.10 Uses of Carboxylic Acids
+
+Methanoic acid is used in rubber, textile, dyeing, leather and electroplating industries. Ethanoic acid is used as solvent and as vinegar in food industry. Hexanedioic acid is used in the manufacture of nylon-6, 6. Esters of benzoic acid are used in perfumery. Sodium benzoate is used as a food preservative. Higher fatty acids are used for the manufacture of soaps and detergents.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Summary
+Aldehydes, ketones and carboxylic acids are some of the important classes of organic compounds containing carbonyl group. These are highly polar molecules. Therefore, they boil at higher temperatures than the hydrocarbons and weakly polar compounds such as ethers of comparable molecular masses. The lower members are more soluble in water because they form hydrogen bonds with water. The higher members, because of large size of hydrophobic chain of carbon atoms, are insoluble in water but soluble in common organic solvents. Aldehydes are prepared by dehydrogenation or controlled oxidation of primary alcohols and controlled or selective reduction of acyl halides. Aromatic aldehydes may also be prepared by oxidation of (i) methylbenzene with chromyl chloride or CrO3 in the presence of acetic anhydride, (ii) formylation of arenes with carbon monoxide and hydrochloric acid in the presence of anhydrous aluminium chloride, and (iii) cuprous chloride or by hydrolysis of benzal chloride. Ketones are prepared by oxidation of secondary alcohols and hydration of alkynes. Ketones are also prepared by reaction of acyl chloride with dialkylcadmium. A good method for the preparation of aromatic ketones is the Friedel-Crafts acylation of aromatic hydrocarbons with acyl chlorides or anhydrides. Both aldehydes and ketones can be prepared by ozonolysis of alkenes. Aldehydes and ketones undergo nucleophilic addition reactions onto the carbonyl group with a number of nucleophiles such as, HCN, NaHSO3, alcohols (or diols), ammonia derivatives, and Grignard reagents. The α-hydrogens in aldehydes and ketones are acidic. Therefore, aldehydes and ketones having at least one α-hydrogen, undergo Aldol condensation in the presence of a base to give α-hydroxyaldehydes (aldol) and α-hydroxyketones(ketol), respectively. Aldehydes having no α-hydrogen undergo Cannizzaro reaction in the presence of concentrated alkali. Aldehydes and ketones are reduced to alcohols with NaBH4, LiAlH4, or by catalytic hydrogenation. The carbonyl group of aldehydes and ketones can be reduced to a methylene group by Clemmensen reduction or Wolff-Kishner reduction. Aldehydes are easily oxidised to carboxylic acids by mild oxidising reagents such as Tollens’ reagent and Fehling’s reagent. These oxidation reactions are used to distinguish aldehydes from ketones. Carboxylic acids are prepared by the oxidation of primary alcohols, aldehydes and alkenes by hydrolysis of nitriles, and by treatment of Grignard reagents with carbon dioxide. Aromatic carboxylic acids are also prepared by side-chain oxidation of alkylbenzenes. Carboxylic acids are considerably more acidic than alcohols and most of simple phenols. Carboxylic acids are reduced to primary alcohols with LiAlH4, or better with diborane in ether solution and also undergo α-halogenation with Cl2 and Br2 in the presence of red phosphorus (Hell-Volhard Zelinsky reaction). Methanal, ethanal, propanone, benzaldehyde, formic acid, acetic acid and benzoic acid are highly useful compounds in industry.
+
+
+
+
+
+
+
+
+
+Exercises
+
+12.1 What is meant by the following terms ? Give an example of the reaction in each case.
+(i) Cyanohydrin (ii) Acetal (iii) Semicarbazone 
+(iv) Aldol (v) Hemiacetal (vi) Oxime
+(vii) Ketal (vii) Imine (ix) 2,4-DNP-derivative
+ (x) Schiff’s base
+12.2 Name the following compounds according to IUPAC system of nomenclature:
+(i) CH3CH(CH3)CH2CH2CHO (ii) CH3CH2COCH(C2H5)CH2CH2Cl
+(iii) CH3CH=CHCHO (iv) CH3COCH2COCH3
+(v) CH3CH(CH3)CH2C(CH3)2COCH3 (vi) (CH3)3CCH2COOH
+(vii) OHCC6H4CHO-p
+12.3 Draw the structures of the following compounds.
+(i) 3-Methylbutanal (ii) p-Nitropropiophenone
+(iii) p-Methylbenzaldehyde (iv) 4-Methylpent-3-en-2-one
+(v) 4-Chloropentan-2-one (vi) 3-Bromo-4-phenylpentanoic acid
+(vii) p,p’-Dihydroxybenzophenone (viii) Hex-2-en-4-ynoic acid
+12.4 Write the IUPAC names of the following ketones and aldehydes. Wherever possible, give also common names.
+(i) CH3CO(CH2)4CH3 (ii) CH3CH2CHBrCH2CH(CH3)CHO
+(iii) CH3(CH2)5CHO (iv) Ph-CH=CH-CHO
+(v)  (vi) PhCOPh
+12.5 Draw structures of the following derivatives.
+(i) The 2,4-dinitrophenylhydrazone of benzaldehyde
+(ii) Cyclopropanone oxime
+(iii) Acetaldehydedimethylacetal
+(iv) The semicarbazone of cyclobutanone
+(v) The ethylene ketal of hexan-3-one
+(vi) The methyl hemiacetal of formaldehyde
+12.6 Predict the products formed when cyclohexanecarbaldehyde reacts with following reagents.
+(i) PhMgBr and then H3O+ (ii) Tollens’ reagent
+(iii) Semicarbazide and weak acid (iv) Excess ethanol and acid
+(v) Zinc amalgam and dilute hydrochloric acid
+12.7 Which of the following compounds would undergo aldol condensation, which the Cannizzaro reaction and which neither? Write the structures of the expected products of aldol condensation and Cannizzaro reaction.
+(i) Methanal (ii) 2-Methylpentanal (iii) Benzaldehyde
+(iv) Benzophenone (v) Cyclohexanone (vi) 1-Phenylpropanone
+(vii) Phenylacetaldehyde (viii) Butan-1-ol (ix) 2,2-Dimethylbutanal
+12.8 How will you convert ethanal into the following compounds?
+(i) Butane-1,3-diol (ii) But-2-enal (iii) But-2-enoic acid
+12.9 Write structural formulas and names of four possible aldol condensation products from propanal and butanal. In each case, indicate which aldehyde acts as nucleophile and which as electrophile.
+12.10 An organic compound with the molecular formula C9H10O forms 2,4-DNP derivative, reduces Tollens’ reagent and undergoes Cannizzaro reaction. On vigorous oxidation, it gives 1,2-benzenedicarboxylic acid. Identify the compound.
+12.11 An organic compound (A) (molecular formula C8H16O2) was hydrolysed with dilute sulphuric acid to give a carboxylic acid (B) and an alcohol (C). Oxidation of (C) with chromic acid produced (B). (C) on dehydration gives but-1-ene. Write equations for the reactions involved.
+12.12 Arrange the following compounds in increasing order of their property as indicated:
+(i) Acetaldehyde, Acetone, Di-tert-butyl ketone, Methyl tert-butyl ketone
+            (reactivity towards HCN)
+(ii) CH3CH2CH(Br)COOH, CH3CH(Br)CH2COOH, (CH3)2CHCOOH,
+CH3CH2CH2COOH (acid strength)
+(iii) Benzoic acid, 4-Nitrobenzoic acid, 3,4-Dinitrobenzoic acid, 4-Methoxybenzoic acid (acid strength)
+12.13 Give simple chemical tests to distinguish between the following pairs of compounds.
+(i) Propanal and Propanone (ii) Acetophenone and Benzophenone
+(iii) Phenol and Benzoic acid (iv) Benzoic acid and Ethyl benzoate
+(v) Pentan-2-one and Pentan-3-one (vi) Benzaldehyde and Acetophenone
+(vii) Ethanal and Propanal
+12.14 How will you prepare the following compounds from benzene? You may use any inorganic reagent and any organic reagent having not more than one carbon atom
+(i) Methyl benzoate (ii) m-Nitrobenzoic acid
+(iii) p-Nitrobenzoic acid (iv) Phenylacetic acid
+(v) p-Nitrobenzaldehyde.
+12.15 How will you bring about the following conversions in not more than two steps?
+(i) Propanone to Propene (ii) Benzoic acid to Benzaldehyde
+(iii) Ethanol to 3-Hydroxybutanal (iv) Benzene to m-Nitroacetophenone
+(v) Benzaldehyde to Benzophenone (vi) Bromobenzene to 1-Phenylethanol
+(vii) Benzaldehyde to 3-Phenylpropan-1-ol
+(viii) Benazaldehyde to α-Hydroxyphenylacetic acid
+(ix) Benzoic acid to m- Nitrobenzyl alcohol
+12.16 Describe the following:
+(i) Acetylation (ii) Cannizzaro reaction
+(iii) Cross aldol condensation (iv) Decarboxylation
+12.17 Complete each synthesis by giving missing starting material, reagent or products
+
+
+12.18 Give plausible explanation for each of the following:
+(i) Cyclohexanone forms cyanohydrin in good yield but 2,2,6-trimethylcyclo-hexanone does not.
+(ii) There are two –NH2 groups in semicarbazide. However, only one is involved in the formation of semicarbazones.
+(iii) During the preparation of esters from a carboxylic acid and an alcohol in the presence of an acid catalyst, the water or the ester should be removed as soon as it is formed.
+12.19 An organic compound contains 69.77% carbon, 11.63% hydrogen and rest oxygen. The molecular mass of the compound is 86. It does not reduce Tollens’ reagent but forms an addition compound with sodium hydrogensulphite and give positive iodoform test. On vigorous oxidation it gives ethanoic and propanoic acid. Write the possible structure of the compound.
+12.20 Although phenoxide ion has more number of resonating structures than carboxylate ion, carboxylic acid is a stronger acid than phenol. Why?
+
+
+
+
+
+
+
+
+Answers to Some Intext Questions
+12.1
+
+
+
+
+
+
+
+
+
+
+
+12.2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+12.3 CH3CH2CH3 < CH3OCH3 < CH3CHO < CH3CH2OH
+
+12.4 (i) Butanone < Propanone < Propanal < Ethanal
+(ii) Acetophenone < p-Tolualdehyde , Benzaldehyde < p-Nitrobenzaldehyde.
+12.5
+
+
+12.6 (i) 3-Phenylpropanoic acid (ii) 3-Methylbut-2-enoic acid
+(iii) 2-Methylcyclopentanecarboxylic acid. (iv) 2,4,6-Trinitrobenzoic acid
+12.7 
+
+
+12.8
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Table of Contents
+
+
+Unit 12
+
+
+Aldehydes, Ketones and Carboxylic Acids
+
+
+12.1 Nomenclature and Structure of Carbonyl Group
+
+
+12.1.1 Nomenclature
+
+
+
+
+
+
+
+12.1.2 Structure of the Carbonyl Group
+
+
+
+
+12.2 Preparation of Aldehydes and Ketones
+
+
+12.2.1 Preparation of Aldehydes and Ketones
+
+
+12.2.2 Preparation of Aldehydes
+
+
+12.2.3 Preparation of Ketones
+
+
+
+
+12.3 Physical Properties
+
+
+12.4 Chemical Reactions
+
+
+12.5 Uses of Aldehydes and Ketones
+
+
+12.6 Nomenclature and Structure of Carboxyl Group
+
+
+12.6.1 Nomenclature
+
+
+12.6.2 Structure of Carboxyl Group
+
+
+
+
+12.7 Methods of Preparation of Carboxylic Acids
+
+
+
+
+
+
+
+12.9 Chemical Reactions
+
+
+12.9.1 Reactions Involving Cleavage of O–H Bond
+
+
+
+
+
+
+
+12.9.2 Reactions Involving Cleavage of C–OH Bond
+
+
+
+
+
+
+
+
+
+
+12.9.3 Reactions Involving –COOH Group
+
+
+
+
+
+
+
+12.10 Uses of Carboxylic Acids
+
+
+Summary
+
+
+Exercises
+
+
+
+
+
+Answers to Some Intext Questions
+
+
+
+
+
+
+
+
+Landmarks
+
+Cover
+
+
+
+
+
+        Chapter 12
+        Aldehydes, Ketones and Carboxylic Acids
+      
+
+
+
+
+
+
diff --git a/12_physics_1.txt b/12_physics_1.txt
new file mode 100644
index 0000000..5d41692
--- /dev/null
+++ b/12_physics_1.txt
@@ -0,0 +1,2085 @@
+
+
+
+
+
+Figure 1.7 
+Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by
+ 
+neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is
+
+Thus the electrostatic force on A, due to B, remains unaltered.
+ 
+1.7  Forces between Multiple Charges 
+The mutual electric force between two charges is given by Coulomb’s law. How to calculate the force on a charge where there are not one but several charges around? Consider a system of n stationary charges q1, q2, q3, ..., qn in vacuum. What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question.  Recall that forces of mechanical origin add according to the parallelogram law of addition. Is the same true for forces of electrostatic origin?
+
+
+
+
+
+Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces are unaffected due to the presence of other charges. This is termed as the principle of superposition. 
+To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in Fig. 1.8(a). The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges. Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq. (1.3)  even though other charges are present. 
+Thus,	F12 
+In the same way, the force on q1 due to q3, denoted by F13, is given by
+
+which again is the Coulomb force on q1 due to q3, even though other charge q2 is present.
+Thus the total force F1 on q1 due to the two charges q2 and q3 is given as
+ (1.4)
+The above calculation of force can be generalised to a system of charges more than three, as shown in Fig. 1.8(b).
+The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn. The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ...,  F1n: 
+i.e.,
+
+
+
+
+
+
+Figure 1.8 A system of (a) three charges (b) multiple charges.
+
+
+
+
+	(1.5)
+The vector sum is obtained as usual by the parallelogram law of addition of vectors. All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.
+Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in Fig. 1.9?
+
+
+
+
+
+Figure 1.9
+Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC,
+AD = AC cos 30º = () l  and the distance AO of the centroid O from A is (2/3) AD = () l. By symmatry AO = BO = CO.
+Thus,
+Force F1 on Q due to charge q at A  =  along AO
+Force F2 on Q due to charge q at B  =  along BO
+Force F3 on Q due to charge q at C  =  along CO
+The resultant of forces F2 and F3 is  along OA, by the parallelogram law. Therefore, the total force on Q =  = 0, where is the unit vector along OA.
+It is clear also by symmetry that the three forces will sum to zero. Suppose that the resultant force was non-zero but in some direction. Consider what would happen if the system was rotated through 60° about O.
+Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in Fig. 1.10. What is the force on each charge?
+
+
+
+
+
+Figure 1.10
+Solution  The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in Fig. 1.10. By the parallelogram law, the total force F1 on the charge q at A is given by
+F1 = F  where  is a unit vector along BC.
+The force of attraction or repulsion for each pair of charges has the same magnitude 
+The total force F2 on charge q at B is thus F2 = F 2, where 2 is a unit vector along AC.
+Similarly the total force on charge –q at C is F3  =  F , where is the unit vector along the direction bisecting the ∠BCA.
+It is interesting to see that the sum of the forces on the three charges is zero, i.e., 
+F1 + F2 + F3 = 0
+The result is not at all surprising. It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law. The proof is left to you as an exercise.
+1.8  Electric Field
+Let us consider a point charge Q placed in vacuum, at the origin O. If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law. We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P. In order to answer such questions, the early scientists introduced the concept of field. According to this, we say that the charge Q produces an electric field everywhere in the surrounding. When another charge q is brought at some point P, the field there acts on it and produces a force. The electric field produced by the charge Q  at a point r is given as
+	(1.6)
+where  r/r, is a unit vector from the origin to the point r. Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r. The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position. The effect of the charge has been incorporated in the existence of the electric field. We obtain the force F exerted by a charge Q on a charge q, as
+	(1.7)
+Note that the charge q also exerts an equal and opposite force on the charge Q. The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa. If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q. Thus,
+F(r) = q E(r)	(1.8)
+Equation (1.8) defines the SI unit of electric field as N/C*. 
+Some important remarks may be made here:
+(i) From Eq. (1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it. Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point. The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge. Note that the source charge Q must remain at its original location. However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move. A way out of this difficulty is to make q negligibly small. The force F is then negligibly small but the ratio F/q is finite and defines the electric field: 
+	(1.9)
+
+
+
+
+
+
+Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.
+
+
+
+A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces!  This may look strange but actually this is what happens in practice. When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet.
+(ii) Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q. This is because F is proportional to q, so the ratio F/q does not depend on q. The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q. Thus, the electric field E due to Q is also dependent on the space coordinate r. For different positions of the charge q all over the space, we get different values of electric field E. The field exists at every point in three-dimensional space. 
+(iii) For a positive charge, the electric field will be directed radially outwards from the charge. On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards.
+(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r. Thus at equal distances from the charge Q, the magnitude of its electric field E is same. The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry.
+1.8.1  Electric field due to a system of charges
+Consider a system of charges q1, q2, ..., qn with  position vectors r1, r2, ..., rn relative  to some origin O. Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn. We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r.
+Electric field E1 at r due to q1 at r1 is given by 
+E1 = 
+where  is a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P.
+In the same manner, electric field E2 at r due to q2 at r2 is 
+E2 = 
+where  is a unit vector in the direction from q2 to P and r2P is the distance between q2 and P. Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn.
+By the superposition principle, the electric field E at r due to the system of charges is (as shown in Fig. 1.12)
+E(r)  =  E1 (r) +  E2 (r)  + …  +  En(r)  
+        = 
+E(r) 	(1.10)
+E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges. 
+1.8.2 Physical significance of electric field
+You may wonder why the notion of electric field has been introduced here at all. After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq. (1.5)]. Why then introduce this intermediate quantity called the electric field?
+For electrostatics, the concept of electric field is convenient, but not really necessary. Electric field is an elegant way of characterising the electrical environment of a system of charges. Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system). Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field. The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point. Electric field is a vector field, since force is a vector quantity.
+
+
+
+
+
+
+Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.	
+
+
+
+The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena. Suppose we consider the force between two distant charges q1, q2 in accelerated motion. Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light. Thus, the effect of any motion of q1 on q2 cannot arise instantaneously. There will be some time delay between the effect (force on q2) and the cause (motion of q1). It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful. The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2. The notion of field elegantly accounts for the time delay. Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs. They have an independent dynamics of their own,  i.e., they evolve according to laws of their own. They can also transport energy. Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy. The concept of field was first introduced by Faraday and is now among the central concepts in physics.
+Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1 [Fig. 1.13(a)]. The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance [Fig. 1.13(b)]. Compute the time of fall in each case. Contrast the situation with that of ‘free fall under gravity’.
+Figure 1.13
+
+
+
+
+
+Solution In Fig. 1.13(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field. The acceleration of the electron is
+ ae  =  eE/me
+where me is the mass of the electron.
+Starting from rest, the time required by the electron to fall through a distance h is given by  
+For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, 
+  E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m,
+	  te = 2.9 × 10–9s
+In Fig. 1.13 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE. The acceleration of the proton is
+ ap  =  eE/mp
+where mp is the mass of the proton; mp = 1.67 × 10–27 kg. The time of fall for the proton is
+
+Thus, the heavier particle (proton) takes a greater time to fall through the same distance. This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body. Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall. To see if this is justified, let us calculate the acceleration of the proton in the given electric field:
+
+ 
+ 
+which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity. The acceleration of the electron is even greater. Thus, the effect of acceleration due to gravity can be ignored in this example.
+Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart. Calculate the electric fields at points A, B and C shown in Fig. 1.14. 
+
+
+
+
+
+Figure 1.14
+Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude
+ =  3.6 × 104  N C–1
+The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude. Hence the magnitude of the total electric field EA at A is
+EA = E1A + E2A = 7.2 × 104 N C–1
+EA is directed toward the right.
+The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude
+= 3.6 × 104 N C–1
+The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude     
+=  4 × 103  N C–1
+The magnitude of the total electric field at B is
+EB = E1B –  E2B = 3.2 × 104 N C–1
+EB is directed towards the left. 
+The magnitude of each electric field vector at point C, due to charge q1 and q2 is
+  =  9 × 103  N C–1
+The directions in which these two vectors point are indicated in Fig. 1.14. The resultant of these two vectors is
+=  9 × 103  N C–1
+EC points towards the right.
+1.9  Electric Field Lines
+We have studied electric field in the last section. It is a vector quantity and can be represented as we represent vectors. Let us try to represent E due to a point charge pictorially. Let the point charge be placed at the origin. Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point. Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward. Figure 1.15 shows such a picture.  In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow.  Connect the arrows pointing in one direction and the resulting figure represents a field line. We thus get many field lines, all pointing outwards from the point charge. Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No. Now the magnitude of the field is represented by the density of field lines. E is strong near the charge, so the density of field lines is more near the charge and the lines are closer. Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines.
+Another person may draw more lines. But the number of lines is not important. In fact, an infinite number of lines can be drawn in any region. It is the relative density of lines in different regions which is important.
+We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions. So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines.  Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge.
+We started by saying that the field lines carry information about the direction of electric field at different points in space. Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points. The field lines crowd where the field is strong and are spaced apart where it is weak. Figure 1.16 shows a set of field lines. We can imagine two equal and small elements of area placed at points R and S normal to the field lines there. The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points. The picture shows that the field at R is stronger than at S.
+To understand the dependence of the field lines on the area, or rather the solid angle subtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions. Recall how a (plane) angle is defined in two-dimensions. Let a small transverse line element ∆l be placed at a distance r from a point O. Then the angle subtended by ∆l  at O can be approximated as ∆θ = ∆l/r. Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2. We know that in a given solid angle the number of radial field lines is the same. In Fig. 1.16, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is ∆Ω at P1 and an element of area ∆Ω at P2, respectively. The number of lines (say n) cutting these area elements are the same. The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively. Since n and ∆Ω are common, the strength of the field clearly has a 1/r2 dependence.
+
+
+
+Figure 1.15 Field of a point charge.
+
+
+
+
+
+
+The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations. Faraday called them lines of force. This term is somewhat misleading, especially in case of magnetic fields. The more appropriate term is field lines (electric or magnetic) that we have adopted in this book.
+Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges. An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point. An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve. A field line is a space curve, i.e., a curve in three dimensions.
+
+
+
+
+
+
+Figure 1.16  Dependence of electric field strength on the distance and its relation to the number of field lines.
+
+
+
+Figure 1.17 shows the field lines around some simple charge configurations. As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane. The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward. The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges. The field lines follow some important general properties:
+(i)	Field lines start from positive charges and end at negative charges. If there is a single charge, they may start or end at infinity.
+(ii)	In a charge-free region, electric field lines can be taken to be continuous curves without any breaks.
+(iii)	Two field lines can never cross each other. (If they did, the field at the point of intersection will not have a unique direction, which is absurd.)
+
+
+
+Figure 1.17 Field lines due to some simple charge configurations.
+
+
+
+
+
+
+(iv)	Electrostatic field lines do not form any closed loops. This follows from the conservative nature of electric field (Chapter 2). 
+1.10  Electric Flux
+Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface. The rate of flow of liquid is given by the volume crossing the area per unit time  v dS and represents the flux of liquid flowing across the plane. If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ. Therefore, the flux going out of the surface dS is v.dS. For the case of  the electric field, we define an analogous quantity and call it electric flux. We should, however, note that there is no flow of a physically observable quantity unlike the case ofliquid flow. 
+In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point. This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S. Now suppose we tilt the area element by angle θ. Clearly, the number of field lines crossing the area element will be smaller. The projection of the area element normal to E is ∆S cosθ. Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ. When θ = 90°, field lines will be parallel to ∆S and will not cross it at all (Fig. 1.18).
+
+
+
+
+
+
+Figure 1.18 Dependence of flux on the inclination θ between E and .
+
+
+
+The orientation of area element and not merely its magnitude is important in many contexts. For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring. If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation. This shows that an area element should be treated as a vector. It has a magnitude and also a direction. How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane. Thus the direction of a planar area vector is along its normal. 
+How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements. Each small area element may be treated as planar and a vector associated with it, as explained before.
+Notice one ambiguity here. The direction of an area element is along its normal. But a normal can point in two directions. Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context. For the case of a closed surface, this convention is very simple. The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal. This is the convention used in Fig. 1.19. Thus, the area element vector ∆S at a point on a closed surface equals ∆S  where ∆S is the magnitude of the area element and  is a unit vector in the direction of outward normal at that point.
+We now come to the definition of electric flux. Electric flux ∆φ through an area element ∆S is defined by
+∆φ = E.∆S = E ∆S  cosθ	(1.11)
+which, as seen before, is proportional to the number of field lines cutting the area element. The angle θ here is the angle between E and ∆S. For a closed surface, with the convention stated already, θ is the angle between E and the outward normal   to the area element. Notice we could look at the expression E ∆S cosθ  in two ways:  E (∆S cosθ ) i.e., E times the projection of area normal to E,  or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element. The unit of electric flux is N C–1 m2.
+The basic definition of electric flux given by Eq. (1.11) can be used, in principle, to calculate the total flux through any given surface. All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up. Thus, the total flux φ through a surface S is
+φ ~ Σ E.∆S	(1.12)
+The approximation sign is put because the electric field E is taken to be constant over the small area element. This is mathematically exact only when you take the limit ∆S → 0  and the sum in Eq. (1.12) is written as an integral. 
+1.11  Electric Dipole
+An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a. The line connecting the two charges defines a direction in space. By convention, the direction from –q to q is said to be the direction of the dipole. The mid-point of locations of –q and q is called the centre of the dipole.
+
+
+
+
+
+
+Figure 1.19 Convention for defining normal  and ∆S.
+
+
+
+The total charge of the electric dipole is obviously zero. This does not mean that the field of the electric dipole is zero. Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out. However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out. The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q). These qualitative ideas are borne out by the explicit calculation as follows:
+1.11.1  The field of an electric dipole
+The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle. The results are simple for the following two cases: (i) when the point is on the dipole axis, and (ii) when it is in the equatorial plane of the dipole, i.e.,  on a plane perpendicular to the dipole axis through its centre.  The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+q due to the charge q, by the parallelogram law of vectors.
+(i) For points on the axis
+Let the point P be at distance r from the centre of the dipole on the side of the charge q,  as shown in Fig. 1.20(a). Then 
+	[1.13(a)]
+where  is the unit vector along the dipole axis (from –q to q). Also
+	[1.13(b)]	The total field at P is
+ 
+ 	(1.14)	For r >> a
+ (r >> a)	(1.15)
+(ii) For points on the equatorial plane
+The magnitudes of the electric fields due to the two charges +q and –q are given by 
+	[1.16(a)]
+	[1.16(b)]	
+and are equal.
+The directions of  E+q and E–q are as shown in Fig. 1.20(b). Clearly, the components normal to the dipole axis cancel away. The components along the dipole axis add up. The total electric field is opposite to . We have 
+E = – (E +q + E –q) cosθ  
+ 	(1.17)
+At large distances (r >> a), this reduces to
+	(1.18)
+From Eqs. (1.15) and (1.18), it is clear that the dipole field at large distances does not involve q and a separately; it depends on the product qa. This suggests the definition of dipole moment. The dipole moment vector p of an electric dipole is defined by
+p  = q × 2a 		(1.19)
+that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q. In terms of p, the electric field of a dipole at large distances takes simple forms:
+At a point on the dipole axis
+ (r >> a)	(1.20)  
+
+
+
+
+
+
+Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole. p is the dipole moment vector of magnitude p = q × 2a  and directed from –q to q.
+
+
+
+At a point on the equatorial plane
+ (r >> a)	(1.21)
+ Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3. Further,  the magnitude and the direction of the dipole field depends not only on the distance r but also on the angle between the position vector r and the dipole moment p. 
+We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite. Such a dipole is referred to as a point dipole. For a point dipole, Eqs. (1.20) and (1.21) are exact, true for any r.
+1.11.2  Physical significance of dipoles
+In most molecules, the centres of positive charges and of negative charges* lie at the same place. Therefore, their dipole moment is zero. CO2 and CH4 are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules, H2O, is an example of this type. Various materials give rise to interesting properties and important applications in the presence or  absence of electric field.
+Example 1.10 Two charges ±10 µC are placed 5.0 mm apart. Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in Fig. 1.21(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in Fig. 1.21(b).  
+
+
+
+
+
+fIGURE 1.21
+Solution  (a) Field at P due to charge +10 µC
+= 
+=  4.13 × 106  N C–1  along BP
+Field at P due to charge –10 µC 
+ 
+=  3.86 × 106  N C–1 along PA
+The resultant electric field at P due to the two charges at A and B is	=  2.7 × 105  N C–1  along BP.
+In this example, the ratio OP/OB is quite large (= 60). Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole. For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude
+ 		(r/a >> 1)
+where p = 2a q is the magnitude of the dipole moment.
+The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q). Here, p =10–5 C × 5 × 10–3 m  = 5 × 10–8 C m
+Therefore,
+E  = = 2.6 × 105  N C–1
+along the dipole moment direction AB, which is close to the result obtained earlier.
+(b) Field at Q due to charge + 10 µC at B
+=
+=  3.99 × 106  N C–1 along BQ
+ 
+Field at Q due to charge –10 µC at A
+=
+=  3.99 × 106  N C–1 along QA.
+Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA. Therefore, the resultant electric field at Q due to the two charges at A and B is
+
+
+*	Centre of a collection of positive point charges is defined much the same way as the centre of mass: .
+
+
+= 2 × along BA
+=  1.33 × 105  N C–1 along BA.
+As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole:
+ 		(r/a >> 1)
+
+=  1.33 × 105  N C–1.
+The direction of electric field in this case is opposite to the direction of the dipole moment vector. Again, the result agrees with that obtained before.
+1.12  Dipole in a Uniform External Field
+Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in Fig. 1.22. (By permanent dipole, we mean that p exists  irrespective of E; it has not been induced by E.)
+There is a force qE on q and a force –qE on –q. The net force on the dipole is zero, since E is uniform. However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole. When the net force is zero, the torque (couple) is independent of the origin. Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).
+Magnitude of torque = q E × 2 a sinθ
+  = 2 q a E sinθ
+Its direction is normal to the plane of the paper, coming out of it.
+The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it. Thus,
+τ  = p × E	(1.22)
+This torque will tend to align the dipole with the field E. When p is aligned with E, the torque is zero.
+What happens if the field is not uniform? In that case, the net force will evidently be non-zero. In addition there will, in general, be a torque on the system as before. The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E. In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform.
+Figure 1.23 is self-explanatory. It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field. When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field. In general, the force depends on the orientation of p with respect to E.
+This brings us to a common observation in frictional electricity. A comb run through dry hair attracts pieces of paper. The comb, as we know, acquires charge through friction. But the paper is not charged. What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field. Further, the electric field due to the comb is not uniform. In this situation, it is easily seen that the paper should move in the direction of the comb! 
+1.13  Continuous Charge Distribution
+We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn. One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus. For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions. For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents. It is more feasible to consider an area element ∆S (Fig. 1.24) on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element. We then define a surface charge density σ at the area element by
+
+
+
+Figure 1.22 Dipole in a uniform electric field.
+
+
+
+
+
+
+	(1.23)
+We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density. The surface charge density σ  so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*. σ  represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically. The units for σ are C/m2.
+Similar considerations apply for a line charge distribution and a volume charge distribution. The linear charge density λ of a wire is defined by
+
+
+
+Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b)  E antiparallel to p.
+
+
+
+
+
+
+	(1.24)
+where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element. The units for λ are C/m. The volume charge density (sometimes simply called charge density) is defined in a similar manner:
+	(1.25)
+where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents. The units for ρ are C/m3.
+The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics. When we refer to the density of a liquid, we are referring to its macroscopic density. We regard it as a continuous fluid and ignore its discrete molecular constitution.
+
+
+
+
+
+
+Figure 1.24 Definition of linear, surface and volume charge densities. In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.
+
+
+
+The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq. (1.10). Suppose a continuous charge distribution in space has a charge density ρ. Choose any convenient origin O and let the position vector of any point in the charge distribution be r. The charge density ρ may vary from point to point, i.e., it is a function of r. Divide the charge distribution into small volume elements of size ∆V. The charge in a volume element ∆V is ρ∆V.
+Now, consider any general point P (inside or outside the distribution) with position vector R (Fig. 1.24). Electric field due to the charge ρ∆V is given by Coulomb’s law:
+	(1.26)	where r′ is the distance between the charge element and P, and  ′ is a unit vector in the direction from the charge element to P. By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements:
+	(1.27)
+Note that ρ, r′,   all can vary from point to point. In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity. In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.
+1.14  Gauss’s Law
+As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre. Divide the sphere into small area elements, as shown in Fig. 1.25.
+The flux through an area element ∆S is
+	(1.28)
+where we have used Coulomb’s law for the electric field due to a single charge q. The unit vector  is along the radius vector from the centre to the area element. Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S and   have the same direction. Therefore, 
+
+
+*	At the microscopic level, charge distribution is discontinuous, because they are discrete charges separated by intervening space where there is no charge.
+
+
+	(1.29)
+since the magnitude of a unit vector is 1.
+The total flux through the sphere is obtained by adding up flux through all the different area elements:
+ 
+Since each area element of the sphere is at the same distance r from the charge,
+
+Now S, the total area of the sphere, equals 4πr2. Thus,
+	(1.30)
+Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law. 
+We state Gauss’s law without proof:
+Electric flux through a closed surface S
+
+
+
+
+
+
+Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.
+
+
+
+= q/ε0 	(1.31)
+q = total charge enclosed by S.
+The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface. We can see that explicitly in the simple situation of Fig. 1.26. 
+Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E. The total flux φ  through the surface is  φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface. Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0. Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E. Therefore,
+	φ1 = –E S1,     φ2 = +E S2
+	S1 = S2 = S
+where S is the area of circular cross-section. Thus, the total flux is zero, as expected by Gauss’s law. Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero. 
+The great significance of Gauss’s law Eq. (1.31), is that it is true in general, and not only for the simple cases we have considered above. Let us note some important points regarding this law:
+(i)	Gauss’s law is true for any closed surface, no matter what its shape or size.
+(ii)	The term q on the right side of Gauss’s law, Eq. (1.31), includes the sum of all charges enclosed by the surface. The charges may be located anywhere inside the surface.
+(iii)	In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq. (1.31)] is due to all the charges, both inside and outside S. The term q on the right side of Gauss’s law, however, represents only the total charge inside S. 
+
+
+
+Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.
+
+
+
+
+
+
+(iv)	The surface that we choose for the application of Gauss’s law is called the Gaussian surface. You may choose any Gaussian surface and apply Gauss’s law. However, take care not to let the Gaussian surface pass through any discrete charge. This is because electric field due to a system of discrete charges is not well defined at the location of any charge. (As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution.
+(v)	Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface. 
+(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law. Any violation of Gauss’s law will indicate departure from the inverse square law.
+Example 1.11 The electric field components in Fig. 1.27 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2. Calculate (a) the flux through the cube, and (b) the charge within the cube. Assume that a = 0.1 m.
+
+
+
+
+
+
+Figure 1.27
+Solution 
+(a)	Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2. Therefore, the flux  φ = E.∆S is separately zero for each face of the cube except the two shaded ones. Now the magnitude of the electric field at the left face is 
+ EL = αx1/2 = αa1/2
+	(x = a at the left face).
+	The magnitude of electric field at the right face is 
+ ER = α x1/2 = α (2a)1/2
+	(x = 2a at the right face).
+	The corresponding fluxes are
+ φL= EL.∆S = =EL ∆S cosθ = –EL ∆S, since θ = 180° 
+	   = –ELa2 
+	φR= ER.∆S = ER ∆S cosθ  = ER ∆S,   since θ = 0° 
+	   = ERa2
+	Net flux through the cube
+= φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2]
+=  αa5/2 
+=  800 (0.1)5/2 
+= 1.05 N m2 C–1
+(b) We can use Gauss’s law to find the total charge q inside the cube. We have φ = q/ε0 or q = φε0. Therefore,
+	  q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C.	
+Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x. It is given that E = 200   N/C for x > 0 and E = –200  N/C for x < 0. A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm (Fig. 1.28). (a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder?
+Solution 
+(a)	We can see from the figure that on the left face E and ∆S are parallel. Therefore, the outward flux is 
+ φL= E.∆S = – 200  
+	=  + 200 ∆S, since  = – ∆S
+	=  + 200 × π (0.05)2 = + 1.57 N m2 C–1
+	On the right face, E and ∆S are parallel and therefore
+ φR =  E.∆S =  + 1.57 N m2 C–1.
+(b)	For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0. Therefore, the flux out of the side of the cylinder is zero.
+(c)	Net outward flux through the cylinder
+ φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1
+
+
+
+
+
+Figure 1.28
+ (d)	The net charge within the cylinder can be found by using Gauss’s law which gives
+ q =  ε0φ
+	   =  3.14 × 8.854 × 10–12  C
+	   =  2.78 × 10–11 C
+1.15  Applications of Gauss’s Law
+The electric field due to a general charge distribution is, as seen above, given by Eq. (1.27). In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space. For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law. This is best understood by some examples.
+1.15.1	Field due to an infinitely long straight uniformly charged wire 
+Consider an infinitely long thin straight wire with uniform linear charge density λ. The wire is obviously an axis of symmetry. Suppose we take the radial vector from O to P and rotate it around the wire. The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire. This implies that the electric field must have the same magnitude at these points. The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0). This is clear from Fig. 1.29.
+Consider a pair of line elements P1 and P2 of the wire, as shown. The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel). This is true for any such pair and hence the total field at any point P is radial. Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire. In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r.
+To calculate the field, imagine a cylindrical Gaussian surface, as shown in the Fig. 1.29(b). Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero. At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r. The surface area of the curved part is  2πrl, where l is the length of the cylinder. 
+Flux through the Gaussian surface
+=	flux through the curved cylindrical part of the surface
+=	E × 2πrl
+The surface includes charge equal to λ l. Gauss’s law then gives
+E × 2πrl = λl/ε0
+i.e.,	E  = 
+Vectorially, E at any point is given by
+	(1.32)
+where  is the radial unit vector in the plane normal to the wire passing through the point. E is directed outward if λ is positive and inward if λ is negative.
+
+
+
+Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density.
+
+
+
+
+
+
+Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A , the scalar A is  an algebraic number. It can be negative or positive. The direction of A will be the same as that of the unit vector if A > 0 and opposite to  if A < 0. When we want to restrict to non-negative values, we use the symbol and call it the modulus of A. Thus, .
+Also note that though only  the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire. Further, the assumption that the wire is infinitely long is crucial. Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface. However, Eq. (1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored. 
+1.15.2  Field due to a uniformly charged infinite plane sheet
+Let σ be the uniform surface charge density of an infinite plane sheet (Fig. 1.30). We take the x-axis normal to the given plane. By symmetry, the electric field will not depend on y and z coordinates and its direction at every point must be parallel to the x-direction. 
+We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional area A, as shown. (A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux. 
+The unit vector normal to surface 1 is in –x direction while  the unit  vector normal to surface 2 is in the +x direction. Therefore, flux  E.∆S through both the surfaces are equal and add up. Therefore the net flux through the Gaussian surface is 2 EA. The charge enclosed by the closed surface is σA. Therefore by Gauss’s law,
+2 EA = σA/ε0
+or,  E = σ/2ε0
+Vectorically, 
+	(1.33)
+where  is a unit vector normal to the plane and going away from it.
+E is directed away from the plate if σ  is positive and toward the plate if σ is negative. Note that the above application of the Gauss’ law has brought out an additional fact: E is independent of x also.
+For a finite large planar sheet, Eq. (1.33) is approximately true in the middle regions of the planar sheet, away from the ends. 
+1.15.3 Field due to a uniformly charged thin spherical shell
+Let σ be the uniform surface charge density of a thin spherical shell of radius R (Fig. 1.31). The situation has obvious spherical symmetry. The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector).
+(i)	Field outside the shell: Consider a point P outside the shell with radius vector r. To calculate E at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P. All points on this sphere are equivalent relative to the given charged configuration. (That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point. Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S. Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2. The charge enclosed is σ × 4 π R2. By Gauss’s law
+E × 4 π r2 = 
+Or,  
+	where q  =  4 π R2 σ is the total charge on the spherical shell.
+Vectorially,
+	(1.34)
+The electric field is directed outward if q > 0 and inward if q < 0. This, however, is exactly the field produced by a charge q placed at the centre O. Thus for points outside the shell, the field due to a uniformly charged shell is as if  the entire charge of the shell is concentrated at its centre.
+
+
+
+
+
+
+Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.
+
+
+
+(ii)	Field inside the shell: In Fig. 1.31(b), the point P is inside the shell. The Gaussian surface is again a sphere through P centred at O. The flux through the Gaussian surface, calculated as before, is E × 4 π r2. However, in this case, the Gaussian surface encloses no charge. Gauss’s law then gives
+E × 4 π r2 =  0	
+i.e.,	E = 0          (r < R )	(1.35)
+that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*. This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law. The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law.
+Example 1.13	An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R. The atom as a whole is neutral. For this model, what is the electric field at a distance r from the nucleus?
+
+
+
+Figure 1.31 Gaussian surfaces for a point with  (a) r > R, (b) r < R.
+
+
+
+
+
+
+
+
+
+
+
+Figure 1.32 
+Solution  The charge distribution for this model of the atom is as shown in Fig. 1.32. The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral. This immediately gives us the negative charge density ρ, since we must have
+or
+To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law. Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r. Its direction is along (or opposite to) the radius vector r from the origin to the point P. The obvious Gaussian surface is a spherical surface centred at the nucleus. We consider two situations, namely, r < R and r > R.
+(i) r < R : The electric flux φ enclosed by the spherical surface is 
+ φ  =  E (r) × 4 π r2 
+where E (r)  is the magnitude of the electric field at r. This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there,  and has the same magnitude at all points on the surface.
+The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r,
+i.e.,
+Substituting for the charge density ρ  obtained earlier, we have
+Gauss’s law then gives,
+The electric field is directed radially outward.
+(ii) r > R:  In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral. Thus, from Gauss’s law,
+E (r) × 4 π  r2 = 0  or  E (r) = 0;    r > R
+At r = R, both cases give the same result: E = 0. 
+Summary
+1.	Electric and magnetic forces determine the properties of atoms, molecules and bulk matter.
+2.	From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract. By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative. 
+3.	Conductors allow movement of electric charge through them, insulators do not. In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile.
+4.	Electric charge has three basic properties: quantisation, additivity and conservation.
+	Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, .... Proton and electron have charges +e, –e, respectively. For macroscopic charges for which n is a very large number, quantisation of charge can be ignored.
+	Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system. 
+	Conservation of electric charges means that the total charge of an isolated system remains unchanged with time. This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge.
+5.	Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21 separating them. Mathematically,
+ F21 = force on q2 due to 
+	where is a unit vector in the direction from q1 to q2 and k =  is the constant of proportionality. 
+	In SI units, the unit of charge is coulomb. The experimental value of the constant ε0 is
+ ε0 = 8.854 × 10–12 C2 N–1 m–2
+	The approximate value of k is
+ k = 9 × 109 N m2 C–2
+6.	The ratio of electric force and gravitational force between a proton and an electron is 
+
+
+*	Compare this with a uniform mass shell discussed in Section 8.5 of Class XI Textbook of Physics.
+
+
+
+7.	Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s). For an assembly of charges q1, q2, q3, ..., the force on any charge, say q1, is the vector sum of the force on  q1 due to q2, the force on q1 due to q3, and so on. For each pair, the force is given by the Coulomb’s law for two charges stated earlier.
+8.	The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge. Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative. Like Coulomb force, electric field also satisfies superposition principle.
+9.	An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point. The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak. In regions of constant electric field, the field lines are uniformly spaced parallel straight lines. 
+
+
+On symmetry operations
+In Physics, we often encounter systems with various symmetries. Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation. Consider, for example an infinite uniform sheet of charge (surface charge density σ) along the y-z plane. This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle. As the system is unchanged under such symmetry operation, so must its properties be. In particular, in this example, the electric field E must be unchanged.
+Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0). Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same. By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction.
+Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate. It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space. The direction, however, is opposite of each other on either side ofthe sheet.
+Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.
+
+
+10.	Some of the important properties of field lines are: (i) Field lines are continuous curves without any breaks. (ii) Two field lines cannot cross each other. (iii) Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops.
+11.	An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a. Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q.
+12.	Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:
+
+
+	Dipole electric field on the axis at a distance r from the centre:
+
+
+	The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2 dependence of electric field due to a point charge.
+13.	In a uniform electric field E, a dipole experiences a torque given by
+  = p × E
+	but experiences no net force. 
+14.	The flux ∆φ of electric field E through a small area element ∆S is given by
+ ∆φ  = E.∆S
+	The vector area element ∆S is 
+ ∆S = ∆S
+	where ∆S is the magnitude of the area element and is normal to the area element, which can be considered planar for sufficiently small ∆S. For an area element of a closed surface, is taken to be the direction of outward normal, by convention.
+15.	Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S. The law is especially useful in determining electric field E, when the source distribution has simple symmetry:
+	(i) Thin infinitely long straight wire of uniform linear charge density λ
+
+	where r is the perpendicular distance of the point from the wire and  is the radial unit vector in the plane normal to the wire passing through the point.
+	(ii) Infinite thin plane sheet of uniform surface charge density σ
+
+	where is a unit vector normal to the plane, outward on either side.
+	(iii)	Thin spherical shell of uniform surface charge density σ 
+
+ E = 0 		     (r   <   R)
+	where r is the distance of the point from the centre of the shell and R the radius of the shell. q is the total charge of the shell:  q = 4πR2σ.
+	The electric field outside the shell is as though the total charge is concentrated at the centre. The same result is true for a solid sphere of uniform volume charge density. The field is zero at all points inside the shell.
+Points to Ponder
+1.	You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus. Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together. The range of distance where this force is effective is, however, very small ~10-14 m. This is precisely the size of the nucleus. Also the electrons are not allowed to sit on top of the protons, i.e. inside the nucleus, due to the laws of quantum mechanics. This gives the atoms their structure as they exist in nature.
+2.	Coulomb force and gravitational force follow the same inverse-square law. But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces. This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature.
+3.	The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law. In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s). In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2 C–2.
+4.	The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on  current–carrying wires) which are generally much weaker than the electric forces. Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects. 
+5.	The additive property of charge is not an ‘obvious’ property. It is related to the fact that electric charge has no direction associated with it; charge is a scalar.
+6.	Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion. This is not always true for every scalar. For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion.
+7.	Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6. Conservation refers to invariance in time in a given frame of reference. A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision). On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system).
+8.	Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass.
+9.	Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors. It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges.
+10.	The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges. For continuous volume charge distribution, it is defined at any point in the distribution. For a surface charge distribution, electric field is discontinuous across the surface.
+11.	The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge. An electric dipole is the simplest example of this fact.
+Exercises
+
+
+Physical quantity	Symbol	Dimensions	Unit	Remarks
+Vector area element	∆S	[L2]	m2 ∆S = ∆S 
+Electric field  	E	[MLT–3A–1]	V m–1 
+Electric flux  	φ	[ML3 T–3A–1]	V m 	∆φ  =  E.∆S
+Dipole moment  	p	[LTA]	C m 	Vector directed 					from negative to 
+				positive charge 
+Charge density: 		
+linear	λ  	[L–1 TA]	C m–1    	 Charge/length
+surface	σ	[L–2 TA]	C m–2     	Charge/area
+volume 	ρ	[L–3 TA]	C m–3      	Charge/volume
+
+
+1.1	What is the force between  two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air?
+1.2	The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first?
+1.3	Check that the ratio ke2/G memp is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?
+1.4	(a)	Explain the meaning of the statement ‘electric charge of a body is quantised’.
+	(b)	Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges?
+1.5	When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge. 
+1.6	Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm. What is the force on a charge of 1 µC placed at the centre of the square?
+1.7	(a)	An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
+	(b)	Explain why two field lines never cross each other at any point? 
+1.8	Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum. 
+	(a)	What is the electric field at the midpoint O of the line AB joining the two charges?
+	(b)	If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge?
+1.9	A system has two charges qA = 2.5 × 10–7 C and qB  =  –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively. What are the total charge and electric dipole moment of the system?
+1.10	An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1. Calculate the magnitude of the torque acting on the dipole.
+1.11	A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C. 
+	(a)	Estimate the number of electrons transferred (from which to which?)
+	(b)	Is there a transfer of mass from wool to polythene?
+1.12	(a)	Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm. What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation. 
+	(b)	What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved?
+1.13	Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?
+1.14	Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?
+
+
+
+
+
+Figure 1.33
+1.15	Consider a uniform electric field E = 3 × 103 î N/C. (a)  What is  the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the  flux through the same square if the normal  to its plane makes a 60° angle with the x-axis?
+1.16	What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes?
+1.17	Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C. (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not?
+1.18	A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)
+
+
+
+
+
+Figure 1.34
+1.19	A point charge of 2.0 µC is at the centre of a  cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface?
+1.20	A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge. (a)  If the radius of the Gaussian surface were doubled, how much flux would  pass through the surface? (b) What is the value of the point charge?
+1.21	A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere?
+1.22	A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2. (a) Find the charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere?
+1.23	An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm. Calculate the linear charge density.
+1.24	Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2. What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?
+Additional Exercises
+1.25	An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment). The density of the oil is 1.26 g cm–3. Estimate the radius of the drop. (g = 9.81 m s–2; e = 1.60 × 10–19 C).
+1.26	Which among the curves shown in Fig. 1.35 cannot possibly represent electrostatic field lines?
+Figure 1.35
+1.27	In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre. What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ?
+1.28	(a)	A conductor A with  a cavity as shown in Fig. 1.36(a) is given a charge Q. Show that the entire charge must appear on the  outer surface of the conductor. (b)  Another conductor B with charge q is inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A is Q + q [Fig. 1.36(b)]. (c)  A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.
+Figure 1.36
+1.29	A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε0), where is the unit vector in the outward normal direction, and σ is the surface charge density near the hole. 
+1.30	Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law. [Hint: Use Coulomb’s law directly and evaluate the necessary integral.]
+1.31	It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks. A proton and a neutron consist of three quarks each. Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter. (Quarks of other types have also been found which give rise to different unusual varieties of matter.)  Suggest a possible quark composition of a proton and neutron. 
+1.32	(a) Consider an arbitrary  electrostatic field configuration. A small test charge is placed at a null point (i.e., where E = 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable.
+	(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart. 
+1.33	A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in Fig. 1.33). The length of plate is L and an uniform electric field E is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is  qEL2/(2m vx2).
+ Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics.
+1.34	Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1. If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)
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+Interactive animation on simple electrostatic experiments:
+http://demoweb.physics.ucla.edu/content/100-simple-electrostatic-experiments
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+ Example 1.1
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+Interactive animation on charging a two-sphere system by induction:
+http://www.physicsclassroom.com/mmedia/estatics/itsn.cfm
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+ Example 1.2
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+ Example 1.3
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+Charles Augustin de Coulomb  (1736 –1806)
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+ Example 1.4
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+Interactive animation on Coulomb’s law:
+http://webphysics.davidson.edu/physlet_resources/bu_semester2/menu_semester2.html
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+ Example 1.5
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+ Example 1.6
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+ Example 1.7
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+ Example 1.7
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+*	An alternate unit V/m will be introduced in the next chapter.
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+ Example 1.8
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+ Example 1.9
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+* Solid angle is a measure of a cone. Consider the intersection of the given cone with a sphere of radius R. The solid angle ∆Ω of the cone is defined to be equal to ∆S/R2, where ∆S is the area on the sphere cut out by the cone.
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+*	It will not be proper to say that the number of field lines is equal to E∆S. The number of field lines is after all, a matter of how many field lines we choose to draw. What is physically significant is the relative number of field lines crossing a given area at different points.
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+Contents
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+Chapter-1
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+Landmarks
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+Cover
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+Chapter One
+
+ELECTRIC CHARGES AND FIELDS
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+1.1  Introduction
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+All of us have the experience of seeing a spark or hearing a crackle when we take off our synthetic clothes or sweater, particularly in dry weather. This is almost inevitable with ladies garments like a polyester saree. Have you ever tried to find any explanation for this phenomenon? Another common example of electric discharge is the lightning that we see in the sky during thunderstorms. We also experience a sensation of an electric shock either while opening the door of a car or holding the iron bar of a bus after sliding from our seat. The reason for these experiences is discharge of electric charges through our body, which were accumulated due to rubbing of insulating surfaces. You might have also heard that this is due to generation of static electricity. This is precisely the topic we are going to discuss in this and the next chapter. Static means anything that does not move or change with time. Electrostatics deals with the study of forces, fields and potentials arising from static charges. 
+
+1.2  Electric Charge
+Historically the credit of discovery of the fact that amber rubbed with wool or silk cloth attracts light objects goes to Thales of Miletus, Greece, around 600 BC. The name electricity is coined from the Greek word elektron meaning amber. Many such pairs of materials were known which on rubbing could attract light objects like straw, pith balls and bits of papers. You can perform the following activity at home to experience such an effect. Cut out long thin strips of white paper and lightly iron them. Take them near a TV screen or computer monitor. You will see that the strips get attracted to the screen. In fact they remain stuck to the screen for a while. 
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+Figure 1.1 Rods and pith balls: like charges repel and unlike charges attract each other.
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+It was observed that if two glass rods rubbed with wool or silk cloth are brought close to each other, they repel each other [Fig. 1.1(a)]. The two strands of wool or two pieces of silk cloth, with which the rods were rubbed, also repel each other. However, the glass rod and wool attracted each other. Similarly, two plastic rods rubbed with cat’s fur repelled each other [Fig. 1.1(b)] but attracted the fur. On the other hand, the plastic rod attracts the glass rod [Fig. 1.1(c)] and repel the silk or wool with which the glass rod is rubbed. The glass rod repels the fur. 
+If a plastic rod rubbed with fur is made to touch two small pith balls (now-a-days we can use polystyrene balls) suspended by silk or nylon thread, then the balls repel each other [Fig. 1.1(d)] and are also repelled by the rod. A similar effect is found if the pith balls are touched with a glass rod rubbed with silk [Fig. 1.1(e)]. A dramatic observation is that a pith ball touched with glass rod attracts another pith ball touched with plastic rod [Fig. 1.1(f)].
+These seemingly simple facts were established from years of efforts and careful experiments and their analyses. It was concluded, after many careful studies by different scientists, that there were only two kinds of an entity which is called the electric charge. We say that the bodies like glass or plastic rods, silk, fur and pith balls are electrified. They acquire an electric charge on rubbing. The experiments on pith balls suggested that there are two kinds of electrification and we find  that (i) like charges repel and (ii) unlike charges attract each other. The experiments also demonstrated that the charges are transferred from the rods to the pith balls on  contact. It is said that the pith balls are electrified or are charged by  contact. The property which differentiates the two kinds of charges is called the polarity of charge.
+When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge. This is true for any pair of objects that are rubbed to be electrified. Now if the electrified glass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other. They also do not attract or repel other light objects as they did on being electrified.
+Thus, the charges acquired after rubbing are lost when the charged bodies are brought in contact. What can you conclude from these observations? It just tells us that unlike charges acquired by the objects neutralise or nullify each other’s effect. Therefore, the charges were named as positive and negative by the American scientist Benjamin Franklin. We know that when we add a positive number to a negative number of the same magnitude, the sum is zero. This might have been the philosophy in naming the charges as positive and negative. By convention, the charge on glass rod or cat’s fur is called positive and that on plastic rod or silk is termed negative. If an object possesses an electric charge, it is said to be electrified or charged. When it has no charge it is said to be electrically neutral.       
+A simple apparatus to detect charge on a body is the gold-leaf electroscope [Fig. 1.2(a)]. It consists of a vertical metal rod housed in a box, with two thin gold leaves attached to its bottom end. When a charged object touches the metal knob at the top of the rod, charge flows on to the leaves and they diverge. The degree of divergance is an indicator of the amount of charge. 
+
+
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+Unification of electricity and magnetism
+In olden days, electricity and magnetism were treated as separate subjects. Electricity dealt with charges on glass rods, cat’s fur, batteries, lightning, etc., while magnetism described interactions of magnets, iron filings, compass needles, etc. In 1820 Danish scientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle. Ampere and Faraday supported this observation by saying that electric charges in motion produce magnetic fields and moving magnets generate electricity. The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Lorentz put forward a theory where they showed the interdependence of these two subjects. This field is called electromagnetism. Most of the phenomena occurring around us can be described under electromagnetism. Virtually every force that we can think of like friction, chemical force between atoms holding the matter together, and even the forces describing processes occurring in cells of living organisms, have its origin in electromagnetic force. Electromagnetic force is one of the fundamental forces of nature. 
+Maxwell put forth four equations that play the same role in classical electromagnetism as Newton’s equations of motion and gravitation law play in mechanics. He also argued that light is electromagnetic in nature and its speed can be found by making purely electric and magnetic measurements. He claimed that the science of optics is intimately related to that of electricity and magnetism.
+The science of electricity and magnetism is the foundation for the modern technological civilisation. Electric power, telecommunication, radio and television, and a wide variety of the practical appliances used in daily life are based on the principles of this science. Although charged particles in motion exert both electric and magnetic forces, in the frame of reference where all the charges are at rest, the forces are purely electrical. You know that gravitational force is a long-range force. Its effect is felt even when the distance between the interacting particles is very large because the force decreases inversely as the square of the distance between the interacting bodies. We will learn in this chapter that electric force is also as pervasive and is in fact stronger than the gravitational force by several orders of magnitude (refer to Chapter 1 of Class XI Physics Textbook).
+
+
+Students can make a simple electroscope as follows [Fig. 1.2(b)]: Take a thin aluminium curtain rod with ball ends fitted for hanging the curtain. Cut out a piece of length about 20 cm with the ball at one end and flatten the cut end. Take a large bottle that can hold this rod and a cork which will fit in the opening of the bottle. Make a hole in the cork sufficient to hold the curtain rod snugly. Slide the rod through the hole in the cork with the cut end on the lower side and ball end projecting above the cork. Fold a small, thin aluminium foil (about 6 cm in length) in the middle and attach it to the flattened end of the rod by cellulose tape. This forms the leaves of your electroscope. Fit the cork in the bottle with about 5 cm of the ball end projecting above the cork. A paper scale may be put inside the bottle in advance to measure the separation of leaves. The separation is a rough measure of the amount of charge on the electroscope.
+
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+Figure 1.2 Electroscopes: (a) The gold leaf electroscope, (b) Schematics of a simple electroscope.
+
+
+
+To understand how the electroscope works, use the white paper strips we used for seeing the attraction of charged bodies. Fold the strips into half so that you make a mark of fold. Open the strip and iron it lightly with the mountain fold up, as shown in Fig. 1.3. Hold the strip by pinching it at the fold. You would notice that the two halves move apart. This shows that the strip has acquired charge on ironing. When you fold it into half, both the halves have the same charge. Hence they repel each other. The same effect is seen in the leaf electroscope. On charging the curtain rod by touching the ball end with an electrified body, charge is transferred to the curtain rod and the attached aluminium foil. Both the halves of the foil get similar charge and therefore repel each other. The divergence in the leaves depends on the amount of charge on them. Let us first try to understand why material bodies acquire charge.
+You know that all matter is made up of atoms and/or molecules. Although normally the materials are electrically neutral,  they do contain charges; but their charges are exactly balanced. Forces that hold the molecules together, forces that hold atoms together in a solid, the adhesive force of glue, forces associated with surface tension, all are basically electrical in nature, arising from the forces between charged particles. Thus the electric force is all pervasive and it encompasses almost each and every field associated with our life. It is therefore  essential that we learn more about such a force.
+
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+
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+Figure 1.3 Paper strip experiment.
+
+
+To electrify a neutral body, we need to add or remove one kind of charge. When we say that a body is charged, we always refer to this excess charge or deficit of charge. In solids,  some of the electrons, being  less tightly bound in the atom, are the charges which are transferred from one body to the other.  A body can thus be charged positively by losing some of its electrons. Similarly, a body can be charged negatively by gaining electrons. When we rub a glass rod with silk, some of the electrons from the rod are transferred to the silk cloth. Thus the rod gets positively charged and the silk gets negatively charged. No new charge is created in the process of rubbing. Also the number of electrons, that are transferred, is a very small fraction of the total number of electrons in the material body. Also only the less tightly bound electrons in a material body can be transferred from it to another by rubbing. Therefore, when a body is rubbed with another, the bodies get charged and that is why we have to stick to certain pairs of materials to notice charging on rubbing the bodies. 
+
+1.3  Conductors and Insulators
+A metal rod held in hand and rubbed with wool will not show any sign of being charged. However, if a metal rod with a wooden or plastic handle is rubbed without touching its metal part, it shows signs of charging. Suppose we connect one end of a copper wire to a neutral pith ball and the other end to a negatively charged plastic rod. We will find that the pith ball acquires a negative charge. If a similar experiment is repeated with a nylon thread or a rubber band, no transfer of charge will take place from the plastic rod to the pith ball. Why does the transfer of charge not take place from the rod to the ball? 
+Some substances readily allow passage of electricity through them, others do not.  Those which allow electricity to pass through them easily are called conductors. They have electric charges (electrons) that are comparatively free to move inside the material. Metals, human and animal bodies and earth are conductors. Most of the non-metals like glass, porcelain, plastic, nylon, wood offer high resistance to the passage of electricity through them. They are called insulators. Most  substances fall into one of the two classes stated above*.
+When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor. In contrast, if some charge is put on an insulator, it stays at the same place. You will learn why this happens in the next chapter.
+This property of the materials tells you why a nylon or plastic comb gets electrified on combing dry hair or on rubbing, but a metal article like spoon does not. The charges on metal leak through our body to the ground as both are conductors of electricity.
+  When we bring a charged body  in contact with the earth,  all the excess charge on the body disappears by causing a momentary current to pass to the ground through the connecting conductor (such as our body). This process of sharing the charges with the earth is called grounding or earthing. Earthing provides a safety measure for electrical circuits and appliances. A thick metal plate is buried deep into the earth and thick wires are drawn from this plate; these are  used in buildings  for the purpose of earthing near the mains supply.  The electric wiring in our houses has three wires: live, neutral and earth. The first two carry electric current from the power station and the third is earthed by connecting it to the buried metal plate. Metallic bodies of the electric appliances such as electric iron, refrigerator, TV are connected to the earth wire. When any fault occurs or live wire touches the metallic body, the charge flows to the earth without damaging the appliance and without causing any injury to the humans; this  would have otherwise been unavoidable since  the human body is a  conductor of electricity.
+
+* There is a third category called semiconductors, which offer resistance to the movement of charges which is intermediate between the conductors and insulators.
+
+
+1.4  Charging by Induction
+When we touch a pith ball with an electrified plastic rod, some of the negative charges on the rod are transferred to the pith ball and it also gets charged. Thus the pith ball is charged by contact. It is then repelled by the plastic rod but is attracted by a glass rod which is oppositely charged. However, why a electrified rod attracts light objects, is  a  question we have still left unanswered. Let us try to understand what could be happening by performing the following experiment.
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+Figure 1.4 Charging by induction.
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+(i)	Bring two metal spheres, A and B, supported on insulating stands, in contact as shown in Fig. 1.4(a). 
+(ii)	Bring a positively charged rod near one of the spheres, say A, taking care that it does not touch the sphere. The free electrons in the spheres are attracted towards the rod. This leaves an excess of positive charge on the rear surface of sphere B. Both kinds of charges are bound in the metal spheres and cannot escape. They, therefore, reside on the surfaces, as shown in Fig. 1.4(b). The left surface of sphere A, has an excess of negative charge and the right surface of sphere B, has an excess of positive charge. However, not all of the electrons in the spheres have accumulated on the left surface of A. As the negative charge starts building up at the left surface of A, other electrons are repelled by these. In a short time, equilibrium is reached under the action of force of attraction of the rod and the force of repulsion due to the accumulated charges. Fig. 1.4(b) shows the equilibrium situation. The process is called induction of charge and happens almost instantly. The accumulated charges remain on the surface, as shown, till the glass rod is held near the sphere. If the rod is removed, the charges are not acted by any outside force and they redistribute to their original neutral state.
+(iii)	Separate the spheres by a small distance while the glass rod is still held near sphere A, as shown in Fig. 1.4(c). The two spheres are found to be oppositely charged and attract each other. 
+(iv)	Remove the rod. The charges on spheres rearrange themselves as shown in Fig. 1.4(d). Now, separate the spheres quite apart. The charges on them get uniformly distributed over them, as shown in Fig. 1.4(e). 
+In this process, the metal spheres will each be equal and oppositely charged. This is charging by induction. The positively charged glass rod does not lose any of its charge, contrary to the process of charging by contact.
+When electrified rods are brought near light objects, a similar effect takes place. The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object. [This happens even when the light object is not a conductor. The mechanism for how this happens is explained later in Sections 1.10 and 2.10.] The centres of the two types of charges are slightly separated. We know that opposite charges attract while similar charges repel. However, the magnitude of force depends on the distance between the charges and in this case the force of attraction overweighs the force of repulsion. As a result the particles like bits of paper or pith balls, being light, are pulled towards the rods.   
+
+Example 1.1 How can you charge a metal sphere positively without touching it?
+Solution Figure 1.5(a) shows an uncharged metallic sphere on an insulating metal stand. Bring a negatively charged rod close to the metallic sphere, as shown in Fig. 1.5(b). As the rod is brought close to the sphere, the free electrons in the sphere move away due to repulsion and start piling up at the farther end. The near end becomes positively charged due to deficit of electrons. This process of charge distribution stops when the net force on the free electrons inside the metal is zero. Connect the sphere to the ground by a conducting wire. The electrons will flow to the ground while the positive charges at the near end will remain held there due to the attractive force of the negative charges on the rod, as shown in Fig. 1.5(c). Disconnect the sphere from the ground. The positive charge continues to be held at the near end [Fig. 1.5(d)]. Remove the electrified rod. The positive charge will spread uniformly over the sphere as shown in Fig. 1.5(e). 
+
+Figure 1.5
+In this experiment, the metal sphere gets charged by the process of induction and the rod does not lose any of its charge.
+
+Similar steps are involved in charging a metal sphere negatively by induction, by bringing a positively charged rod near it. In this case the electrons will flow from the ground to the sphere when the sphere is connected to the ground with a wire. Can you explain why?
+
+
+
+
+1.5  Basic Properties of Electric Charge
+We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other. Here,  we shall now describe some other properties of the electric charge. 
+If the sizes of charged bodies are very small as compared to the distances between them, we treat them as point charges. All the charge content of the body is assumed to be concentrated at one point in space.
+
+1.5.1 Additivity of charges
+We have not as yet given a quantitative definition of a charge; we shall follow it up in the next section. We shall tentatively assume that this can be done and proceed. If a system contains two point charges q1 and q2, the total charge of the system is obtained simply by adding algebraically q1 and q2 , i.e., charges add up like real numbers or they are scalars like the mass of a body. If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn .  Charge has  magnitude but no direction, similar to mass. However, there is one difference between mass and charge. Mass of a body is always positive whereas a charge can be either positive or negative. Proper signs have to be used while adding the charges in a system. For example, the total charge of a system containing five charges +1, +2, –3, +4 and –5, in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the same unit.
+
+1.5.2 Charge is conserved
+We have already  hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed. A picture of particles of electric charge enables us to understand the idea of conservation of charge. When we rub two bodies, what one body gains in charge the other body loses. Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found that the total charge of the isolated system is always conserved. Conservation of charge has been established experimentally.
+It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed in a process. Sometimes nature creates charged particles: a neutron turns into a proton and an electron. The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation. 
+
+1.5.3 Quantisation of charge
+Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted by e. Thus charge q on a body is always given by
+q = ne
+where n is any integer, positive or negative. This basic unit of charge is the charge that an electron or proton carries. By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as –e and that on a proton as +e. 
+The fact that electric charge is always an integral multiple of e is termed as quantisation of charge. There are a large number of situations in physics where certain physical quantities are quantised. The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday. It was experimentally demonstrated by Millikan in 1912. 
+In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C. A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter. In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2 of Class XI, Physics Textbook , Part I). In this system, the value of the basic unit of charge is 
+e = 1.602192 × 10–19 C
+Thus, there are about 6 × 1018 electrons in a charge of  –1C. In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 µC (micro coulomb) = 10–6 C or 1 mC (milli coulomb) = 10–3 C. 
+If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples of e. Thus, if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (–e) = (n2 – n1) e. Since n1 and n2 are integers, their difference is also an integer. Thus the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e. 
+The step size e is, however, very small because at the macroscopic level, we deal with charges of a few µC. At this scale the fact that charge of a body can increase or decrease in units of e is not visible. In this respect, the grainy nature of the charge is lost and it appears to be continuous.
+This situation can be compared with the geometrical concepts of points and lines. A dotted line viewed from a distance appears continuous to us but is not continuous in reality. As many points very close to each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution.
+At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e. Since e = 1.6 × 10–19 C, a charge of magnitude, say 1 µC, contains something like 1013 times the electronic charge. At this scale, the fact that charge can increase or decrease only in units of e is not very different from saying that charge can take continuous values. Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored. However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored. It is the magnitude of scale involved that is very important.
+
+Example 1.2 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 C on the other body?
+Solution In one second 109 electrons move out of the body. Therefore the charge given out in one second is 1.6 × 10–19 × 109 C = 1.6 × 10–10 C. 
+The time required to accumulate a charge of 1 C can then be estimated to be 1 C ÷ (1.6 × 10–10 C/s) = 6.25 × 109 s = 6.25 × 109 ÷ (365 × 24 × 3600) years = 198 years. Thus to collect a charge of one coulomb, from a body from which 109 electrons move out every second, we will need approximately 200 years. One coulomb is, therefore, a very large unit for many practical purposes.
+It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material. A cubic piece of copper of side 1 cm contains about 2.5 × 1024 electrons.
+Example 1.3 How much positive and negative charge is there in a cup of water?
+Solution Let us assume that the mass of one cup of water is 250 g. The molecular mass of water is 18g. Thus, one mole (= 6.02 × 1023 molecules) of water is 18 g. Therefore the number of molecules in one cup of water is (250/18) × 6.02 × 1023.
+
+Each molecule of water contains two hydrogen atoms and one oxygen atom, i.e., 10 electrons and 10 protons. Hence the total positive and total negative charge has the same magnitude. It is equal to (250/18) × 6.02 × 1023 × 10 × 1.6 × 10–19 C = 1.34 × 107 C.
+
+
+1.6  Coulomb’s Law
+Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges. Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges. Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by  
+	(1.1)
+How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance*  for measuring the force between two charged metallic spheres. When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges. However, the charges on the spheres were unknown, to begin with. How then could he discover a relation like Eq. (1.1)? Coulomb thought of the following simple way: Suppose the charge on a metallic sphere is q. If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres. By symmetry, the charge on each sphere will be q/2*. Repeating this process, we can get charges q/2, q/4, etc. Coulomb varied the distance for a fixed pair of charges and measured the force for different separations. He then varied the charges in pairs, keeping the distance fixed for each pair. Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation, Eq. (1.1). 
+
+
+* A torsion balance is a sensitive device to measure force. It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.
+
+Coulomb’s law, a simple mathematical statement,  was initially experimentally arrived at  in the manner described above. While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level  (r ~ 10–10 m).
+Coulomb discovered his law without knowing the explicit magnitude of the charge. In fact, it is the other way round: Coulomb’s law can now be employed to furnish a definition for a unit of charge. In the relation, Eq. (1.1), k is so far arbitrary. We can choose any positive value of k. The choice of k determines the size of the unit of charge. In SI units, the value of k is about 9 × 109 . The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4. Putting this value of k in Eq. (1.1), we see that for q1 = q2 = 1 C, r = 1 m 
+F = 9 × 109 N
+
+
+Charles Augustin de Coulomb (1736 – 1806) Coulomb, a French physicist, began his career as a military engineer in the West Indies. In 1776, he returned to Paris and retired to a small estate to do his scientific research. He invented a torsion balance to measure the quantity of a force and used it for determination of forces of electric attraction or repulsion between small charged spheres. He thus arrived in 1785 at the inverse square law relation, now known as Coulomb’s law. The law had been anticipated by Priestley and also by Cavendish earlier, though Cavendish never published his results. Coulomb also found the inverse square law of force between unlike and like magnetic poles.
+
+That is, 1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N. One coulomb is evidently too big a unit to be used. In practice, in electrostatics, one uses smaller units like 1 mC or 1 µC.
+The constant k in Eq. (1.1) is usually put as k = 1/4πε0 for later convenience, so that Coulomb’s law is written as
+	(1.2)
+ε0 is called the permittivity of free space . The value of ε0 in SI units is
+= 8.854 × 10–12 C2 N–1m–2
+
+
+* Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q.
+
+Since force is a vector, it is better to write Coulomb’s law in the vector notation. Let the position vectors of charges q1 and q2 be r1 and r2 respectively [see Fig.1.6(a)]. We denote force on q1 due to q2 by F12 and force on q2 due to q1 by F21. The two point charges q1 and q2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21: 
+r21 = r2 – r1
+In the same way, the vector leading from 2 to 1 is denoted by r12:
+r12 = r1 – r2 = – r21
+The magnitude of the vectors r21 and r12 is denoted by r21 and r12, respectively (r12 = r21). The direction of a vector is specified by a unit vector along the vector. To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors:
+, 
+Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as
+	(1.3)
+Some remarks on Eq. (1.3) are relevant:
+
+
+
+
+
+
+Figure 1.6 (a) Geometry and (b) Forces between charges. 
+
+
+
+•	Equation (1.3) is valid for any sign of q1 and q2 whether positive or negative. If q1 and q2 are of the same sign (either both positive or both negative), F21 is along 21, which denotes repulsion, as it should be for like charges. If q1 and q2 are of opposite signs, F21 is along –21(=12), which denotes attraction, as expected for unlike charges. Thus, we do not have to write separate equations for the cases of like and unlike charges. Equation (1.3) takes care of both cases correctly [Fig. 1.6(b)]. 
+•	The force F12  on charge q1 due to charge q2, is obtained from Eq. (1.3),  by simply interchanging 1 and 2, i.e.,
+ 
+ Thus, Coulomb’s law agrees with the Newton’s third law.
+•	Coulomb’s law [Eq. (1.3)] gives the force between two charges q1 and q2 in vacuum. If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter. We shall consider electrostatics in matter in the next chapter.
+
+Example 1.4 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges and masses respectively.(a) Compare the strength of these forces by determining the ratio of their magnitudes (i) for an electron and a proton and (ii) for two protons. (b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are1 Å (= 10-10 m) apart? (mp = 1.67 × 10–27 kg, me = 9.11 × 10–31 kg)
+Solution
+(a)	(i) The electric force between an electron and a proton at a distance r apart is:
+
+where the negative sign indicates that the force is attractive. The corresponding gravitational force (always attractive) is:
+
+where mp and me are the masses of a proton and an electron respectively.
+
+(ii) On similar lines, the ratio of the magnitudes of electric force to the gravitational force between two protons at a distance r apart is:
+1.3 × 1036
+However, it may be mentioned here that the signs of the two forces are different. For two protons, the gravitational force is attractive in nature and the Coulomb force is repulsive. The actual values of these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N, whereas, FG ~ 1.9 × 10–34 N.
+The (dimensionless) ratio of the two forces shows that electrical forces are enormously stronger than the gravitational forces.
+(b) The electric force F exerted by a proton on an electron is same in magnitude to the force exerted by an electron on a proton; however, the masses of an electron and a proton are different. Thus, the magnitude of force is
+|F| == 8.987 × 109 Nm2/C2 × (1.6 ×10–19C)2 / (10–10m)2 = 2.3 × 10–8 N
+Using Newton’s second law of motion, F = ma, the acceleration that an electron will undergo is
+a = 2.3×10–8 N / 9.11 ×10–31 kg = 2.5 × 1022 m/s2
+Comparing this with the value of acceleration due to gravity, we can conclude that the effect of gravitational field is negligible on the motion of electron and it undergoes very large accelerations under the action of Coulomb force due to a proton.
+The value for acceleration of the proton is
+
+2.3 × 10–8 N / 1.67 × 10–27 kg = 1.4 × 1019 m/s2
+Example 1.5 A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B held by an insulating handle is brought close to A such that the distance between their centres is 10 cm, as shown in Fig. 1.7(a). The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen). Spheres A and B are touched by uncharged spheres C and D respectively, as shown in Fig. 1.7(b). C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in Fig. 1.7(c).What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes. Ignore the sizes of A and B in comparison to the separation between their centres.
+
+Figure 1.7
+Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by
+
+neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is
+
+Thus the electrostatic force on A, due to B, remains unaltered.
+
+
+
+1.7 Forces between Multiple Charges
+The mutual electric force between two charges is given by Coulomb’s law. How to calculate the force on a charge where there are not one but several charges around? Consider a system of nstationary charges q1, q2, q3, ..., qn in vacuum. What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question. Recall that forces of mechanical origin add according to the parallelogram law of addition. Is the same true for forces of electrostatic origin?
+
+
+Figure 1.8 A system of (a) three charges (b) multiple charges.
+
+Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces are unaffected due to the presence of other charges. This is termed as theprinciple of superposition.
+To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in Fig. 1.8(a). The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges. Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq. (1.3) even though other charges are present. Thus,	F12
+
+In the same way, the force on q1 due to q3, denoted by F13, is given by
+
+which again is the Coulomb force on q1 due to q3, even though other charge q2 is present.
+Thus the total force F1 on q1 due to the two charges q2 and q3 is given as
+(1.4)
+The above calculation of force can be generalised to a system of charges more than three, as shown in Fig. 1.8(b).
+The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn. The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n:
+i.e.,
+
+
+
+
+
+
+(1.5)
+The vector sum is obtained as usual by the parallelogram law of addition of vectors. All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.
+
+Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in Fig. 1.9?
+
+Figure 1.9
+Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC,
+AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l. By symmatry AO = BO = CO.
+Thus,
+Force F1 on Q due to charge q at A =along AO
+Force F2 on Q due to charge q at B =along BO
+Force F3 on Q due to charge q at C =along CO
+The resultant of forces F2 and F3 isalong OA, by the parallelogram law. Therefore, the total force on Q == 0, whereis the unit vector along OA.
+
+It is clear also by symmetry that the three forces will sum to zero. Suppose that the resultant force was non-zero but in some direction. Consider what would happen if the system was rotated through 60° about O.
+Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in Fig. 1.10. What is the force on each charge?
+
+
+Figure 1.10
+Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in Fig. 1.10. By the parallelogram law, the total force F1 on the charge q at A is given by
+F1 = Fwhereis a unit vector along BC.
+The force of attraction or repulsion for each pair of charges has the same magnitude
+The total force F2 on charge q at B is thus F2 = F2, where2 is a unit vector along AC.
+Similarly the total force on charge –q at C is F3 =F, whereis the unit vector along the direction bisecting the ∠BCA.
+It is interesting to see that the sum of the forces on the three charges is zero, i.e.,
+F1 + F2 + F3 = 0
+
+The result is not at all surprising. It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law. The proof is left to you as an exercise.
+
+
+1.8 Electric Field
+Let us consider a point charge Q placed in vacuum, at the origin O. If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law. We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P. In order to answer such questions, the early scientists introduced the concept offield. According to this, we say that the charge Q produces an electric field everywhere in the surrounding. When another charge q is brought at some point P, the field there acts on it and produces a force. The electric field produced by the charge Q at a point r is given as
+(1.6)
+wherer/r, is a unit vector from the origin to the point r. Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r. The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position. The effect of the charge has been incorporated in the existence of the electric field. We obtain the force F exerted by a charge Q on a charge q, as
+(1.7)
+Note that the charge q also exerts an equal and opposite force on the charge Q. The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa. If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q. Thus,
+F(r) = q E(r)	(1.8)
+Equation (1.8) defines the SI unit of electric field as N/C*.
+Some important remarks may be made here:
+(i) From Eq. (1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it. Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point. The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge. Note that the source charge Q must remain at its original location. However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move. A way out of this difficulty is to make q negligibly small. The force F is then negligibly small but the ratio F/q is finite and defines the electric field:
+(1.9)
+
+Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.
+
+
+
+A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice. When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet.
+(ii) Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q. This is because F is proportional to q, so the ratio F/q does not depend on q. The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q. Thus, the electric field E due to Q is also dependent on the space coordinate r. For different positions of the charge q all over the space, we get different values of electric field E. The field exists at every point in three-dimensional space.
+(iii) For a positive charge, the electric field will be directed radially outwards from the charge. On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards.
+(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r. Thus at equal distances from the charge Q, the magnitude of its electric field E is same. The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry.
+
+1.8.1 Electric field due to a system of charges
+Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O. Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn. We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r.
+Electric field E1 at r due to q1 at r1 is given by
+E1 =
+whereis a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P.
+In the same manner, electric field E2 at r due to q2 at r2 is E2 =
+
+whereis a unit vector in the direction from q2 to P and r2P is the distance between q2 and P. Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn.
+By the superposition principle, the electric field E at r due to the system of charges is (as shown in Fig. 1.12)
+
+
+Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.
+
+E(r) = E1 (r) + E2 (r) + … + En(r)
+=
+E(r)(1.10)
+E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges.
+
+1.8.2 Physical significance of electric field
+You may wonder why the notion of electric field has been introduced here at all. After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq. (1.5)]. Why then introduce this intermediate quantity called the electric field?
+For electrostatics, the concept of electric field is convenient, but not really necessary. Electric field is an elegant way of characterising the electrical environment of a system of charges. Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system). Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field. The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point. Electric field is a vector field, since force is a vector quantity.
+
+
+
+
+The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena. Suppose we consider the force between two distant charges q1, q2 in accelerated motion. Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light. Thus, the effect of any motion of q1 on q2 cannot arise instantaneously. There will be some time delay between the effect (force on q2) and the cause (motion of q1). It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful. The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2. The notion of field elegantly accounts for the time delay. Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs. They have an independent dynamics of their own, i.e., they evolve according to laws of their own. They can also transport energy. Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy. The concept of field was first introduced by Faraday and is now among the central concepts in physics.
+
+Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1 [Fig. 1.13(a)]. The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance [Fig. 1.13(b)]. Compute the time of fall in each case. Contrast the situation with that of ‘free fall under gravity’.
+
+Figure 1.13
+
+Solution In Fig. 1.13(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field. The acceleration of the electron is
+ae = eE/me
+where me is the mass of the electron.
+Starting from rest, the time required by the electron to fall through a distance h is given by
+For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg,
+E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m,
+te = 2.9 × 10–9s
+In Fig. 1.13 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE. The acceleration of the proton is
+ap = eE/mp
+where mp is the mass of the proton; mp = 1.67 × 10–27 kg. The time of fall for the proton is
+
+
+Thus, the heavier particle (proton) takes a greater time to fall through the same distance. This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body. Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall. To see if this is justified, let us calculate the acceleration of the proton in the given electric field:
+
+
+
+which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity. The acceleration of the electron is even greater. Thus, the effect of acceleration due to gravity can be ignored in this example.
+Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart. Calculate the electric fields at points A, B and C shown in Fig. 1.14.
+
+
+Figure 1.14
+Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude
+= 3.6 × 104 N C–1
+The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude. Hence the magnitude of the total electric field EA at A is
+EA = E1A + E2A = 7.2 × 104 N C–1
+EA is directed toward the right.
+The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude
+= 3.6 × 104 N C–1
+The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude
+= 4 × 103 N C–1
+The magnitude of the total electric field at B is
+EB = E1B – E2B = 3.2 × 104 N C–1
+EB is directed towards the left.
+The magnitude of each electric field vector at point C, due to charge q1 and q2 is
+= 9 × 103 N C–1
+The directions in which these two vectors point are indicated in Fig. 1.14. The resultant of these two vectors is
+= 9 × 103 N C–1
+
+EC points towards the right.
+
+
+
+1.9 Electric Field Lines
+We have studied electric field in the last section. It is a vector quantity and can be represented as we represent vectors. Let us try to represent E due to a point charge pictorially. Let the point charge be placed at the origin. Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point. Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward. Figure 1.15 shows such a picture. In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow. Connect the arrows pointing in one direction and the resulting figure represents a field line. We thus get many field lines, all pointing outwards from the point charge. Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No. Now the magnitude of the field is represented by the density of field lines. E is strong near the charge, so the density of field lines is more near the charge and the lines are closer. Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines.
+Another person may draw more lines. But the number of lines is not important. In fact, an infinite number of lines can be drawn in any region. It is the relative density of lines in different regions which is important.
+We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions. So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines. Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge.
+We started by saying that the field lines carry information about the direction of electric field at different points in space. Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points. The field lines crowd where the field is strong and are spaced apart where it is weak. Figure 1.16 shows a set of field lines. We can imagine two equal and small elements of area placed at points R and S normal to the field lines there. The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points. The picture shows that the field at R is stronger than at S.
+
+Figure 1.15 Field of a point charge.
+To understand the dependence of the field lines on the area, or rather the solid anglesubtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions. Recall how a (plane) angle is defined in two-dimensions. Let a small transverse line element ∆l be placed at a distance r from a point O. Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r. Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2. We know that in a given solid angle the number of radial field lines is the same. In Fig. 1.16, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is∆Ω at P1 and an element of area
+
+
+Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.
+
+∆Ω at P2, respectively. The number of lines (say n) cutting these area elements are the same. The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively. Since n and ∆Ω are common, the strength of the field clearly has a 1/r2dependence.
+The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations. Faraday called them lines of force. This term is somewhat misleading, especially in case of magnetic fields. The more appropriate term is field lines (electric or magnetic) that we have adopted in this book.
+Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges. An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point. An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve. A field line is a space curve, i.e., a curve in three dimensions.
+
+Figure 1.17 Field lines due to some simple charge configurations.
+
+
+
+
+Figure 1.17 shows the field lines around some simple charge configurations. As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane. The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward. The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges. The field lines follow some important general properties:
+(i)	Field lines start from positive charges and end at negative charges. If there is a single charge, they may start or end at infinity.
+(ii)	In a charge-free region, electric field lines can be taken to be continuous curves without any breaks.
+(iii)	Two field lines can never cross each other. (If they did, the field at the point of intersection will not have a unique direction, which is absurd.)
+(iv)	Electrostatic field lines do not form any closed loops. This follows from the conservative nature of electric field (Chapter 2).
+
+1.10 Electric Flux
+Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface. The rate of flow of liquid is given by the volume crossing the area per unit time v dSand represents the flux of liquid flowing across the plane. If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ. Therefore, the flux going out of the surface dSis v.dS. For the case of the electric field, we define an analogous quantity and call it electric flux. We should, however, note that there is no flow of a physically observable quantity unlike the case of liquid flow.
+In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point. This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S. Now suppose we tilt the area element by angle θ. Clearly, the number of field lines crossing the area element will be smaller. The projection of the area element normal to E is ∆S cosθ. Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ. When θ = 90°, field lines will be parallel to ∆Sand will not cross it at all (Fig. 1.18).
+
+
+Figure 1.18 Dependence of flux on the inclination θ between E and n
+
+
+
+
+The orientation of area element and not merely its magnitude is important in many contexts. For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring. If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation. This shows that an area element should be treated as a vector. It has a magnitude and also a direction. How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane. Thus the direction of a planar area vector is along its normal.
+How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements. Each small area element may be treated as planar and a vector associated with it, as explained before.
+Notice one ambiguity here. The direction of an area element is along its normal. But a normal can point in two directions. Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context. For the case of a closed surface, this convention is very simple. The vector associated with every area element of a closed surface is taken to be in the direction of theoutward normal. This is the convention used in Fig. 1.19. Thus, the area element vector ∆S at a point on a closed surface equals ∆Sn where ∆S is the magnitude of the area element and n is a unit vector in the direction of outward normal at that point.
+We now come to the definition of electric flux. Electric flux ∆φ through an area element ∆S is defined by
+∆φ = E.∆S = E ∆S cosθ	(1.11)
+which, as seen before, is proportional to the number of field lines cutting the area element. The angle θ here is the angle between E and ∆S. For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element. Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element. The unit of electric flux is N C–1 m2.
+The basic definition of electric flux given by Eq. (1.11) can be used, in principle, to calculate the total flux through any given surface. All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up. Thus, the total flux φ through a surface S is
+φ ~ Σ E.∆S	(1.12)
+The approximation sign is put because the electric field E is taken to be constant over the small area element. This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq. (1.12) is written as an integral.
+
+Figure 1.19 Convention for defining normal 
+
+1.11 Electric Dipole
+An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a. The line connecting the two charges defines a direction in space. By convention, the direction from –q to q is said to be the direction of the dipole. The mid-point of locations of –q and q is called the centre of the dipole.
+The total charge of the electric dipole is obviously zero. This does not mean that the field of the electric dipole is zero. Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out. However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out. The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q). These qualitative ideas are borne out by the explicit calculation as follows:
+
+1.11.1 The field of an electric dipole
+The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle. The results are simple for the following two cases: (i) when the point is on the dipole axis, and (ii) when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre. The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+qdue to the charge q, by the parallelogram law of vectors.
+(i) For points on the axis
+Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in Fig. 1.20(a). Then
+[1.13(a)]
+whereis the unit vector along the dipole axis (from –q to q). Also
+[1.13(b)]	The total field at P is
+
+(1.14)	For r >> a
+(r >> a)	(1.15)
+
+
+Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole. 
+p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q.
+(ii) For points on the equatorial plane
+The magnitudes of the electric fields due to the two charges +q and –q are given by
+[1.16(a)]
+[1.16(b)]
+and are equal.
+The directions of E+q and E–q are as shown in Fig. 1.20(b). Clearly, the components normal to the dipole axis cancel away. The components along the dipole axis add up. The total electric field is opposite to. We have
+E = – (E +q + E –q) cosθ
+(1.17)
+At large distances (r >> a), this reduces to
+(1.18)
+From Eqs. (1.15) and (1.18), it is clear that the dipole field at large distances does not involveq and a separately; it depends on the product qa. This suggests the definition of dipole moment. The dipole moment vector p of an electric dipole is defined by
+p = q × 2a(1.19)
+that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q. In terms of p, the electric field of a dipole at large distances takes simple forms:
+At a point on the dipole axis
+(r >> a)	(1.20)
+
+
+
+
+
+At a point on the equatorial plane
+(r >> a)	(1.21)
+Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3. Further, the magnitude and the direction of the dipole field depends not only on the distance rbut also on the angle between the position vector r and the dipole moment p.
+We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite. Such a dipole is referred to as a point dipole. For a point dipole, Eqs. (1.20) and (1.21) are exact, true for any r.
+
+
+1.11.2 Physical significance of dipoles
+
+In most molecules, the centres of positive charges and of negative charges* lie at the same place. Therefore, their dipole moment is zero. CO2 and CH4 are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules, H2O, is an example of this type. Various materials give rise to interesting properties and important applications in the presence or absence of electric field.
+
+Example 1.10 Two charges ±10 µC are placed 5.0 mm apart. Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in Fig. 1.21(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in Fig. 1.21(b).
+
+figure 1.21
+Solution (a) Field at P due to charge +10 µC
+=
+= 4.13 × 106 N C–1 along BP
+Field at P due to charge –10 µC
+
+= 3.86 × 106 N C–1 along PA
+The resultant electric field at P due to the two charges at A and B is	= 2.7 × 105 N C–1 along BP.
+In this example, the ratio OP/OB is quite large (= 60). Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole. For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude
+(r/a >> 1)
+where p = 2a q is the magnitude of the dipole moment.
+The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q). Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m
+Therefore,
+E == 2.6 × 105 N C–1
+along the dipole moment direction AB, which is close to the result obtained earlier.
+(b) Field at Q due to charge + 10 µC at B
+=
+= 3.99 × 106 N C–1 along BQ
+
+Field at Q due to charge –10 µC at A
+=
+= 3.99 × 106 N C–1 along QA.
+
+Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA. Therefore, the resultant electric field at Q due to the two charges at A and B is
+
+.
+
+
+= 2 ×along BA
+= 1.33 × 105 N C–1 along BA.
+As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole:
+(r/a >> 1)
+
+= 1.33 × 105 N C–1.
+
+The direction of electric field in this case is opposite to the direction of the dipole moment vector. Again, the result agrees with that obtained before.
+
+
+
+
+1.12 Dipole in a Uniform External Field
+Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in Fig. 1.22. (By permanent dipole, we mean that p exists irrespective of E; it has not been induced byE.)
+There is a force qE on q and a force –qE on –q. The net force on the dipole is zero, since E is uniform. However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole. When the net force is zero, the torque (couple) is independent of the origin. Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).
+
+Figure 1.22 Dipole in a uniform electric field.
+Magnitude of torque = q E × 2 a sinθ
+= 2 q a E sinθ
+Its direction is normal to the plane of the paper, coming out of it.
+The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it. Thus,
+τ = p × E	(1.22)
+This torque will tend to align the dipole with the field E. When p is aligned with E, the torque is zero.
+What happens if the field is not uniform? In that case, the net force will evidently be non-zero. In addition there will, in general, be a torque on the system as before. The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E. In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform.
+Figure 1.23 is self-explanatory. It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field. When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field. In general, the force depends on the orientation of p with respect to E.
+
+This brings us to a common observation in frictional electricity. A comb run through dry hair attracts pieces of paper. The comb, as we know, acquires charge through friction. But the paper is not charged. What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field. Further, the electric field due to the comb is not uniform. In this situation, it is easily seen that the paper should move in the direction of the comb!
+
+1.13 Continuous Charge Distribution
+We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn. One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus. For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions. For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents. It is more feasible to consider an area element ∆S (Fig. 1.24) on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element. We then define a surface charge density σ at the area element by
+
+
+
+
+
+(1.23)
+We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density. The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*. σrepresents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically. The units for σ are C/m2.
+Similar considerations apply for a line charge distribution and a volume charge distribution. The linear charge density λ of a wire is defined by
+Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.
+
+
+
+
+(1.24)
+where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element. The units for λ are C/m. The volume charge density (sometimes simply called charge density) is defined in a similar manner:
+(1.25)
+where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents. The units for ρ are C/m3.
+The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics. When we refer to the density of a liquid, we are referring to its macroscopic density. We regard it as a continuous fluid and ignore its discrete molecular constitution.
+
+
+Figure 1.24 Definition of linear, surface and volume charge densities. 
+In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.
+
+
+
+The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq. (1.10). Suppose a continuous charge distribution in space has a charge density ρ. Choose any convenient origin O and let the position vector of any point in the charge distribution be r. The charge density ρ may vary from point to point, i.e., it is a function of r. Divide the charge distribution into small volume elements of size ∆V. The charge in a volume element ∆V is ρ∆V.
+Now, consider any general point P (inside or outside the distribution) with position vector R(Fig. 1.24). Electric field due to the charge ρ∆V is given by Coulomb’s law:
+(1.26)	where r′ is the distance between the charge element and P, and′ is a unit vector in the direction from the charge element to P. By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements:
+(1.27)
+Note that ρ, r′,all can vary from point to point. In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity. In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.
+
+1.14 Gauss’s Law
+As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre. Divide the sphere into small area elements, as shown in Fig. 1.25.
+
+
+Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.
+
+The flux through an area element ∆S is
+(1.28)
+where we have used Coulomb’s law for the electric field due to a single charge q. The unit vectoris along the radius vector from the centre to the area element. Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S andhave the same direction. Therefore,
+(1.29)
+since the magnitude of a unit vector is 1.
+The total flux through the sphere is obtained by adding up flux through all the different area elements:
+
+Since each area element of the sphere is at the same distance r from the charge,
+
+Now S, the total area of the sphere, equals 4πr2. Thus,
+(1.30)
+
+
+
+
+Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.
+
+Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law.
+We state Gauss’s law without proof:
+Electric flux through a closed surface S
+
+
+
+
+= q/ε0 (1.31)
+q = total charge enclosed by S.
+The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface. We can see that explicitly in the simple situation of Fig. 1.26.
+Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E. The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface. Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0. Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E. Therefore,
+φ1 = –E S1, φ2 = +E S2
+S1 = S2 = S
+where S is the area of circular cross-section. Thus, the total flux is zero, as expected by Gauss’s law. Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero.
+The great significance of Gauss’s law Eq. (1.31), is that it is true in general, and not only for the simple cases we have considered above. Let us note some important points regarding this law:
+(i)	Gauss’s law is true for any closed surface, no matter what its shape or size.
+(ii)	The term q on the right side of Gauss’s law, Eq. (1.31), includes the sum of all charges enclosed by the surface. The charges may be located anywhere inside the surface.
+(iii)	In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq. (1.31)] is due to all the charges, both inside and outside S. The term q on the right side of Gauss’s law, however, represents only the total charge inside S.
+(iv)	The surface that we choose for the application of Gauss’s law is called the Gaussian surface. You may choose any Gaussian surface and apply Gauss’s law. However, take care not to let the Gaussian surface pass through any discrete charge. This is because electric field due to a system of discrete charges is not well defined at the location of any charge. (As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution.
+(v)	Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface.
+(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law. Any violation of Gauss’s law will indicate departure from the inverse square law.
+
+Example 1.11 The electric field components in Fig. 1.27 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2. Calculate (a) the flux through the cube, and (b) the charge within the cube. Assume that a = 0.1 m.
+
+
+Figure 1.27
+Solution
+(a)	Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2. Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones. Now the magnitude of the electric field at the left face is
+EL = αx1/2 = αa1/2
+(x = a at the left face).
+The magnitude of electric field at the right face is
+ER = α x1/2 = α (2a)1/2
+(x = 2a at the right face).
+The corresponding fluxes are
+φL= EL.∆S ==EL ∆S cosθ = –EL ∆S, since θ = 180°
+= –ELa2
+φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0°
+= ERa2
+Net flux through the cube
+= φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2]
+= αa5/2
+= 800 (0.1)5/2
+
+= 1.05 N m2 C–1
+(b) We can use Gauss’s law to find the total charge q inside the cube. We have φ = q/ε0 or q =φε0. Therefore,
+
+q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C.
+Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x. It is given that E= 200N/C for x > 0 and E = –200N/C for x < 0. A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm (Fig. 1.28). (a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder?
+Solution
+(a)	We can see from the figure that on the left face E and ∆S are parallel. Therefore, the outward flux is
+φL= E.∆S = – 200
+= + 200 ∆S, since= – ∆S
+= + 200 × π (0.05)2 = + 1.57 N m2 C–1
+On the right face, E and ∆S are parallel and therefore
+φR = E.∆S = + 1.57 N m2 C–1.
+(b)	For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0. Therefore, the flux out of the side of the cylinder is zero.
+(c)	Net outward flux through the cylinder
+φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1
+
+
+Figure 1.28
+(d)	The net charge within the cylinder can be found by using Gauss’s law which gives
+q = ε0φ
+= 3.14 × 8.854 × 10–12 C
+
+= 2.78 × 10–11 C
+
+
+1.15 Applications of Gauss’s Law
+The electric field due to a general charge distribution is, as seen above, given by Eq. (1.27). In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space. For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law. This is best understood by some examples.
+
+1.15.1 Field due to an infinitely long straight uniformly charged wire
+Consider an infinitely long thin straight wire with uniform linear charge density λ. The wire is obviously an axis of symmetry. Suppose we take the radial vector from O to P and rotate it around the wire. The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire. This implies that the electric field must have the same magnitude at these points. The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0). This is clear from Fig. 1.29.
+
+Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density.
+Consider a pair of line elements P1 and P2 of the wire, as shown. The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel). This is true for any such pair and hence the total field at any point P is radial. Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire. In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r.
+To calculate the field, imagine a cylindrical Gaussian surface, as shown in the Fig. 1.29(b).Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero. At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r. The surface area of the curved part is 2πrl, where l is the length of the cylinder.
+Flux through the Gaussian surface
+=	flux through the curved cylindrical part of the surface
+=	E × 2πrl
+The surface includes charge equal to λ l. Gauss’s law then gives
+E × 2πrl = λl/ε0
+i.e.,	E =
+Vectorially, E at any point is given by
+(1.32)
+whereis the radial unit vector in the plane normal to the wire passing through the point. E is directed outward if λ is positive and inward if λ is negative.
+
+
+
+
+
+Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A, the scalar A is an algebraic number. It can be negative or positive. The direction of A will be the same as that of the unit vectorif A > 0 and opposite toif A < 0. When we want to restrict to non-negative values, we use the symboland call it the modulus of A. Thus,.
+Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire. Further, the assumption that the wire is infinitely long is crucial. Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface. However, Eq. (1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored.
+
+1.15.2 Field due to a uniformly charged infinite plane sheet
+Let σ be the uniform surface charge density of an infinite plane sheet (Fig. 1.30). We take thex-axis normal to the given plane. By symmetry, the electric field will not depend on y and zcoordinates and its direction at every point must be parallel to the x-direction.
+We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional areaA, as shown. (A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux.
+
+
+Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.
+
+The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction. Therefore, flux E.∆S through both the surfaces are equal and add up. Therefore the net flux through the Gaussian surface is 2 EA. The charge enclosed by the closed surface is σA. Therefore by Gauss’s law,
+2 EA = σA/ε0
+or, E = σ/2ε0
+Vectorically,
+(1.33)
+where n is a unit vector normal to the plane and going away from it.
+E is directed away from the plate if σ is positive and toward the plate if σ is negative. Note that the above application of the Gauss’ law has brought out an additional fact: E is independent ofx also.
+For a finite large planar sheet, Eq. (1.33) is approximately true in the middle regions of the planar sheet, away from the ends.
+
+1.15.3 Field due to a uniformly charged thin spherical shell
+Let σ be the uniform surface charge density of a thin spherical shell of radius R (Fig. 1.31). The situation has obvious spherical symmetry. The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector).
+(i)	Field outside the shell: Consider a point P outside the shell with radius vector r. To calculateE at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P. All points on this sphere are equivalent relative to the given charged configuration. (That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point. Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S. Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2. The charge enclosed is  σ × 4 π R2. By Gauss’s law E × 4 π r2 =
+
+Or,
+where q = 4 π R2 σ is the total charge on the spherical shell.
+Vectorially,
+(1.34)
+
+Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R.
+The electric field is directed outward if q > 0 and inward if q < 0. This, however, is exactly the field produced by a charge q placed at the centre O. Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.
+
+
+
+
+(ii)	Field inside the shell: In Fig. 1.31(b), the point P is inside the shell. The Gaussian surface is again a sphere through P centred at O. The flux through the Gaussian surface, calculated as before, is  E × 4 π r2. However, in this case, the Gaussian surface encloses no charge. Gauss’s law then gives
+E × 4 π r2 = 0
+i.e.,	E = 0 (r < R )	(1.35)
+that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*. This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law. The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law.
+
+Example 1.13	An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R. The atom as a whole is neutral. For this model, what is the electric field at a distance r from the nucleus?
+
+
+Figure 1.32
+Solution The charge distribution for this model of the atom is as shown in Fig. 1.32. The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral. This immediately gives us the negative charge density ρ, since we must have
+or
+To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law. Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r. Its direction is along (or opposite to) the radius vector r from the origin to the point P. The obvious Gaussian surface is a spherical surface centred at the nucleus. We consider two situations, namely, r < R and r > R.
+(i) r < R : The electric flux φ enclosed by the spherical surface is
+φ = E (r) × 4 π r2
+where E (r) is the magnitude of the electric field at r. This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface.
+The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r,
+i.e.,
+Substituting for the charge density ρ obtained earlier, we have
+Gauss’s law then gives,
+
+The electric field is directed radially outward.
+(ii) r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral. Thus, from Gauss’s law,
+E (r) × 4 π r2 = 0 or E (r) = 0; r > R
+
+At r = R, both cases give the same result: E = 0.
+
+
+
+On symmetry operations
+In Physics, we often encounter systems with various symmetries. Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation. Consider, for example an infinite uniform sheet of charge (surface charge densityσ) along the y-z plane. This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle. As the system is unchanged under such symmetry operation, so must its properties be. In particular, in this example, the electric field E must be unchanged.
+Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0). Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same. By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction.
+Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate. It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space. The direction, however, is opposite of each other on either side ofthe sheet.
+Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.
+
+
+
+
+
+
+Summary
+1.	Electric and magnetic forces determine the properties of atoms, molecules and bulk matter.
+2.	From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract. By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative.
+3.	Conductors allow movement of electric charge through them, insulators do not. In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile.
+4.	Electric charge has three basic properties: quantisation, additivity and conservation.
+Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, .... Proton and electron have charges +e, –e, respectively. For macroscopic charges for which n is a very large number, quantisation of charge can be ignored.
+Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system.
+Conservation of electric charges means that the total charge of an isolated system remains unchanged with time. This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge.
+5.	Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21separating them. Mathematically,
+F21 = force on q2 due to
+whereis a unit vector in the direction from q1 to q2 and k =is the constant of proportionality.
+In SI units, the unit of charge is coulomb. The experimental value of the constant ε0 is
+ε0 = 8.854 × 10–12 C2 N–1 m–2
+The approximate value of k is
+k = 9 × 109 N m2 C–2
+
+6.	The ratio of electric force and gravitational force between a proton and an electron is 
+
+7.	Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s). For an assembly of charges q1, q2, q3, ..., the force on any charge, sayq1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on. For each pair, the force is given by the Coulomb’s law for two charges stated earlier.
+8.	The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge. Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative. Like Coulomb force, electric field also satisfies superposition principle.
+9.	An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point. The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak. In regions of constant electric field, the field lines are uniformly spaced parallel straight lines.
+
+
+
+10.	Some of the important properties of field lines are: (i) Field lines are continuous curves without any breaks. (ii) Two field lines cannot cross each other. (iii) Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops.
+11.	An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a. Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q.
+12.	Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:
+
+
+Dipole electric field on the axis at a distance r from the centre:
+
+
+The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2dependence of electric field due to a point charge.
+13.	In a uniform electric field E, a dipole experiences a torquegiven by
+= p × E
+but experiences no net force.
+14.	The flux ∆φ of electric field E through a small area element ∆S is given by
+∆φ = E.∆S
+The vector area element ∆S is
+∆S = ∆S
+where ∆S is the magnitude of the area element andis normal to the area element, which can be considered planar for sufficiently small ∆S. For an area element of a closed surface,is taken to be the direction of outward normal, by convention.
+15.	Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S. The law is especially useful in determining electric field E, when the source distribution has simple symmetry:
+(i) Thin infinitely long straight wire of uniform linear charge density λ
+
+where r is the perpendicular distance of the point from the wire andis the radial unit vector in the plane normal to the wire passing through the point.
+(ii) Infinite thin plane sheet of uniform surface charge density σ
+
+whereis a unit vector normal to the plane, outward on either side.
+(iii)	Thin spherical shell of uniform surface charge density σ
+
+E = 0 (r < R)
+where r is the distance of the point from the centre of the shell and R the radius of the shell. q is the total charge of the shell: q = 4πR2σ.
+The electric field outside the shell is as though the total charge is concentrated at the centre. The same result is true for a solid sphere of uniform volume charge density. The field is zero at all points inside the shell.
+
+
+
+
+Points to Ponder
+1.	You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus. Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together. The range of distance where this force is effective is, however, very small ~10-14 m. This is precisely the size of the nucleus. Also the electrons are not allowed to sit on top of the protons, i.e. inside the nucleus, due to the laws of quantum mechanics. This gives the atoms their structure as they exist in nature.
+2.	Coulomb force and gravitational force follow the same inverse-square law. But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces. This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature.
+3.	The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law. In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s). In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2C–2.
+4.	The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces. Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects.
+5.	The additive property of charge is not an ‘obvious’ property. It is related to the fact that electric charge has no direction associated with it; charge is a scalar.
+6.	Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion. This is not always true for every scalar. For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion.
+7.	Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6. Conservation refers to invariance in time in a given frame of reference. A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision). On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system).
+8.	Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass.
+9.	Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors. It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges.
+10.	The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges. For continuous volume charge distribution, it is defined at any point in the distribution. For a surface charge distribution, electric field is discontinuous across the surface.
+
+11.	The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge. An electric dipole is the simplest example of this fact.
+
+
+
+Exercises
+1.1	What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air?
+
+1.2	The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first?
+1.3	Check that the ratio ke2/G memp is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?
+1.4	(a)	Explain the meaning of the statement ‘electric charge of a body is quantised’.
+(b)	Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges?
+1.5	When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge.
+1.6	Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm. What is the force on a charge of 1 µC placed at the centre of the square?
+1.7	(a)	An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
+(b)	Explain why two field lines never cross each other at any point?
+1.8	Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum.
+(a)	What is the electric field at the midpoint O of the line AB joining the two charges?
+(b)	If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge?
+1.9	A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively. What are the total charge and electric dipole moment of the system?
+1.10	An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1. Calculate the magnitude of the torque acting on the dipole.
+1.11	A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C.
+(a)	Estimate the number of electrons transferred (from which to which?)
+(b)	Is there a transfer of mass from wool to polythene?
+1.12	(a)	Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm. What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation.
+(b)	What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved?
+1.13	Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?
+1.14	Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?
+
+
+Figure 1.33
+1.15	Consider a uniform electric field E = 3 × 103 î N/C. (a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?
+1.16	What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes?
+1.17	Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C. (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not?
+1.18	A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint:Think of the square as one face of a cube with edge 10 cm.)
+
+
+Figure 1.34
+1.19	A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface?
+1.20	A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge?
+1.21	A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere?
+1.22	A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2. (a) Find the charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere?
+1.23	An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm. Calculate the linear charge density.
+
+1.24	Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2. What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?
+
+Additional Exercises
+1.25	An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment). The density of the oil is 1.26 g cm–3. Estimate the radius of the drop. (g = 9.81 m s–2; e = 1.60 × 10–19 C).
+1.26	Which among the curves shown in Fig. 1.35 cannot possibly represent electrostatic field lines?
+
+
+
+
+
+
+
+
+
+
+Figure 1.35
+1.27	In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre. What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ?
+1.28	(a)	A conductor A with a cavity as shown in Fig. 1.36(a) is given a charge Q. Show that the entire charge must appear on the outer surface of the conductor. (b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A is Q + q [Fig. 1.36(b)]. (c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.
+
+Figure 1.36
+1.29	A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε0), whereis the unit vector in the outward normal direction, and σ is the surface charge density near the hole.
+1.30	Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law. [Hint: Use Coulomb’s law directly and evaluate the necessary integral.]
+1.31	It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks. A proton and a neutron consist of three quarks each. Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter. (Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron.
+1.32	(a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., where E = 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable.
+(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.
+1.33	A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in Fig. 1.33). The length of plate is Land an uniform electric field E is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2).
+Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics.
+
+1.34	Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1. If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)
+
+
+
+
+
+
+
+
+
+  Chapter 1
+ELECTRIC CHARGES AND FIELDS
+
+
+
+
+
+
+
+
+
diff --git a/Generate Features.csv.ipynb b/Generate Features.csv.ipynb
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@@ -0,0 +1,145 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "[nltk_data] Downloading package punkt to\n",
+      "[nltk_data]     /home/pranav_kirsur/nltk_data...\n",
+      "[nltk_data]   Package punkt is already up-to-date!\n",
+      "[nltk_data] Downloading package averaged_perceptron_tagger to\n",
+      "[nltk_data]     /home/pranav_kirsur/nltk_data...\n",
+      "[nltk_data]   Package averaged_perceptron_tagger is already up-to-\n",
+      "[nltk_data]       date!\n",
+      "[nltk_data] Downloading package punkt to\n",
+      "[nltk_data]     /home/pranav_kirsur/nltk_data...\n",
+      "[nltk_data]   Package punkt is already up-to-date!\n"
+     ]
+    }
+   ],
+   "source": [
+    "import nltk\n",
+    "import pranav\n",
+    "import ahish\n",
+    "import discourse_connector\n",
+    "# import SimToSumm\n",
+    "import SimToTitle\n",
+    "import csv\n",
+    "import textprocessing"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "textDict = textprocessing.getTextDict()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[{'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.7 Solution Let the original charge on sphere A be q and that on B be q′', 'At a distance r between their centres, the magnitude of the electrostatic force on each is given by neglecting the sizes of spheres A and B in comparison to r', 'When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2', 'Similarly, after D touches B, the redistributed charge on each is q′/2', 'Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus the electrostatic force on A, due to B, remains unaltered', '7 Forces between Multiple Charges The mutual electric force between two charges is given by Coulomb’s law', 'How to calculate the force on a charge where there are not one but several charges around? Consider a system of n stationary charges q1, q2, q3, ..., qn in vacuum', 'What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question', 'Recall that forces of mechanical origin add according to the parallelogram law of addition', 'Is the same true for forces of electrostatic origin?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time', 'The individual forces are unaffected due to the presence of other charges', 'This is termed as the principle of superposition', 'To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in (a)', 'The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges', 'Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq', '(1.3) even though other charges are present', 'Thus, F12 In the same way, the force on q1 due to q3, denoted by F13, is given by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which again is the Coulomb force on q1 due to q3, even though other charge q2 is present', 'Thus the total force F1 on q1 due to the two charges q2 and q3 is given as (1.4) The above calculation of force can be generalised to a system of charges more than three, as shown in (b)', 'The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn', 'The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n: i.e.,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.8 A system of (a) three charges (b) multiple charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.5) The vector sum is obtained as usual by the parallelogram law of addition of vectors', 'All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle', 'Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l', 'What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in ?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.9 Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC, AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l', 'By symmatry AO = BO = CO', 'Thus, Force F1 on Q due to charge q at A = along AO Force F2 on Q due to charge q at B = along BO Force F3 on Q due to charge q at C = along CO The resultant of forces F2 and F3 is along OA, by the parallelogram law', 'Therefore, the total force on Q = = 0, where is the unit vector along OA', 'It is clear also by symmetry that the three forces will sum to zero', 'Suppose that the resultant force was non-zero but in some direction', 'Consider what would happen if the system was rotated through 60° about O', 'Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in 0', 'What is the force on each charge?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.10 Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in 0', 'By the parallelogram law, the total force F1 on the charge q at A is given by F1 = F where is a unit vector along BC', 'The force of attraction or repulsion for each pair of charges has the same magnitude The total force F2 on charge q at B is thus F2 = F 2, where 2 is a unit vector along AC', 'Similarly the total force on charge –q at C is F3 = F , where is the unit vector along the direction bisecting the ∠BCA', 'It is interesting to see that the sum of the forces on the three charges is zero, i.e., F1 + F2 + F3 = 0 The result is not at all surprising', 'It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law', 'The proof is left to you as an exercise', '1.8 Electric Field Let us consider a point charge Q placed in vacuum, at the origin O', 'If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law', 'We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P', 'In order to answer such questions, the early scientists introduced the concept of field', 'According to this, we say that the charge Q produces an electric field everywhere in the surrounding', 'When another charge q is brought at some point P, the field there acts on it and produces a force', 'The electric field produced by the charge Q at a point r is given as (1.6) where r/r, is a unit vector from the origin to the point r', 'Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r', 'The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position', 'The effect of the charge has been incorporated in the existence of the electric field', 'We obtain the force F exerted by a charge Q on a charge q, as (1.7) Note that the charge q also exerts an equal and opposite force on the charge Q', 'The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa', 'If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q', 'Thus, F(r) = q E(r) (1.8) Equation (1.8) defines the SI unit of electric field as N/C*', 'Some important remarks may be made here: From Eq', '(1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it', 'Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point', 'The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge', 'Note that the source charge Q must remain at its original location', 'However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move', 'A way out of this difficulty is to make q negligibly small', 'The force F is then negligibly small but the ratio F/q is finite and defines the electric field: (1.9)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice', 'When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet', 'Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q', 'This is because F is proportional to q, so the ratio F/q does not depend on q', 'The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q', 'Thus, the electric field E due to Q is also dependent on the space coordinate r', 'For different positions of the charge q all over the space, we get different values of electric field E', 'The field exists at every point in three-dimensional space', 'For a positive charge, the electric field will be directed radially outwards from the charge', 'On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards', '(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r', 'Thus at equal distances from the charge Q, the magnitude of its electric field E is same', 'The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry', '1.8.1 Electric field due to a system of charges Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O', 'Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn', 'We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r', 'Electric field E1 at r due to q1 at r1 is given by E1 = where is a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P', 'In the same manner, electric field E2 at r due to q2 at r2 is E2 = where is a unit vector in the direction from q2 to P and r2P is the distance between q2 and P', 'Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn', 'By the superposition principle, the electric field E at r due to the system of charges is (as shown in 2) E(r) = E1 (r) + E2 (r) + … + En(r) = E(r) (1.10) E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges', '8.2 Physical significance of electric field You may wonder why the notion of electric field has been introduced here at all', 'After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq', '(1.5)]', 'Why then introduce this intermediate quantity called the electric field? For electrostatics, the concept of electric field is convenient, but not really necessary', 'Electric field is an elegant way of characterising the electrical environment of a system of charges', 'Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system)', 'Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field', 'The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point', 'Electric field is a vector field, since force is a vector quantity.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena', 'Suppose we consider the force between two distant charges q1, q2 in accelerated motion', 'Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light', 'Thus, the effect of any motion of q1 on q2 cannot arise instantaneously', 'There will be some time delay between the effect (force on q2) and the cause (motion of q1)', 'It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful', 'The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2', 'The notion of field elegantly accounts for the time delay', 'Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs', 'They have an independent dynamics of their own, i.e., they evolve according to laws of their own', 'They can also transport energy', 'Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy', 'The concept of field was first introduced by Faraday and is now among the central concepts in physics', 'Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1', 'The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance', 'Compute the time of fall in each case', 'Contrast the situation with that of ‘free fall under gravity’', 'Figure 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Solution In 3(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field', 'The acceleration of the electron is ae = eE/me where me is the mass of the electron', 'Starting from rest, the time required by the electron to fall through a distance h is given by For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m, te = 2.9 × 10–9s In 3 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE', 'The acceleration of the proton is ap = eE/mp where mp is the mass of the proton; mp = 1.67 × 10–27 kg', 'The time of fall for the proton is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus, the heavier particle (proton) takes a greater time to fall through the same distance', 'This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body', 'Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall', 'To see if this is justified, let us calculate the acceleration of the proton in the given electric field:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity', 'The acceleration of the electron is even greater', 'Thus, the effect of acceleration due to gravity can be ignored in this example', 'Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart', 'Calculate the electric fields at points A, B and C shown in 4']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.14 Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude', 'Hence the magnitude of the total electric field EA at A is EA = E1A + E2A = 7.2 × 104 N C–1 EA is directed toward the right', 'The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude = 4 × 103 N C–1 The magnitude of the total electric field at B is EB = E1B – E2B = 3.2 × 104 N C–1 EB is directed towards the left', 'The magnitude of each electric field vector at point C, due to charge q1 and q2 is = 9 × 103 N C–1 The directions in which these two vectors point are indicated in 4', 'The resultant of these two vectors is = 9 × 103 N C–1 EC points towards the right', '1.9 Electric Field Lines We have studied electric field in the last section', 'It is a vector quantity and can be represented as we represent vectors', 'Let us try to represent E due to a point charge pictorially', 'Let the point charge be placed at the origin', 'Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point', 'Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward', 'Figure 1.15 shows such a picture', 'In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow', 'Connect the arrows pointing in one direction and the resulting figure represents a field line', 'We thus get many field lines, all pointing outwards from the point charge', 'Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No', 'Now the magnitude of the field is represented by the density of field lines', 'E is strong near the charge, so the density of field lines is more near the charge and the lines are closer', 'Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines', 'Another person may draw more lines', 'But the number of lines is not important', 'In fact, an infinite number of lines can be drawn in any region', 'It is the relative density of lines in different regions which is important', 'We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions', 'So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines', 'Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge', 'We started by saying that the field lines carry information about the direction of electric field at different points in space', 'Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points', 'The field lines crowd where the field is strong and are spaced apart where it is weak', 'Figure 1.16 shows a set of field lines', 'We can imagine two equal and small elements of area placed at points R and S normal to the field lines there', 'The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points', 'The picture shows that the field at R is stronger than at S', 'To understand the dependence of the field lines on the area, or rather the solid angle subtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions', 'Recall how a (plane) angle is defined in two-dimensions', 'Let a small transverse line element ∆l be placed at a distance r from a point O', 'Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r', 'Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2', 'We know that in a given solid angle the number of radial field lines is the same', 'In 6, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is ∆Ω at P1 and an element of area ∆Ω at P2, respectively', 'The number of lines (say n) cutting these area elements are the same', 'The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively', 'Since n and ∆Ω are common, the strength of the field clearly has a 1/r2 dependence.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.15 Field of a point charge.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations', 'Faraday called them lines of force', 'This term is somewhat misleading, especially in case of magnetic fields', 'The more appropriate term is field lines (electric or magnetic) that we have adopted in this book', 'Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges', 'An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point', 'An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve', 'A field line is a space curve, i.e., a curve in three dimensions.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 shows the field lines around some simple charge configurations', 'As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane', 'The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward', 'The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges', 'The field lines follow some important general properties: Field lines start from positive charges and end at negative charges', 'If there is a single charge, they may start or end at infinity', 'In a charge-free region, electric field lines can be taken to be continuous curves without any breaks', 'Two field lines can never cross each other', '(If they did, the field at the point of intersection will not have a unique direction, which is absurd.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 Field lines due to some simple charge configurations.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(iv) Electrostatic field lines do not form any closed loops', 'This follows from the conservative nature of electric field (Chapter 2)', '10 Electric Flux Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface', 'The rate of flow of liquid is given by the volume crossing the area per unit time v dS and represents the flux of liquid flowing across the plane', 'If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ', 'Therefore, the flux going out of the surface dS is v.dS', 'For the case of the electric field, we define an analogous quantity and call it electric flux', 'We should, however, note that there is no flow of a physically observable quantity unlike the case ofliquid flow', 'In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point', 'This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S', 'Now suppose we tilt the area element by angle θ', 'Clearly, the number of field lines crossing the area element will be smaller', 'The projection of the area element normal to E is ∆S cosθ', 'Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ', 'When θ = 90°, field lines will be parallel to ∆S and will not cross it at all .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.18 Dependence of flux on the inclination θ between E and .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The orientation of area element and not merely its magnitude is important in many contexts', 'For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring', 'If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation', 'This shows that an area element should be treated as a vector', 'It has a magnitude and also a direction', 'How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane', 'Thus the direction of a planar area vector is along its normal', 'How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements', 'Each small area element may be treated as planar and a vector associated with it, as explained before', 'Notice one ambiguity here', 'The direction of an area element is along its normal', 'But a normal can point in two directions', 'Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context', 'For the case of a closed surface, this convention is very simple', 'The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal', 'This is the convention used in 9', 'Thus, the area element vector ∆S at a point on a closed surface equals ∆S where ∆S is the magnitude of the area element and is a unit vector in the direction of outward normal at that point', 'We now come to the definition of electric flux', 'Electric flux ∆φ through an area element ∆S is defined by ∆φ = E.∆S = E ∆S cosθ (1.11) which, as seen before, is proportional to the number of field lines cutting the area element', 'The angle θ here is the angle between E and ∆S', 'For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element', 'Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element', 'The unit of electric flux is N C–1 m2', 'The basic definition of electric flux given by Eq', '(1.11) can be used, in principle, to calculate the total flux through any given surface', 'All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up', 'Thus, the total flux φ through a surface S is φ ~ Σ E.∆S (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element', 'This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq', '(1.12) is written as an integral', '11 Electric Dipole An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a', 'The line connecting the two charges defines a direction in space', 'By convention, the direction from –q to q is said to be the direction of the dipole', 'The mid-point of locations of –q and q is called the centre of the dipole.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.19 Convention for defining normal and ∆S.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The total charge of the electric dipole is obviously zero', 'This does not mean that the field of the electric dipole is zero', 'Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out', 'However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out', 'The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q)', 'These qualitative ideas are borne out by the explicit calculation as follows: 1.11.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle', 'The results are simple for the following two cases: when the point is on the dipole axis, and when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre', 'The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+q due to the charge q, by the parallelogram law of vectors', 'For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in 0(a)', 'Then [1.13(a)] where is the unit vector along the dipole axis (from –q to q)', 'Also [1.13(b)] The total field at P is (1.14) For r >> a (r >> a) (1.15) For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by [1.16(a)] [1.16(b)] and are equal', 'The directions of E+q and E–q are as shown in 0(b)', 'Clearly, the components normal to the dipole axis cancel away', 'The components along the dipole axis add up', 'The total electric field is opposite to', 'We have E = – (E +q + E –q) cosθ (1.17) At large distances (r >> a), this reduces to (1.18) From Eqs', '(1.15) and (1.18), it is clear that the dipole field at large distances does not involve q and a separately; it depends on the product qa', 'This suggests the definition of dipole moment', 'The dipole moment vector p of an electric dipole is defined by p = q × 2a (1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q', 'In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis (r >> a) (1.20)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole', 'p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['At a point on the equatorial plane (r >> a) (1.21) Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3', 'Further, the magnitude and the direction of the dipole field depends not only on the distance r but also on the angle between the position vector r and the dipole moment p', 'We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite', 'Such a dipole is referred to as a point dipole', 'For a point dipole, Eqs', '(1.20) and (1.21) are exact, true for any r', '1.11.2 Physical significance of dipoles In most molecules, the centres of positive charges and of negative charges* lie at the same place', 'Therefore, their dipole moment is zero', 'CO2 and CH4 are of this type of molecules', 'However, they develop a dipole moment when an electric field is applied', 'But in some molecules, the centres of negative charges and of positive charges do not coincide', 'Therefore they have a permanent electric dipole moment, even in the absence of an electric field', 'Such molecules are called polar molecules', 'Water molecules, H2O, is an example of this type', 'Various materials give rise to interesting properties and important applications in the presence or absence of electric field', 'Example 1.10 Two charges ±10 µC are placed 5.0 mm apart', 'Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in 1(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in 1(b)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['fIGURE 1.21 Solution (a) Field at P due to charge +10 µC = = 4.13 × 106 N C–1 along BP Field at P due to charge –10 µC = 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP', 'In this example, the ratio OP/OB is quite large (= 60)', 'Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole', 'For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude (r/a >> 1) where p = 2a q is the magnitude of the dipole moment', 'The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q)', 'Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, E = = 2.6 × 105 N C–1 along the dipole moment direction AB, which is close to the result obtained earlier', '(b) Field at Q due to charge + 10 µC at B = = 3.99 × 106 N C–1 along BQ Field at Q due to charge –10 µC at A = = 3.99 × 106 N C–1 along QA', 'Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA', 'Therefore, the resultant electric field at Q due to the two charges at A and B is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Centre of a collection of positive point charges is defined much the same way as the centre of mass: .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 2 × along BA = 1.33 × 105 N C–1 along BA', 'As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole: (r/a >> 1)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 1.33 × 105 N C–1', 'The direction of electric field in this case is opposite to the direction of the dipole moment vector', 'Again, the result agrees with that obtained before', '1.12 Dipole in a Uniform External Field Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in 2', '(By permanent dipole, we mean that p exists irrespective of E; it has not been induced by E.) There is a force qE on q and a force –qE on –q', 'The net force on the dipole is zero, since E is uniform', 'However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole', 'When the net force is zero, the torque (couple) is independent of the origin', 'Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces)', 'Magnitude of torque = q E × 2 a sinθ = 2 q a E sinθ Its direction is normal to the plane of the paper, coming out of it', 'The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it', 'Thus, τ = p × E (1.22) This torque will tend to align the dipole with the field E', 'When p is aligned with E, the torque is zero', 'What happens if the field is not uniform? In that case, the net force will evidently be non-zero', 'In addition there will, in general, be a torque on the system as before', 'The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E', 'In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform', 'Figure 1.23 is self-explanatory', 'It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field', 'When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field', 'In general, the force depends on the orientation of p with respect to E', 'This brings us to a common observation in frictional electricity', 'A comb run through dry hair attracts pieces of paper', 'The comb, as we know, acquires charge through friction', 'But the paper is not charged', 'What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field', 'Further, the electric field due to the comb is not uniform', 'In this situation, it is easily seen that the paper should move in the direction of the comb! 13 Continuous Charge Distribution We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn', 'One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus', 'For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions', 'For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents', 'It is more feasible to consider an area element ∆S on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element', 'We then define a surface charge density σ at the area element by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.22 Dipole in a uniform electric field.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.23) We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density', 'The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*', 'σ represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically', 'The units for σ are C/m2', 'Similar considerations apply for a line charge distribution and a volume charge distribution', 'The linear charge density λ of a wire is defined by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.24) where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element', 'The units for λ are C/m', 'The volume charge density (sometimes simply called charge density) is defined in a similar manner: (1.25) where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents', 'The units for ρ are C/m3', 'The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics', 'When we refer to the density of a liquid, we are referring to its macroscopic density', 'We regard it as a continuous fluid and ignore its discrete molecular constitution.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.24 Definition of linear, surface and volume charge densities', 'In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq', '(1.10)', 'Suppose a continuous charge distribution in space has a charge density ρ', 'Choose any convenient origin O and let the position vector of any point in the charge distribution be r', 'The charge density ρ may vary from point to point, i.e., it is a function of r', 'Divide the charge distribution into small volume elements of size ∆V', 'The charge in a volume element ∆V is ρ∆V', 'Now, consider any general point P (inside or outside the distribution) with position vector R', 'Electric field due to the charge ρ∆V is given by Coulomb’s law: (1.26) where r′ is the distance between the charge element and P, and ′ is a unit vector in the direction from the charge element to P', 'By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements: (1.27) Note that ρ, r′, all can vary from point to point', 'In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity', 'In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous', '1.14 Gauss’s Law As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre', 'Divide the sphere into small area elements, as shown in 5', 'The flux through an area element ∆S is (1.28) where we have used Coulomb’s law for the electric field due to a single charge q', 'The unit vector is along the radius vector from the centre to the area element', 'Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S and have the same direction', 'Therefore,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* At the microscopic level, charge distribution is discontinuous, because they are discrete charges separated by intervening space where there is no charge.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.29) since the magnitude of a unit vector is 1', 'The total flux through the sphere is obtained by adding up flux through all the different area elements: Since each area element of the sphere is at the same distance r from the charge,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Now S, the total area of the sphere, equals 4πr2', 'Thus, (1.30) Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law', 'We state Gauss’s law without proof: Electric flux through a closed surface S']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= q/ε0 (1.31) q = total charge enclosed by S', 'The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface', 'We can see that explicitly in the simple situation of 6', 'Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E', 'The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface', 'Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0', 'Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E', 'Therefore, φ1 = –E S1, φ2 = +E S2 S1 = S2 = S where S is the area of circular cross-section', 'Thus, the total flux is zero, as expected by Gauss’s law', 'Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero', 'The great significance of Gauss’s law Eq', '(1.31), is that it is true in general, and not only for the simple cases we have considered above', 'Let us note some important points regarding this law: Gauss’s law is true for any closed surface, no matter what its shape or size', 'The term q on the right side of Gauss’s law, Eq', '(1.31), includes the sum of all charges enclosed by the surface', 'The charges may be located anywhere inside the surface', 'In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq', '(1.31)] is due to all the charges, both inside and outside S', 'The term q on the right side of Gauss’s law, however, represents only the total charge inside S']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface', 'You may choose any Gaussian surface and apply Gauss’s law', 'However, take care not to let the Gaussian surface pass through any discrete charge', 'This is because electric field due to a system of discrete charges is not well defined at the location of any charge', '(As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution', '(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry', 'This is facilitated by the choice of a suitable Gaussian surface', '(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law', 'Any violation of Gauss’s law will indicate departure from the inverse square law', 'Example 1.11 The electric field components in 7 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2', 'Calculate (a) the flux through the cube, and (b) the charge within the cube', 'Assume that a = 0.1 m.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.27 Solution (a) Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2', 'Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones', 'Now the magnitude of the electric field at the left face is EL = αx1/2 = αa1/2 (x = a at the left face)', 'The magnitude of electric field at the right face is ER = α x1/2 = α (2a)1/2 (x = 2a at the right face)', 'The corresponding fluxes are φL= EL.∆S = =EL ∆S cosθ = –EL ∆S, since θ = 180° = –ELa2 φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0° = ERa2 Net flux through the cube = φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2] = αa5/2 = 800 (0.1)5/2 = 1.05 N m2 C–1 (b) We can use Gauss’s law to find the total charge q inside the cube', 'We have φ = q/ε0 or q = φε0', 'Therefore, q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C. Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x', 'It is given that E = 200 N/C for x > 0 and E = –200 N/C for x < 0', 'A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm', '(a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder? Solution (a) We can see from the figure that on the left face E and ∆S are parallel', 'Therefore, the outward flux is φL= E.∆S = – 200 = + 200 ∆S, since = – ∆S = + 200 × π (0.05)2 = + 1.57 N m2 C–1 On the right face, E and ∆S are parallel and therefore φR = E.∆S = + 1.57 N m2 C–1', '(b) For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0', 'Therefore, the flux out of the side of the cylinder is zero', '(c) Net outward flux through the cylinder φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.28 (d) The net charge within the cylinder can be found by using Gauss’s law which gives q = ε0φ = 3.14 × 8.854 × 10–12 C = 2.78 × 10–11 C 1.15 Applications of Gauss’s Law The electric field due to a general charge distribution is, as seen above, given by Eq', '(1.27)', 'In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space', 'For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law', 'This is best understood by some examples', '1.15.1 Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ', 'The wire is obviously an axis of symmetry', 'Suppose we take the radial vector from O to P and rotate it around the wire', 'The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire', 'This implies that the electric field must have the same magnitude at these points', 'The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0)', 'This is clear from 9', 'Consider a pair of line elements P1 and P2 of the wire, as shown', 'The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel)', 'This is true for any such pair and hence the total field at any point P is radial', 'Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire', 'In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r', 'To calculate the field, imagine a cylindrical Gaussian surface, as shown in the 9(b)', 'Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero', 'At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r', 'The surface area of the curved part is 2πrl, where l is the length of the cylinder', 'Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E × 2πrl The surface includes charge equal to λ l', 'Gauss’s law then gives E × 2πrl = λl/ε0 i.e., E = Vectorially, E at any point is given by (1.32) where is the radial unit vector in the plane normal to the wire passing through the point', 'E is directed outward if λ is positive and inward if λ is negative.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A , the scalar A is an algebraic number', 'It can be negative or positive', 'The direction of A will be the same as that of the unit vector if A > 0 and opposite to if A < 0', 'When we want to restrict to non-negative values, we use the symbol and call it the modulus of A', 'Thus,', 'Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire', 'Further, the assumption that the wire is infinitely long is crucial', 'Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface', 'However, Eq', '(1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored', '15.2 Field due to a uniformly charged infinite plane sheet Let σ be the uniform surface charge density of an infinite plane sheet', 'We take the x-axis normal to the given plane', 'By symmetry, the electric field will not depend on y and z coordinates and its direction at every point must be parallel to the x-direction', 'We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional area A, as shown', '(A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux', 'The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction', 'Therefore, flux E.∆S through both the surfaces are equal and add up', 'Therefore the net flux through the Gaussian surface is 2 EA', 'The charge enclosed by the closed surface is σA', 'Therefore by Gauss’s law, 2 EA = σA/ε0 or, E = σ/2ε0 Vectorically, (1.33) where is a unit vector normal to the plane and going away from it', 'E is directed away from the plate if σ is positive and toward the plate if σ is negative', 'Note that the above application of the Gauss’ law has brought out an additional fact: E is independent of x also', 'For a finite large planar sheet, Eq', '(1.33) is approximately true in the middle regions of the planar sheet, away from the ends', '15.3 Field due to a uniformly charged thin spherical shell Let σ be the uniform surface charge density of a thin spherical shell of radius R', 'The situation has obvious spherical symmetry', 'The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector)', 'Field outside the shell: Consider a point P outside the shell with radius vector r', 'To calculate E at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P', 'All points on this sphere are equivalent relative to the given charged configuration', '(That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point', 'Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S', 'Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2', 'The charge enclosed is σ × 4 π R2', 'By Gauss’s law E × 4 π r2 = Or, where q = 4 π R2 σ is the total charge on the spherical shell', 'Vectorially, (1.34) The electric field is directed outward if q > 0 and inward if q < 0', 'This, however, is exactly the field produced by a charge q placed at the centre O', 'Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Field inside the shell: In 1(b), the point P is inside the shell', 'The Gaussian surface is again a sphere through P centred at O', 'The flux through the Gaussian surface, calculated as before, is E × 4 π r2', 'However, in this case, the Gaussian surface encloses no charge', 'Gauss’s law then gives E × 4 π r2 = 0 i.e., E = 0 (r < R ) (1.35) that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*', 'This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law', 'The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law', 'Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R', 'The atom as a whole is neutral', 'For this model, what is the electric field at a distance r from the nucleus?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.32 Solution The charge distribution for this model of the atom is as shown in 2', 'The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral', 'This immediately gives us the negative charge density ρ, since we must have or To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law', 'Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r', 'Its direction is along (or opposite to) the radius vector r from the origin to the point P', 'The obvious Gaussian surface is a spherical surface centred at the nucleus', 'We consider two situations, namely, r < R and r > R', 'r < R : The electric flux φ enclosed by the spherical surface is φ = E (r) × 4 π r2 where E (r) is the magnitude of the electric field at r', 'This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface', 'The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r, i.e., Substituting for the charge density ρ obtained earlier, we have Gauss’s law then gives, The electric field is directed radially outward', 'r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral', 'Thus, from Gauss’s law, E (r) × 4 π r2 = 0 or E (r) = 0; r > R At r = R, both cases give the same result: E = 0', 'Summary 1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter', '2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract', 'By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative', 'Conductors allow movement of electric charge through them, insulators do not', 'In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile', '4. Electric charge has three basic properties: quantisation, additivity and conservation', 'Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, ...', 'Proton and electron have charges +e, –e, respectively', 'For macroscopic charges for which n is a very large number, quantisation of charge can be ignored', 'Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system', 'Conservation of electric charges means that the total charge of an isolated system remains unchanged with time', 'This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge', '5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21 separating them', 'Mathematically, F21 = force on q2 due to where is a unit vector in the direction from q1 to q2 and k = is the constant of proportionality', 'In SI units, the unit of charge is coulomb', 'The experimental value of the constant ε0 is ε0 = 8.854 × 10–12 C2 N–1 m–2 The approximate value of k is k = 9 × 109 N m2 C–2 6. The ratio of electric force and gravitational force between a proton and an electron is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Compare this with a uniform mass shell discussed in Section 8.5 of Class XI Textbook of Physics.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s)', 'For an assembly of charges q1, q2, q3, ..., the force on any charge, say q1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on', 'For each pair, the force is given by the Coulomb’s law for two charges stated earlier', '8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge', 'Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative', 'Like Coulomb force, electric field also satisfies superposition principle', '9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point', 'The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak', 'In regions of constant electric field, the field lines are uniformly spaced parallel straight lines']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['On symmetry operations In Physics, we often encounter systems with various symmetries', 'Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation', 'Consider, for example an infinite uniform sheet of charge (surface charge density σ) along the y-z plane', 'This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle', 'As the system is unchanged under such symmetry operation, so must its properties be', 'In particular, in this example, the electric field E must be unchanged', 'Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0)', 'Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same', 'By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction', 'Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate', 'It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space', 'The direction, however, is opposite of each other on either side ofthe sheet', 'Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['10. Some of the important properties of field lines are: Field lines are continuous curves without any breaks', 'Two field lines cannot cross each other', 'Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops', '11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a', 'Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q', '12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Dipole electric field on the axis at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2 dependence of electric field due to a point charge', '13. In a uniform electric field E, a dipole experiences a torque given by = p × E but experiences no net force', '. The flux ∆φ of electric field E through a small area element ∆S is given by ∆φ = E.∆S The vector area element ∆S is ∆S = ∆S where ∆S is the magnitude of the area element and is normal to the area element, which can be considered planar for sufficiently small ∆S', 'For an area element of a closed surface, is taken to be the direction of outward normal, by convention', '15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S', 'The law is especially useful in determining electric field E, when the source distribution has simple symmetry: Thin infinitely long straight wire of uniform linear charge density λ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where r is the perpendicular distance of the point from the wire and is the radial unit vector in the plane normal to the wire passing through the point', 'Infinite thin plane sheet of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where is a unit vector normal to the plane, outward on either side', 'Thin spherical shell of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['E = 0 (r < R) where r is the distance of the point from the centre of the shell and R the radius of the shell', 'q is the total charge of the shell: q = 4πR2σ', 'The electric field outside the shell is as though the total charge is concentrated at the centre', 'The same result is true for a solid sphere of uniform volume charge density', 'The field is zero at all points inside the shell', 'Points to Ponder 1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus', 'Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together', 'The range of distance where this force is effective is, however, very small ~10-14 m', 'This is precisely the size of the nucleus', 'Also the electrons are not allowed to sit on top of the protons, i.e', 'inside the nucleus, due to the laws of quantum mechanics', 'This gives the atoms their structure as they exist in nature', '2. Coulomb force and gravitational force follow the same inverse-square law', 'But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces', 'This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature', '3. The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law', 'In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s)', 'In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2 C–2', '4. The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces', 'Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects', 'The additive property of charge is not an ‘obvious’ property', 'It is related to the fact that electric charge has no direction associated with it; charge is a scalar', '6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion', 'This is not always true for every scalar', 'For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion', '7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6', 'Conservation refers to invariance in time in a given frame of reference', 'A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision)', 'On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system)', '8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass', '9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors', 'It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges', '10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges', 'For continuous volume charge distribution, it is defined at any point in the distribution', 'For a surface charge distribution, electric field is discontinuous across the surface', '11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge', 'An electric dipole is the simplest example of this fact', 'Exercises']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Physical quantity Symbol Dimensions Unit Remarks Vector area element ∆S [L2] m2 ∆S = ∆S Electric field E [MLT–3A–1] V m–1 Electric flux φ [ML3 T–3A–1] V m ∆φ = E.∆S Dipole moment p [LTA] C m Vector directed from negative to positive charge Charge density: linear λ [L–1 TA] C m–1 Charge/length surface σ [L–2 TA] C m–2 Charge/area volume ρ [L–3 TA] C m–3 Charge/volume']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.1 What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air? 1.2 The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N', '(a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? 1.3 Check that the ratio ke2/G memp is dimensionless', 'Look up a Table of Physical Constants and determine the value of this ratio', 'What does the ratio signify? 1.4 (a) Explain the meaning of the statement ‘electric charge of a body is quantised’', '(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? 1.5 When a glass rod is rubbed with a silk cloth, charges appear on both', 'A similar phenomenon is observed with many other pairs of bodies', 'Explain how this observation is consistent with the law of conservation of charge', '6 Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm', 'What is the force on a charge of 1 µC placed at the centre of the square? 1.7 (a) An electrostatic field line is a continuous curve', 'That is, a field line cannot have sudden breaks', 'Why not? (b) Explain why two field lines never cross each other at any point? 8 Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum', '(a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge? 1.9 A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively', 'What are the total charge and electric dipole moment of the system? 1.10 An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1', 'Calculate the magnitude of the torque acting on the dipole', '1.11 A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C', '(a) Estimate the number of electrons transferred (from which to which?) (b) Is there a transfer of mass from wool to polythene? 1.12 (a) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm', 'What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation', '(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? 1.13 Suppose the spheres A and B in Exercise 1.12 have identical sizes', 'A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both', 'What is the new force of repulsion between A and B? 1.14 Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field', 'Give the signs of the three charges', 'Which particle has the highest charge to mass ratio?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.33 1.15 Consider a uniform electric field E = 3 × 103 î N/C', '(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis? 1.16 What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes? 1.17 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C', '(a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? 1.18 A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in 4', 'What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.34 1.19 A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge', 'What is the net electric flux through the surface? 1.20 A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge', '(a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 1.21 A conducting sphere of radius 10 cm has an unknown charge', 'If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere? 1.22 A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2', '(a) Find the charge on the sphere', '(b) What is the total electric flux leaving the surface of the sphere? 1.23 An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm', 'Calculate the linear charge density', '1.24 Two large, thin metal plates are parallel and close to each other', 'On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2', 'What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates? Additional Exercises 1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment)', 'The density of the oil is 1.26 g cm–3', 'Estimate the radius of the drop', '(g = 9.81 m s–2; e = 1.60 × 10–19 C)', '1.26 Which among the curves shown in 5 cannot possibly represent electrostatic field lines? Figure 1.35 1.27 In a certain region of space, electric field is along the z-direction throughout', 'The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre', 'What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ? 1.28 (a) A conductor A with a cavity as shown in 6(a) is given a charge Q', 'Show that the entire charge must appear on the outer surface of the conductor', '(b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A', 'Show that the total charge on the outside surface of A is Q + q', '(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment', 'Suggest a possible way', 'Figure 1.36 1.29 A hollow charged conductor has a tiny hole cut into its surface', 'Show that the electric field in the hole is (σ/2ε0), where is the unit vector in the outward normal direction, and σ is the surface charge density near the hole', '30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law', '[Hint: Use Coulomb’s law directly and evaluate the necessary integral.] 1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks', 'A proton and a neutron consist of three quarks each', 'Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter', '(Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron', '32 (a) Consider an arbitrary electrostatic field configuration', 'A small test charge is placed at a null point (i.e., where E = 0) of the configuration', 'Show that the equilibrium of the test charge is necessarily unstable', '(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart', '33 A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in 3)', 'The length of plate is L and an uniform electric field E is maintained between the plates', 'Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2)', 'Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics', '1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1', 'If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Interactive animation on simple electrostatic experiments: http://demoweb.physics.ucla.edu/content/100-simple-electrostatic-experiments']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Interactive animation on charging a two-sphere system by induction: http://www.physicsclassroom.com/mmedia/estatics/itsn.cfm']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.3']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Charles Augustin de Coulomb (1736 –1806)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.4']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Interactive animation on Coulomb’s law: http://webphysics.davidson.edu/physlet_resources/bu_semester2/menu_semester2.html']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.5']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.5']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.6']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.6']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.7']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.7']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* An alternate unit V/m will be introduced in the next chapter.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.8']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.8']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.9']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.9']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Solid angle is a measure of a cone', 'Consider the intersection of the given cone with a sphere of radius R', 'The solid angle ∆Ω of the cone is defined to be equal to ∆S/R2, where ∆S is the area on the sphere cut out by the cone.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* It will not be proper to say that the number of field lines is equal to E∆S', 'The number of field lines is after all, a matter of how many field lines we choose to draw', 'What is physically significant is the relative number of field lines crossing a given area at different points.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.11']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.11']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.12']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Contents']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Chapter-1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Landmarks']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Cover']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Chapter One']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['ELECTRIC CHARGES AND FIELDS']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.1 Introduction']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['All of us have the experience of seeing a spark or hearing a crackle when we take off our synthetic clothes or sweater, particularly in dry weather', 'This is almost inevitable with ladies garments like a polyester saree', 'Have you ever tried to find any explanation for this phenomenon? Another common example of electric discharge is the lightning that we see in the sky during thunderstorms', 'We also experience a sensation of an electric shock either while opening the door of a car or holding the iron bar of a bus after sliding from our seat', 'The reason for these experiences is discharge of electric charges through our body, which were accumulated due to rubbing of insulating surfaces', 'You might have also heard that this is due to generation of static electricity', 'This is precisely the topic we are going to discuss in this and the next chapter', 'Static means anything that does not move or change with time', 'Electrostatics deals with the study of forces, fields and potentials arising from static charges']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.2 Electric Charge Historically the credit of discovery of the fact that amber rubbed with wool or silk cloth attracts light objects goes to Thales of Miletus, Greece, around 600 BC', 'The name electricity is coined from the Greek word elektron meaning amber', 'Many such pairs of materials were known which on rubbing could attract light objects like straw, pith balls and bits of papers', 'You can perform the following activity at home to experience such an effect', 'Cut out long thin strips of white paper and lightly iron them', 'Take them near a TV screen or computer monitor', 'You will see that the strips get attracted to the screen', 'In fact they remain stuck to the screen for a while']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.1 Rods and pith balls: like charges repel and unlike charges attract each other.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['It was observed that if two glass rods rubbed with wool or silk cloth are brought close to each other, they repel each other', 'The two strands of wool or two pieces of silk cloth, with which the rods were rubbed, also repel each other', 'However, the glass rod and wool attracted each other', 'Similarly, two plastic rods rubbed with cat’s fur repelled each other but attracted the fur', 'On the other hand, the plastic rod attracts the glass rod and repel the silk or wool with which the glass rod is rubbed', 'The glass rod repels the fur', 'If a plastic rod rubbed with fur is made to touch two small pith balls (now-a-days we can use polystyrene balls) suspended by silk or nylon thread, then the balls repel each other and are also repelled by the rod', 'A similar effect is found if the pith balls are touched with a glass rod rubbed with silk', 'A dramatic observation is that a pith ball touched with glass rod attracts another pith ball touched with plastic rod', 'These seemingly simple facts were established from years of efforts and careful experiments and their analyses', 'It was concluded, after many careful studies by different scientists, that there were only two kinds of an entity which is called the electric charge', 'We say that the bodies like glass or plastic rods, silk, fur and pith balls are electrified', 'They acquire an electric charge on rubbing', 'The experiments on pith balls suggested that there are two kinds of electrification and we find that like charges repel and unlike charges attract each other', 'The experiments also demonstrated that the charges are transferred from the rods to the pith balls on contact', 'It is said that the pith balls are electrified or are charged by contact', 'The property which differentiates the two kinds of charges is called the polarity of charge', 'When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge', 'This is true for any pair of objects that are rubbed to be electrified', 'Now if the electrified glass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other', 'They also do not attract or repel other light objects as they did on being electrified', 'Thus, the charges acquired after rubbing are lost when the charged bodies are brought in contact', 'What can you conclude from these observations? It just tells us that unlike charges acquired by the objects neutralise or nullify each other’s effect', 'Therefore, the charges were named as positive and negative by the American scientist Benjamin Franklin', 'We know that when we add a positive number to a negative number of the same magnitude, the sum is zero', 'This might have been the philosophy in naming the charges as positive and negative', 'By convention, the charge on glass rod or cat’s fur is called positive and that on plastic rod or silk is termed negative', 'If an object possesses an electric charge, it is said to be electrified or charged', 'When it has no charge it is said to be electrically neutral', 'A simple apparatus to detect charge on a body is the gold-leaf electroscope', 'It consists of a vertical metal rod housed in a box, with two thin gold leaves attached to its bottom end', 'When a charged object touches the metal knob at the top of the rod, charge flows on to the leaves and they diverge', 'The degree of divergance is an indicator of the amount of charge']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Unification of electricity and magnetism In olden days, electricity and magnetism were treated as separate subjects', 'Electricity dealt with charges on glass rods, cat’s fur, batteries, lightning, etc., while magnetism described interactions of magnets, iron filings, compass needles, etc', 'In 1820 Danish scientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle', 'Ampere and Faraday supported this observation by saying that electric charges in motion produce magnetic fields and moving magnets generate electricity', 'The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Lorentz put forward a theory where they showed the interdependence of these two subjects', 'This field is called electromagnetism', 'Most of the phenomena occurring around us can be described under electromagnetism', 'Virtually every force that we can think of like friction, chemical force between atoms holding the matter together, and even the forces describing processes occurring in cells of living organisms, have its origin in electromagnetic force', 'Electromagnetic force is one of the fundamental forces of nature', 'Maxwell put forth four equations that play the same role in classical electromagnetism as Newton’s equations of motion and gravitation law play in mechanics', 'He also argued that light is electromagnetic in nature and its speed can be found by making purely electric and magnetic measurements', 'He claimed that the science of optics is intimately related to that of electricity and magnetism', 'The science of electricity and magnetism is the foundation for the modern technological civilisation', 'Electric power, telecommunication, radio and television, and a wide variety of the practical appliances used in daily life are based on the principles of this science', 'Although charged particles in motion exert both electric and magnetic forces, in the frame of reference where all the charges are at rest, the forces are purely electrical', 'You know that gravitational force is a long-range force', 'Its effect is felt even when the distance between the interacting particles is very large because the force decreases inversely as the square of the distance between the interacting bodies', 'We will learn in this chapter that electric force is also as pervasive and is in fact stronger than the gravitational force by several orders of magnitude (refer to Chapter 1 of Class XI Physics Textbook).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Students can make a simple electroscope as follows : Take a thin aluminium curtain rod with ball ends fitted for hanging the curtain', 'Cut out a piece of length about 20 cm with the ball at one end and flatten the cut end', 'Take a large bottle that can hold this rod and a cork which will fit in the opening of the bottle', 'Make a hole in the cork sufficient to hold the curtain rod snugly', 'Slide the rod through the hole in the cork with the cut end on the lower side and ball end projecting above the cork', 'Fold a small, thin aluminium foil (about 6 cm in length) in the middle and attach it to the flattened end of the rod by cellulose tape', 'This forms the leaves of your electroscope', 'Fit the cork in the bottle with about 5 cm of the ball end projecting above the cork', 'A paper scale may be put inside the bottle in advance to measure the separation of leaves', 'The separation is a rough measure of the amount of charge on the electroscope.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.2 Electroscopes: (a) The gold leaf electroscope, (b) Schematics of a simple electroscope.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['To understand how the electroscope works, use the white paper strips we used for seeing the attraction of charged bodies', 'Fold the strips into half so that you make a mark of fold', 'Open the strip and iron it lightly with the mountain fold up, as shown in', 'Hold the strip by pinching it at the fold', 'You would notice that the two halves move apart', 'This shows that the strip has acquired charge on ironing', 'When you fold it into half, both the halves have the same charge', 'Hence they repel each other', 'The same effect is seen in the leaf electroscope', 'On charging the curtain rod by touching the ball end with an electrified body, charge is transferred to the curtain rod and the attached aluminium foil', 'Both the halves of the foil get similar charge and therefore repel each other', 'The divergence in the leaves depends on the amount of charge on them', 'Let us first try to understand why material bodies acquire charge', 'You know that all matter is made up of atoms and/or molecules', 'Although normally the materials are electrically neutral, they do contain charges; but their charges are exactly balanced', 'Forces that hold the molecules together, forces that hold atoms together in a solid, the adhesive force of glue, forces associated with surface tension, all are basically electrical in nature, arising from the forces between charged particles', 'Thus the electric force is all pervasive and it encompasses almost each and every field associated with our life', 'It is therefore essential that we learn more about such a force.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.3 Paper strip experiment.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['To electrify a neutral body, we need to add or remove one kind of charge', 'When we say that a body is charged, we always refer to this excess charge or deficit of charge', 'In solids, some of the electrons, being less tightly bound in the atom, are the charges which are transferred from one body to the other', 'A body can thus be charged positively by losing some of its electrons', 'Similarly, a body can be charged negatively by gaining electrons', 'When we rub a glass rod with silk, some of the electrons from the rod are transferred to the silk cloth', 'Thus the rod gets positively charged and the silk gets negatively charged', 'No new charge is created in the process of rubbing', 'Also the number of electrons, that are transferred, is a very small fraction of the total number of electrons in the material body', 'Also only the less tightly bound electrons in a material body can be transferred from it to another by rubbing', 'Therefore, when a body is rubbed with another, the bodies get charged and that is why we have to stick to certain pairs of materials to notice charging on rubbing the bodies']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.3 Conductors and Insulators A metal rod held in hand and rubbed with wool will not show any sign of being charged', 'However, if a metal rod with a wooden or plastic handle is rubbed without touching its metal part, it shows signs of charging', 'Suppose we connect one end of a copper wire to a neutral pith ball and the other end to a negatively charged plastic rod', 'We will find that the pith ball acquires a negative charge', 'If a similar experiment is repeated with a nylon thread or a rubber band, no transfer of charge will take place from the plastic rod to the pith ball', 'Why does the transfer of charge not take place from the rod to the ball? Some substances readily allow passage of electricity through them, others do not', 'Those which allow electricity to pass through them easily are called conductors', 'They have electric charges (electrons) that are comparatively free to move inside the material', 'Metals, human and animal bodies and earth are conductors', 'Most of the non-metals like glass, porcelain, plastic, nylon, wood offer high resistance to the passage of electricity through them', 'They are called insulators', 'Most substances fall into one of the two classes stated above*', 'When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor', 'In contrast, if some charge is put on an insulator, it stays at the same place', 'You will learn why this happens in the next chapter', 'This property of the materials tells you why a nylon or plastic comb gets electrified on combing dry hair or on rubbing, but a metal article like spoon does not', 'The charges on metal leak through our body to the ground as both are conductors of electricity', 'When we bring a charged body in contact with the earth, all the excess charge on the body disappears by causing a momentary current to pass to the ground through the connecting conductor (such as our body)', 'This process of sharing the charges with the earth is called grounding or earthing', 'Earthing provides a safety measure for electrical circuits and appliances', 'A thick metal plate is buried deep into the earth and thick wires are drawn from this plate; these are used in buildings for the purpose of earthing near the mains supply', 'The electric wiring in our houses has three wires: live, neutral and earth', 'The first two carry electric current from the power station and the third is earthed by connecting it to the buried metal plate', 'Metallic bodies of the electric appliances such as electric iron, refrigerator, TV are connected to the earth wire', 'When any fault occurs or live wire touches the metallic body, the charge flows to the earth without damaging the appliance and without causing any injury to the humans; this would have otherwise been unavoidable since the human body is a conductor of electricity.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* There is a third category called semiconductors, which offer resistance to the movement of charges which is intermediate between the conductors and insulators.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.4 Charging by Induction When we touch a pith ball with an electrified plastic rod, some of the negative charges on the rod are transferred to the pith ball and it also gets charged', 'Thus the pith ball is charged by contact', 'It is then repelled by the plastic rod but is attracted by a glass rod which is oppositely charged', 'However, why a electrified rod attracts light objects, is a question we have still left unanswered', 'Let us try to understand what could be happening by performing the following experiment.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.4 Charging by induction.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Bring two metal spheres, A and B, supported on insulating stands, in contact as shown in (a)', 'Bring a positively charged rod near one of the spheres, say A, taking care that it does not touch the sphere', 'The free electrons in the spheres are attracted towards the rod', 'This leaves an excess of positive charge on the rear surface of sphere B', 'Both kinds of charges are bound in the metal spheres and cannot escape', 'They, therefore, reside on the surfaces, as shown in (b)', 'The left surface of sphere A, has an excess of negative charge and the right surface of sphere B, has an excess of positive charge', 'However, not all of the electrons in the spheres have accumulated on the left surface of A', 'As the negative charge starts building up at the left surface of A, other electrons are repelled by these', 'In a short time, equilibrium is reached under the action of force of attraction of the rod and the force of repulsion due to the accumulated charges', '(b) shows the equilibrium situation', 'The process is called induction of charge and happens almost instantly', 'The accumulated charges remain on the surface, as shown, till the glass rod is held near the sphere', 'If the rod is removed, the charges are not acted by any outside force and they redistribute to their original neutral state', 'Separate the spheres by a small distance while the glass rod is still held near sphere A, as shown in (c)', 'The two spheres are found to be oppositely charged and attract each other', '(iv) Remove the rod', 'The charges on spheres rearrange themselves as shown in (d)', 'Now, separate the spheres quite apart', 'The charges on them get uniformly distributed over them, as shown in (e)', 'In this process, the metal spheres will each be equal and oppositely charged', 'This is charging by induction', 'The positively charged glass rod does not lose any of its charge, contrary to the process of charging by contact', 'When electrified rods are brought near light objects, a similar effect takes place', 'The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object', '[This happens even when the light object is not a conductor', 'The mechanism for how this happens is explained later in Sections 1.10 and 2.10.] The centres of the two types of charges are slightly separated', 'We know that opposite charges attract while similar charges repel', 'However, the magnitude of force depends on the distance between the charges and in this case the force of attraction overweighs the force of repulsion', 'As a result the particles like bits of paper or pith balls, being light, are pulled towards the rods']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.1 How can you charge a metal sphere positively without touching it? Solution Figure 1.5(a) shows an uncharged metallic sphere on an insulating metal stand', 'Bring a negatively charged rod close to the metallic sphere, as shown in (b)', 'As the rod is brought close to the sphere, the free electrons in the sphere move away due to repulsion and start piling up at the farther end', 'The near end becomes positively charged due to deficit of electrons', 'This process of charge distribution stops when the net force on the free electrons inside the metal is zero', 'Connect the sphere to the ground by a conducting wire', 'The electrons will flow to the ground while the positive charges at the near end will remain held there due to the attractive force of the negative charges on the rod, as shown in (c)', 'Disconnect the sphere from the ground', 'The positive charge continues to be held at the near end', 'Remove the electrified rod', 'The positive charge will spread uniformly over the sphere as shown in (e)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.5 In this experiment, the metal sphere gets charged by the process of induction and the rod does not lose any of its charge.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Similar steps are involved in charging a metal sphere negatively by induction, by bringing a positively charged rod near it', 'In this case the electrons will flow from the ground to the sphere when the sphere is connected to the ground with a wire', 'Can you explain why?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5 Basic Properties of Electric Charge We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other', 'Here, we shall now describe some other properties of the electric charge', 'If the sizes of charged bodies are very small as compared to the distances between them, we treat them as point charges', 'All the charge content of the body is assumed to be concentrated at one point in space.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5.1 Additivity of charges We have not as yet given a quantitative definition of a charge; we shall follow it up in the next section', 'We shall tentatively assume that this can be done and proceed', 'If a system contains two point charges q1 and q2, the total charge of the system is obtained simply by adding algebraically q1 and q2 , i.e., charges add up like real numbers or they are scalars like the mass of a body', 'If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn', 'Charge has magnitude but no direction, similar to mass', 'However, there is one difference between mass and charge', 'Mass of a body is always positive whereas a charge can be either positive or negative', 'Proper signs have to be used while adding the charges in a system', 'For example, the total charge of a system containing five charges +1, +2, –3, +4 and –5, in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the same unit.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5.2 Charge is conserved We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed', 'A picture of particles of electric charge enables us to understand the idea of conservation of charge', 'When we rub two bodies, what one body gains in charge the other body loses', 'Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found that the total charge of the isolated system is always conserved', 'Conservation of charge has been established experimentally', 'It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed in a process', 'Sometimes nature creates charged particles: a neutron turns into a proton and an electron', 'The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.5.3 Quantisation of charge Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted by e', 'Thus charge q on a body is always given by q = ne where n is any integer, positive or negative', 'This basic unit of charge is the charge that an electron or proton carries', 'By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as –e and that on a proton as +e', 'The fact that electric charge is always an integral multiple of e is termed as quantisation of charge', 'There are a large number of situations in physics where certain physical quantities are quantised', 'The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday', 'It was experimentally demonstrated by Millikan in 1912', 'In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C', 'A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter', 'In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2 of Class XI, Physics Textbook , Part I)', 'In this system, the value of the basic unit of charge is e = 1.602192 × 10–19 C Thus, there are about 6 × 1018 electrons in a charge of –1C', 'In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 µC (micro coulomb) = 10–6 C or 1 mC (milli coulomb) = 10–3 C', 'If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples of e', 'Thus, if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (–e) = (n2 – n1) e', 'Since n1 and n2 are integers, their difference is also an integer', 'Thus the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e', 'The step size e is, however, very small because at the macroscopic level, we deal with charges of a few µC', 'At this scale the fact that charge of a body can increase or decrease in units of e is not visible', 'In this respect, the grainy nature of the charge is lost and it appears to be continuous', 'This situation can be compared with the geometrical concepts of points and lines', 'A dotted line viewed from a distance appears continuous to us but is not continuous in reality', 'As many points very close to each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution', 'At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e', 'Since e = 1.6 × 10–19 C, a charge of magnitude, say 1 µC, contains something like 1013 times the electronic charge', 'At this scale, the fact that charge can increase or decrease only in units of e is not very different from saying that charge can take continuous values', 'Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored', 'However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored', 'It is the magnitude of scale involved that is very important.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.2 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 C on the other body? Solution In one second 109 electrons move out of the body', 'Therefore the charge given out in one second is 1.6 × 10–19 × 109 C = 1.6 × 10–10 C', 'The time required to accumulate a charge of 1 C can then be estimated to be 1 C ÷ (1.6 × 10–10 C/s) = 6.25 × 109 s = 6.25 × 109 ÷ (365 × 24 × 3600) years = 198 years', 'Thus to collect a charge of one coulomb, from a body from which 109 electrons move out every second, we will need approximately 200 years', 'One coulomb is, therefore, a very large unit for many practical purposes', 'It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material', 'A cubic piece of copper of side 1 cm contains about 2.5 × 1024 electrons', 'Example 1.3 How much positive and negative charge is there in a cup of water? Solution Let us assume that the mass of one cup of water is 250 g', 'The molecular mass of water is 18g', 'Thus, one mole (= 6.02 × 1023 molecules) of water is 18 g', 'Therefore the number of molecules in one cup of water is (250/18) × 6.02 × 1023.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Each molecule of water contains two hydrogen atoms and one oxygen atom, i.e., 10 electrons and 10 protons', 'Hence the total positive and total negative charge has the same magnitude', 'It is equal to (250/18) × 6.02 × 1023 × 10 × 1.6 × 10–19 C = 1.34 × 107 C.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.6 Coulomb’s Law Coulomb’s law is a quantitative statement about the force between two point charges', 'When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges', 'Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges', 'Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by (1.1) How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres', 'When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges', 'However, the charges on the spheres were unknown, to begin with', 'How then could he discover a relation like Eq', '(1.1)? Coulomb thought of the following simple way: Suppose the charge on a metallic sphere is q', 'If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres', 'By symmetry, the charge on each sphere will be q/2*', 'Repeating this process, we can get charges q/2, q/4, etc', 'Coulomb varied the distance for a fixed pair of charges and measured the force for different separations', 'He then varied the charges in pairs, keeping the distance fixed for each pair', 'Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation, Eq', '(1.1)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* A torsion balance is a sensitive device to measure force', 'It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Coulomb’s law, a simple mathematical statement, was initially experimentally arrived at in the manner described above', 'While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level (r ~ 10–10 m)', 'Coulomb discovered his law without knowing the explicit magnitude of the charge', 'In fact, it is the other way round: Coulomb’s law can now be employed to furnish a definition for a unit of charge', 'In the relation, Eq', '(1.1), k is so far arbitrary', 'We can choose any positive value of k', 'The choice of k determines the size of the unit of charge', 'In SI units, the value of k is about 9 × 109', 'The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4', 'Putting this value of k in Eq', '(1.1), we see that for q1 = q2 = 1 C, r = 1 m F = 9 × 109 N']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Charles Augustin de Coulomb (1736 – 1806) Coulomb, a French physicist, began his career as a military engineer in the West Indies', 'In 1776, he returned to Paris and retired to a small estate to do his scientific research', 'He invented a torsion balance to measure the quantity of a force and used it for determination of forces of electric attraction or repulsion between small charged spheres', 'He thus arrived in 1785 at the inverse square law relation, now known as Coulomb’s law', 'The law had been anticipated by Priestley and also by Cavendish earlier, though Cavendish never published his results', 'Coulomb also found the inverse square law of force between unlike and like magnetic poles.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['That is, 1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N', 'One coulomb is evidently too big a unit to be used', 'In practice, in electrostatics, one uses smaller units like 1 mC or 1 µC', 'The constant k in Eq', '(1.1) is usually put as k = 1/4πε0 for later convenience, so that Coulomb’s law is written as (1.2) ε0 is called the permittivity of free space', 'The value of ε0 in SI units is = 8.854 × 10–12 C2 N–1m–2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['* Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Since force is a vector, it is better to write Coulomb’s law in the vector notation', 'Let the position vectors of charges q1 and q2 be r1 and r2 respectively [see Fig.1.6(a)]', 'We denote force on q1 due to q2 by F12 and force on q2 due to q1 by F21', 'The two point charges q1 and q2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21: r21 = r2 – r1 In the same way, the vector leading from 2 to 1 is denoted by r12: r12 = r1 – r2 = – r21 The magnitude of the vectors r21 and r12 is denoted by r21 and r12, respectively (r12 = r21)', 'The direction of a vector is specified by a unit vector along the vector', 'To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors: , Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as (1.3) Some remarks on Eq', '(1.3) are relevant:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.6 (a) Geometry and (b) Forces between charges']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['• Equation (1.3) is valid for any sign of q1 and q2 whether positive or negative', 'If q1 and q2 are of the same sign (either both positive or both negative), F21 is along 21, which denotes repulsion, as it should be for like charges', 'If q1 and q2 are of opposite signs, F21 is along –21(=12), which denotes attraction, as expected for unlike charges', 'Thus, we do not have to write separate equations for the cases of like and unlike charges', 'Equation (1.3) takes care of both cases correctly', '• The force F12 on charge q1 due to charge q2, is obtained from Eq', '(1.3), by simply interchanging 1 and 2, i.e., Thus, Coulomb’s law agrees with the Newton’s third law', '• Coulomb’s law [Eq', '(1.3)] gives the force between two charges q1 and q2 in vacuum', 'If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter', 'We shall consider electrostatics in matter in the next chapter.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.4 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges and masses respectively.(a) Compare the strength of these forces by determining the ratio of their magnitudes for an electron and a proton and for two protons', '(b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are1 Å (= 10-10 m) apart? (mp = 1.67 × 10–27 kg, me = 9.11 × 10–31 kg) Solution (a) The electric force between an electron and a proton at a distance r apart is:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where the negative sign indicates that the force is attractive', 'The corresponding gravitational force (always attractive) is:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where mp and me are the masses of a proton and an electron respectively.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['On similar lines, the ratio of the magnitudes of electric force to the gravitational force between two protons at a distance r apart is: 1.3 × 1036 However, it may be mentioned here that the signs of the two forces are different', 'For two protons, the gravitational force is attractive in nature and the Coulomb force is repulsive', 'The actual values of these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N, whereas, FG ~ 1.9 × 10–34 N', 'The (dimensionless) ratio of the two forces shows that electrical forces are enormously stronger than the gravitational forces', '(b) The electric force F exerted by a proton on an electron is same in magnitude to the force exerted by an electron on a proton; however, the masses of an electron and a proton are different', 'Thus, the magnitude of force is |F| == 8.987 × 109 Nm2/C2 × (1.6 ×10–19C)2 / (10–10m)2 = 2.3 × 10–8 N Using Newton’s second law of motion, F = ma, the acceleration that an electron will undergo is a = 2.3×10–8 N / 9.11 ×10–31 kg = 2.5 × 1022 m/s2 Comparing this with the value of acceleration due to gravity, we can conclude that the effect of gravitational field is negligible on the motion of electron and it undergoes very large accelerations under the action of Coulomb force due to a proton', 'The value for acceleration of the proton is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['2.3 × 10–8 N / 1.67 × 10–27 kg = 1.4 × 1019 m/s2 Example 1.5 A charged metallic sphere A is suspended by a nylon thread', 'Another charged metallic sphere B held by an insulating handle is brought close to A such that the distance between their centres is 10 cm, as shown in (a)', 'The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen)', 'Spheres A and B are touched by uncharged spheres C and D respectively, as shown in (b)', 'C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in (c).What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes', 'Ignore the sizes of A and B in comparison to the separation between their centres.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.7 Solution Let the original charge on sphere A be q and that on B be q′', 'At a distance r between their centres, the magnitude of the electrostatic force on each is given by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['neglecting the sizes of spheres A and B in comparison to r', 'When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2', 'Similarly, after D touches B, the redistributed charge on each is q′/2', 'Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus the electrostatic force on A, due to B, remains unaltered.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.7 Forces between Multiple Charges The mutual electric force between two charges is given by Coulomb’s law', 'How to calculate the force on a charge where there are not one but several charges around? Consider a system of nstationary charges q1, q2, q3, ..., qn in vacuum. What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question', 'Recall that forces of mechanical origin add according to the parallelogram law of addition', 'Is the same true for forces of electrostatic origin?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.8 A system of (a) three charges (b) multiple charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time', 'The individual forces are unaffected due to the presence of other charges', 'This is termed as theprinciple of superposition', 'To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in (a). The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges', 'Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq', '(1.3) even though other charges are present. Thus, F12']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['In the same way, the force on q1 due to q3, denoted by F13, is given by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which again is the Coulomb force on q1 due to q3, even though other charge q2 is present', 'Thus the total force F1 on q1 due to the two charges q2 and q3 is given as (1.4) The above calculation of force can be generalised to a system of charges more than three, as shown in (b)', 'The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn', 'The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n: i.e.,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.5) The vector sum is obtained as usual by the parallelogram law of addition of vectors', 'All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in ?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.9 Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC, AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l. By symmatry AO = BO = CO', 'Thus, Force F1 on Q due to charge q at A =along AO Force F2 on Q due to charge q at B =along BO Force F3 on Q due to charge q at C =along CO The resultant of forces F2 and F3 isalong OA, by the parallelogram law', 'Therefore, the total force on Q == 0, whereis the unit vector along OA.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['It is clear also by symmetry that the three forces will sum to zero', 'Suppose that the resultant force was non-zero but in some direction', 'Consider what would happen if the system was rotated through 60° about O', 'Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in 0', 'What is the force on each charge?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.10 Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in 0', 'By the parallelogram law, the total force F1 on the charge q at A is given by F1 = Fwhereis a unit vector along BC', 'The force of attraction or repulsion for each pair of charges has the same magnitude The total force F2 on charge q at B is thus F2 = F2, where2 is a unit vector along AC', 'Similarly the total force on charge –q at C is F3 =F, whereis the unit vector along the direction bisecting the ∠BCA', 'It is interesting to see that the sum of the forces on the three charges is zero, i.e., F1 + F2 + F3 = 0']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The result is not at all surprising', 'It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law', 'The proof is left to you as an exercise.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.8 Electric Field Let us consider a point charge Q placed in vacuum, at the origin O', 'If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law', 'We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P', 'In order to answer such questions, the early scientists introduced the concept offield', 'According to this, we say that the charge Q produces an electric field everywhere in the surrounding', 'When another charge q is brought at some point P, the field there acts on it and produces a force', 'The electric field produced by the charge Q at a point r is given as (1.6) wherer/r, is a unit vector from the origin to the point r', 'Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r', 'The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position', 'The effect of the charge has been incorporated in the existence of the electric field', 'We obtain the force F exerted by a charge Q on a charge q, as (1.7) Note that the charge q also exerts an equal and opposite force on the charge Q. The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa. If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q. Thus, F(r) = q E(r) (1.8) Equation (1.8) defines the SI unit of electric field as N/C*', 'Some important remarks may be made here: From Eq', '(1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it', 'Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point', 'The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge', 'Note that the source charge Q must remain at its original location', 'However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move', 'A way out of this difficulty is to make q negligibly small', 'The force F is then negligibly small but the ratio F/q is finite and defines the electric field: (1.9)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice', 'When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet', 'Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q', 'This is because F is proportional to q, so the ratio F/q does not depend on q', 'The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q. Thus, the electric field E due to Q is also dependent on the space coordinate r. For different positions of the charge q all over the space, we get different values of electric field E', 'The field exists at every point in three-dimensional space', 'For a positive charge, the electric field will be directed radially outwards from the charge', 'On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards', '(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r', 'Thus at equal distances from the charge Q, the magnitude of its electric field E is same. The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.8.1 Electric field due to a system of charges Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O', 'Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn', 'We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r', 'Electric field E1 at r due to q1 at r1 is given by E1 = whereis a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P', 'In the same manner, electric field E2 at r due to q2 at r2 is E2 =']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['whereis a unit vector in the direction from q2 to P and r2P is the distance between q2 and P', 'Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn', 'By the superposition principle, the electric field E at r due to the system of charges is (as shown in 2)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['E(r) = E1 (r) + E2 (r) + … + En(r) = E(r)(1.10) E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.8.2 Physical significance of electric field You may wonder why the notion of electric field has been introduced here at all', 'After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq', '(1.5)]', 'Why then introduce this intermediate quantity called the electric field? For electrostatics, the concept of electric field is convenient, but not really necessary', 'Electric field is an elegant way of characterising the electrical environment of a system of charges', 'Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system)', 'Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field', 'The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point', 'Electric field is a vector field, since force is a vector quantity.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena', 'Suppose we consider the force between two distant charges q1, q2 in accelerated motion', 'Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light', 'Thus, the effect of any motion of q1 on q2 cannot arise instantaneously', 'There will be some time delay between the effect (force on q2) and the cause (motion of q1)', 'It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful. The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2', 'The notion of field elegantly accounts for the time delay', 'Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs', 'They have an independent dynamics of their own, i.e., they evolve according to laws of their own', 'They can also transport energy', 'Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy', 'The concept of field was first introduced by Faraday and is now among the central concepts in physics.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1', 'The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance', 'Compute the time of fall in each case', 'Contrast the situation with that of ‘free fall under gravity’.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.13']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Solution In 3(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field', 'The acceleration of the electron is ae = eE/me where me is the mass of the electron', 'Starting from rest, the time required by the electron to fall through a distance h is given by For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m, te = 2.9 × 10–9s In 3 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE', 'The acceleration of the proton is ap = eE/mp where mp is the mass of the proton; mp = 1.67 × 10–27 kg', 'The time of fall for the proton is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Thus, the heavier particle (proton) takes a greater time to fall through the same distance', 'This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body', 'Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall', 'To see if this is justified, let us calculate the acceleration of the proton in the given electric field:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity', 'The acceleration of the electron is even greater', 'Thus, the effect of acceleration due to gravity can be ignored in this example', 'Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart', 'Calculate the electric fields at points A, B and C shown in 4.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.14 Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude', 'Hence the magnitude of the total electric field EA at A is EA = E1A + E2A = 7.2 × 104 N C–1 EA is directed toward the right', 'The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude = 4 × 103 N C–1 The magnitude of the total electric field at B is EB = E1B – E2B = 3.2 × 104 N C–1 EB is directed towards the left', 'The magnitude of each electric field vector at point C, due to charge q1 and q2 is = 9 × 103 N C–1 The directions in which these two vectors point are indicated in 4', 'The resultant of these two vectors is = 9 × 103 N C–1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['EC points towards the right.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.9 Electric Field Lines We have studied electric field in the last section', 'It is a vector quantity and can be represented as we represent vectors', 'Let us try to represent E due to a point charge pictorially', 'Let the point charge be placed at the origin', 'Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point', 'Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward', 'Figure 1.15 shows such a picture', 'In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow', 'Connect the arrows pointing in one direction and the resulting figure represents a field line', 'We thus get many field lines, all pointing outwards from the point charge', 'Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No', 'Now the magnitude of the field is represented by the density of field lines. E is strong near the charge, so the density of field lines is more near the charge and the lines are closer', 'Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines', 'Another person may draw more lines', 'But the number of lines is not important', 'In fact, an infinite number of lines can be drawn in any region', 'It is the relative density of lines in different regions which is important', 'We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions', 'So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines', 'Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge', 'We started by saying that the field lines carry information about the direction of electric field at different points in space', 'Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points', 'The field lines crowd where the field is strong and are spaced apart where it is weak', 'Figure 1.16 shows a set of field lines', 'We can imagine two equal and small elements of area placed at points R and S normal to the field lines there', 'The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points', 'The picture shows that the field at R is stronger than at S.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.15 Field of a point charge', 'To understand the dependence of the field lines on the area, or rather the solid anglesubtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions', 'Recall how a (plane) angle is defined in two-dimensions', 'Let a small transverse line element ∆l be placed at a distance r from a point O', 'Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r', 'Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2', 'We know that in a given solid angle the number of radial field lines is the same', 'In 6, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is∆Ω at P1 and an element of area']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['∆Ω at P2, respectively', 'The number of lines (say n) cutting these area elements are the same', 'The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively', 'Since n and ∆Ω are common, the strength of the field clearly has a 1/r2dependence', 'The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations', 'Faraday called them lines of force', 'This term is somewhat misleading, especially in case of magnetic fields', 'The more appropriate term is field lines (electric or magnetic) that we have adopted in this book', 'Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges', 'An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point', 'An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve', 'A field line is a space curve, i.e., a curve in three dimensions.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 Field lines due to some simple charge configurations.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.17 shows the field lines around some simple charge configurations', 'As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane', 'The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward', 'The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges', 'The field lines follow some important general properties: Field lines start from positive charges and end at negative charges', 'If there is a single charge, they may start or end at infinity', 'In a charge-free region, electric field lines can be taken to be continuous curves without any breaks', 'Two field lines can never cross each other', '(If they did, the field at the point of intersection will not have a unique direction, which is absurd.) (iv) Electrostatic field lines do not form any closed loops', 'This follows from the conservative nature of electric field (Chapter 2).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.10 Electric Flux Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface', 'The rate of flow of liquid is given by the volume crossing the area per unit time v dSand represents the flux of liquid flowing across the plane', 'If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ', 'Therefore, the flux going out of the surface dSis v.dS. For the case of the electric field, we define an analogous quantity and call it electric flux', 'We should, however, note that there is no flow of a physically observable quantity unlike the case of liquid flow', 'In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point', 'This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S', 'Now suppose we tilt the area element by angle θ', 'Clearly, the number of field lines crossing the area element will be smaller', 'The projection of the area element normal to E is ∆S cosθ', 'Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ', 'When θ = 90°, field lines will be parallel to ∆Sand will not cross it at all .']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.18 Dependence of flux on the inclination θ between E and n']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The orientation of area element and not merely its magnitude is important in many contexts', 'For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring', 'If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation', 'This shows that an area element should be treated as a vector', 'It has a magnitude and also a direction', 'How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane', 'Thus the direction of a planar area vector is along its normal', 'How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements', 'Each small area element may be treated as planar and a vector associated with it, as explained before', 'Notice one ambiguity here', 'The direction of an area element is along its normal', 'But a normal can point in two directions', 'Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context', 'For the case of a closed surface, this convention is very simple', 'The vector associated with every area element of a closed surface is taken to be in the direction of theoutward normal', 'This is the convention used in 9', 'Thus, the area element vector ∆S at a point on a closed surface equals ∆Sn where ∆S is the magnitude of the area element and n is a unit vector in the direction of outward normal at that point', 'We now come to the definition of electric flux', 'Electric flux ∆φ through an area element ∆S is defined by ∆φ = E.∆S = E ∆S cosθ (1.11) which, as seen before, is proportional to the number of field lines cutting the area element', 'The angle θ here is the angle between E and ∆S', 'For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element', 'Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element', 'The unit of electric flux is N C–1 m2', 'The basic definition of electric flux given by Eq', '(1.11) can be used, in principle, to calculate the total flux through any given surface', 'All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up', 'Thus, the total flux φ through a surface S is φ ~ Σ E.∆S (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element', 'This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq', '(1.12) is written as an integral.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.19 Convention for defining normal']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.11 Electric Dipole An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a', 'The line connecting the two charges defines a direction in space', 'By convention, the direction from –q to q is said to be the direction of the dipole', 'The mid-point of locations of –q and q is called the centre of the dipole', 'The total charge of the electric dipole is obviously zero', 'This does not mean that the field of the electric dipole is zero', 'Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out', 'However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out', 'The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q)', 'These qualitative ideas are borne out by the explicit calculation as follows:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.11.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle', 'The results are simple for the following two cases: when the point is on the dipole axis, and when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre', 'The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+qdue to the charge q, by the parallelogram law of vectors', 'For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in 0(a)', 'Then [1.13(a)] whereis the unit vector along the dipole axis (from –q to q)', 'Also [1.13(b)] The total field at P is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.14) For r >> a (r >> a) (1.15)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole. p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q', 'For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by [1.16(a)] [1.16(b)] and are equal', 'The directions of E+q and E–q are as shown in 0(b)', 'Clearly, the components normal to the dipole axis cancel away', 'The components along the dipole axis add up', 'The total electric field is opposite to', 'We have E = – (E +q + E –q) cosθ (1.17) At large distances (r >> a), this reduces to (1.18) From Eqs', '(1.15) and (1.18), it is clear that the dipole field at large distances does not involveq and a separately; it depends on the product qa', 'This suggests the definition of dipole moment', 'The dipole moment vector p of an electric dipole is defined by p = q × 2a(1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q', 'In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis (r >> a) (1.20)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['At a point on the equatorial plane (r >> a) (1.21) Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3', 'Further, the magnitude and the direction of the dipole field depends not only on the distance rbut also on the angle between the position vector r and the dipole moment p', 'We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite', 'Such a dipole is referred to as a point dipole', 'For a point dipole, Eqs', '(1.20) and (1.21) are exact, true for any r.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.11.2 Physical significance of dipoles']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['In most molecules, the centres of positive charges and of negative charges* lie at the same place', 'Therefore, their dipole moment is zero', 'CO2 and CH4 are of this type of molecules', 'However, they develop a dipole moment when an electric field is applied', 'But in some molecules, the centres of negative charges and of positive charges do not coincide', 'Therefore they have a permanent electric dipole moment, even in the absence of an electric field', 'Such molecules are called polar molecules', 'Water molecules, H2O, is an example of this type', 'Various materials give rise to interesting properties and important applications in the presence or absence of electric field.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.10 Two charges ±10 µC are placed 5.0 mm apart', 'Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in 1(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in 1(b).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['figure 1.21 Solution (a) Field at P due to charge +10 µC = = 4.13 × 106 N C–1 along BP Field at P due to charge –10 µC']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP', 'In this example, the ratio OP/OB is quite large (= 60)', 'Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole', 'For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude (r/a >> 1) where p = 2a q is the magnitude of the dipole moment', 'The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q)', 'Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, E == 2.6 × 105 N C–1 along the dipole moment direction AB, which is close to the result obtained earlier', '(b) Field at Q due to charge + 10 µC at B = = 3.99 × 106 N C–1 along BQ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Field at Q due to charge –10 µC at A = = 3.99 × 106 N C–1 along QA.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA', 'Therefore, the resultant electric field at Q due to the two charges at A and B is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 2 ×along BA = 1.33 × 105 N C–1 along BA', 'As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole: (r/a >> 1)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 1.33 × 105 N C–1.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The direction of electric field in this case is opposite to the direction of the dipole moment vector', 'Again, the result agrees with that obtained before.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.12 Dipole in a Uniform External Field Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in 2', '(By permanent dipole, we mean that p exists irrespective of E; it has not been induced byE.) There is a force qE on q and a force –qE on –q', 'The net force on the dipole is zero, since E is uniform', 'However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole', 'When the net force is zero, the torque (couple) is independent of the origin', 'Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.22 Dipole in a uniform electric field', 'Magnitude of torque = q E × 2 a sinθ = 2 q a E sinθ Its direction is normal to the plane of the paper, coming out of it', 'The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it', 'Thus, τ = p × E (1.22) This torque will tend to align the dipole with the field E', 'When p is aligned with E, the torque is zero', 'What happens if the field is not uniform? In that case, the net force will evidently be non-zero', 'In addition there will, in general, be a torque on the system as before', 'The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E', 'In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform', 'Figure 1.23 is self-explanatory', 'It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field', 'When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field', 'In general, the force depends on the orientation of p with respect to E.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['This brings us to a common observation in frictional electricity', 'A comb run through dry hair attracts pieces of paper', 'The comb, as we know, acquires charge through friction', 'But the paper is not charged', 'What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field', 'Further, the electric field due to the comb is not uniform', 'In this situation, it is easily seen that the paper should move in the direction of the comb!']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.13 Continuous Charge Distribution We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn', 'One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus', 'For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions', 'For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents', 'It is more feasible to consider an area element ∆S on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element', 'We then define a surface charge density σ at the area element by']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.23) We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density', 'The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*. σrepresents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically', 'The units for σ are C/m2', 'Similar considerations apply for a line charge distribution and a volume charge distribution', 'The linear charge density λ of a wire is defined by Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['(1.24) where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element', 'The units for λ are C/m', 'The volume charge density (sometimes simply called charge density) is defined in a similar manner: (1.25) where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents', 'The units for ρ are C/m3', 'The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics', 'When we refer to the density of a liquid, we are referring to its macroscopic density', 'We regard it as a continuous fluid and ignore its discrete molecular constitution.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.24 Definition of linear, surface and volume charge densities. In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq', '(1.10)', 'Suppose a continuous charge distribution in space has a charge density ρ', 'Choose any convenient origin O and let the position vector of any point in the charge distribution be r', 'The charge density ρ may vary from point to point, i.e., it is a function of r', 'Divide the charge distribution into small volume elements of size ∆V', 'The charge in a volume element ∆V is ρ∆V', 'Now, consider any general point P (inside or outside the distribution) with position vector R', 'Electric field due to the charge ρ∆V is given by Coulomb’s law: (1.26) where r′ is the distance between the charge element and P, and′ is a unit vector in the direction from the charge element to P', 'By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements: (1.27) Note that ρ, r′,all can vary from point to point', 'In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity', 'In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.14 Gauss’s Law As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre', 'Divide the sphere into small area elements, as shown in 5.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The flux through an area element ∆S is (1.28) where we have used Coulomb’s law for the electric field due to a single charge q', 'The unit vectoris along the radius vector from the centre to the area element', 'Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S andhave the same direction', 'Therefore, (1.29) since the magnitude of a unit vector is 1', 'The total flux through the sphere is obtained by adding up flux through all the different area elements:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Since each area element of the sphere is at the same distance r from the charge,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Now S, the total area of the sphere, equals 4πr2', 'Thus, (1.30)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law', 'We state Gauss’s law without proof: Electric flux through a closed surface S']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= q/ε0 (1.31) q = total charge enclosed by S', 'The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface', 'We can see that explicitly in the simple situation of 6', 'Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E', 'The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface', 'Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0', 'Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E', 'Therefore, φ1 = –E S1, φ2 = +E S2 S1 = S2 = S where S is the area of circular cross-section', 'Thus, the total flux is zero, as expected by Gauss’s law', 'Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero', 'The great significance of Gauss’s law Eq', '(1.31), is that it is true in general, and not only for the simple cases we have considered above', 'Let us note some important points regarding this law: Gauss’s law is true for any closed surface, no matter what its shape or size', 'The term q on the right side of Gauss’s law, Eq', '(1.31), includes the sum of all charges enclosed by the surface', 'The charges may be located anywhere inside the surface', 'In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq', '(1.31)] is due to all the charges, both inside and outside S', 'The term q on the right side of Gauss’s law, however, represents only the total charge inside S', '(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface', 'You may choose any Gaussian surface and apply Gauss’s law', 'However, take care not to let the Gaussian surface pass through any discrete charge', 'This is because electric field due to a system of discrete charges is not well defined at the location of any charge', '(As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution', '(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface', '(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law', 'Any violation of Gauss’s law will indicate departure from the inverse square law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.11 The electric field components in 7 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2', 'Calculate (a) the flux through the cube, and (b) the charge within the cube', 'Assume that a = 0.1 m.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.27 Solution (a) Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2', 'Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones', 'Now the magnitude of the electric field at the left face is EL = αx1/2 = αa1/2 (x = a at the left face)', 'The magnitude of electric field at the right face is ER = α x1/2 = α (2a)1/2 (x = 2a at the right face)', 'The corresponding fluxes are φL= EL.∆S ==EL ∆S cosθ = –EL ∆S, since θ = 180° = –ELa2 φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0° = ERa2 Net flux through the cube = φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2] = αa5/2 = 800 (0.1)5/2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 1.05 N m2 C–1 (b) We can use Gauss’s law to find the total charge q inside the cube', 'We have φ = q/ε0 or q =φε0', 'Therefore,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C', 'Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x', 'It is given that E= 200N/C for x > 0 and E = –200N/C for x < 0', 'A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm', '(a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder? Solution (a) We can see from the figure that on the left face E and ∆S are parallel', 'Therefore, the outward flux is φL= E.∆S = – 200 = + 200 ∆S, since= – ∆S = + 200 × π (0.05)2 = + 1.57 N m2 C–1 On the right face, E and ∆S are parallel and therefore φR = E.∆S = + 1.57 N m2 C–1', '(b) For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0', 'Therefore, the flux out of the side of the cylinder is zero', '(c) Net outward flux through the cylinder φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.28 (d) The net charge within the cylinder can be found by using Gauss’s law which gives q = ε0φ = 3.14 × 8.854 × 10–12 C']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['= 2.78 × 10–11 C']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15 Applications of Gauss’s Law The electric field due to a general charge distribution is, as seen above, given by Eq', '(1.27)', 'In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space', 'For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law', 'This is best understood by some examples.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15.1 Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ', 'The wire is obviously an axis of symmetry', 'Suppose we take the radial vector from O to P and rotate it around the wire', 'The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire', 'This implies that the electric field must have the same magnitude at these points', 'The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0)', 'This is clear from 9.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density', 'Consider a pair of line elements P1 and P2 of the wire, as shown', 'The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel)', 'This is true for any such pair and hence the total field at any point P is radial', 'Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire', 'In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r', 'To calculate the field, imagine a cylindrical Gaussian surface, as shown in the 9(b).Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero', 'At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r', 'The surface area of the curved part is 2πrl, where l is the length of the cylinder', 'Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E × 2πrl The surface includes charge equal to λ l', 'Gauss’s law then gives E × 2πrl = λl/ε0 i.e., E = Vectorially, E at any point is given by (1.32) whereis the radial unit vector in the plane normal to the wire passing through the point. E is directed outward if λ is positive and inward if λ is negative.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A, the scalar A is an algebraic number', 'It can be negative or positive', 'The direction of A will be the same as that of the unit vectorif A > 0 and opposite toif A < 0', 'When we want to restrict to non-negative values, we use the symboland call it the modulus of A', 'Thus,', 'Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire', 'Further, the assumption that the wire is infinitely long is crucial', 'Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface', 'However, Eq', '(1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15.2 Field due to a uniformly charged infinite plane sheet Let σ be the uniform surface charge density of an infinite plane sheet', 'We take thex-axis normal to the given plane', 'By symmetry, the electric field will not depend on y and zcoordinates and its direction at every point must be parallel to the x-direction', 'We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional areaA, as shown', '(A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction', 'Therefore, flux E.∆S through both the surfaces are equal and add up', 'Therefore the net flux through the Gaussian surface is 2 EA', 'The charge enclosed by the closed surface is σA', 'Therefore by Gauss’s law, 2 EA = σA/ε0 or, E = σ/2ε0 Vectorically, (1.33) where n is a unit vector normal to the plane and going away from it', 'E is directed away from the plate if σ is positive and toward the plate if σ is negative', 'Note that the above application of the Gauss’ law has brought out an additional fact: E is independent ofx also', 'For a finite large planar sheet, Eq', '(1.33) is approximately true in the middle regions of the planar sheet, away from the ends.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.15.3 Field due to a uniformly charged thin spherical shell Let σ be the uniform surface charge density of a thin spherical shell of radius R', 'The situation has obvious spherical symmetry', 'The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector)', 'Field outside the shell: Consider a point P outside the shell with radius vector r', 'To calculateE at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P', 'All points on this sphere are equivalent relative to the given charged configuration', '(That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point', 'Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S. Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2', 'The charge enclosed is σ × 4 π R2', 'By Gauss’s law E × 4 π r2 =']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Or, where q = 4 π R2 σ is the total charge on the spherical shell', 'Vectorially, (1.34)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R', 'The electric field is directed outward if q > 0 and inward if q < 0. This, however, is exactly the field produced by a charge q placed at the centre O', 'Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Field inside the shell: In 1(b), the point P is inside the shell', 'The Gaussian surface is again a sphere through P centred at O. The flux through the Gaussian surface, calculated as before, is E × 4 π r2', 'However, in this case, the Gaussian surface encloses no charge', 'Gauss’s law then gives E × 4 π r2 = 0 i.e., E = 0 (r < R ) (1.35) that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*', 'This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law', 'The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R', 'The atom as a whole is neutral', 'For this model, what is the electric field at a distance r from the nucleus?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.32 Solution The charge distribution for this model of the atom is as shown in 2', 'The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral', 'This immediately gives us the negative charge density ρ, since we must have or To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law', 'Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r', 'Its direction is along (or opposite to) the radius vector r from the origin to the point P', 'The obvious Gaussian surface is a spherical surface centred at the nucleus', 'We consider two situations, namely, r < R and r > R', 'r < R : The electric flux φ enclosed by the spherical surface is φ = E (r) × 4 π r2 where E (r) is the magnitude of the electric field at r', 'This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface', 'The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r, i.e., Substituting for the charge density ρ obtained earlier, we have Gauss’s law then gives,']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The electric field is directed radially outward', 'r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral', 'Thus, from Gauss’s law, E (r) × 4 π r2 = 0 or E (r) = 0; r > R']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['At r = R, both cases give the same result: E = 0.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['On symmetry operations In Physics, we often encounter systems with various symmetries', 'Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation', 'Consider, for example an infinite uniform sheet of charge (surface charge densityσ) along the y-z plane', 'This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle', 'As the system is unchanged under such symmetry operation, so must its properties be', 'In particular, in this example, the electric field E must be unchanged', 'Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0)', 'Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same', 'By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction', 'Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate', 'It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space', 'The direction, however, is opposite of each other on either side ofthe sheet', 'Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Summary 1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter', '2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract', 'By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative', '3. Conductors allow movement of electric charge through them, insulators do not', 'In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile', '4. Electric charge has three basic properties: quantisation, additivity and conservation', 'Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, ...', 'Proton and electron have charges +e, –e, respectively', 'For macroscopic charges for which n is a very large number, quantisation of charge can be ignored', 'Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system', 'Conservation of electric charges means that the total charge of an isolated system remains unchanged with time', 'This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge', '5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21separating them', 'Mathematically, F21 = force on q2 due to whereis a unit vector in the direction from q1 to q2 and k =is the constant of proportionality', 'In SI units, the unit of charge is coulomb', 'The experimental value of the constant ε0 is ε0 = 8.854 × 10–12 C2 N–1 m–2 The approximate value of k is k = 9 × 109 N m2 C–2']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['6. The ratio of electric force and gravitational force between a proton and an electron is']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s)', 'For an assembly of charges q1, q2, q3, ..., the force on any charge, sayq1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on', 'For each pair, the force is given by the Coulomb’s law for two charges stated earlier', '8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge', 'Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative', 'Like Coulomb force, electric field also satisfies superposition principle', '9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point', 'The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak', 'In regions of constant electric field, the field lines are uniformly spaced parallel straight lines.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['10. Some of the important properties of field lines are: Field lines are continuous curves without any breaks', 'Two field lines cannot cross each other', 'Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops', '11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a', 'Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q', '12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Dipole electric field on the axis at a distance r from the centre:']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2dependence of electric field due to a point charge', '13. In a uniform electric field E, a dipole experiences a torquegiven by = p × E but experiences no net force', '14. The flux ∆φ of electric field E through a small area element ∆S is given by ∆φ = E.∆S The vector area element ∆S is ∆S = ∆S where ∆S is the magnitude of the area element andis normal to the area element, which can be considered planar for sufficiently small ∆S', 'For an area element of a closed surface,is taken to be the direction of outward normal, by convention', '15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S', 'The law is especially useful in determining electric field E, when the source distribution has simple symmetry: Thin infinitely long straight wire of uniform linear charge density λ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['where r is the perpendicular distance of the point from the wire andis the radial unit vector in the plane normal to the wire passing through the point', 'Infinite thin plane sheet of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['whereis a unit vector normal to the plane, outward on either side', 'Thin spherical shell of uniform surface charge density σ']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['E = 0 (r < R) where r is the distance of the point from the centre of the shell and R the radius of the shell. q is the total charge of the shell: q = 4πR2σ', 'The electric field outside the shell is as though the total charge is concentrated at the centre', 'The same result is true for a solid sphere of uniform volume charge density', 'The field is zero at all points inside the shell.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Points to Ponder 1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus', 'Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together', 'The range of distance where this force is effective is, however, very small ~10-14 m', 'This is precisely the size of the nucleus', 'Also the electrons are not allowed to sit on top of the protons, i.e', 'inside the nucleus, due to the laws of quantum mechanics', 'This gives the atoms their structure as they exist in nature', '2. Coulomb force and gravitational force follow the same inverse-square law', 'But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces', 'This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature', '3. The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law', 'In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s)', 'In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2C–2', '4. The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces', 'Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects', '5. The additive property of charge is not an ‘obvious’ property', 'It is related to the fact that electric charge has no direction associated with it; charge is a scalar', '6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion', 'This is not always true for every scalar', 'For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion', '7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6', 'Conservation refers to invariance in time in a given frame of reference', 'A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision)', 'On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system)', '8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass', '9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors', 'It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges', '10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges', 'For continuous volume charge distribution, it is defined at any point in the distribution', 'For a surface charge distribution, electric field is discontinuous across the surface.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge', 'An electric dipole is the simplest example of this fact.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Exercises 1.1 What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.2 The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N', '(a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? 1.3 Check that the ratio ke2/G memp is dimensionless', 'Look up a Table of Physical Constants and determine the value of this ratio', 'What does the ratio signify? 1.4 (a) Explain the meaning of the statement ‘electric charge of a body is quantised’', '(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? 1.5 When a glass rod is rubbed with a silk cloth, charges appear on both', 'A similar phenomenon is observed with many other pairs of bodies', 'Explain how this observation is consistent with the law of conservation of charge', '1.6 Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm', 'What is the force on a charge of 1 µC placed at the centre of the square? 1.7 (a) An electrostatic field line is a continuous curve', 'That is, a field line cannot have sudden breaks', 'Why not? (b) Explain why two field lines never cross each other at any point? 1.8 Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum', '(a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge? 1.9 A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively', 'What are the total charge and electric dipole moment of the system? 1.10 An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1', 'Calculate the magnitude of the torque acting on the dipole', '1.11 A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C', '(a) Estimate the number of electrons transferred (from which to which?) (b) Is there a transfer of mass from wool to polythene? 1.12 (a) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm', 'What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation', '(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? 1.13 Suppose the spheres A and B in Exercise 1.12 have identical sizes', 'A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both', 'What is the new force of repulsion between A and B? 1.14 Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field', 'Give the signs of the three charges', 'Which particle has the highest charge to mass ratio?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.33 1.15 Consider a uniform electric field E = 3 × 103 î N/C', '(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis? 1.16 What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes? 1.17 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C', '(a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? 1.18 A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in 4', 'What is the magnitude of the electric flux through the square? (Hint:Think of the square as one face of a cube with edge 10 cm.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.34 1.19 A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge', 'What is the net electric flux through the surface? 1.20 A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge', '(a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 1.21 A conducting sphere of radius 10 cm has an unknown charge', 'If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere? 1.22 A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2', '(a) Find the charge on the sphere', '(b) What is the total electric flux leaving the surface of the sphere? 1.23 An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm', 'Calculate the linear charge density.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.24 Two large, thin metal plates are parallel and close to each other', 'On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2', 'What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Additional Exercises 1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment)', 'The density of the oil is 1.26 g cm–3', 'Estimate the radius of the drop. (g = 9.81 m s–2; e = 1.60 × 10–19 C)', '1.26 Which among the curves shown in 5 cannot possibly represent electrostatic field lines?']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.35 1.27 In a certain region of space, electric field is along the z-direction throughout', 'The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre', 'What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ? 1.28 (a) A conductor A with a cavity as shown in 6(a) is given a charge Q', 'Show that the entire charge must appear on the outer surface of the conductor', '(b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A', 'Show that the total charge on the outside surface of A is Q + q', '(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment', 'Suggest a possible way.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Figure 1.36 1.29 A hollow charged conductor has a tiny hole cut into its surface', 'Show that the electric field in the hole is (σ/2ε0), whereis the unit vector in the outward normal direction, and σ is the surface charge density near the hole', '1.30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law', '[Hint: Use Coulomb’s law directly and evaluate the necessary integral.] 1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks', 'A proton and a neutron consist of three quarks each', 'Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter', '(Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron', '1.32 (a) Consider an arbitrary electrostatic field configuration', 'A small test charge is placed at a null point (i.e., where E = 0) of the configuration', 'Show that the equilibrium of the test charge is necessarily unstable', '(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart', '1.33 A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in 3)', 'The length of plate is Land an uniform electric field E is maintained between the plates', 'Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2)', 'Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics.']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1', 'If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)']}, {'title': 'Figure 1.7  Solution Let the original charge on sphere A be q and that on B be q′. At a distance r between their centres, the magnitude of the electrostatic force on each is given by   neglecting the sizes of spheres A and B in comparison to r. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is', 'text': ['Chapter 1 ELECTRIC CHARGES AND FIELDS']}]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(textDict)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "all_sentences = []\n",
+    "\n",
+    "for i in textDict:\n",
+    "    for j in i['text']:\n",
+    "        all_sentences.append(j)\n",
+    "with open('sentences-test.txt', 'a') as f:\n",
+    "    f.write('\\n'.join(all_sentences))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "for j in textDict:\n",
+    "    inputTextArray = j['text']\n",
+    "    inputTextTitle = j['title']\n",
+    "#     print(j)\n",
+    "    nFeatures = 10\n",
+    "\n",
+    "    data = [ [] for i in range(len(inputTextArray))]\n",
+    "\n",
+    "#     similarityFeatures = ahish.getCosSims(inputTextArray)\n",
+    "    similarityFeatures = [-1 for i in range(len(inputTextArray))]\n",
+    "#     simToSum = SimToSumm.getSimtoSum(inputTextArray, ' '.join(inputTextArray))\n",
+    "    simToTitle = SimToTitle.SimToTitle(inputTextTitle, inputTextArray)\n",
+    "\n",
+    "    maxLen = 0\n",
+    "\n",
+    "    for i in inputTextArray:\n",
+    "        curr = len(i.split())\n",
+    "        if curr > maxLen:\n",
+    "            maxLen = curr\n",
+    "\n",
+    "    for i in range(len(inputTextArray)):\n",
+    "        data[i] += [len(inputTextArray[i].split())/maxLen, i/len(inputTextArray)]\n",
+    "#         data[i] += [similarityFeatures[i]]\n",
+    "#         data[i] += [simToSum[i]]\n",
+    "        data[i] += [simToTitle[i]]\n",
+    "        curr = inputTextArray[i]\n",
+    "        data[i]+= [pranav.keyword_present(curr) , pranav.has_number(curr), pranav.number_of_superlatives(curr), pranav.abbrevs_per_length(curr), pranav.nouns_per_length(curr), pranav.pronouns_per_length(curr), discourse_connector.discourse_connector(curr)]\n",
+    "                \n",
+    "    with open(\"features-test.csv\", \"a\", newline=\"\") as f:\n",
+    "        writer = csv.writer(f)\n",
+    "        writer.writerows(data)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/SimToSumm.py b/SimToSumm.py
index 0931f51..dac0a74 100644
--- a/SimToSumm.py
+++ b/SimToSumm.py
@@ -109,29 +109,10 @@ def _generate_summary(sentences, sentenceValue, threshold):
 
 
 
-<<<<<<< HEAD
 def getSimtoSum(p, paragraph):
     l=_create_frequency_table(paragraph)
     # i = sent_tokenize(paragraph)
     scoredict = _score_sentences(p,l)
-=======
-paragraph="""A group of cells that are similar in structure and/or work together to achieve a particular function forms a tissue.
-6.1	Are Plants and Animals Made of Same Types of Tissues?
-Let us compare their structure and functions. Do plants and animals have the same structure? Do they both perform similar functions?
-There are noticeable differences between the two. Plants are stationary or fixed – they don’t move. Since they have to be upright, they have a large quantity of supportive tissue. The supportive tissue generally has dead cells.
-Animals on the other hand move around in search of food, mates and shelter. They consume more energy as compared to plants. Most of the tissues they contain are living.
-Another difference between animals and plants is in the pattern of growth. The growth in plants is limited to certain regions, while this is not so in animals. There are some tissues in plants that divide throughout their life. These tissues are localised in certain regions. Based on the dividing capacity of the tissues, various plant tissues can be classified as growing or meristematic tissue and permanent tissue. Cell growth in animals is more uniform. So, there is no such demarcation of dividing and non-dividing regions in animals.
-The structural organisation of organs and organ systems is far more specialised and localised in complex animals than even in very complex plants. This fundamental difference reflects the different modes of life pursued by these two major groups of organisms, particularly in their different feeding methods. Also, they are differently adapted for a sedentary existence on one hand (plants) and active locomotion on the other (animals), contributing to this difference in organ system design.
-It is with reference to these complex animal and plant bodies that we will now talk about the concept of tissues in some detail.
-"""
-
-
-def getSimtoSum(paragraph):
-    l=_create_frequency_table(paragraph)
-    i=sent_tokenize(paragraph)
-    print(type(i))
-    scoredict = _score_sentences(i,l)
->>>>>>> 8280de6fdf2ac3253e75acec876d83e15ecfcbe7
     threshold = _find_average_score(scoredict)
     summary = _generate_summary(p, scoredict, 1.5 * threshold)
     #print(scoredict.values()/len(scoredict.keys().split()))
diff --git a/__pycache__/textprocessing.cpython-36.pyc b/__pycache__/textprocessing.cpython-36.pyc
index d306f2a..8f1ec5a 100644
Binary files a/__pycache__/textprocessing.cpython-36.pyc and b/__pycache__/textprocessing.cpython-36.pyc differ
diff --git a/features-test.csv b/features-test.csv
index 719b199..73c8e10 100644
--- a/features-test.csv
+++ b/features-test.csv
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diff --git a/features.csv b/features.csv
index 9eac86b..719b199 100644
--- a/features.csv
+++ b/features.csv
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diff --git a/final.json b/final.json
index da62052..e36ad27 100644
--- a/final.json
+++ b/final.json
@@ -1,295 +1,408 @@
-{"text": "Classification of Elements and Periodicity in Properties ", "fib": "Classification"}
-{"text": "After studying this Unit, you will be able to \u2022 appreciate how the concept of grouping elements in accordance to their properties led to the development of Periodic Table ", "fib": "Periodic"}
-{"text": "It is a remarkable demonstration of the fact that the chemical elements are not a random cluster of entities but instead display trends and lie together in families ", "fib": "chemical"}
-{"text": "Seaborg In this Unit, we will study the historical development of the Periodic Table as it stands today and the Modern Periodic Law ", "fib": "Law"}
-{"text": "We will also learn how the periodic classification follows as a logical consequence of the electronic configuration of atoms ", "fib": "atoms"}
-{"text": "Finally, we shall examine some of the periodic trends in the physical and chemical properties of the elements ", "fib": "chemical"}
-{"text": "3.1 WHY DO WE NEED TO CLASSIFY ELEMENTS ? We know by now that the elements are the basic units of all types of matter ", "fib": "3.1"}
-{"text": "In 1800, only 31 elements were known ", "fib": "1800"}
-{"text": "By 1865, the number of identified elements had more than doubled to 63 ", "fib": "1865"}
-{"text": "At present 114 elements are known ", "fib": "114"}
-{"text": "Not only that it would rationalize known chemical facts about elements, but even predict new ones for undertaking further study. 3.2 GENESIS OF PERIODIC CLASSIFICATION Classification of elements into groups and development of Periodic Law and Periodic Table are the consequences of systematising the knowledge gained by a number of scientists through their observations and experiments ", "fib": ""}
-{"text": "In each case, he noticed that the middle element of each of the Triads had an atomic weight about half way between the atomic weights of the other two (Table 3.1) ", "fib": "3.1"}
+{"text": "\u2022 understand the Periodic Law; \u2022 understand the significance of atomic number and electronic configuration as the basis for periodic classification; \u2022 name the elements with Z >100 according to IUPAC nomenclature; \u2022 classify elements into s, p, d, f blocks and learn their main characteristics; \u2022 recognise the periodic trends in physical and chemical properties of elements; \u2022 compare the reactivity of elements and correlate it with their occurrence in nature; \u2022 explain the relationship between ionization enthalpy and metallic character; \u2022 use scientific vocabulary appropriately to communicate ideas related to certain important properties of atoms e.g., atomic/ ionic radii, ionization enthalpy, electron gain enthalpy, electronegativity, valence of elements. ", "fib": "100"}
+{"text": "The Periodic Table is arguably the most important concept in chemistry, both in principle and in practice ", "fib": "Periodic"}
+{"text": "It is the everyday support for students, it suggests new avenues of research to professionals, and it provides a succinct organization of the whole of chemistry ", "fib": "chemistry"}
+{"text": "Glenn T ", "fib": "Glenn"}
+{"text": "Efforts to synthesise new elements are continuing ", "fib": "Efforts"}
+{"text": "With such a large number of elements it is very difficult to study individually the chemistry of all these elements and their innumerable compounds individually ", "fib": "chemistry"}
 {"text": "Also the properties of the middle element were in between those of the other two members ", "fib": "members"}
-{"text": "Since Dobereiner\u2019s relationship, referred to as the Law of Triads, seemed to work only for a few elements, it was dismissed as coincidence ", "fib": "Dobereiner"}
-{"text": "The next reported attempt to classify elements was made by a French geologist, A.E.B ", "fib": "A.E.B"}
-{"text": "de Chancourtois in 1862 ", "fib": "1862"}
-{"text": "He arranged the then known elements in order of increasing atomic weights and made a cylindrical table of elements to display the periodic recurrence of properties ", "fib": "atomic"}
 {"text": "This also did not attract much attention ", "fib": "much"}
-{"text": "The relationship was just like every eighth note that resembles the first in octaves of music ", "fib": "eighth"}
+{"text": "The English chemist, John Alexander Newlands in 1865 profounded the Law of Octaves ", "fib": "1865"}
 {"text": "Newlands\u2019s Law of Octaves seemed to be true only for elements up to calcium ", "fib": "Newlands"}
 {"text": "Although his idea was not widely accepted at that time, he, for his work, was later awarded Davy Medal in 1887 by the Royal Society, London ", "fib": "1887"}
 {"text": "Table 3.1 Dobereiner\u2019s Triads ", "fib": "3.1"}
+{"text": "The Periodic Law, as we know it today owes its development to the Russian chemist, Dmitri Mendeleev (1834-1907) and the German chemist, Lothar Meyer (1830-1895) ", "fib": "1830-1895"}
+{"text": "Working independently, both the chemists in 1869 proposed that on arranging elements in the increasing order of their atomic weights, similarities appear in physical and chemical properties at regular intervals ", "fib": "1869"}
 {"text": "Lothar Meyer plotted the physical properties such as atomic volume, melting point and boiling point against atomic weight and obtained a periodically repeated pattern ", "fib": "Lothar"}
 {"text": "Unlike Newlands, Lothar Meyer observed a change in length of that repeating pattern ", "fib": "Lothar"}
+{"text": "By 1868, Lothar Meyer had developed a table of the elements that closely resembles the Modern Periodic Table ", "fib": "1868"}
+{"text": "However, his work was not published until after the work of Dmitri Mendeleev, the scientist who is generally credited with the development of the Modern Periodic Table. ", "fib": "Dmitri"}
+{"text": "Table 3.2 Newlands\u2019 Octaves ", "fib": "3.2"}
+{"text": "While Dobereiner initiated the study of periodic relationship, it was Mendeleev who was responsible for publishing the Periodic Law for the first time ", "fib": "Dobereiner"}
 {"text": "It states as follows : The properties of the elements are a periodic function of their atomic weights ", "fib": "atomic"}
+{"text": "Mendeleev arranged elements in horizontal rows and vertical columns of a table in order of their increasing atomic weights in such a way that the elements with similar properties occupied the same vertical column or group ", "fib": "Mendeleev"}
 {"text": "Mendeleev\u2019s system of classifying elements was more elaborate than that of Lothar Meyer\u2019s ", "fib": "Lothar"}
 {"text": "He fully recognized the significance of periodicity and used broader range of physical and chemical properties to classify the elements ", "fib": "broader"}
 {"text": "In particular, Mendeleev relied on the similarities in the empirical formulas and properties of the compounds formed by the elements ", "fib": "Mendeleev"}
 {"text": "He realized that some of the elements did not fit in with his scheme of classification if the order of atomic weight was strictly followed ", "fib": "atomic"}
-{"text": "He ignored the order of atomic weights, thinking that the atomic measurements might be incorrect, and placed the elements with similar properties together ", "fib": "atomic"}
 {"text": "For example, iodine with lower atomic weight than that of tellurium (Group VI) was placed in Group VII along with fluorine, chlorine, bromine because of similarities in properties ", "fib": "Group"}
-{"text": "For example, both gallium and germanium were unknown at the time Mendeleev published his Periodic Table ", "fib": "Mendeleev"}
+{"text": "He left the gap under aluminium and a gap under silicon, and called these elements Eka-Aluminium and Eka-Silicon ", "fib": "Eka-Aluminium"}
+{"text": "Mendeleev predicted not only the existence of gallium and germanium, but also described some of their general physical properties ", "fib": "Mendeleev"}
+{"text": "These elements were discovered later ", "fib": "elements"}
 {"text": "Some of the properties predicted by Mendeleev for these elements and those found experimentally are listed in Table 3.3. ", "fib": "3.3"}
-{"text": "Table 3.3 Mendeleev\u2019s Predictions for the Elements Eka-aluminium (Gallium) and Eka-silicon (Germanium) ", "fib": "3.3"}
-{"text": "Mendeleev\u2019s Periodic Table published in 1905 is shown in . ", "fib": "1905"}
+{"text": "The boldness of Mendeleev\u2019s quantitative predictions and their eventual success made him and his Periodic Table famous ", "fib": "Mendeleev"}
 {"text": "PERIODIC SYSTEM OF THE ELEMENTS IN GROUPS AND SERIES ", "fib": ""}
 {"text": "A plot of (whereis frequency of X-rays emitted) against atomic number (Z ) gave a straight line and not the plot of vs atomic mass ", "fib": "X-rays"}
+{"text": "He thereby showed that the atomic number is a more fundamental property of an element than its atomic mass ", "fib": "atomic"}
 {"text": "Mendeleev\u2019s Periodic Law was, therefore, accordingly modified ", "fib": "Law"}
-{"text": "This is known as the Modern Periodic Law and can be stated as : The physical and chemical properties of the elements are periodic functions of their atomic numbers ", "fib": "Law"}
 {"text": "The Periodic Law revealed important analogies among the 94 naturally occurring elements (neptunium and plutonium like actinium and protoactinium are also found in pitch blende \u2013 an ore of uranium) ", "fib": "94"}
-{"text": "It stimulated renewed interest in Inorganic Chemistry and has carried into the present with the creation of artificially produced short-lived elements ", "fib": "Chemistry"}
-{"text": "You may recall that the atomic number is equal to the nuclear charge (i.e., number of protons) or the number of electrons in a neutral atom ", "fib": "atomic"}
-{"text": "It is then easy to visualize the significance of quantum numbers and electronic configurations in periodicity of elements ", "fib": "electronic"}
-{"text": "A modern version, the so-called \u201clong form\u201d of the Periodic Table of the elements , is the most convenient and widely used ", "fib": "Periodic"}
-{"text": "The period number corresponds to the highest principal quantum number (n) of the elements in the period ", "fib": "corresponds"}
-{"text": "The seventh period is incomplete and like the sixth period would have a theoretical maximum (on the basis of quantum numbers) of 32 elements ", "fib": "32"}
+{"text": "Numerous forms of Periodic Table have been devised from time to time ", "fib": "Numerous"}
+{"text": "According to the recommendation of International Union of Pure and Applied Chemistry (IUPAC), the groups are numbered from 1 to 18 replacing the older notation of groups IA \u2026 VIIA, VIII, IB \u2026 VIIB and 0 ", "fib": "0"}
 {"text": "Their synthesis and characterisation, therefore, require highly sophisticated costly equipment and laboratory ", "fib": "characterisation"}
-{"text": "Such work is carried out with competitive spirit only in some laboratories in the world ", "fib": "Such"}
-{"text": "Scientists, before collecting the reliable data on the new element, at times get tempted to claim for its discovery ", "fib": "Scientists"}
-{"text": "For example, both American and Soviet scientists claimed credit for discovering element 104 ", "fib": "104"}
-{"text": "The Americans named it Rutherfordium whereas Soviets named it Kurchatovium ", "fib": "Americans"}
-{"text": "To avoid such problems, the IUPAC has made recommendation that until a new element\u2019s discovery is proved, and its name is officially recognised, a systematic nomenclature be derived directly from the atomic number of the element using the numerical roots for 0 and numbers 1-9 ", "fib": "0"}
-{"text": "These are shown in Table 3.4 ", "fib": "3.4"}
-{"text": "The roots are put together in order of digits which make up the atomic number and \u201cium\u201d is added at the end ", "fib": "atomic"}
-{"text": "The IUPAC names for elements with Z above 100 are shown in Table 3.5. ", "fib": "100"}
-{"text": "* Glenn T ", "fib": "*"}
-{"text": "In 1951, Seaborg was awarded the Nobel Prize in chemistry for his work ", "fib": ""}
 {"text": "Element 106 has been named Seaborgium (Sg) in his honour. ", "fib": "106"}
+{"text": "Table 3.4 Notation for IUPAC Nomenclature of Elements ", "fib": "3.4"}
 {"text": "Table 3.5 Nomenclature of Elements with Atomic Number Above 100 ", "fib": ""}
+{"text": "Thus, the new element first gets a temporary name, with symbol consisting of three letters ", "fib": ""}
+{"text": "The permanent name might reflect the country (or state of the country) in which the element was discovered, or pay tribute to a notable scientist ", "fib": "country"}
+{"text": "As of now, elements with atomic numbers up to 118 have been discovered ", "fib": "118"}
 {"text": "Problem 3.1 What would be the IUPAC name and symbol for the element with atomic number 120? Solution From Table 3.4, the roots for 1, 2 and 0 are un, bi and nil, respectively ", "fib": "0"}
-{"text": "3.5 ELECTRONIC CONFIGURATIONS OF ELEMENTS AND THE PERIODIC TABLE In the preceding unit we have learnt that an electron in an atom is characterised by a set of four quantum numbers, and the principal quantum number (n ) defines the main energy level known as shell ", "fib": ""}
 {"text": "We have also studied about the filling of electrons into different subshells, also referred to as orbitals (s, p, d, f) in an atom ", "fib": ""}
-{"text": "In this section we will observe a direct connection between the electronic configurations of the elements and the long form of the Periodic Table ", "fib": "Periodic"}
+{"text": "The distribution of electrons into orbitals of an atom is called its electronic configuration ", "fib": "atom"}
+{"text": "An element\u2019s location in the Periodic Table reflects the quantum numbers of the last orbital filled ", "fib": "Periodic"}
 {"text": "(a) Electronic Configurations in Periods The period indicates the value of n for the outermost or valence shell ", "fib": "Electronic"}
-{"text": "In other words, successive period in the Periodic Table is associated with the filling of the next higher principal energy level (n = 1, n = 2, etc.) ", "fib": ""}
-{"text": "It can be readily seen that the number of elements in each period is twice the number of atomic orbitals available in the energy level that is being filled ", "fib": "atomic"}
 {"text": "The first period (n = 1) starts with the filling of the lowest level (1s) and therefore has two elements \u2014 hydrogen (ls1) and helium (ls2) when the first shell (K) is completed ", "fib": ""}
+{"text": "The next element, beryllium has four electrons and has the electronic configuration 1s22s2 ", "fib": "1s22s2"}
 {"text": "Starting from the next element boron, the 2p orbitals are filled with electrons when the L shell is completed at neon (2s22p6) ", "fib": "2p"}
 {"text": "Thus there are 8 elements in the second period ", "fib": "8"}
-{"text": "The fourth period ends at krypton with the filling up of the 4p orbitals ", "fib": "4p"}
-{"text": "Altogether we have 18 elements in this fourth period ", "fib": "18"}
+{"text": "The third period (n = 3) begins at sodium, and the added electron enters a 3s orbital ", "fib": "3"}
+{"text": "The fourth period (n = 4) starts at potassium, and the added electrons fill up the 4s orbital ", "fib": "4"}
+{"text": "Now you may note that before the 4p orbital is filled, filling up of 3d orbitals becomes energetically favourable and we come across the so called 3d transition series of elements ", "fib": "3d"}
+{"text": "This starts from scandium (Z = 21) which has the electronic configuration 3d14s2 ", "fib": "21"}
+{"text": "The 3d orbitals are filled at zinc (Z=30) with electronic configuration 3d104s2 ", "fib": "3d"}
 {"text": "The fifth period (n = 5) beginning with rubidium is similar to the fourth period and contains the 4d transition series starting at yttrium (Z = 39) ", "fib": "39"}
 {"text": "This period ends at xenon with the filling up of the 5p orbitals ", "fib": "5p"}
-{"text": "The sixth period (n = 6) contains 32 elements and successive electrons enter 6s, 4f, 5d and 6p orbitals, in the order \u2014 filling up of the 4f orbitals begins with cerium (Z = 58) and ends at lutetium (Z = 71) to give the 4f-inner transition series which is called the lanthanoid series ", "fib": "32"}
-{"text": "The seventh period (n = 7) is similar to the sixth period with the successive filling up of the 7s, 5f, 6d and 7p orbitals and includes most of the man-made radioactive elements ", "fib": "5f"}
-{"text": "Problem 3.2 How would you justify the presence of 18 elements in the 5th period of the Periodic Table? Solution When n = 5, l = 0, 1, 2, 3 ", "fib": ""}
-{"text": "The total number of orbitals available are 9 ", "fib": "9"}
+{"text": "The order in which the energy of the available orbitals 4d, 5s and 5p increases is 5s < 4d < 5p ", "fib": "4d"}
 {"text": "The maximum number of electrons that can be accommodated is 18; and therefore 18 elements are there in the 5th period. ", "fib": "18"}
-{"text": "3.6 ELECTRONIC CONFIGURATIONS AND TYPES OF ELEMENTS: s-, p-, d-, f- BLOCKS The aufbau (build up) principle and the electronic configuration of atoms provide a theoretical foundation for the periodic classification ", "fib": ""}
-{"text": "The elements in a vertical column of the Periodic Table constitute a group or family and exhibit similar chemical behaviour ", "fib": "Periodic"}
-{"text": "This similarity arises because these elements have the same number and same distribution of electrons in their outermost orbitals ", "fib": "electrons"}
-{"text": "We notice two exceptions to this categorisation ", "fib": "categorisation"}
+{"text": "For example, the Group 1 elements (alkali metals) all have ns1 valence shell electronic configuration as shown below ", "fib": "1"}
+{"text": "We can classify the elements into four blocks viz., s-block, p-block, d-block and f-block depending on the type of atomic orbitals that are being filled with electrons ", "fib": "atomic"}
 {"text": "The other exception is hydrogen ", "fib": "hydrogen"}
 {"text": "3.6.1 The s-Block Elements ", "fib": "3.6.1"}
-{"text": "The elements of Group 1 (alkali metals) and Group 2 (alkaline earth metals) which have ns1 and ns2 outermost electronic configuration belong to the s-Block Elements ", "fib": "1"}
-{"text": "They are all reactive metals with low ionization enthalpies ", "fib": "enthalpies"}
-{"text": "They lose the outermost electron(s) readily to form 1+ ion (in the case of alkali metals) or 2+ ion (in the case of alkaline earth metals) ", "fib": "1+"}
-{"text": "The metallic character and the reactivity increase as we go down the group ", "fib": "group"}
-{"text": "The compounds of the s-block elements, with the exception of those of lithium and beryllium are predominantly ionic ", "fib": "beryllium"}
-{"text": "They are all metals ", "fib": "metals"}
+{"text": "3.6.3 The d-Block Elements (Transition Elements) These are the elements of Group 3 to 12 in the centre of the Periodic Table ", "fib": "12"}
 {"text": "They mostly form coloured ions, exhibit variable valence (oxidation states), paramagnetism and oftenly used as catalysts ", "fib": "catalysts"}
-{"text": "However, Zn, Cd and Hg which have the electronic configuration, (n-1) d10ns2 do not show most of the properties of transition elements ", "fib": "Cd"}
-{"text": "In a way, transition metals form a bridge between the chemically active metals of s-block elements and the less active elements of Groups 13 and 14 and thus take their familiar name \u201cTransition Elements\u201d ", "fib": "13"}
-{"text": "Problem 3.4 Considering the atomic number and position in the periodic table, arrange the following elements in the increasing order of metallic character : Si, Be, Mg, Na, P ", "fib": ""}
-{"text": "Hence the order of increasing metallic character is: P < Si < Be < Mg < Na. ", "fib": ""}
-{"text": "3.7 PERIODIC TRENDS IN PROPERTIES OF ELEMENTS There are many observable patterns in the physical and chemical properties of elements as we descend in a group or move across a period in the Periodic Table ", "fib": ""}
-{"text": "For example, within a period, chemical reactivity tends to be high in Group 1 metals, lower in elements towards the middle of the table, and increases to a maximum in the Group 17 non-metals ", "fib": "1"}
-{"text": "In this section we shall discuss the periodic trends in certain physical and chemical properties and try to explain them in terms of number of electrons and energy levels ", "fib": "chemical"}
-{"text": "3.7.1 Trends in Physical Properties There are numerous physical properties of elements such as melting and boiling points, heats of fusion and vaporization, energy of atomization, etc ", "fib": "3.7.1"}
-{"text": "which show periodic variations ", "fib": "periodic"}
-{"text": "In other words, there is no practical way by which the size of an individual atom can be measured ", "fib": "individual"}
-{"text": "However, an estimate of the atomic size can be made by knowing the distance between the atoms in the combined state ", "fib": "atomic"}
+{"text": "The chemistry of the early actinoids is more complicated than the corresponding lanthanoids, due to the large number of oxidation states possible for these actinoid elements ", "fib": "actinoid"}
+{"text": "3.6.5 Metals, Non-metals and Metalloids In addition to displaying the classification of elements into s-, p-, d-, and f-blocks, shows another broad classification of elements based on their properties ", "fib": "3.6.5"}
+{"text": "However, we shall discuss the periodic trends with respect to atomic and ionic radii, ionization enthalpy, electron gain enthalpy and electronegativity ", "fib": "electronegativity"}
+{"text": "(a) Atomic Radius You can very well imagine that finding the size of an atom is a lot more complicated than measuring the radius of a ball ", "fib": "Atomic"}
+{"text": "Secondly, since the electron cloud surrounding the atom does not have a sharp boundary, the determination of the atomic size cannot be precise ", "fib": "atom"}
+{"text": "We can explain these trends in terms of nuclear charge and energy level ", "fib": "energy"}
+{"text": "The atomic size generally decreases across a period as illustrated in (a) for the elements of the second period ", "fib": "atomic"}
 {"text": "Within a family or vertical column of the periodic table, the atomic radius increases regularly with atomic number as illustrated in (b) ", "fib": "atomic"}
-{"text": "The size of an anion will be larger than that of the parent atom because the addition of one or more electrons would result in increased repulsion among the electrons and a decrease in effective nuclear charge ", "fib": "addition"}
-{"text": "For example, the ionic radius of fluoride ion (F\u2013 ) is 136 pm whereas the atomic radius of fluorine is only 64 pm ", "fib": "136"}
-{"text": "* Two or more species with same number of atoms, same number of valence electrons and same structure, regardless of the nature of elements involved. ", "fib": "*"}
+{"text": "In fact radii of noble gases should be compared not with the covalent radii but with the van der Waals radii of other elements ", "fib": "Waals"}
+{"text": "In this case, the net repulsion of the electrons will outweigh the nuclear charge and the ion will expand in size. ", "fib": "case"}
+{"text": "Hence the largest species is Mg; the smallest one is Al3+. ", "fib": "Al3+"}
+{"text": "(c) Ionization Enthalpy A quantitative measure of the tendency of an element to lose electron is given by its Ionization Enthalpy ", "fib": ""}
+{"text": "It represents the energy required to remove an electron from an isolated gaseous atom (X) in its ground state ", "fib": "X"}
+{"text": "X(g) \u2192 X+(g) + e\u2013 (3.1) The ionization enthalpy is expressed in units of kJ mol\u20131 ", "fib": "+"}
+{"text": "X+(g) \u2192 X2+(g) + e\u2013 (3.2) Energy is always required to remove electrons from an atom and hence ionization enthalpies are always positive ", "fib": "+"}
 {"text": "The first ionization enthalpies of elements having atomic numbers up to 60 are plotted in ", "fib": "60"}
-{"text": "This situation occurs in the case of alkali metals which have single outermost ns-electron preceded by a noble gas electronic configuration. ", "fib": "alkali"}
-{"text": "In this case, increase in shielding outweighs the increasing nuclear charge and the removal of the outermost electron requires less energy down a group ", "fib": "case"}
-{"text": "From (a), you will also notice that the first ionization enthalpy of boron (Z = 5) is slightly less than that of beryllium (Z = 4) even though the former has a greater nuclear charge ", "fib": "4"}
+{"text": "The periodicity of the graph is quite striking ", "fib": "graph"}
+{"text": "In addition, you will notice two trends the first ionization enthalpy generally increases as we go across a period and decreases as we descend in a group ", "fib": "addition"}
+{"text": "The effective nuclear charge experienced by a valence electron in an atom will be less than the actual charge on the nucleus because of \u201cshielding\u201d or \u201cscreening\u201d of the valence electron from the nucleus by the intervening core electrons ", "fib": ""}
+{"text": "For example, the 2s electron in lithium is shielded from the nucleus by the inner core of 1s electrons ", "fib": "1s"}
 {"text": "When we consider the same principal quantum level, an s-electron is attracted to the nucleus more than a p-electron ", "fib": "level"}
 {"text": "In beryllium, the electron removed during the ionization is an s-electron whereas the electron removed during ionization of boron is a p-electron ", "fib": "beryllium"}
-{"text": "Predict whether the first \u2206i H value for Al will be more close to 575 or 760 kJ mol\u20131 ? Justify your answer ", "fib": "575"}
-{"text": "For many elements energy is released when an electron is added to the atom and the electron gain enthalpy is negative ", "fib": "atom"}
-{"text": "We should also expect electron gain enthalpy to become less negative as we go down a group because the size of the atom increases and the added electron would be farther from the nucleus ", "fib": "atom"}
-{"text": "This is generally the case (Table 3.7) ", "fib": "3.7"}
-{"text": "However, electron gain enthalpy of O or F is less negative than that of the succeeding element ", "fib": ""}
-{"text": "This is because when an electron is added to O or F, the added electron goes to the smaller n = 2 quantum level and suffers significant repulsion from the other electrons present in this level ", "fib": "2"}
-{"text": "For the n = 3 quantum level (S or Cl), the added electron occupies a larger region of space and the electron-electron repulsion is much less. ", "fib": "S"}
-{"text": "All these periodic trends are summarised in figure 3.7. ", "fib": "3.7"}
-{"text": "(a) Periodicity of Valence or Oxidation States The valence is the most characteristic property of the elements and can be understood in terms of their electronic configurations ", "fib": "Oxidation"}
-{"text": "The valence of representative elements is usually (though not necessarily) equal to the number of electrons in the outermost orbitals and / or equal to eight minus the number of outermost electrons as shown below ", "fib": "/"}
-{"text": "Solution (a) Silicon is group 14 element with a valence of 4; bromine belongs to the halogen family with a valence of 1 ", "fib": "1"}
-{"text": "(b) Aluminium belongs to group 13 with a valence of 3; sulphur belongs to group 16 elements with a valence of 2 ", "fib": "13"}
-{"text": "Hence, the formula of the compound formed would be Al2S3. ", "fib": "Al2S3"}
-{"text": "Some periodic trends observed in the valence of elements (hydrides and oxides) are shown in Table 3.9 ", "fib": "3.9"}
-{"text": "(b) Anomalous Properties of Second Period Elements The first element of each of the groups 1 (lithium) and 2 (beryllium) and groups 13-17 (boron to fluorine) differs in many respects from the other members of their respective group ", "fib": "1"}
+{"text": "Therefore, it is easier to remove the 2p-electron from boron compared to the removal of a 2s- electron from beryllium ", "fib": "2p-electron"}
+{"text": "Thus, boron has a smaller first ionization enthalpy than beryllium ", "fib": "beryllium"}
+{"text": "Problem 3.7 Which of the following will have the most negative electron gain enthalpy and which the least negative? P, S, Cl, F ", "fib": "S"}
+{"text": "Explain your answer ", "fib": "answer"}
+{"text": "Solution Electron gain enthalpy generally becomes more negative across a period as we move from left to right ", "fib": "Electron"}
+{"text": "However, adding an electron to the 2p-orbital leads to greater repulsion than adding an electron to the larger 3p-orbital ", "fib": "2p-orbital"}
+{"text": "Hence the element with most negative electron gain enthalpy is chlorine; the one with the least negative electron gain enthalpy is phosphorus. ", "fib": "element"}
+{"text": "(e) Electronegativity A qualitative measure of the ability of an atom in a chemical compound to attract shared electrons to itself is called electronegativity ", "fib": ""}
+{"text": "Unlike ionization enthalpy and electron gain enthalpy, it is not a measureable quantity ", "fib": "electron"}
+{"text": "The one which is the most widely used is the Pauling scale ", "fib": "one"}
+{"text": "If energy is released when an electron is added to an atom, the electron affinity is taken as positive, contrary to thermodynamic convention ", "fib": "affinity"}
+{"text": "If energy has to be supplied to add an electron to an atom, then the electron affinity of the atom is assigned a negative sign ", "fib": "affinity"}
+{"text": "However, electron affinity is defined as absolute zero and, therefore at any other temperature (T) heat capacities of the reactants and the products have to be taken into account in \u2206egH = \u2013Ae \u2013 5/2 RT. ", "fib": "5/2"}
+{"text": "The trend is similar to that of ionization enthalpy. ", "fib": "enthalpy"}
+{"text": "Table 3.8(a) Electronegativity Values (on Pauling scale) Across the Periods ", "fib": ""}
+{"text": "Table 3.8(b) Electronegativity Values (on Pauling scale) Down a Family 3.7.2 Periodic Trends in Chemical Properties Most of the trends in chemical properties of elements, such as diagonal relationships, inert pair effect, effects of lanthanoid contraction etc ", "fib": ""}
+{"text": "Nowadays the term oxidation state is frequently used for valence ", "fib": "oxidation"}
+{"text": "Problem 3.8 Using the Periodic Table, predict the formulas of compounds which might be formed by the following pairs of elements; (a) silicon and bromine (b) aluminium and sulphur ", "fib": "3.8"}
 {"text": "For example, lithium unlike other alkali metals, and beryllium unlike other alkaline earth metals, form compounds with pronounced covalent character; the other members of these groups predominantly form ionic compounds ", "fib": "beryllium"}
-{"text": "Problem 3.9 Are the oxidation state and covalency of Al in [AlCl(H2O)5]2+ same ? Solution No ", "fib": ""}
-{"text": "Oxides of elements in the centre are amphoteric (e.g., Al2O3, As2O3) or neutral (e.g., CO, NO, N2O) ", "fib": ""}
-{"text": "Solution Na2O with water forms a strong base whereas Cl2O7 forms strong acid ", "fib": "Cl2O7"}
-{"text": "Na2O + H2O \u2192 2NaOH Cl2O7 + H2O \u2192 2HClO4 Their basic or acidic nature can be qualitatively tested with litmus paper. ", "fib": ""}
-{"text": "Thus, the metallic character increases down the group and non-metallic character decreases ", "fib": "decreases"}
-{"text": "Non-metals, which are located at the top of the periodic table, are less than twenty in number ", "fib": "Non-metals"}
-{"text": "The atomic radii decrease while going from left to right in a period and increase with atomic number in a group ", "fib": "atomic"}
+{"text": "In fact the behaviour of lithium and beryllium is more similar with the second element of the following group i.e., magnesium and aluminium, respectively ", "fib": "aluminium"}
+{"text": "That is, all chemical and physical properties are a manifestation of the electronic configuration of elements ", "fib": "chemical"}
+{"text": "This results into high chemical reactivity at the two extremes and the lowest in the centre ", "fib": "centre"}
+{"text": "This property can be related with the reducing and oxidizing behaviour of the elements which you will learn later ", "fib": ""}
+{"text": "Thus, the metallic character of an element, which is highest at the extremely left decreases and the non-metallic character increases while moving from left to right across the period ", "fib": "element"}
+{"text": "As a consequence, they are less electropositive than group 1 and 2 metals ", "fib": "1"}
+{"text": "Four types of elements can be recognized in the periodic table on the basis of their electronic configurations ", "fib": "Four"}
+{"text": "These are s-block, p-block, d-block and f-block elements ", "fib": "d-block"}
+{"text": "Hydrogen with one electron in the 1s orbital occupies a unique position in the periodic table ", "fib": "1s"}
+{"text": "Oxides formed of the elements on the left are basic and of the elements on the right are acidic in nature ", "fib": "Oxides"}
 {"text": "Oxides of elements in the centre are amphoteric or neutral. ", "fib": "Oxides"}
+{"text": "3.15 Energy of an electron in the ground state of the hydrogen atom is \u20132.18\u00d710\u201318J ", "fib": "3.15"}
+{"text": "3.22 What is the basic difference between the terms electron gain enthalpy and electronegativity? 3.23 How would you react to the statement that the electronegativity of N on Pauling scale is 3.0 in all the nitrogen compounds? 3.24 Describe the theory associated with the radius of an atom as it (a) gains an electron (b) loses an electron 3.25 Would you expect the first ionization enthalpies for two isotopes of the same element to be the same or different? Justify your answer ", "fib": ""}
 {"text": "(b) Identify an element that would tend to lose two electrons ", "fib": "b"}
-{"text": "(c) Identify an element that would tend to gain two electrons ", "fib": "c"}
-{"text": "(d) Identify the group having metal, non-metal, liquid as well as gas at the room temperature ", "fib": "d"}
-{"text": "Table of Contents ", "fib": "Contents"}
-{"text": "Unit 3 ", "fib": "3"}
-{"text": "3.2 GENESIS OF PERIODIC CLASSIFICATION ", "fib": ""}
-{"text": "3.7.2 Periodic Trends in Chemical Properties ", "fib": "3.7.2"}
-{"text": "3.7.3 Periodic Trends and Chemical Reactivity ", "fib": "3.7.3"}
-{"text": "A thousand years later, a Polish monk named Nicolas Copernicus (1473-1543) proposed a definitive model in which the planets moved in circles around a fixed central sun ", "fib": "Copernicus"}
-{"text": "His theory was discredited by the church, but notable amongst its supporters was Galileo who had to face prosecution from the state for his beliefs ", "fib": "Galileo"}
-{"text": "The ellipse, of which the circle is a special case, is a closed curve which can be drawn very simply as follows. ", "fib": "circle"}
-{"text": "Join the points F1 and F2 and extend the line to intersect the ellipse at points P and A as shown in (b) ", "fib": ""}
+{"text": "Which of the above elements is likely to be : (a) the least reactive element ", "fib": "elements"}
+{"text": "(c) the most reactive non-metal ", "fib": "c"}
+{"text": "(c) Each block contains a number of columns equal to the number of electrons that can occupy that subshell ", "fib": "block"}
+{"text": "3.35 Anything that influences the valence electrons will affect the chemistry of the element ", "fib": "3.35"}
+{"text": "Classification of Elements and Periodicity in Properties ", "fib": "Classification"}
+{"text": "3.4 NOMENCLATURE OF ELEMENTS WITH ATOMIC NUMBERS > 100 ", "fib": "100"}
+{"text": "3.6.1 The s-Block Elements ", "fib": "3.6.1"}
+{"text": "3.6.2 The p-Block Elements ", "fib": "3.6.2"}
+{"text": "3.7.1 Trends in Physical Properties ", "fib": "3.7.1"}
+{"text": "SUMMARY ", "fib": "SUMMARY"}
+{"text": "Exercises ", "fib": "Exercises"}
+{"text": "8.1 Introduction Early in our lives, we become aware of the tendency of all material objects to be attracted towards the earth ", "fib": "8.1"}
+{"text": "Anything thrown up falls down towards the earth, going uphill is lot more tiring than going downhill, raindrops from the clouds above fall towards the earth and there are many other such phenomena ", "fib": "clouds"}
+{"text": "Historically it was the Italian Physicist Galileo (1564-1642) who recognised the fact that all bodies, irrespective of their masses, are accelerated towards the earth with a constant acceleration ", "fib": "Galileo"}
+{"text": "It is said that he made a public demonstration of this fact ", "fib": "fact"}
+{"text": "The only motion that was thought to be possible for celestial objects was motion in a circle ", "fib": "celestial"}
+{"text": "Similar theories were also advanced by Indian astronomers some 400 years later ", "fib": "400"}
+{"text": "His compiled data were analysed later by his assistant Johannes Kepler (1571-1640) ", "fib": "1571-1640"}
+{"text": "Take a length of a string and fix its ends at F1 and F2 by pins ", "fib": "F1"}
+{"text": "For a circle, the two focii merge into one and the semi-major axis becomes the radius of the circle ", "fib": "circle"}
+{"text": "This law comes from the observations that planets appear to move slower when they are farther from the sun than when they are nearer. ", "fib": "law"}
+{"text": "Table 8.1 Data from measurement of planetary motions given below confirm Kepler\u2019s Law of Periods (a \u2261 Semi-major axis in units of 1010 m ", "fib": "1010"}
 {"text": "T \u2261 Time period of revolution of the planet in years(y) ", "fib": "T"}
-{"text": "Let the Sun be at the origin and let the position and momentum of the planet be denoted by r and p respectively ", "fib": "Sun"}
+{"text": "A central force is such that the force on the planet is along the vector joining the Sun and the planet ", "fib": "Sun"}
 {"text": "This is the law of areas ", "fib": "areas"}
-{"text": "Example 8.1 Let the speed of the planet at the perihelion P in (a) be vP and the Sun-planet distance SP be rP ", "fib": "8.1"}
-{"text": "Similarly, LA = mp rA vA ", "fib": "="}
-{"text": "The two results (1) and (2) described above, therefore, apply to the motion of the planet ", "fib": "1"}
-{"text": "In fact, the result (2) is the well-known second law of Kepler. ", "fib": "2"}
-{"text": "Tr is the trejectory of the particle under the central force ", "fib": "Tr"}
-{"text": "At a position P, the force is directed along OP, O is the centre of the force taken as the origin ", "fib": "O"}
-{"text": "The area swept in \u2206t is the area of sector POP\u2032 \u2248 (r sin \u03b1 ) PP\u2032/2 = (r v sin a) \u2206t/2.) ", "fib": "="}
-{"text": "(8.3) ", "fib": "8.3"}
-{"text": "(a) What is the force acting on a mass 2m placed at the centroid G of the triangle? (b) What is the force if the mass at the vertex A is doubled ? Take AG = BG = CG = 1 m (see ) ", "fib": ""}
-{"text": "Each point mass in the extended object will exert a force on the given point mass and these force will not all be in the same direction ", "fib": "force"}
+{"text": "He formulated the three laws of planetary motion based on the painstaking observations of Tycho Brahe and coworkers ", "fib": "Brahe"}
+{"text": "Kepler himself was an assistant to Brahe and it took him sixteen long years to arrive at the three planetary laws ", "fib": "Brahe"}
+{"text": "Hence the planet will take a longer time to traverse BAC than CPB. ", "fib": "BAC"}
+{"text": "The time period T is about 27.3 days and Rm was already known then to be about 3.84 \u00d7 10\u00ad8m ", "fib": "10\u00ad8m"}
+{"text": "The angular momentum of the particle is conserved, if the torquedue to the force F on it vanishes ", "fib": "F"}
+{"text": "Central forces satisfy this condition ", "fib": "Central"}
+{"text": "Two important results follow from this: (1) The motion of a particle under the central force is always confined to a plane ", "fib": "1"}
+{"text": "Stated Mathematically, Newton\u2019s gravitation law reads : The force F on a point mass m2 due to another point mass m1 has the magnitude (8.5) ", "fib": ""}
+{"text": "Alternatively, one expects on the basis of symmetry that the resultant force ought to be zero ", "fib": "basis"}
+{"text": "(b) Now if the mass at vertex A is doubled then ", "fib": "b"}
 {"text": "This is easily done using calculus ", "fib": "calculus"}
-{"text": "Kepler had formulated his third law by 1619 ", "fib": "1619"}
+{"text": "For two special cases, a simple law results when you do that : (1) The force of attraction between a hollow spherical shell of uniform density and a point mass situated outside is just as if the entire mass of the shell is concentrated at the centre of the shell ", "fib": "1"}
+{"text": "Qualitatively this can be understood as follows: Gravitational forces caused by the various regions of the shell have components along the line joining the point mass to the centre as well as along a direction prependicular to this line ", "fib": "Gravitational"}
+{"text": "These forces cancel each other completely. ", "fib": "forces"}
+{"text": "A point outside the earth is obviously outside all the shells ", "fib": "earth"}
 {"text": "The total mass of all the shells combined is just the mass of the earth ", "fib": "earth"}
 {"text": "If the mass m is situated on the surface of earth, then r = RE and the gravitational force on it is, from Eq ", "fib": "="}
-{"text": "(8.10) (8.11) ", "fib": "8.10"}
-{"text": "8.6 Acceleration due to gravity below and above the surface of earth Consider a point mass m at a height h above the surface of the earth as shown in (a) ", "fib": "8.6"}
-{"text": "The radius of the earth is denoted by RE ", "fib": "RE"}
-{"text": "Since this point is outside the earth, its distance from the centre of the earth is (RE + h ) ", "fib": "+"}
+{"text": "(8.12) ", "fib": "8.12"}
+{"text": "As far as the smaller sphere of radius ( RE \u2013 d ) is concerned, the point mass is outside it and hence according to the result quoted earlier, the force due to this smaller sphere is just as if the entire mass of the smaller sphere is concentrated at the centre ", "fib": "RE"}
+{"text": "If Ms is the mass of the smaller sphere, then, Ms/ME = ( RE \u2013 d)3 / RE3 ( 8.16) Since mass of a sphere is proportional to be cube of its radius. ", "fib": ""}
+{"text": "(b) g at a depth d ", "fib": "b"}
+{"text": "In this case only the smaller sphere of radius (RE\u2013d) contributes to g ", "fib": "RE\u2013d"}
+{"text": "Thus the force on the point mass is F (d) = G Ms m / (RE \u2013 d ) 2 (8.17) Substituting for Ms from above , we get F (d) = G ME m ( RE \u2013 d ) / RE 3 (8.18) and hence the acceleration due to gravity at a depth d, g(d) = is (8.19) ", "fib": "/"}
 {"text": "Thus, as we go down below earth\u2019s surface, the acceleration due gravity decreases by a factor The remarkable thing about acceleration due to earth\u2019s gravity is that it is maximum on its surface decreasing whether you go up or down. ", "fib": "acceleration"}
-{"text": "8.7 Gravitational potential energy We had discussed earlier the notion of potential energy as being the energy stored in the body at its given position ", "fib": "8.7"}
 {"text": "If the position of the particle changes on account of forces acting on it, then the change in its potential energy is just the amount of work done on the body by the force ", "fib": "account"}
-{"text": "Of course using machines we can shoot an object with much greater speeds and with greater and greater initial speed, the object scales higher and higher heights ", "fib": "course"}
-{"text": "This is the reason that moon has no atmosphere ", "fib": ""}
-{"text": "Thereafter, the greater gravitational pull of 4M would suffice ", "fib": "4M"}
-{"text": "The mechanical energy at the surface of M is ", "fib": "M"}
-{"text": "At the neutral point N, the speed approaches zero ", "fib": "N"}
-{"text": "From the principle of conservation of mechanical energy or A point to note is that the speed of the projectile is zero at N, but is nonzero when it strikes the heavier sphere 4 M ", "fib": "4"}
-{"text": "(8.35) ", "fib": "8.35"}
-{"text": "(8.37), we get T 2 = k ( RE + h)3 (where k = 4 \u03c02 / GME) (8.38) which is Kepler\u2019s law of periods, as applied to motion of satellites around the earth ", "fib": "+"}
-{"text": "If we substitute the numerical values g \u2245 9.8 m s-2 and RE = 6400 km., we get ", "fib": "6400"}
-{"text": "Which is approximately 85 minutes. ", "fib": "85"}
-{"text": "Example 8.5 The planet Mars has two moons, phobos and delmos ", "fib": "8.5"}
-{"text": "phobos has a period 7 hours, 39 minutes and an orbital radius of 9.4 \u00d7103 km ", "fib": "39"}
-{"text": "What is the length of the martian year in days ? ", "fib": "days"}
-{"text": "\u2234 TM = (1.52)3/2 \u00d7 365 = 684 days We note that the orbits of all planets except Mercury, Mars and Pluto* are very close to being circular ", "fib": "1.52"}
-{"text": "From the derivation of Kepler\u2019s third law [see Eq ", "fib": "Eq"}
-{"text": "The moon is at a distance of 3.84 \u00d7 105 km from the earth ", "fib": "105"}
+{"text": "As we had discussed earlier, forces for which the work done is independent of the path are the conservative forces ", "fib": "conservative"}
+{"text": "The force of gravity is a conservative force and we can calculate the potential energy of a body arising out of this force, called the gravitational potential energy ", "fib": "body"}
+{"text": "Consider points close to the surface of earth, at distances from the surface much smaller than the radius of the earth ", "fib": "distances"}
+{"text": "If we consider a point at a height h1 from the surface of the earth and another point vertically above it at a height h2 from the surface, the work done in lifting the particle of mass m from the first to the second position is denoted by W12 W12 = Force \u00d7 displacement = mg (h2 \u2013 h1) (8.20) If we associate a potential energy W(h) at a point at a height h above the surface such that W(h) = mgh + Wo (8.21) (where Wo = constant) ; then it is clear that W12 = W(h2) \u2013 W(h1) (8.22) The work done in moving the particle is just the difference of potential energy between its final and initial positions.Observe that the constant Wo cancels out in Eq ", "fib": "8.20"}
+{"text": "If we now calculate the work done in lifting a particle from r = r1 to r = r2 (r2 > r1) along a vertical path, we get instead of Eq ", "fib": "="}
+{"text": "(8.20) (8.24) ", "fib": "8.20"}
+{"text": "(8.22) and (8.24) ", "fib": "8.22"}
+{"text": "This is an example of the application of the superposition principle. ", "fib": "application"}
+{"text": "Answer Consider four masses each of mass m at the corners of a square of side l; See ", "fib": ""}
+{"text": "8.8 Escape Speed If a stone is thrown by hand, we see it falls back to the earth ", "fib": "8.8"}
+{"text": "A natural query that arises in our mind is the following: \u2018can we throw an object with such high initial speeds that it does not fall back to the earth?\u2019 The principle of conservation of energy helps us to answer this question ", "fib": "conservation"}
+{"text": "By the principle of energy conservation Eqs ", "fib": "Eqs"}
+{"text": "Thus, an object can reach infinity as long as Vi is such that ", "fib": "Vi"}
+{"text": "(8.29) ", "fib": "8.29"}
+{"text": "of Eq ", "fib": "Eq"}
+{"text": "Moon is the only natural satellite of the earth with a near circular orbit with a time period of approximately 27.3 days which is also roughly equal to the rotational period of the moon about its own axis ", "fib": "27.3"}
+{"text": "Equating R.H.S of Eqs ", "fib": "Eqs"}
+{"text": "From ,the speed V for h = 0 is (8.36) ", "fib": "0"}
+{"text": "where we have used the relation g = ", "fib": "="}
+{"text": "Squaring both sides of Eq ", "fib": "Eq"}
+{"text": "For a satellite very close to the surface of earth h can be neglected in comparison to RE in Eq ", "fib": "Eq"}
+{"text": "(8.38) ", "fib": "8.38"}
+{"text": "= 5.97\u00d7 1024 kg ", "fib": "1024"}
+{"text": "The moon is a satellite of the Earth ", "fib": "Earth"}
+{"text": "Example 8.7 Express the constant k of Eq ", "fib": "8.7"}
+{"text": "(8.38) and the given value of k, the time period of the moon is T2 = (1.33 \u00d7 10-14)(3.84 \u00d7 105)3 T = 27.3 d Note that Eq ", "fib": "1.33"}
+{"text": "(8.38) also holds for elliptical orbits if we replace (RE+h) by the semi-major axis of the ellipse ", "fib": "8.38"}
 {"text": "(8.40) ", "fib": "8.40"}
-{"text": "However, in magnitude the K.E ", "fib": "K.E"}
-{"text": "How much energy is required to transfer it to a circular orbit of radius 4RE ? What are the changes in the kinetic and potential energies ? ", "fib": "4RE"}
-{"text": "8.11 Geostationary and Polar Satellites An interesting phenomenon arises if in we arrange the value of (RE+ h) such that T in Eq ", "fib": "8.11"}
+{"text": "Considering gravitational potential energy at infinity to be zero, the potential energy at distance (RE+h) from the centre of the earth is (8.41) ", "fib": "8.41"}
+{"text": "The K.E is positive whereas the P.E is negative ", "fib": "K.E"}
+{"text": "When the orbit of a satellite becomes elliptic, both the K.E ", "fib": "K.E"}
+{"text": "Satellites are always at finite distance from the earth and hence their energies cannot be positive or zero ", "fib": "Satellites"}
 {"text": "(8.37) becomes equal to 24 hours ", "fib": "24"}
+{"text": "and for T = 24 hours, h works out to be 35800 km ", "fib": "24"}
+{"text": "Satellites in a circular orbits around the earth in the equatorial plane with T = 24 hours are called Geostationery Satellites ", "fib": "24"}
+{"text": "Clearly, since the earth rotates with the same period, the satellite would appear fixed from any point on earth ", "fib": "earth"}
+{"text": "It takes very powerful rockets to throw up a satellite to such large heights above the earth but this has been done in view of the several benefits of many practical applications. ", "fib": "earth"}
+{"text": "India\u2019s Leap into Space India started its space programme in 1962 when Indian National Committee for Space Research was set up by the Government of India which was superseded by the Indian Space Research Organisation (ISRO) in 1969 ", "fib": ""}
+{"text": "ISRO identified the role and importance of space technology in nation\u2019s development and bringing space to the service of the common man ", "fib": "ISRO"}
 {"text": "India launched its first low orbit satellite Aryabhata in 1975, for which the launch vehicle was provided by the erstwhile Soviet Union ", "fib": "1975"}
-{"text": "Various research centers and autonomous institutions for remote sensing, astronomy and astrophysics, atmospheric sciences and space research are functioning under the aegis of the Department of Space, Government of India ", "fib": "Department"}
-{"text": "A Geostationery satellite, appearing fixed above the broadcasting station can however receive these signals and broadcast them back to a wide area on earth ", "fib": "Geostationery"}
+{"text": "ISRO started employing its indigenous launching vehicle in 1979 by sending Rohini series of satellites into space from its main launch site at Satish Dhawan Space Center, Sriharikota, Andhra Pradesh ", "fib": "1979"}
+{"text": "The tremendous progress in India\u2019s space programme has made ISRO one of the six largest space agencies in the world ", "fib": "ISRO"}
+{"text": "In order to achieve complete self-reliance in these applications, cost effective and reliable Polar Satellite Launch Vehicle (PSLV) was developed in early 1990s ", "fib": "1990s"}
+{"text": "PSLV has thus become a favoured carrier for satellites of various countries, promoting unprecedented international collaboration ", "fib": "PSLV"}
+{"text": "Another class of satellites are called the Polar satellites ", "fib": "Polar"}
 {"text": "These are low altitude (h \u2248 500 to 800 km) satellites, but they go around the poles of the earth in a north-south direction whereas the earth rotates around its axis in an east-west direction ", "fib": "500"}
-{"text": "Information gathered from such satellites is extremely useful for remote sensing, meterology as well as for environmental studies of the earth. ", "fib": "Information"}
-{"text": "This is because the spring is pulled down a little by the gravitational pull of the object and in turn the spring exerts a force on the object vertically upwards ", "fib": "force"}
+{"text": "8.12 Weightlessness Weight of an object is the force with which the earth attracts it ", "fib": "8.12"}
+{"text": "The same principle holds good when we measure the weight of an object by a spring balance hung from a fixed point e.g ", "fib": "balance"}
+{"text": "This is exactly what the spring exerts on the object ", "fib": "exerts"}
 {"text": "Now, imagine that the top end of the balance is no longer held fixed to the top ceiling of the room ", "fib": "balance"}
-{"text": "Thus, when an object is in free fall, it is weightless and this phenomenon is usually called the phenomenon of weightlessness ", "fib": "free"}
-{"text": "The resultant force FR is then found by vector addition FR = F1 + F2 + \u2026\u2026+ Fn = where the symbol \u2018\u03a3\u2019 stands for summation ", "fib": "+"}
-{"text": "Kepler\u2019s laws of planetary motion state that (a) All planets move in elliptical orbits with the Sun at one of the focal points (b) The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals ", "fib": "Kepler"}
+{"text": "The spring is not stretched and does not exert any upward force on the object which is moving down with acceleration g due to gravity ", "fib": "acceleration"}
+{"text": "The reading recorded in the spring balance is zero since the spring is not stretched at all ", "fib": "balance"}
+{"text": "This is just as if we were falling towards the earth from a height ", "fib": "earth"}
+{"text": "Summary 1. Newton\u2019s law of universal gravitation states that the gravitational force of attraction between any two particles of masses m1 and m2 separated by a distance r has the magnitude where G is the universal gravitational constant, which has the value 6.672 \u00d710\u201311 N m2 kg\u20132 ", "fib": "1"}
+{"text": "Let F1, F2, \u2026.Fn be the individual forces due to M1, M2, \u2026.Mn, each given by the law of gravitation ", "fib": "F1"}
+{"text": "From the principle of superposition each force acts independently and uninfluenced by the other bodies ", "fib": "force"}
+{"text": "Most planets have nearly circular orbits about the Sun ", "fib": "Most"}
 {"text": "For elliptical orbits, the above equation is valid if R is replaced by the semi-major axis, a ", "fib": "R"}
-{"text": "4. The acceleration due to gravity ", "fib": "4"}
-{"text": "(a) at a height h above the earth\u2019s surface for h << RE (b) at depth d below the earth\u2019s surface is 5. The gravitational force is a conservative force, and therefore a potential energy function can be defined ", "fib": "5"}
-{"text": "The gravitational potential energy associated with two particles separated by a distance r is given by where V is taken to be zero at r \u2192 \u221e ", "fib": "V"}
-{"text": "The total potential energy for a system of particles is the sum of energies for all pairs of particles, with each pair represented by a term of the form given by above equation ", "fib": "energies"}
+{"text": "The total energy is negative for any bound system, that is, one in which the orbit is closed, such as an elliptical orbit ", "fib": "bound"}
+{"text": "The kinetic and potential energies are The escape speed from the surface of the earth is = and has a value of 11.2 km s\u20131 ", "fib": "11.2"}
+{"text": "If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attracts the particle as though the mass of the sphere or shell were concentrated at the centre of the sphere ", "fib": ""}
+{"text": "10. If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero ", "fib": "10"}
+{"text": "If a particle is inside a homogeneous solid sphere, the force on the particle acts toward the centre of the sphere ", "fib": ""}
 {"text": ". A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at a approximate distance of 4.22 \u00d7 104 km from the earth\u2019s centre ", "fib": "104"}
+{"text": "Points to Ponder 1. In considering motion of an object under the gravitational influence of another object the following quantities are conserved: (a) Angular momentum (b) Total mechanical energy Linear momentum is not conserved 2. Angular momentum conservation leads to Kepler\u2019s second law ", "fib": "1"}
+{"text": "However, it is not special to the inverse square law of gravitation ", "fib": "gravitation"}
+{"text": "It holds for any central force ", "fib": "central"}
+{"text": "3. In Kepler\u2019s third law (see Eq ", "fib": "Eq"}
 {"text": "(8.1) and T2 = KS R3 ", "fib": "8.1"}
-{"text": "Exercises 8.1 Answer the following : (a) You can shield a charge from electrical forces by putting it inside a hollow conductor ", "fib": ""}
+{"text": "The constant KS is the same for all planets in circular orbits ", "fib": "circular"}
+{"text": "This applies to satellites orbiting the Earth [(Eq ", "fib": "Earth"}
+{"text": "(8.38)] ", "fib": "8.38"}
+{"text": "4. An astronaut experiences weightlessness in a space satellite ", "fib": "4"}
+{"text": "This is not because the gravitational force is small at that location in space ", "fib": "gravitational"}
+{"text": "It is because both the astronaut and the satellite are in \u201cfree fall\u201d towards the Earth ", "fib": "Earth"}
+{"text": "5. The gravitational potential energy associated with two particles separated by a distance r is given by The constant can be given any value ", "fib": "5"}
+{"text": "With this choice This choice implies that V \u2192 0 as r \u2192 \u221e ", "fib": "0"}
+{"text": "Note that the gravitational force is not altered by the choice of this constant ", "fib": "Note"}
+{"text": "Relative to infinity (i.e ", "fib": "i.e"}
+{"text": "if we presume that the potential energy of the object at infinity is zero), the gravitational potential energy of an object is negative ", "fib": "gravitational"}
+{"text": "The total energy of a satellite is negative ", "fib": "satellite"}
+{"text": "8. Although the gravitational force between two particles is central, the force between two finite rigid bodies is not necessarily along the line joining their centre of mass ", "fib": "8"}
+{"text": "The gravitational force on a particle inside a spherical shell is zero ", "fib": "gravitational"}
+{"text": "Gravitational shielding is not possible. ", "fib": ""}
 {"text": "Why ? 8.2 Choose the correct alternative : (a) Acceleration due to gravity increases/decreases with increasing altitude ", "fib": ""}
-{"text": "8.5 Let us assume that our galaxy consists of 2.5 \u00d7 1011 stars each of one solar mass ", "fib": ""}
+{"text": "(b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density) ", "fib": ""}
+{"text": "(c) Acceleration due to gravity is independent of mass of the earth/mass of the body ", "fib": "Acceleration"}
+{"text": "(d) The formula \u2013G Mm(1/r2 \u2013 1/r1) is more/less accurate than the formulamg(r2 \u2013 r1) for the difference of potential energy between two points r2 and r1 distance away from the centre of the earth ", "fib": "1/r1"}
+{"text": "8.7 Does the escape speed of a body from the earth depend on (a) the mass of the body, (b) the location from where it is projected, (c) the direction of projection, (d) the height of the location from where the body is launched? 8 A comet orbits the sun in a highly elliptical orbit ", "fib": ""}
 {"text": "8.9 Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem ", "fib": ""}
+{"text": "8.10 In the following two exercises, choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12) a, b,c, (iv) 0. ", "fib": "0"}
+{"text": "APPENDIX 8.1 : LIST OF INDIAN SATELLITES ", "fib": "8.1"}
 {"text": "India has so far also launched 239 foreign satellites of 28 countries from Satish Dhawan Space Center, Sriharikota, Andhra Pradesh: May 26, 1999 (02); Oct ", "fib": "02"}
-{"text": "15, 2017 (101) and thus setting a world record; June 23, 2017 (29) ", "fib": "101"}
+{"text": "23, 2007 (01); Jan ", "fib": "01"}
+{"text": "20, 2011 (01) Sep ", "fib": "01"}
+{"text": "09, 2012 (02); Feb ", "fib": "02"}
+{"text": "25, 2013 (06); June 30, 2014 (05); July 10, 2015 (05); Sep ", "fib": "05"}
+{"text": "26, 2016 (05); Feb ", "fib": "05"}
+{"text": "Jan 12, 2018 (28); Sep ", "fib": "12"}
 {"text": "Table of Contents ", "fib": "Contents"}
+{"text": "Chapter Eight ", "fib": "Chapter"}
+{"text": "Gravitation ", "fib": "Gravitation"}
+{"text": "8.1 Introduction ", "fib": "8.1"}
+{"text": "8.2 Kepler\u2019s laws ", "fib": "8.2"}
+{"text": "8.4 The Gravitational Constant ", "fib": "8.4"}
+{"text": "8.5 Acceleration due to gravity of the earth ", "fib": "8.5"}
 {"text": "8.6 Acceleration due to gravity below and above the surface of earth ", "fib": "8.6"}
+{"text": "8.7 Gravitational potential energy ", "fib": "8.7"}
+{"text": "8.8 Escape Speed ", "fib": "8.8"}
 {"text": "8.9 Earth Satellites ", "fib": "8.9"}
+{"text": "8.10 Energy of an orbiting Satellite ", "fib": "8.10"}
 {"text": "8.11 Geostationary and Polar Satellites ", "fib": "8.11"}
-{"text": "For example, in Amoeba, a single cell carries out movement, intake of food, gaseous exchange and excretion ", "fib": "Amoeba"}
-{"text": "Each specialised function is taken up by a different group of cells ", "fib": "cells"}
-{"text": "In plants, vascular tissues conduct food and water from one part of the plant to other parts ", "fib": "food"}
-{"text": "6.2 Plant Tissues ", "fib": "6.2"}
+{"text": "8.12 Weightlessness ", "fib": "8.12"}
+{"text": "Summary ", "fib": "Summary"}
+{"text": "Cover ", "fib": "Cover"}
+{"text": "From the last chapter, we recall that all living organisms are made of cells ", "fib": "cells"}
+{"text": "In unicellular organisms, a single cell performs all basic functions ", "fib": "basic"}
+{"text": "Most of these cells are specialised to carry out specific functions ", "fib": "cells"}
+{"text": "This means that a particular function is carried out by a cluster of cells at a definite place in the body ", "fib": "body"}
+{"text": "Tissues A group of cells that are similar in structure and/or work together to achieve a particular function forms a tissue ", "fib": "and/or"}
+{"text": "6.1 Are Plants and Animals Made of Same Types of Tissues? Let us compare their structure and functions ", "fib": ""}
+{"text": "Do plants and animals have the same structure? Do they both perform similar functions? There are noticeable differences between the two ", "fib": "animals"}
+{"text": "These tissues are localised in certain regions ", "fib": "certain"}
+{"text": "Based on the dividing capacity of the tissues, various plant tissues can be classified as growing or meristematic tissue and permanent tissue ", "fib": "capacity"}
+{"text": "Cell growth in animals is more uniform ", "fib": "Cell"}
+{"text": "This fundamental difference reflects the different modes of life pursued by these two major groups of organisms, particularly in their different feeding methods ", "fib": "different"}
 {"text": "Jar 1 Jar 2 : Growth of roots in onion bulbs ", "fib": "1"}
-{"text": "The girth of the stem or root increases due to lateral meristem (cambium) ", "fib": "cambium"}
-{"text": "Activity 6.3 \u2022 Take a freshly plucked leaf of Rhoeo ", "fib": "6.3"}
+{"text": "\u2022 Now, take two onion bulbs and place one on each jar, as shown in ", "fib": "bulbs"}
+{"text": "\u2022 Observe the growth of roots in both the bulbs for a few days ", "fib": "Observe"}
+{"text": "After this, observe the growth of roots in both the jars and measure their lengths each day for five more days and record the observations in tables, like the table below: Length Day 1 Day 2 Day 3 Day 4 Day 5 Jar 1 Jar 2 ", "fib": "1"}
+{"text": "The growth of plants occurs only in certain specific regions ", "fib": "certain"}
+{"text": "This is because the dividing tissue, also known as meristematic tissue, is located only at these points ", "fib": ""}
+{"text": "Depending on the region where they are present, meristematic tissues are classified as apical, lateral and intercalary ", "fib": "meristematic"}
+{"text": "New cells produced by meristem are initially like those of meristem itself, but as they grow and mature, their characteristics slowly change and they become differentiated as components of other tissues. ", "fib": "New"}
+{"text": "6.2.2 Permanent tissue What happens to the cells formed by meristematic tissue? They take up a specific role and lose the ability to divide ", "fib": "6.2.2"}
+{"text": "This process of taking up a permanent shape, size, and a function is called differentiation ", "fib": "differentiation"}
+{"text": "Place one neatly cut section on a slide, and put a drop of glycerine ", "fib": "Place"}
+{"text": "We can even try cutting sections of root and stem of different plants. ", "fib": "different"}
+{"text": "6.2.2 Simple permanent tissue A few layers of cells beneath the epidermis are generally simple permanent tissue ", "fib": ""}
+{"text": "It consists of relatively unspecialised cells with thin cell walls ", "fib": "thin"}
+{"text": "They are living cells ", "fib": "cells"}
+{"text": "They are usually loosely arranged, thus large spaces between cells (intercellular spaces) are found in this tissue ", "fib": "cells"}
+{"text": "This tissue generally stores food ", "fib": "food"}
+{"text": "In some situations, it contains chlorophyll and performs photosynthesis, and then it is called chlorenchyma ", "fib": "chlorenchyma"}
+{"text": "Such a parenchyma type is called aerenchyma ", "fib": "aerenchyma"}
+{"text": "It allows bending of various parts of a plant like tendrils and stems of climbers without breaking ", "fib": "bending"}
+{"text": "It also provides mechanical support ", "fib": "mechanical"}
+{"text": "We can find this tissue in leaf stalks below the epidermis ", "fib": "epidermis"}
+{"text": "The cells of this tissue are living, elongated and irregularly thickened at the corners ", "fib": "cells"}
+{"text": "There is very little intercellular space . ", "fib": "little"}
 {"text": "\u2022 Stretch and break it by applying pressure ", "fib": "pressure"}
 {"text": "\u2022 While breaking it, keep it stretched gently so that some peel or skin projects out from the cut ", "fib": "cut"}
-{"text": "(a) (b) : Guard cells and epidermal cells: (a) lateral view, (b) surface view ", "fib": "Guard"}
-{"text": "These pores are called stomata ", "fib": "pores"}
-{"text": "Transpiration (loss of water in the form ofwater vapour) also takes place through stomata ", "fib": "Transpiration"}
+{"text": "\u2022 Remove this peel and put it in a petri dish filled with water ", "fib": ""}
+{"text": "\u2022 Add a few drops of safranin ", "fib": ""}
+{"text": "\u2022 Wait for a couple of minutes and then transfer it onto a slide ", "fib": "couple"}
+{"text": "Gently place a cover slip over it ", "fib": "cover"}
+{"text": "\u2022 Observe under microscope. ", "fib": "microscope"}
+{"text": "The epidermis is usually made of a single layer of cells ", "fib": "cells"}
+{"text": "The entire surface of a plant has an outer covering epidermis ", "fib": "entire"}
+{"text": "They are necessary for exchanging gases with the atmosphere ", "fib": ""}
 {"text": "Find out the role of transpiration in plants ", "fib": "plants"}
-{"text": "Epidermal cells of the roots, whose function is water absorption, commonly bear long hair-like parts that greatly increase the total absorptive surface area ", "fib": "Epidermal"}
-{"text": "This allows them to transport water and minerals vertically ", "fib": "minerals"}
-{"text": "The parenchyma stores food ", "fib": "food"}
-{"text": "Sieve tubes are tubular cells with perforated walls ", "fib": "Sieve"}
-{"text": "Phloem transports food from leaves to other parts of the plant ", "fib": "Phloem"}
-{"text": "1. Name types of simple tissues ", "fib": "1"}
-{"text": "When we breathe we can actually feel the movement of our chest ", "fib": "chest"}
-{"text": "They have only a small amount of cementing material between them and almost no intercellular spaces ", "fib": "intercellular"}
-{"text": "Obviously, anything entering or leaving the body must cross at least one layer of epithelium ", "fib": "anything"}
-{"text": "Different epithelia show differing structures that correlate with their unique functions ", "fib": "Different"}
+{"text": "Can we think of a reason for this? Is the outer layer of a branch of a tree different from the outer layer of a young stem? As plants grow older, the outer protective tissue undergoes certain changes ", "fib": "branch"}
+{"text": "A strip of secondary meristem located in the cortex forms layers of cells which constitute the cork ", "fib": "cells"}
+{"text": "Cells of cork are dead and compactly arranged without intercellular spaces ", "fib": "Cells"}
+{"text": "Such tissues are called simple permanent tissue ", "fib": "Such"}
+{"text": "Complex tissues are made of more than one type of cells ", "fib": "Complex"}
+{"text": "All these cells coordinate to perform a common function ", "fib": "cells"}
+{"text": "Xylem and phloem are examples of such complex tissues ", "fib": "Xylem"}
+{"text": "They are both conducting tissues and constitute a vascular bundle ", "fib": "tissues"}
+{"text": "Vascular tissue is a distinctive feature of the complex plants, one that has made possible their survival in the terrestrial environment ", "fib": "Vascular"}
+{"text": "In showing a section of stem, can you see different types of cells in the vascular bundle? Xylem consists of tracheids, vessels, xylem parenchyma and xylem fibres ", "fib": "Xylem"}
+{"text": "Questions ", "fib": "Questions"}
+{"text": "Why would cells need oxygen? The functions of mitochondria we studied earlier provide a clue to this question ", "fib": "cells"}
+{"text": "For example, it carries oxygen and food to all cells ", "fib": "cells"}
+{"text": "6.3.1 Epithelial tissue The covering or protective tissues in the animal body are epithelial tissues ", "fib": ""}
+{"text": "Epithelium covers most organs and cavities within the body ", "fib": "Epithelium"}
+{"text": "As a result, the permeability of the cells of various epithelia play an important role in regulating the exchange of materials between the body and the external environment and also between different parts of the body ", "fib": "body"}
+{"text": "The oesophagus and the lining of the mouth are also covered with squamous epithelium ", "fib": ""}
+{"text": "The skin, which protects the body, is also made of squamous epithelium ", "fib": "body"}
+{"text": "This columnar (meaning \u2018pillar-like\u2019) epithelium facilitates movement across the epithelial barrier ", "fib": "columnar"}
+{"text": "This is glandular epithelium. ", "fib": "glandular"}
+{"text": "It forms the framework that supports the body ", "fib": "body"}
+{"text": "Bone cells are embedded in a hard matrix that is composed of calcium and phosphorus compounds ", "fib": "Bone"}
+{"text": "Two bones can be connected to each other by another type of connective tissue called the ligament ", "fib": "Two"}
+{"text": "This tissue is very elastic ", "fib": "tissue"}
+{"text": "It has considerable strength ", "fib": "considerable"}
+{"text": "Tendons connect muscles to bones and are another type of connective tissue ", "fib": "Tendons"}
+{"text": "Tendons are fibrous tissue with great strength but limited flexibility ", "fib": "Tendons"}
+{"text": "Another type of connective tissue, cartilage, has widely spaced cells ", "fib": "cartilage"}
+{"text": "The solid matrix is composed of proteins and sugars ", "fib": "proteins"}
+{"text": "It fills the space inside the organs, supports internal organs and helps in repair of tissues ", "fib": "internal"}
+{"text": "Where are fats stored in our body? Fat-storing adipose tissue is found below the skin and between internal organs ", "fib": "adipose"}
+{"text": "The cells of this tissue are filled with fat globules ", "fib": "cells"}
+{"text": "Storage of fats also lets it act as an insulator. ", "fib": "Storage"}
+{"text": "The cells of this tissue are filled with fat globules ", "fib": "cells"}
 {"text": "We can move some muscles by conscious will ", "fib": "muscles"}
-{"text": "Muscles present in our limbs move when we want them to, and stop when we so decide ", "fib": "Muscles"}
 {"text": "Such muscles are called voluntary muscles ", "fib": "Such"}
 {"text": "These muscles are also called skeletal muscles as they are mostly attached to bones and help in body movement ", "fib": "body"}
-{"text": "Under the microscope, these muscles show alternate light and dark bands or striations when stained appropriately ", "fib": "alternate"}
 {"text": "As a result, they are also called striated muscles ", "fib": "muscles"}
 {"text": "The cells of this tissue are long, cylindrical, unbranched and multinucleate (having many nuclei) ", "fib": "cells"}
+{"text": "The movement of food in the alimentary canal or the contraction and relaxation of blood vessels are involuntary movements ", "fib": "alimentary"}
 {"text": "We cannot really start them or stop them simply by wanting to do so! Smooth muscles or involuntary muscles control such movements ", "fib": "Smooth"}
 {"text": "They are also found in the iris of the eye, in ureters and in the bronchi of the lungs ", "fib": "bronchi"}
 {"text": "The cells are long with pointed ends (spindle-shaped) and uninucleate (having a single nucleus) ", "fib": "cells"}
-{"text": "They are also called unstriated muscles \u2013 why would they be called that? Muscular tissue consists of elongated cells, also called muscle fibres ", "fib": "Muscular"}
-{"text": "This tissue is responsible for movement in our body. Muscles contain special proteins called contractile proteins, which contract and relax to cause movement ", "fib": "Muscles"}
 {"text": "We can move some muscles by conscious will ", "fib": "muscles"}
+{"text": "Muscles present in our limbs move when we want them to, and stop when we so decide ", "fib": "Muscles"}
 {"text": "Such muscles are called voluntary muscles ", "fib": "Such"}
-{"text": "Under the microscope, these muscles show alternate light and dark bands or striations when stained appropriately ", "fib": "alternate"}
 {"text": "As a result, they are also called striated muscles ", "fib": "muscles"}
-{"text": "The cells of this tissue are long, cylindrical, unbranched and multinucleate (having many nuclei). The movement of food in the alimentary canal or the contraction and relaxation of blood vessels are involuntary movements ", "fib": "alimentary"}
 {"text": "We cannot really start them or stop them simply by wanting to do so! Smooth muscles or involuntary muscles control such movements ", "fib": "Smooth"}
-{"text": "They are also found in the iris of the eye, in ureters and in the bronchi of the lungs ", "fib": "bronchi"}
 {"text": "The cells are long with pointed ends (spindle-shaped) and uninucleate (having a single nucleus) ", "fib": "cells"}
-{"text": "They are also called unstriated muscles \u2013 why would they be called that? The muscles of the heart show rhythmic contraction and relaxation throughout life ", "fib": "heart"}
 {"text": "These involuntary muscles are called cardiac muscles ", "fib": "cardiac"}
-{"text": "Heart muscle cells are cylindrical, branched and uninucleate ", "fib": "Heart"}
-{"text": "Activity 6.5 Compare the structures of different types of muscular tissues ", "fib": "6.5"}
 {"text": "Note down their shape, number of nuclei and position of nuclei within the cell in the Table 6.1. ", "fib": "6.1"}
-{"text": "However, cells of the nervous tissue are highly specialised for being stimulated and then transmitting the stimulus very rapidly from one place to another within the body ", "fib": "body"}
-{"text": "The brain, spinal cord and nerves are all composed of the nervous tissue ", "fib": "brain"}
-{"text": "Questions 1. Name the tissue responsible for movement in our body ", "fib": "1"}
-{"text": "2. What does a neuron look like? 3. Give three features of cardiac muscles ", "fib": "2"}
-{"text": "\u2022 Permanent tissues are derived from meristematic tissue once they lose the ability to divide ", "fib": "ability"}
-{"text": "They are classified as simple and complex tissues ", "fib": "complex"}
-{"text": "\u2022 Parenchyma, collenchyma and sclerenchyma are three types of simple tissues ", "fib": "Parenchyma"}
-{"text": "\u2022 Depending on shape and function, epithelial tissue is classified as squamous, cuboidal, columnar, ciliated and glandular ", "fib": "columnar"}
-{"text": "\u2022 The different types of connective tissues in our body include areolar tissue, adipose tissue, bone, tendon, ligament, cartilage and blood ", "fib": "adipose"}
-{"text": "\u2022 Nervous tissue is made of neurons that receive and conduct impulses. ", "fib": "Nervous"}
+{"text": "The cells of this tissue are called nerve cells or neurons ", "fib": "cells"}
+{"text": "4. What are the functions of areolar tissue? ", "fib": "4"}
+{"text": "\u2022 Plant tissues are of two main types \u2013 meristematic and permanent ", "fib": "Plant"}
+{"text": "Xylem and phloem are types of complex tissues ", "fib": "Xylem"}
+{"text": "\u2022 Striated, unstriated and cardiac are three types of muscle tissues ", "fib": ""}
+{"text": "Exercises 1. Define the term \u201ctissue\u201d ", "fib": "1"}
+{"text": "2. How many types of elements together make up the xylem tissue? Name them ", "fib": "2"}
 {"text": "3. How are simple tissues different from complex tissues in plants? 4. Differentiate between parenchyma, collenchyma and sclerenchyma on the basis of their cell wall ", "fib": "4"}
 {"text": "5. What are the functions of the stomata? 6. Diagrammatically show the difference between the three types of muscle fibres ", "fib": ""}
-{"text": "10. Name the following ", "fib": ""}
 {"text": "(a) Tissue that forms the inner lining of our mouth ", "fib": "Tissue"}
-{"text": "(b) Tissue that connects muscle to bone in humans ", "fib": "Tissue"}
-{"text": "(c) Tissue that transports food in plants ", "fib": "Tissue"}
-{"text": "(d) Tissue that stores fat in our body ", "fib": "Tissue"}
-{"text": "12. Name the regions in which parenchyma tissue is present ", "fib": "12"}
-{"text": "13. What is the role of epidermis in plants? ", "fib": "13"}
+{"text": "(e) Connective tissue with a fluid matrix ", "fib": "Connective"}
+{"text": "(f) Tissue present in the brain ", "fib": "Tissue"}
+{"text": "11. Identify the type of tissue in the following: skin, bark of tree, bone, lining of kidney tubule, vascular bundle ", "fib": ""}
 {"text": "14. How does the cork act as a protective tissue? 15. Complete the following chart: ", "fib": "14"}
-{"text": "Chapter 6 ", "fib": "6"}
-{"text": "6.2 Plant Tissues ", "fib": "6.2"}
-{"text": "6.3 Animal Tissues ", "fib": "6.3"}
-{"text": "6.3.2 Connective tissue ", "fib": "6.3.2"}
-{"text": "6.3.4 Nervous tissue ", "fib": "6.3.4"}
+{"text": "Table of Contents ", "fib": "Contents"}
+{"text": "Tissues ", "fib": "Tissues"}
+{"text": "6.2.2 Permanent tissue ", "fib": "6.2.2"}
+{"text": "6.3.1 Epithelial tissue ", "fib": "6.3.1"}
+{"text": "Landmarks ", "fib": "Landmarks"}
 {"text": "Cover ", "fib": "Cover"}
 {"text": "Chapter 6Tissues", "fib": "6Tissues"}
diff --git a/model.ipynb b/model.ipynb
index 6ec376d..646a6d7 100644
--- a/model.ipynb
+++ b/model.ipynb
@@ -117,145 +117,145 @@
       "WARNING:tensorflow:From /home/pranav_kirsur/.pyenv/versions/3.6.8/envs/megathon2019/lib/python3.6/site-packages/keras/backend/tensorflow_backend.py:422: The name tf.global_variables is deprecated. Please use tf.compat.v1.global_variables instead.\n",
       "\n",
       "Epoch 1/300\n",
-      "1100/1100 [==============================] - 0s 165us/step - loss: 1.3875 - mean_absolute_error: 0.4967\n",
+      "1100/1100 [==============================] - 0s 154us/step - loss: 1.3119 - mean_absolute_error: 0.4619\n",
       "Epoch 2/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.3781 - mean_absolute_error: 0.4943\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.3004 - mean_absolute_error: 0.4579\n",
       "Epoch 3/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3686 - mean_absolute_error: 0.4919\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.2886 - mean_absolute_error: 0.4537\n",
       "Epoch 4/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3590 - mean_absolute_error: 0.4894\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.2767 - mean_absolute_error: 0.4493\n",
       "Epoch 5/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.3498 - mean_absolute_error: 0.4870\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 1.2646 - mean_absolute_error: 0.4448\n",
       "Epoch 6/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.3406 - mean_absolute_error: 0.4847\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2522 - mean_absolute_error: 0.4401\n",
       "Epoch 7/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 1.3314 - mean_absolute_error: 0.4823\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.2398 - mean_absolute_error: 0.4352\n",
       "Epoch 8/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.3225 - mean_absolute_error: 0.4799\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 1.2272 - mean_absolute_error: 0.4301\n",
       "Epoch 9/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3133 - mean_absolute_error: 0.4775\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2143 - mean_absolute_error: 0.4248\n",
       "Epoch 10/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.3040 - mean_absolute_error: 0.4749\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.2013 - mean_absolute_error: 0.4193\n",
       "Epoch 11/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2945 - mean_absolute_error: 0.4723\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1881 - mean_absolute_error: 0.4137\n",
       "Epoch 12/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.2850 - mean_absolute_error: 0.4695\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 1.1749 - mean_absolute_error: 0.4079\n",
       "Epoch 13/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2753 - mean_absolute_error: 0.4667\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1616 - mean_absolute_error: 0.4019\n",
       "Epoch 14/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2655 - mean_absolute_error: 0.4637\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1482 - mean_absolute_error: 0.3958\n",
       "Epoch 15/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2557 - mean_absolute_error: 0.4607\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1348 - mean_absolute_error: 0.3895\n",
       "Epoch 16/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.2458 - mean_absolute_error: 0.4576\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1213 - mean_absolute_error: 0.3831\n",
       "Epoch 17/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.2359 - mean_absolute_error: 0.4545\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1078 - mean_absolute_error: 0.3766\n",
       "Epoch 18/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.2259 - mean_absolute_error: 0.4512\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0942 - mean_absolute_error: 0.3699\n",
       "Epoch 19/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2157 - mean_absolute_error: 0.4478\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0807 - mean_absolute_error: 0.3631\n",
       "Epoch 20/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.2055 - mean_absolute_error: 0.4444\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0672 - mean_absolute_error: 0.3562\n",
       "Epoch 21/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1953 - mean_absolute_error: 0.4408\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0536 - mean_absolute_error: 0.3492\n",
       "Epoch 22/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1849 - mean_absolute_error: 0.4372\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0401 - mean_absolute_error: 0.3420\n",
       "Epoch 23/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1746 - mean_absolute_error: 0.4334\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0267 - mean_absolute_error: 0.3348\n",
       "Epoch 24/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.1642 - mean_absolute_error: 0.4296\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0132 - mean_absolute_error: 0.3274\n",
       "Epoch 25/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.1537 - mean_absolute_error: 0.4257\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9998 - mean_absolute_error: 0.3200\n",
       "Epoch 26/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 1.1431 - mean_absolute_error: 0.4217\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9865 - mean_absolute_error: 0.3124\n",
       "Epoch 27/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1325 - mean_absolute_error: 0.4176\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9733 - mean_absolute_error: 0.3048\n",
       "Epoch 28/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1218 - mean_absolute_error: 0.4134\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9602 - mean_absolute_error: 0.2971\n",
       "Epoch 29/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1110 - mean_absolute_error: 0.4090\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9472 - mean_absolute_error: 0.2893\n",
       "Epoch 30/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.1001 - mean_absolute_error: 0.4046\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9344 - mean_absolute_error: 0.2816\n",
       "Epoch 31/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0892 - mean_absolute_error: 0.4000\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.9217 - mean_absolute_error: 0.2738\n",
       "Epoch 32/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 1.0780 - mean_absolute_error: 0.3952\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.9093 - mean_absolute_error: 0.2659\n",
       "Epoch 33/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0669 - mean_absolute_error: 0.3904\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.8970 - mean_absolute_error: 0.2581\n",
       "Epoch 34/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0557 - mean_absolute_error: 0.3854\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.8849 - mean_absolute_error: 0.2504\n",
       "Epoch 35/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0444 - mean_absolute_error: 0.3803\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8731 - mean_absolute_error: 0.2426\n",
       "Epoch 36/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0330 - mean_absolute_error: 0.3750\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8616 - mean_absolute_error: 0.2350\n",
       "Epoch 37/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 1.0216 - mean_absolute_error: 0.3697\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8503 - mean_absolute_error: 0.2274\n",
       "Epoch 38/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 1.0100 - mean_absolute_error: 0.3641\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8394 - mean_absolute_error: 0.2199\n",
       "Epoch 39/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.9985 - mean_absolute_error: 0.3585\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8288 - mean_absolute_error: 0.2126\n",
       "Epoch 40/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.9868 - mean_absolute_error: 0.3527\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.8185 - mean_absolute_error: 0.2054\n",
       "Epoch 41/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.9750 - mean_absolute_error: 0.3467\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.8086 - mean_absolute_error: 0.1983\n",
       "Epoch 42/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.9633 - mean_absolute_error: 0.3406\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7990 - mean_absolute_error: 0.1915\n",
       "Epoch 43/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.9516 - mean_absolute_error: 0.3344\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7898 - mean_absolute_error: 0.1849\n",
       "Epoch 44/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.9398 - mean_absolute_error: 0.3281\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7810 - mean_absolute_error: 0.1784\n",
       "Epoch 45/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.9280 - mean_absolute_error: 0.3217\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7725 - mean_absolute_error: 0.1723\n",
       "Epoch 46/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.9162 - mean_absolute_error: 0.3151\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7645 - mean_absolute_error: 0.1663\n",
       "Epoch 47/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.9044 - mean_absolute_error: 0.3084\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.7568 - mean_absolute_error: 0.1606\n",
       "Epoch 48/300\n",
-      "1100/1100 [==============================] - 0s 8us/step - loss: 0.8926 - mean_absolute_error: 0.3017\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7495 - mean_absolute_error: 0.1552\n",
       "Epoch 49/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.8809 - mean_absolute_error: 0.2948\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7425 - mean_absolute_error: 0.1501\n",
       "Epoch 50/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.8693 - mean_absolute_error: 0.2879\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7358 - mean_absolute_error: 0.1452\n",
       "Epoch 51/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8577 - mean_absolute_error: 0.2809\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.7295 - mean_absolute_error: 0.1406\n",
       "Epoch 52/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8463 - mean_absolute_error: 0.2738\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7235 - mean_absolute_error: 0.1363\n",
       "Epoch 53/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8349 - mean_absolute_error: 0.2667\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7178 - mean_absolute_error: 0.1322\n",
       "Epoch 54/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.8238 - mean_absolute_error: 0.2596\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7124 - mean_absolute_error: 0.1284\n",
       "Epoch 55/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.8127 - mean_absolute_error: 0.2524\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.7072 - mean_absolute_error: 0.1248\n",
       "Epoch 56/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.8019 - mean_absolute_error: 0.2454\n",
+      "1100/1100 [==============================] - 0s 7us/step - loss: 0.7022 - mean_absolute_error: 0.1215\n",
       "Epoch 57/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7913 - mean_absolute_error: 0.2383\n",
+      "1100/1100 [==============================] - 0s 8us/step - loss: 0.6975 - mean_absolute_error: 0.1184\n",
       "Epoch 58/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.7809 - mean_absolute_error: 0.2313\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6930 - mean_absolute_error: 0.1156\n",
       "Epoch 59/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7707 - mean_absolute_error: 0.2243\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6886 - mean_absolute_error: 0.1130\n",
       "Epoch 60/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7609 - mean_absolute_error: 0.2175\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6844 - mean_absolute_error: 0.1106\n",
       "Epoch 61/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7512 - mean_absolute_error: 0.2107\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6803 - mean_absolute_error: 0.1084\n",
       "Epoch 62/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.7419 - mean_absolute_error: 0.2041\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6764 - mean_absolute_error: 0.1063\n",
       "Epoch 63/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7329 - mean_absolute_error: 0.1976\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6725 - mean_absolute_error: 0.1045\n",
       "Epoch 64/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7241 - mean_absolute_error: 0.1912\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6688 - mean_absolute_error: 0.1028\n",
       "Epoch 65/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.7157 - mean_absolute_error: 0.1851\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6651 - mean_absolute_error: 0.1013\n",
       "Epoch 66/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.7076 - mean_absolute_error: 0.1791\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6615 - mean_absolute_error: 0.0999\n",
       "Epoch 67/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6999 - mean_absolute_error: 0.1733\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6580 - mean_absolute_error: 0.0987\n",
       "Epoch 68/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6924 - mean_absolute_error: 0.1677\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6545 - mean_absolute_error: 0.0975\n",
       "Epoch 69/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6853 - mean_absolute_error: 0.1623\n",
+      "1100/1100 [==============================] - 0s 8us/step - loss: 0.6511 - mean_absolute_error: 0.0966\n",
       "Epoch 70/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6784 - mean_absolute_error: 0.1572\n"
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6477 - mean_absolute_error: 0.0957\n"
      ]
     },
     {
@@ -263,147 +263,147 @@
      "output_type": "stream",
      "text": [
       "Epoch 71/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6719 - mean_absolute_error: 0.1522\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6443 - mean_absolute_error: 0.0949\n",
       "Epoch 72/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6657 - mean_absolute_error: 0.1475\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6410 - mean_absolute_error: 0.0942\n",
       "Epoch 73/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6598 - mean_absolute_error: 0.1430\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6377 - mean_absolute_error: 0.0936\n",
       "Epoch 74/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6541 - mean_absolute_error: 0.1388\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6345 - mean_absolute_error: 0.0931\n",
       "Epoch 75/300\n",
-      "1100/1100 [==============================] - 0s 9us/step - loss: 0.6487 - mean_absolute_error: 0.1347\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6312 - mean_absolute_error: 0.0927\n",
       "Epoch 76/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6436 - mean_absolute_error: 0.1309\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6280 - mean_absolute_error: 0.0923\n",
       "Epoch 77/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6387 - mean_absolute_error: 0.1274\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.6248 - mean_absolute_error: 0.0920\n",
       "Epoch 78/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6340 - mean_absolute_error: 0.1240\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6217 - mean_absolute_error: 0.0918\n",
       "Epoch 79/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6295 - mean_absolute_error: 0.1209\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6185 - mean_absolute_error: 0.0916\n",
       "Epoch 80/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.6252 - mean_absolute_error: 0.1179\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6154 - mean_absolute_error: 0.0915\n",
       "Epoch 81/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6211 - mean_absolute_error: 0.1152\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6123 - mean_absolute_error: 0.0914\n",
       "Epoch 82/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6172 - mean_absolute_error: 0.1127\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6092 - mean_absolute_error: 0.0913\n",
       "Epoch 83/300\n",
-      "1100/1100 [==============================] - 0s 6us/step - loss: 0.6134 - mean_absolute_error: 0.1103\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6062 - mean_absolute_error: 0.0913\n",
       "Epoch 84/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6097 - mean_absolute_error: 0.1081\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6031 - mean_absolute_error: 0.0913\n",
       "Epoch 85/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6061 - mean_absolute_error: 0.1061\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6001 - mean_absolute_error: 0.0913\n",
       "Epoch 86/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.6027 - mean_absolute_error: 0.1042\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5971 - mean_absolute_error: 0.0914\n",
       "Epoch 87/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5993 - mean_absolute_error: 0.1025\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5941 - mean_absolute_error: 0.0915\n",
       "Epoch 88/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5960 - mean_absolute_error: 0.1010\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5912 - mean_absolute_error: 0.0916\n",
       "Epoch 89/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5928 - mean_absolute_error: 0.0996\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5883 - mean_absolute_error: 0.0917\n",
       "Epoch 90/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5897 - mean_absolute_error: 0.0983\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5854 - mean_absolute_error: 0.0919\n",
       "Epoch 91/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5866 - mean_absolute_error: 0.0971\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5825 - mean_absolute_error: 0.0920\n",
       "Epoch 92/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5836 - mean_absolute_error: 0.0960\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5796 - mean_absolute_error: 0.0922\n",
       "Epoch 93/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5806 - mean_absolute_error: 0.0950\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5768 - mean_absolute_error: 0.0923\n",
       "Epoch 94/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5776 - mean_absolute_error: 0.0942\n",
+      "1100/1100 [==============================] - 0s 7us/step - loss: 0.5740 - mean_absolute_error: 0.0925\n",
       "Epoch 95/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5747 - mean_absolute_error: 0.0934\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5712 - mean_absolute_error: 0.0926\n",
       "Epoch 96/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5719 - mean_absolute_error: 0.0927\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5684 - mean_absolute_error: 0.0928\n",
       "Epoch 97/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5690 - mean_absolute_error: 0.0921\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5657 - mean_absolute_error: 0.0930\n",
       "Epoch 98/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5662 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5630 - mean_absolute_error: 0.0931\n",
       "Epoch 99/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5634 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5603 - mean_absolute_error: 0.0933\n",
       "Epoch 100/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5606 - mean_absolute_error: 0.0908\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5576 - mean_absolute_error: 0.0934\n",
       "Epoch 101/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5578 - mean_absolute_error: 0.0904\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5549 - mean_absolute_error: 0.0936\n",
       "Epoch 102/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5551 - mean_absolute_error: 0.0901\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5523 - mean_absolute_error: 0.0937\n",
       "Epoch 103/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5523 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5497 - mean_absolute_error: 0.0938\n",
       "Epoch 104/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5496 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5471 - mean_absolute_error: 0.0940\n",
       "Epoch 105/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5469 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5445 - mean_absolute_error: 0.0941\n",
       "Epoch 106/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5442 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5420 - mean_absolute_error: 0.0942\n",
       "Epoch 107/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5416 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5394 - mean_absolute_error: 0.0943\n",
       "Epoch 108/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5389 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5369 - mean_absolute_error: 0.0943\n",
       "Epoch 109/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5363 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.5344 - mean_absolute_error: 0.0944\n",
       "Epoch 110/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5337 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5319 - mean_absolute_error: 0.0945\n",
       "Epoch 111/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5311 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5294 - mean_absolute_error: 0.0945\n",
       "Epoch 112/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5285 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5270 - mean_absolute_error: 0.0945\n",
       "Epoch 113/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5260 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5246 - mean_absolute_error: 0.0946\n",
       "Epoch 114/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5235 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5222 - mean_absolute_error: 0.0946\n",
       "Epoch 115/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5210 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5198 - mean_absolute_error: 0.0946\n",
       "Epoch 116/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5185 - mean_absolute_error: 0.0898\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5174 - mean_absolute_error: 0.0946\n",
       "Epoch 117/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5160 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5150 - mean_absolute_error: 0.0946\n",
       "Epoch 118/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5136 - mean_absolute_error: 0.0901\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5127 - mean_absolute_error: 0.0945\n",
       "Epoch 119/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5111 - mean_absolute_error: 0.0902\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5103 - mean_absolute_error: 0.0945\n",
       "Epoch 120/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5087 - mean_absolute_error: 0.0903\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5080 - mean_absolute_error: 0.0945\n",
       "Epoch 121/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5064 - mean_absolute_error: 0.0904\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5057 - mean_absolute_error: 0.0944\n",
       "Epoch 122/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5040 - mean_absolute_error: 0.0905\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.5035 - mean_absolute_error: 0.0944\n",
       "Epoch 123/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.5016 - mean_absolute_error: 0.0906\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.5012 - mean_absolute_error: 0.0943\n",
       "Epoch 124/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4993 - mean_absolute_error: 0.0908\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4990 - mean_absolute_error: 0.0943\n",
       "Epoch 125/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4970 - mean_absolute_error: 0.0909\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4967 - mean_absolute_error: 0.0942\n",
       "Epoch 126/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4947 - mean_absolute_error: 0.0910\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4945 - mean_absolute_error: 0.0941\n",
       "Epoch 127/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4924 - mean_absolute_error: 0.0910\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4924 - mean_absolute_error: 0.0940\n",
       "Epoch 128/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4902 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4902 - mean_absolute_error: 0.0940\n",
       "Epoch 129/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4879 - mean_absolute_error: 0.0912\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4880 - mean_absolute_error: 0.0939\n",
       "Epoch 130/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4857 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4859 - mean_absolute_error: 0.0938\n",
       "Epoch 131/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4835 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4838 - mean_absolute_error: 0.0937\n",
       "Epoch 132/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4813 - mean_absolute_error: 0.0914\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4816 - mean_absolute_error: 0.0936\n",
       "Epoch 133/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4791 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4795 - mean_absolute_error: 0.0936\n",
       "Epoch 134/300\n",
-      "1100/1100 [==============================] - 0s 7us/step - loss: 0.4770 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4775 - mean_absolute_error: 0.0935\n",
       "Epoch 135/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4749 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4754 - mean_absolute_error: 0.0934\n",
       "Epoch 136/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4727 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4734 - mean_absolute_error: 0.0933\n",
       "Epoch 137/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4706 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4713 - mean_absolute_error: 0.0932\n",
       "Epoch 138/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4686 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4693 - mean_absolute_error: 0.0931\n",
       "Epoch 139/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4665 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4673 - mean_absolute_error: 0.0931\n",
       "Epoch 140/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4644 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4653 - mean_absolute_error: 0.0930\n",
       "Epoch 141/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4624 - mean_absolute_error: 0.0916\n"
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4633 - mean_absolute_error: 0.0929\n"
      ]
     },
     {
@@ -411,147 +411,147 @@
      "output_type": "stream",
      "text": [
       "Epoch 142/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4604 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4614 - mean_absolute_error: 0.0928\n",
       "Epoch 143/300\n",
-      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4584 - mean_absolute_error: 0.0916\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4594 - mean_absolute_error: 0.0927\n",
       "Epoch 144/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4564 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4575 - mean_absolute_error: 0.0927\n",
       "Epoch 145/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4544 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4556 - mean_absolute_error: 0.0926\n",
       "Epoch 146/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4524 - mean_absolute_error: 0.0915\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4537 - mean_absolute_error: 0.0925\n",
       "Epoch 147/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4505 - mean_absolute_error: 0.0914\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4518 - mean_absolute_error: 0.0925\n",
       "Epoch 148/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4485 - mean_absolute_error: 0.0914\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4499 - mean_absolute_error: 0.0924\n",
       "Epoch 149/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4466 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4480 - mean_absolute_error: 0.0923\n",
       "Epoch 150/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4447 - mean_absolute_error: 0.0913\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4462 - mean_absolute_error: 0.0923\n",
       "Epoch 151/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4428 - mean_absolute_error: 0.0912\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4443 - mean_absolute_error: 0.0922\n",
       "Epoch 152/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4409 - mean_absolute_error: 0.0912\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4425 - mean_absolute_error: 0.0921\n",
       "Epoch 153/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4391 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4407 - mean_absolute_error: 0.0921\n",
       "Epoch 154/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4372 - mean_absolute_error: 0.0911\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4389 - mean_absolute_error: 0.0920\n",
       "Epoch 155/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4354 - mean_absolute_error: 0.0910\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4371 - mean_absolute_error: 0.0919\n",
       "Epoch 156/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4336 - mean_absolute_error: 0.0909\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4354 - mean_absolute_error: 0.0919\n",
       "Epoch 157/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4318 - mean_absolute_error: 0.0909\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4336 - mean_absolute_error: 0.0918\n",
       "Epoch 158/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4300 - mean_absolute_error: 0.0908\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4318 - mean_absolute_error: 0.0918\n",
       "Epoch 159/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4282 - mean_absolute_error: 0.0907\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4301 - mean_absolute_error: 0.0917\n",
       "Epoch 160/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4264 - mean_absolute_error: 0.0907\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4284 - mean_absolute_error: 0.0917\n",
       "Epoch 161/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4247 - mean_absolute_error: 0.0906\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4267 - mean_absolute_error: 0.0916\n",
       "Epoch 162/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4229 - mean_absolute_error: 0.0905\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4250 - mean_absolute_error: 0.0916\n",
       "Epoch 163/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4212 - mean_absolute_error: 0.0905\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4233 - mean_absolute_error: 0.0915\n",
       "Epoch 164/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4195 - mean_absolute_error: 0.0904\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.4216 - mean_absolute_error: 0.0915\n",
       "Epoch 165/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4178 - mean_absolute_error: 0.0903\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4200 - mean_absolute_error: 0.0914\n",
       "Epoch 166/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4161 - mean_absolute_error: 0.0903\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4183 - mean_absolute_error: 0.0914\n",
       "Epoch 167/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4144 - mean_absolute_error: 0.0902\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4167 - mean_absolute_error: 0.0913\n",
       "Epoch 168/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.4128 - mean_absolute_error: 0.0902\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4151 - mean_absolute_error: 0.0913\n",
       "Epoch 169/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4111 - mean_absolute_error: 0.0901\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4135 - mean_absolute_error: 0.0913\n",
       "Epoch 170/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4095 - mean_absolute_error: 0.0900\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.4119 - mean_absolute_error: 0.0912\n",
       "Epoch 171/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4079 - mean_absolute_error: 0.0900\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4103 - mean_absolute_error: 0.0912\n",
       "Epoch 172/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4063 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4087 - mean_absolute_error: 0.0911\n",
       "Epoch 173/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4047 - mean_absolute_error: 0.0899\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4071 - mean_absolute_error: 0.0911\n",
       "Epoch 174/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4031 - mean_absolute_error: 0.0898\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4056 - mean_absolute_error: 0.0910\n",
       "Epoch 175/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4015 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4040 - mean_absolute_error: 0.0910\n",
       "Epoch 176/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3999 - mean_absolute_error: 0.0897\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.4025 - mean_absolute_error: 0.0910\n",
       "Epoch 177/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3984 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.4010 - mean_absolute_error: 0.0909\n",
       "Epoch 178/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3968 - mean_absolute_error: 0.0896\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3995 - mean_absolute_error: 0.0909\n",
       "Epoch 179/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3953 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3980 - mean_absolute_error: 0.0909\n",
       "Epoch 180/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3938 - mean_absolute_error: 0.0895\n",
+      "1100/1100 [==============================] - 0s 6us/step - loss: 0.3965 - mean_absolute_error: 0.0908\n",
       "Epoch 181/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3923 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3950 - mean_absolute_error: 0.0908\n",
       "Epoch 182/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3908 - mean_absolute_error: 0.0894\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.3936 - mean_absolute_error: 0.0907\n",
       "Epoch 183/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3893 - mean_absolute_error: 0.0893\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3921 - mean_absolute_error: 0.0907\n",
       "Epoch 184/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3878 - mean_absolute_error: 0.0893\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3907 - mean_absolute_error: 0.0907\n",
       "Epoch 185/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3864 - mean_absolute_error: 0.0893\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3892 - mean_absolute_error: 0.0906\n",
       "Epoch 186/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3849 - mean_absolute_error: 0.0892\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3878 - mean_absolute_error: 0.0906\n",
       "Epoch 187/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3835 - mean_absolute_error: 0.0892\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3864 - mean_absolute_error: 0.0906\n",
       "Epoch 188/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3821 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3850 - mean_absolute_error: 0.0905\n",
       "Epoch 189/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3807 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.3836 - mean_absolute_error: 0.0905\n",
       "Epoch 190/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3793 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3823 - mean_absolute_error: 0.0905\n",
       "Epoch 191/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3779 - mean_absolute_error: 0.0891\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3809 - mean_absolute_error: 0.0904\n",
       "Epoch 192/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3765 - mean_absolute_error: 0.0890\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3795 - mean_absolute_error: 0.0904\n",
       "Epoch 193/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3751 - mean_absolute_error: 0.0890\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3782 - mean_absolute_error: 0.0904\n",
       "Epoch 194/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3737 - mean_absolute_error: 0.0890\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3768 - mean_absolute_error: 0.0903\n",
       "Epoch 195/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3724 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3755 - mean_absolute_error: 0.0903\n",
       "Epoch 196/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3711 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3742 - mean_absolute_error: 0.0903\n",
       "Epoch 197/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3697 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3729 - mean_absolute_error: 0.0902\n",
       "Epoch 198/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3684 - mean_absolute_error: 0.0889\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3716 - mean_absolute_error: 0.0902\n",
       "Epoch 199/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3671 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3703 - mean_absolute_error: 0.0902\n",
       "Epoch 200/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3658 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3690 - mean_absolute_error: 0.0901\n",
       "Epoch 201/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3645 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3678 - mean_absolute_error: 0.0901\n",
       "Epoch 202/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3632 - mean_absolute_error: 0.0888\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3665 - mean_absolute_error: 0.0901\n",
       "Epoch 203/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3620 - mean_absolute_error: 0.0887\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3653 - mean_absolute_error: 0.0900\n",
       "Epoch 204/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3607 - mean_absolute_error: 0.0887\n",
+      "1100/1100 [==============================] - 0s 5us/step - loss: 0.3640 - mean_absolute_error: 0.0900\n",
       "Epoch 205/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3594 - mean_absolute_error: 0.0887\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3628 - mean_absolute_error: 0.0900\n",
       "Epoch 206/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3582 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3616 - mean_absolute_error: 0.0900\n",
       "Epoch 207/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3570 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3604 - mean_absolute_error: 0.0899\n",
       "Epoch 208/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3557 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3592 - mean_absolute_error: 0.0899\n",
       "Epoch 209/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3545 - mean_absolute_error: 0.0886\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3580 - mean_absolute_error: 0.0899\n",
       "Epoch 210/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3533 - mean_absolute_error: 0.0885\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3568 - mean_absolute_error: 0.0898\n",
       "Epoch 211/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3521 - mean_absolute_error: 0.0885\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3556 - mean_absolute_error: 0.0898\n",
       "Epoch 212/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3509 - mean_absolute_error: 0.0885\n"
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3544 - mean_absolute_error: 0.0898\n"
      ]
     },
     {
@@ -559,147 +559,147 @@
      "output_type": "stream",
      "text": [
       "Epoch 213/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3498 - mean_absolute_error: 0.0885\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3533 - mean_absolute_error: 0.0897\n",
       "Epoch 214/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3486 - mean_absolute_error: 0.0884\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3521 - mean_absolute_error: 0.0897\n",
       "Epoch 215/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3474 - mean_absolute_error: 0.0884\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3510 - mean_absolute_error: 0.0897\n",
       "Epoch 216/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3463 - mean_absolute_error: 0.0884\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3499 - mean_absolute_error: 0.0897\n",
       "Epoch 217/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3451 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3487 - mean_absolute_error: 0.0896\n",
       "Epoch 218/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3440 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3476 - mean_absolute_error: 0.0896\n",
       "Epoch 219/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3429 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3465 - mean_absolute_error: 0.0896\n",
       "Epoch 220/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3418 - mean_absolute_error: 0.0883\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3454 - mean_absolute_error: 0.0895\n",
       "Epoch 221/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3407 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3443 - mean_absolute_error: 0.0895\n",
       "Epoch 222/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3396 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3432 - mean_absolute_error: 0.0895\n",
       "Epoch 223/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3385 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3422 - mean_absolute_error: 0.0895\n",
       "Epoch 224/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3374 - mean_absolute_error: 0.0882\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3411 - mean_absolute_error: 0.0894\n",
       "Epoch 225/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3363 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3401 - mean_absolute_error: 0.0894\n",
       "Epoch 226/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3353 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3390 - mean_absolute_error: 0.0894\n",
       "Epoch 227/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3342 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3380 - mean_absolute_error: 0.0894\n",
       "Epoch 228/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3332 - mean_absolute_error: 0.0881\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3369 - mean_absolute_error: 0.0893\n",
       "Epoch 229/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3321 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3359 - mean_absolute_error: 0.0893\n",
       "Epoch 230/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3311 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3349 - mean_absolute_error: 0.0893\n",
       "Epoch 231/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3301 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3339 - mean_absolute_error: 0.0892\n",
       "Epoch 232/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3290 - mean_absolute_error: 0.0880\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3329 - mean_absolute_error: 0.0892\n",
       "Epoch 233/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3280 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3319 - mean_absolute_error: 0.0892\n",
       "Epoch 234/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3270 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3309 - mean_absolute_error: 0.0892\n",
       "Epoch 235/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3260 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3299 - mean_absolute_error: 0.0892\n",
       "Epoch 236/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3251 - mean_absolute_error: 0.0879\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3290 - mean_absolute_error: 0.0891\n",
       "Epoch 237/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3241 - mean_absolute_error: 0.0878\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3280 - mean_absolute_error: 0.0891\n",
       "Epoch 238/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3231 - mean_absolute_error: 0.0878\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3270 - mean_absolute_error: 0.0891\n",
       "Epoch 239/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3221 - mean_absolute_error: 0.0878\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3261 - mean_absolute_error: 0.0891\n",
       "Epoch 240/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3212 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3251 - mean_absolute_error: 0.0890\n",
       "Epoch 241/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3202 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3242 - mean_absolute_error: 0.0890\n",
       "Epoch 242/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3193 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3233 - mean_absolute_error: 0.0890\n",
       "Epoch 243/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3184 - mean_absolute_error: 0.0877\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3224 - mean_absolute_error: 0.0890\n",
       "Epoch 244/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3174 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3214 - mean_absolute_error: 0.0889\n",
       "Epoch 245/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3165 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3205 - mean_absolute_error: 0.0889\n",
       "Epoch 246/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3156 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3196 - mean_absolute_error: 0.0889\n",
       "Epoch 247/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3147 - mean_absolute_error: 0.0876\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3187 - mean_absolute_error: 0.0889\n",
       "Epoch 248/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3138 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3179 - mean_absolute_error: 0.0889\n",
       "Epoch 249/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3129 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3170 - mean_absolute_error: 0.0888\n",
       "Epoch 250/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3120 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3161 - mean_absolute_error: 0.0888\n",
       "Epoch 251/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3111 - mean_absolute_error: 0.0875\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3152 - mean_absolute_error: 0.0888\n",
       "Epoch 252/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3103 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3144 - mean_absolute_error: 0.0888\n",
       "Epoch 253/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3094 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3135 - mean_absolute_error: 0.0887\n",
       "Epoch 254/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3085 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3127 - mean_absolute_error: 0.0887\n",
       "Epoch 255/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3077 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3118 - mean_absolute_error: 0.0887\n",
       "Epoch 256/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3069 - mean_absolute_error: 0.0874\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3110 - mean_absolute_error: 0.0887\n",
       "Epoch 257/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3060 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3102 - mean_absolute_error: 0.0886\n",
       "Epoch 258/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3052 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3094 - mean_absolute_error: 0.0886\n",
       "Epoch 259/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3044 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3085 - mean_absolute_error: 0.0886\n",
       "Epoch 260/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3035 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3077 - mean_absolute_error: 0.0886\n",
       "Epoch 261/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3027 - mean_absolute_error: 0.0873\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3069 - mean_absolute_error: 0.0886\n",
       "Epoch 262/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3019 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3061 - mean_absolute_error: 0.0885\n",
       "Epoch 263/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3011 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3053 - mean_absolute_error: 0.0885\n",
       "Epoch 264/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3003 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3045 - mean_absolute_error: 0.0885\n",
       "Epoch 265/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2995 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3038 - mean_absolute_error: 0.0885\n",
       "Epoch 266/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2988 - mean_absolute_error: 0.0872\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3030 - mean_absolute_error: 0.0884\n",
       "Epoch 267/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2980 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3022 - mean_absolute_error: 0.0884\n",
       "Epoch 268/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2972 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.3015 - mean_absolute_error: 0.0884\n",
       "Epoch 269/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2964 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.3007 - mean_absolute_error: 0.0884\n",
       "Epoch 270/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2957 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.3000 - mean_absolute_error: 0.0883\n",
       "Epoch 271/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2949 - mean_absolute_error: 0.0871\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2992 - mean_absolute_error: 0.0883\n",
       "Epoch 272/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2942 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2985 - mean_absolute_error: 0.0883\n",
       "Epoch 273/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2934 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2977 - mean_absolute_error: 0.0883\n",
       "Epoch 274/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2927 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2970 - mean_absolute_error: 0.0882\n",
       "Epoch 275/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2920 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2963 - mean_absolute_error: 0.0882\n",
       "Epoch 276/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2913 - mean_absolute_error: 0.0870\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2956 - mean_absolute_error: 0.0882\n",
       "Epoch 277/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2905 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2948 - mean_absolute_error: 0.0882\n",
       "Epoch 278/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2898 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2941 - mean_absolute_error: 0.0881\n",
       "Epoch 279/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2891 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2934 - mean_absolute_error: 0.0881\n",
       "Epoch 280/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2884 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2927 - mean_absolute_error: 0.0881\n",
       "Epoch 281/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2877 - mean_absolute_error: 0.0869\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2920 - mean_absolute_error: 0.0881\n",
       "Epoch 282/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2870 - mean_absolute_error: 0.0868\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2914 - mean_absolute_error: 0.0881\n",
       "Epoch 283/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2863 - mean_absolute_error: 0.0868\n"
+      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2907 - mean_absolute_error: 0.0880\n"
      ]
     },
     {
@@ -707,45 +707,45 @@
      "output_type": "stream",
      "text": [
       "Epoch 284/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2856 - mean_absolute_error: 0.0868\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2900 - mean_absolute_error: 0.0880\n",
       "Epoch 285/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2850 - mean_absolute_error: 0.0867\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2893 - mean_absolute_error: 0.0880\n",
       "Epoch 286/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2843 - mean_absolute_error: 0.0867\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2887 - mean_absolute_error: 0.0880\n",
       "Epoch 287/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2836 - mean_absolute_error: 0.0866\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2880 - mean_absolute_error: 0.0879\n",
       "Epoch 288/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2830 - mean_absolute_error: 0.0866\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2873 - mean_absolute_error: 0.0879\n",
       "Epoch 289/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2823 - mean_absolute_error: 0.0866\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2867 - mean_absolute_error: 0.0879\n",
       "Epoch 290/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2816 - mean_absolute_error: 0.0865\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2860 - mean_absolute_error: 0.0879\n",
       "Epoch 291/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2810 - mean_absolute_error: 0.0865\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2854 - mean_absolute_error: 0.0879\n",
       "Epoch 292/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2803 - mean_absolute_error: 0.0865\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2848 - mean_absolute_error: 0.0878\n",
       "Epoch 293/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2797 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2841 - mean_absolute_error: 0.0878\n",
       "Epoch 294/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2791 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2835 - mean_absolute_error: 0.0878\n",
       "Epoch 295/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2784 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2829 - mean_absolute_error: 0.0878\n",
       "Epoch 296/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2778 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2823 - mean_absolute_error: 0.0878\n",
       "Epoch 297/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2772 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2817 - mean_absolute_error: 0.0877\n",
       "Epoch 298/300\n",
-      "1100/1100 [==============================] - 0s 4us/step - loss: 0.2766 - mean_absolute_error: 0.0864\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2810 - mean_absolute_error: 0.0877\n",
       "Epoch 299/300\n",
-      "1100/1100 [==============================] - 0s 2us/step - loss: 0.2760 - mean_absolute_error: 0.0863\n",
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2804 - mean_absolute_error: 0.0877\n",
       "Epoch 300/300\n",
-      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2754 - mean_absolute_error: 0.0863\n"
+      "1100/1100 [==============================] - 0s 3us/step - loss: 0.2798 - mean_absolute_error: 0.0877\n"
      ]
     },
     {
      "data": {
       "text/plain": [
-       "<keras.callbacks.callbacks.History at 0x7f1f34672518>"
+       "<keras.callbacks.callbacks.History at 0x7f9e41c6ffd0>"
       ]
      },
      "execution_count": 6,
@@ -766,8 +766,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "1100/1100 [==============================] - 0s 50us/step\n",
-      "Mean Absolute Error Percentage: 0.006589039818931172\n"
+      "1100/1100 [==============================] - 0s 51us/step\n",
+      "Mean Absolute Error Percentage: 0.006692540895847874\n"
      ]
     }
    ],
@@ -794,13 +794,216 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[0.0627239 ]\n",
-      " [0.07016328]\n",
-      " [0.08498865]\n",
-      " ...\n",
-      " [0.03661832]\n",
-      " [0.03941888]\n",
-      " [0.03665248]]\n"
+      "[[0.05987275]\n",
+      " [0.07047856]\n",
+      " [0.08760342]\n",
+      " [0.04979923]\n",
+      " [0.05166233]\n",
+      " [0.04934117]\n",
+      " [0.07812735]\n",
+      " [0.05524576]\n",
+      " [0.04840016]\n",
+      " [0.0965617 ]\n",
+      " [0.10365805]\n",
+      " [0.08732992]\n",
+      " [0.10197255]\n",
+      " [0.07169345]\n",
+      " [0.08316922]\n",
+      " [0.08781105]\n",
+      " [0.06089982]\n",
+      " [0.05451554]\n",
+      " [0.04658008]\n",
+      " [0.06053025]\n",
+      " [0.07305136]\n",
+      " [0.06277847]\n",
+      " [0.03665629]\n",
+      " [0.098389  ]\n",
+      " [0.0868187 ]\n",
+      " [0.09067354]\n",
+      " [0.10324144]\n",
+      " [0.0848611 ]\n",
+      " [0.09628972]\n",
+      " [0.06930184]\n",
+      " [0.04741266]\n",
+      " [0.0641183 ]\n",
+      " [0.07043156]\n",
+      " [0.08346471]\n",
+      " [0.06404978]\n",
+      " [0.0701538 ]\n",
+      " [0.04513595]\n",
+      " [0.05924329]\n",
+      " [0.06592053]\n",
+      " [0.06626862]\n",
+      " [0.03470093]\n",
+      " [0.04479468]\n",
+      " [0.02683493]\n",
+      " [0.02569604]\n",
+      " [0.07547492]\n",
+      " [0.06036592]\n",
+      " [0.06471756]\n",
+      " [0.10437256]\n",
+      " [0.07415253]\n",
+      " [0.07223606]\n",
+      " [0.08091393]\n",
+      " [0.08793288]\n",
+      " [0.06360474]\n",
+      " [0.0709784 ]\n",
+      " [0.04671657]\n",
+      " [0.05989459]\n",
+      " [0.04279709]\n",
+      " [0.08662194]\n",
+      " [0.08666131]\n",
+      " [0.06101134]\n",
+      " [0.08401617]\n",
+      " [0.07692432]\n",
+      " [0.06763563]\n",
+      " [0.0451411 ]\n",
+      " [0.06105676]\n",
+      " [0.08719212]\n",
+      " [0.05750605]\n",
+      " [0.0865581 ]\n",
+      " [0.08127308]\n",
+      " [0.0760203 ]\n",
+      " [0.08066425]\n",
+      " [0.07768643]\n",
+      " [0.07854018]\n",
+      " [0.06418386]\n",
+      " [0.06303421]\n",
+      " [0.05167019]\n",
+      " [0.0704079 ]\n",
+      " [0.06521657]\n",
+      " [0.04750559]\n",
+      " [0.05712697]\n",
+      " [0.06134903]\n",
+      " [0.07936773]\n",
+      " [0.04882196]\n",
+      " [0.05709308]\n",
+      " [0.08182999]\n",
+      " [0.06822082]\n",
+      " [0.05949762]\n",
+      " [0.03348103]\n",
+      " [0.05904639]\n",
+      " [0.08303881]\n",
+      " [0.07341632]\n",
+      " [0.09511766]\n",
+      " [0.09009409]\n",
+      " [0.08403727]\n",
+      " [0.07363299]\n",
+      " [0.07215634]\n",
+      " [0.07517365]\n",
+      " [0.10830417]\n",
+      " [0.0831027 ]\n",
+      " [0.06821123]\n",
+      " [0.07043773]\n",
+      " [0.06828451]\n",
+      " [0.0673196 ]\n",
+      " [0.07388246]\n",
+      " [0.05345121]\n",
+      " [0.06180677]\n",
+      " [0.04424304]\n",
+      " [0.05934063]\n",
+      " [0.05631447]\n",
+      " [0.07077795]\n",
+      " [0.06338841]\n",
+      " [0.0643796 ]\n",
+      " [0.07952979]\n",
+      " [0.06385201]\n",
+      " [0.03907123]\n",
+      " [0.0926834 ]\n",
+      " [0.07305819]\n",
+      " [0.09337723]\n",
+      " [0.07430211]\n",
+      " [0.08875009]\n",
+      " [0.06230545]\n",
+      " [0.06216493]\n",
+      " [0.08424464]\n",
+      " [0.06766856]\n",
+      " [0.05280164]\n",
+      " [0.05075416]\n",
+      " [0.05293822]\n",
+      " [0.0629158 ]\n",
+      " [0.04674768]\n",
+      " [0.04449037]\n",
+      " [0.10647008]\n",
+      " [0.09053552]\n",
+      " [0.09864247]\n",
+      " [0.07641321]\n",
+      " [0.07405573]\n",
+      " [0.09431157]\n",
+      " [0.05055869]\n",
+      " [0.06393123]\n",
+      " [0.06474605]\n",
+      " [0.07344761]\n",
+      " [0.05588779]\n",
+      " [0.07900479]\n",
+      " [0.06947374]\n",
+      " [0.0356732 ]\n",
+      " [0.02500269]\n",
+      " [0.04629824]\n",
+      " [0.06867462]\n",
+      " [0.08204171]\n",
+      " [0.07958868]\n",
+      " [0.05840129]\n",
+      " [0.05575734]\n",
+      " [0.07071397]\n",
+      " [0.06452814]\n",
+      " [0.06130123]\n",
+      " [0.06218821]\n",
+      " [0.05575269]\n",
+      " [0.0510855 ]\n",
+      " [0.03636277]\n",
+      " [0.02983409]\n",
+      " [0.03601119]\n",
+      " [0.07719433]\n",
+      " [0.07829916]\n",
+      " [0.06654462]\n",
+      " [0.07042193]\n",
+      " [0.09790522]\n",
+      " [0.06097481]\n",
+      " [0.07226038]\n",
+      " [0.0603691 ]\n",
+      " [0.05554828]\n",
+      " [0.04249924]\n",
+      " [0.05688986]\n",
+      " [0.0637235 ]\n",
+      " [0.0360972 ]\n",
+      " [0.04375321]\n",
+      " [0.03207108]\n",
+      " [0.03577149]\n",
+      " [0.02954644]\n",
+      " [0.04560512]\n",
+      " [0.04873624]\n",
+      " [0.08011109]\n",
+      " [0.07926044]\n",
+      " [0.07943588]\n",
+      " [0.06822565]\n",
+      " [0.057161  ]\n",
+      " [0.06316799]\n",
+      " [0.02389175]\n",
+      " [0.03852141]\n",
+      " [0.0356386 ]\n",
+      " [0.03183255]\n",
+      " [0.04871386]\n",
+      " [0.03614715]\n",
+      " [0.0496394 ]\n",
+      " [0.03470093]\n",
+      " [0.02884576]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.0327442 ]\n",
+      " [0.0327442 ]\n",
+      " [0.03470093]\n",
+      " [0.02569604]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.03192452]\n",
+      " [0.03470093]\n",
+      " [0.06807911]\n",
+      " [0.03470093]\n",
+      " [0.0341661 ]\n",
+      " [0.03470089]\n",
+      " [0.03035369]]\n"
      ]
     }
    ],
@@ -830,25 +1033,29 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 16,
    "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "54\n"
-     ]
-    }
-   ],
+   "outputs": [],
+   "source": [
+    "test_X = loadtxt(\"features-test.csv\", delimiter=',')\n",
+    "T_X = test_X[:, 0:nFeatures]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "predictions = model.predict(T_X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [],
    "source": [
-    "count = 0\n",
-    "for i in range(len(predictions)):\n",
-    "    if predictions[i][0] > 0.065 and CV_Y[i][0] == 1:\n",
-    "        count+=1\n",
-    "#     elif predictions[i][0] < 0.04 and CV_Y[i][0] == 0:\n",
-    "#         count+=1\n",
-    "print(count)\n",
     "a = [0 for i in range(len(predictions))]\n",
     "for i in range(len(predictions)):\n",
     "    a[i] = predictions[i][0] > 0.065\n"
@@ -856,7 +1063,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 99,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -937,131 +1144,95 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 101,
-   "metadata": {
-    "collapsed": true
-   },
+   "execution_count": 22,
+   "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[('Classification', 'NN')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('Periodicity', 'NN')]\n",
-      "[('Properties', 'NNS')]\n",
-      "[('Unit', 'NN')]\n",
-      "[('concept', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('accordance', 'NN')]\n",
-      "[('properties', 'NNS')]\n",
-      "[('development', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('remarkable', 'JJ'), ('demonstration', 'NN')]\n",
-      "[('fact', 'NN')]\n",
-      "[('chemical', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('random', 'NN'), ('cluster', 'NN')]\n",
-      "[('entities', 'NNS')]\n",
-      "[('trends', 'NNS')]\n",
-      "[('families', 'NNS')]\n",
-      "[('Seaborg', 'NN')]\n",
-      "[('Unit', 'NN')]\n",
-      "[('historical', 'JJ'), ('development', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('today', 'NN')]\n",
-      "[('Modern', 'JJ')]\n",
+      "[('•', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Law', 'NN')]\n",
-      "[('periodic', 'JJ'), ('classification', 'NN')]\n",
-      "[('logical', 'JJ'), ('consequence', 'NN')]\n",
+      "[('•', 'NN')]\n",
+      "[('significance', 'NN')]\n",
+      "[('atomic', 'JJ'), ('number', 'NN')]\n",
       "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('3.1', 'CD')]\n",
-      "[('WE', 'NN')]\n",
-      "[('NEED', 'NN')]\n",
-      "[('TO', 'NN')]\n",
-      "[('CLASSIFY', 'NN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
+      "[('basis', 'NN')]\n",
+      "[('periodic', 'JJ'), ('classification', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('basic', 'JJ'), ('units', 'NNS')]\n",
-      "[('types', 'NNS')]\n",
-      "[('matter', 'NN')]\n",
-      "[('1800', 'CD')]\n",
-      "[('31', 'CD')]\n",
+      "[('Z', 'NN')]\n",
+      "[('100', 'CD')]\n",
+      "[('IUPAC', 'NN')]\n",
+      "[('nomenclature', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('1865', 'CD')]\n",
-      "[('number', 'NN')]\n",
-      "[('identified', 'VBN'), ('elements', 'NNS')]\n",
-      "[('63', 'CD')]\n",
-      "[('present', 'JJ'), ('114', 'CD')]\n",
+      "[('s', 'NN')]\n",
+      "[('p', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('f', 'NN')]\n",
+      "[('blocks', 'NNS')]\n",
+      "[('main', 'JJ'), ('characteristics', 'NNS')]\n",
+      "[('•', 'NN')]\n",
+      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
+      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
-      "[('known', 'VBN'), ('chemical', 'JJ'), ('facts', 'NNS')]\n",
+      "[('•', 'NN')]\n",
+      "[('reactivity', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('new', 'JJ'), ('ones', 'NNS')]\n",
-      "[('further', 'RB'), ('study', 'NN')]\n",
-      "[('3.2', 'CD')]\n",
-      "[('GENESIS', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('CLASSIFICATION', 'NN')]\n",
-      "[('Classification', 'NN')]\n",
+      "[('occurrence', 'NN')]\n",
+      "[('nature', 'NN')]\n",
+      "[('relationship', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('metallic', 'JJ'), ('character', 'NN')]\n",
+      "[('scientific', 'JJ'), ('vocabulary', 'NN')]\n",
+      "[('ideas', 'NNS')]\n",
+      "[('certain', 'JJ'), ('important', 'JJ'), ('properties', 'NNS')]\n",
+      "[('atoms', 'NNS')]\n",
+      "[('atomic/', 'NN'), ('ionic', 'JJ'), ('radii', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('valence', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('groups', 'NNS')]\n",
-      "[('development', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Law', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
-      "[('consequences', 'NNS')]\n",
-      "[('knowledge', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('scientists', 'NNS')]\n",
-      "[('observations', 'NNS')]\n",
-      "[('experiments', 'NNS')]\n",
-      "[('case', 'NN')]\n",
-      "[('middle', 'JJ'), ('element', 'NN')]\n",
-      "[('Triads', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('weight', 'NN')]\n",
-      "[('way', 'NN')]\n",
-      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
-      "[('other', 'JJ'), ('two', 'CD')]\n",
-      "[('Table', 'JJ'), ('3.1', 'CD')]\n",
+      "[('important', 'JJ'), ('concept', 'NN')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('practice', 'NN')]\n",
+      "[('everyday', 'JJ'), ('support', 'NN')]\n",
+      "[('students', 'NNS')]\n",
+      "[('new', 'JJ'), ('avenues', 'NNS')]\n",
+      "[('research', 'NN')]\n",
+      "[('professionals', 'NNS')]\n",
+      "[('succinct', 'JJ'), ('organization', 'NN')]\n",
+      "[('whole', 'JJ')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('Glenn', 'NNP')]\n",
+      "[('T', 'NN')]\n",
+      "[('Efforts', 'NNS')]\n",
+      "[('new', 'JJ'), ('elements', 'NNS')]\n",
+      "[('large', 'JJ'), ('number', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('innumerable', 'JJ'), ('compounds', 'NNS')]\n",
       "[('properties', 'NNS')]\n",
       "[('middle', 'JJ'), ('element', 'NN')]\n",
       "[('other', 'JJ'), ('two', 'CD')]\n",
       "[('members', 'NNS')]\n",
-      "[('Dobereiner', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN')]\n",
-      "[('relationship', 'NN')]\n",
-      "[('Law', 'NN')]\n",
-      "[('Triads', 'NNS')]\n",
-      "[('few', 'JJ'), ('elements', 'NNS')]\n",
-      "[('coincidence', 'NN')]\n",
-      "[('attempt', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('French', 'JJ'), ('geologist', 'NN')]\n",
-      "[('A.E.B', 'NN')]\n",
-      "[('Chancourtois', 'NN')]\n",
-      "[('1862', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('order', 'NN')]\n",
-      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
-      "[('cylindrical', 'JJ'), ('table', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('periodic', 'JJ'), ('recurrence', 'NN')]\n",
-      "[('properties', 'NNS')]\n",
       "[('much', 'JJ'), ('attention', 'NN')]\n",
-      "[('relationship', 'NN')]\n",
-      "[('eighth', 'JJ'), ('note', 'NN')]\n",
-      "[('octaves', 'NNS')]\n",
-      "[('music', 'NN')]\n",
+      "[('English', 'JJ')]\n",
+      "[('chemist', 'NN')]\n",
+      "[('John', 'NNP')]\n",
+      "[('Alexander', 'NN')]\n",
+      "[('Newlands', 'NNS')]\n",
+      "[('1865', 'CD')]\n",
+      "[('Law', 'NN')]\n",
+      "[('Octaves', 'NNS')]\n",
       "[('Newlands', 'NNS')]\n",
       "[('s', 'NN'), ('Law', 'NN')]\n",
       "[('Octaves', 'NNS')]\n",
@@ -1079,6 +1250,25 @@
       "[('Dobereiner', 'NN')]\n",
       "[('’', 'NN')]\n",
       "[('Triads', 'NNS')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Law', 'NN')]\n",
+      "[('today', 'NN')]\n",
+      "[('development', 'NN')]\n",
+      "[('Russian', 'JJ'), ('chemist', 'NN')]\n",
+      "[('Dmitri', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('German', 'JJ'), ('chemist', 'NN')]\n",
+      "[('Lothar', 'NN')]\n",
+      "[('Meyer', 'NN')]\n",
+      "[('1830-1895', 'JJ')]\n",
+      "[('chemists', 'NNS')]\n",
+      "[('1869', 'CD')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('order', 'NN')]\n",
+      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
+      "[('similarities', 'NNS')]\n",
+      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
+      "[('regular', 'JJ'), ('intervals', 'NNS')]\n",
       "[('Lothar', 'NN')]\n",
       "[('Meyer', 'NN')]\n",
       "[('physical', 'JJ'), ('properties', 'NNS')]\n",
@@ -1093,11 +1283,50 @@
       "[('change', 'NN')]\n",
       "[('length', 'NN')]\n",
       "[('pattern', 'NN')]\n",
+      "[('1868', 'CD')]\n",
+      "[('Lothar', 'NN')]\n",
+      "[('Meyer', 'NN')]\n",
+      "[('table', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('Modern', 'JJ')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('Dmitri', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('scientist', 'NN')]\n",
+      "[('development', 'NN')]\n",
+      "[('Modern', 'JJ')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('Table', 'JJ'), ('3.2', 'CD')]\n",
+      "[('Newlands', 'NNS')]\n",
+      "[('’', 'NN'), ('Octaves', 'NNS')]\n",
+      "[('Dobereiner', 'NN')]\n",
+      "[('study', 'NN')]\n",
+      "[('periodic', 'NN'), ('relationship', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Law', 'NN')]\n",
+      "[('first', 'JJ'), ('time', 'NN')]\n",
       "[('properties', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
       "[('periodic', 'JJ'), ('function', 'NN')]\n",
       "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
       "[('Mendeleev', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('horizontal', 'NN'), ('rows', 'NNS')]\n",
+      "[('vertical', 'JJ'), ('columns', 'NN')]\n",
+      "[('table', 'NN')]\n",
+      "[('order', 'NN')]\n",
+      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
+      "[('way', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('similar', 'JJ'), ('properties', 'NNS')]\n",
+      "[('same', 'JJ'), ('vertical', 'JJ'), ('column', 'NN')]\n",
+      "[('group', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
       "[('’', 'NN')]\n",
       "[('s', 'NN')]\n",
       "[('system', 'NN')]\n",
@@ -1122,11 +1351,6 @@
       "[('classification', 'NN')]\n",
       "[('order', 'NN')]\n",
       "[('atomic', 'JJ'), ('weight', 'NN')]\n",
-      "[('order', 'NN')]\n",
-      "[('atomic', 'JJ'), ('weights', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('measurements', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('similar', 'JJ'), ('properties', 'NNS')]\n",
       "[('example', 'NN')]\n",
       "[('iodine', 'NN')]\n",
       "[('lower', 'JJR'), ('atomic', 'NN')]\n",
@@ -1141,31 +1365,30 @@
       "[('bromine', 'NN')]\n",
       "[('similarities', 'NNS')]\n",
       "[('properties', 'NNS')]\n",
-      "[('example', 'NN')]\n",
+      "[('gap', 'NN')]\n",
+      "[('aluminium', 'NN')]\n",
+      "[('gap', 'NN')]\n",
+      "[('silicon', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('Eka-Aluminium', 'NN')]\n",
+      "[('Eka-Silicon', 'NN')]\n",
+      "[('Mendeleev', 'NN')]\n",
+      "[('existence', 'NN')]\n",
       "[('gallium', 'NN')]\n",
       "[('germanium', 'NN')]\n",
-      "[('time', 'NN')]\n",
-      "[('Mendeleev', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
+      "[('general', 'JJ'), ('physical', 'NN'), ('properties', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
       "[('properties', 'NNS')]\n",
       "[('Mendeleev', 'NN')]\n",
       "[('elements', 'NNS')]\n",
       "[('Table', 'JJ'), ('3.3', 'CD')]\n",
-      "[('Table', 'JJ'), ('3.3', 'CD')]\n",
-      "[('Mendeleev', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'JJ'), ('Predictions', 'NNS')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('Eka-aluminium', 'NN')]\n",
-      "[('Gallium', 'NN')]\n",
-      "[('Eka-silicon', 'NN')]\n",
-      "[('Germanium', 'NN')]\n",
+      "[('boldness', 'NN')]\n",
       "[('Mendeleev', 'NN')]\n",
       "[('’', 'NN')]\n",
+      "[('quantitative', 'JJ'), ('predictions', 'NNS')]\n",
+      "[('eventual', 'JJ'), ('success', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
-      "[('1905', 'CD')]\n",
       "[('PERIODIC', 'NN')]\n",
       "[('SYSTEM', 'NN')]\n",
       "[('THE', 'DT')]\n",
@@ -1182,17 +1405,14 @@
       "[('straight', 'JJ'), ('line', 'NN')]\n",
       "[('plot', 'NN')]\n",
       "[('vs', 'JJ'), ('atomic', 'JJ'), ('mass', 'NN')]\n",
+      "[('atomic', 'JJ'), ('number', 'NN')]\n",
+      "[('fundamental', 'JJ'), ('property', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('atomic', 'JJ'), ('mass', 'NN')]\n",
       "[('Mendeleev', 'NN')]\n",
       "[('’', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Law', 'NN')]\n",
-      "[('Modern', 'JJ')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Law', 'NN')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('periodic', 'JJ'), ('functions', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Law', 'NN')]\n",
       "[('important', 'JJ'), ('analogies', 'NNS')]\n",
@@ -1206,108 +1426,45 @@
       "[('blende', 'NN')]\n",
       "[('ore', 'NN')]\n",
       "[('uranium', 'NN')]\n",
-      "[('interest', 'NN')]\n",
-      "[('Inorganic', 'JJ')]\n",
-      "[('Chemistry', 'NN')]\n",
-      "[('creation', 'NN')]\n",
-      "[('short-lived', 'JJ'), ('elements', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('nuclear', 'JJ'), ('charge', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('protons', 'NNS')]\n",
-      "[('number', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('neutral', 'JJ'), ('atom', 'NN')]\n",
-      "[('significance', 'NN')]\n",
-      "[('quantum', 'NN')]\n",
-      "[('numbers', 'NNS')]\n",
-      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
-      "[('periodicity', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('modern', 'JJ'), ('version', 'NN')]\n",
-      "[('so-called', 'JJ'), ('“', 'NN')]\n",
-      "[('long', 'RB'), ('form', 'NN')]\n",
-      "[('”', 'NN')]\n",
+      "[('Numerous', 'JJ'), ('forms', 'NNS')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('period', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('corresponds', 'NNS')]\n",
-      "[('highest', 'JJS'), ('principal', 'JJ'), ('quantum', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('period', 'NN')]\n",
-      "[('seventh', 'JJ'), ('period', 'NN')]\n",
-      "[('sixth', 'JJ'), ('period', 'NN')]\n",
-      "[('theoretical', 'JJ'), ('maximum', 'NN')]\n",
-      "[('basis', 'NN')]\n",
-      "[('quantum', 'NN'), ('numbers', 'NNS')]\n",
-      "[('32', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
+      "[('time', 'NN')]\n",
+      "[('time', 'NN')]\n",
+      "[('recommendation', 'NN')]\n",
+      "[('International', 'NNP')]\n",
+      "[('Union', 'NNP')]\n",
+      "[('Pure', 'NN')]\n",
+      "[('Applied', 'NNP')]\n",
+      "[('Chemistry', 'NN')]\n",
+      "[('IUPAC', 'NN')]\n",
+      "[('groups', 'NNS')]\n",
+      "[('1', 'CD')]\n",
+      "[('18', 'CD')]\n",
+      "[('older', 'JJR'), ('notation', 'NN')]\n",
+      "[('groups', 'NNS')]\n",
+      "[('IA', 'NN')]\n",
+      "[('…', 'NN')]\n",
+      "[('VIIA', 'NN')]\n",
+      "[('VIII', 'NN')]\n",
+      "[('IB', 'NN')]\n",
+      "[('…', 'NN')]\n",
+      "[('VIIB', 'NN')]\n",
+      "[('0', 'CD')]\n",
       "[('synthesis', 'NN')]\n",
       "[('characterisation', 'NN')]\n",
       "[('sophisticated', 'VBN'), ('costly', 'JJ'), ('equipment', 'NN')]\n",
       "[('laboratory', 'NN')]\n",
-      "[('Such', 'JJ'), ('work', 'NN')]\n",
-      "[('competitive', 'JJ'), ('spirit', 'NN')]\n",
-      "[('laboratories', 'NNS')]\n",
-      "[('world', 'NN')]\n",
-      "[('Scientists', 'NNS')]\n",
-      "[('reliable', 'JJ'), ('data', 'NNS')]\n",
-      "[('new', 'JJ'), ('element', 'NN')]\n",
-      "[('times', 'NNS')]\n",
-      "[('discovery', 'NN')]\n",
-      "[('example', 'NN')]\n",
-      "[('American', 'JJ')]\n",
-      "[('Soviet', 'JJ'), ('scientists', 'NNS')]\n",
-      "[('credit', 'NN')]\n",
-      "[('element', 'NN')]\n",
-      "[('104', 'CD')]\n",
-      "[('Americans', 'NNS')]\n",
-      "[('Rutherfordium', 'NN')]\n",
-      "[('whereas', 'NNS'), ('Soviets', 'NNS')]\n",
-      "[('Kurchatovium', 'NN')]\n",
-      "[('such', 'JJ'), ('problems', 'NNS')]\n",
-      "[('IUPAC', 'NN')]\n",
-      "[('recommendation', 'NN')]\n",
-      "[('new', 'JJ'), ('element', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN'), ('discovery', 'NN')]\n",
-      "[('name', 'NN')]\n",
-      "[('systematic', 'JJ'), ('nomenclature', 'NN')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('element', 'NN')]\n",
-      "[('numerical', 'JJ'), ('roots', 'NNS')]\n",
-      "[('0', 'CD')]\n",
-      "[('numbers', 'NNS')]\n",
-      "[('Table', 'JJ'), ('3.4', 'CD')]\n",
-      "[('roots', 'NNS')]\n",
-      "[('order', 'NN')]\n",
-      "[('digits', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('“', 'NN'), ('ium', 'NN')]\n",
-      "[('”', 'NN')]\n",
-      "[('end', 'NN')]\n",
-      "[('IUPAC', 'NN')]\n",
-      "[('names', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('Z', 'NN')]\n",
-      "[('100', 'CD')]\n",
-      "[('Table', 'JJ'), ('3.5', 'CD')]\n",
-      "[('*', 'NN'), ('Glenn', 'NNP')]\n",
-      "[('T', 'NN')]\n",
-      "[('1951', 'CD')]\n",
-      "[('Seaborg', 'NN')]\n",
-      "[('Nobel', 'NN')]\n",
-      "[('Prize', 'VB')]\n",
-      "[('chemistry', 'NN')]\n",
-      "[('work', 'NN')]\n",
       "[('Element', 'NN')]\n",
       "[('106', 'CD')]\n",
       "[('Seaborgium', 'NN')]\n",
       "[('Sg', 'NN')]\n",
       "[('honour', 'NN')]\n",
+      "[('Table', 'JJ'), ('3.4', 'CD')]\n",
+      "[('Notation', 'NN')]\n",
+      "[('IUPAC', 'NN')]\n",
+      "[('Nomenclature', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
       "[('Table', 'JJ'), ('3.5', 'CD')]\n",
       "[('Nomenclature', 'NN')]\n",
       "[('Elements', 'NNS')]\n",
@@ -1315,6 +1472,22 @@
       "[('Number', 'NNP')]\n",
       "[('Above', 'IN')]\n",
       "[('100', 'CD')]\n",
+      "[('new', 'JJ'), ('element', 'NN')]\n",
+      "[('temporary', 'JJ'), ('name', 'NN')]\n",
+      "[('symbol', 'NN')]\n",
+      "[('consisting', 'VBG')]\n",
+      "[('three', 'CD')]\n",
+      "[('letters', 'NNS')]\n",
+      "[('permanent', 'JJ'), ('name', 'NN')]\n",
+      "[('country', 'NN')]\n",
+      "[('state', 'NN')]\n",
+      "[('country', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('tribute', 'NN')]\n",
+      "[('notable', 'JJ'), ('scientist', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
+      "[('118', 'CD')]\n",
       "[('Problem', 'NN')]\n",
       "[('3.1', 'CD')]\n",
       "[('IUPAC', 'NN')]\n",
@@ -1331,26 +1504,6 @@
       "[('0', 'CD')]\n",
       "[('bi', 'NN')]\n",
       "[('nil', 'NN')]\n",
-      "[('3.5', 'CD')]\n",
-      "[('ELECTRONIC', 'NN')]\n",
-      "[('CONFIGURATIONS', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
-      "[('THE', 'DT')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('TABLE', 'NN')]\n",
-      "[('unit', 'NN')]\n",
-      "[('electron', 'NN')]\n",
-      "[('atom', 'NN')]\n",
-      "[('set', 'NN')]\n",
-      "[('four', 'CD')]\n",
-      "[('quantum', 'NN')]\n",
-      "[('numbers', 'NNS')]\n",
-      "[('principal', 'JJ'), ('quantum', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('main', 'JJ'), ('energy', 'NN')]\n",
-      "[('level', 'NN')]\n",
-      "[('shell', 'NN')]\n",
       "[('filling', 'VBG')]\n",
       "[('electrons', 'NNS')]\n",
       "[('different', 'JJ'), ('subshells', 'NNS')]\n",
@@ -1360,13 +1513,19 @@
       "[('d', 'NN')]\n",
       "[('f', 'NN')]\n",
       "[('atom', 'NN')]\n",
-      "[('section', 'NN')]\n",
-      "[('direct', 'JJ'), ('connection', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('long', 'RB'), ('form', 'NN')]\n",
+      "[('distribution', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('orbitals', 'NNS')]\n",
+      "[('atom', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('s', 'NN')]\n",
+      "[('location', 'NN')]\n",
       "[('Periodic', 'NN')]\n",
       "[('Table', 'NN')]\n",
+      "[('quantum', 'NN'), ('numbers', 'NNS')]\n",
+      "[('last', 'JJ'), ('orbital', 'JJ')]\n",
       "[('Electronic', 'JJ'), ('Configurations', 'NNS')]\n",
       "[('Periods', 'NNS')]\n",
       "[('period', 'NN')]\n",
@@ -1375,23 +1534,6 @@
       "[('outermost', 'NN')]\n",
       "[('valence', 'NN')]\n",
       "[('shell', 'NN')]\n",
-      "[('other', 'JJ'), ('words', 'NNS')]\n",
-      "[('successive', 'JJ'), ('period', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('filling', 'VBG')]\n",
-      "[('next', 'JJ'), ('higher', 'JJR'), ('principal', 'JJ'), ('energy', 'NN')]\n",
-      "[('level', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('1', 'CD')]\n",
-      "[('=', '$'), ('2', 'CD')]\n",
-      "[('number', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('period', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('atomic', 'JJ'), ('orbitals', 'NNS')]\n",
-      "[('energy', 'NN')]\n",
-      "[('level', 'NN')]\n",
       "[('first', 'JJ'), ('period', 'NN')]\n",
       "[('n', 'NNS'), ('=', 'VBP')]\n",
       "[('1', 'CD')]\n",
@@ -1407,6 +1549,12 @@
       "[('first', 'RB'), ('shell', 'NN')]\n",
       "[('K', 'NNP')]\n",
       "[('next', 'JJ'), ('element', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('four', 'CD')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('1s22s2', 'CD')]\n",
+      "[('next', 'JJ'), ('element', 'NN')]\n",
       "[('boron', 'NN')]\n",
       "[('2p', 'CD')]\n",
       "[('orbitals', 'NNS')]\n",
@@ -1418,13 +1566,41 @@
       "[('8', 'CD')]\n",
       "[('elements', 'NNS')]\n",
       "[('second', 'JJ'), ('period', 'NN')]\n",
+      "[('third', 'JJ'), ('period', 'NN')]\n",
+      "[('n', 'NNS'), ('=', 'VBP')]\n",
+      "[('3', 'CD')]\n",
+      "[('sodium', 'NN')]\n",
+      "[('added', 'VBD'), ('electron', 'NN')]\n",
+      "[('3s', 'CD')]\n",
+      "[('orbital', 'NN')]\n",
       "[('fourth', 'JJ'), ('period', 'NN')]\n",
-      "[('krypton', 'NN')]\n",
+      "[('n', 'NNS'), ('=', 'VBP')]\n",
+      "[('4', 'CD')]\n",
+      "[('starts', 'NNS')]\n",
+      "[('potassium', 'NN')]\n",
+      "[('added', 'VBD'), ('electrons', 'NNS')]\n",
+      "[('4s', 'CD')]\n",
+      "[('orbital', 'NN')]\n",
       "[('4p', 'CD')]\n",
+      "[('orbital', 'NN')]\n",
+      "[('3d', 'CD')]\n",
       "[('orbitals', 'NNS')]\n",
-      "[('18', 'CD')]\n",
+      "[('called', 'VBN'), ('3d', 'CD')]\n",
+      "[('transition', 'NN')]\n",
+      "[('series', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('fourth', 'JJ'), ('period', 'NN')]\n",
+      "[('scandium', 'NN')]\n",
+      "[('Z', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('21', 'CD')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('3d14s2', 'CD')]\n",
+      "[('3d', 'CD')]\n",
+      "[('orbitals', 'NNS')]\n",
+      "[('zinc', 'NN')]\n",
+      "[('Z=30', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('3d104s2', 'CD')]\n",
       "[('fifth', 'JJ'), ('period', 'NN')]\n",
       "[('n', 'NNS'), ('=', 'VBP')]\n",
       "[('5', 'CD')]\n",
@@ -1438,263 +1614,117 @@
       "[('=', 'NN')]\n",
       "[('39', 'CD')]\n",
       "[('period', 'NN')]\n",
-      "[('xenon', 'NN')]\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
+      "[('xenon', 'NN')]\n",
       "[('5p', 'CD')]\n",
       "[('orbitals', 'NNS')]\n",
-      "[('sixth', 'JJ'), ('period', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('6', 'CD')]\n",
-      "[('32', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('successive', 'JJ'), ('electrons', 'NNS')]\n",
-      "[('6s', 'CD')]\n",
-      "[('4f', 'CD')]\n",
-      "[('5d', 'CD')]\n",
-      "[('6p', 'CD')]\n",
-      "[('orbitals', 'NNS')]\n",
       "[('order', 'NN')]\n",
-      "[('—', 'NN')]\n",
-      "[('4f', 'CD')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('cerium', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('58', 'CD')]\n",
-      "[('lutetium', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('71', 'CD')]\n",
-      "[('4f-inner', 'JJ'), ('transition', 'NN')]\n",
-      "[('series', 'NN')]\n",
-      "[('lanthanoid', 'NN')]\n",
-      "[('series', 'NN')]\n",
-      "[('seventh', 'JJ'), ('period', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('7', 'CD')]\n",
-      "[('sixth', 'JJ'), ('period', 'NN')]\n",
-      "[('successive', 'NN')]\n",
-      "[('7s', 'CD')]\n",
-      "[('5f', 'CD')]\n",
-      "[('6d', 'CD')]\n",
-      "[('7p', 'CD')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('man-made', 'JJ'), ('radioactive', 'JJ'), ('elements', 'NNS')]\n",
-      "[('Problem', 'NN')]\n",
-      "[('3.2', 'CD')]\n",
-      "[('How', 'WRB')]\n",
-      "[('presence', 'NN')]\n",
-      "[('18', 'CD')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('5th', 'CD'), ('period', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('n', 'NN')]\n",
-      "[('5', 'CD')]\n",
-      "[('l', 'NN')]\n",
-      "[('0', 'CD')]\n",
-      "[('1', 'CD')]\n",
-      "[('2', 'CD')]\n",
-      "[('3', 'CD')]\n",
-      "[('total', 'JJ'), ('number', 'NN')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('9', 'CD')]\n",
+      "[('energy', 'NN')]\n",
+      "[('available', 'JJ'), ('orbitals', 'NNS')]\n",
+      "[('4d', 'CD')]\n",
+      "[('5s', 'CD')]\n",
+      "[('5p', 'CD')]\n",
+      "[('increases', 'NNS')]\n",
+      "[('5s', 'CD')]\n",
+      "[('<', '$'), ('4d', 'CD')]\n",
+      "[('5p', 'CD')]\n",
       "[('maximum', 'JJ'), ('number', 'NN')]\n",
       "[('electrons', 'NNS')]\n",
       "[('18', 'CD')]\n",
       "[('18', 'CD')]\n",
       "[('elements', 'NNS')]\n",
       "[('5th', 'CD'), ('period', 'NN')]\n",
-      "[('3.6', 'CD')]\n",
-      "[('ELECTRONIC', 'NN')]\n",
-      "[('CONFIGURATIONS', 'NN')]\n",
-      "[('AND', 'CC')]\n",
-      "[('TYPES', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
-      "[('f-', 'NN'), ('BLOCKS', 'NNP')]\n",
-      "[('aufbau', 'NN')]\n",
-      "[('principle', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('theoretical', 'JJ'), ('foundation', 'NN')]\n",
-      "[('periodic', 'JJ'), ('classification', 'NN')]\n",
+      "[('example', 'NN')]\n",
+      "[('Group', 'NNP')]\n",
+      "[('1', 'CD')]\n",
       "[('elements', 'NNS')]\n",
-      "[('vertical', 'JJ'), ('column', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('family', 'NN')]\n",
-      "[('exhibit', 'NN')]\n",
-      "[('similar', 'JJ'), ('chemical', 'NN')]\n",
-      "[('behaviour', 'NN')]\n",
-      "[('similarity', 'NN')]\n",
+      "[('alkali', 'JJ'), ('metals', 'NNS')]\n",
+      "[('valence', 'NN')]\n",
+      "[('shell', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('same', 'JJ'), ('number', 'NN')]\n",
-      "[('same', 'JJ'), ('distribution', 'NN')]\n",
+      "[('four', 'CD')]\n",
+      "[('blocks', 'NNS')]\n",
+      "[('f-block', 'NN'), ('depending', 'VBG')]\n",
+      "[('type', 'NN')]\n",
+      "[('atomic', 'JJ'), ('orbitals', 'NNS')]\n",
       "[('electrons', 'NNS')]\n",
-      "[('outermost', 'NN')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('two', 'CD')]\n",
-      "[('exceptions', 'NNS')]\n",
-      "[('categorisation', 'NN')]\n",
-      "[('other', 'JJ'), ('exception', 'NN')]\n",
+      "[('other', 'JJ'), ('exception', 'NN')]\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
       "[('hydrogen', 'NN')]\n",
       "[('3.6.1', 'CD')]\n",
       "[('s-Block', 'NN')]\n",
       "[('Elements', 'NNS')]\n",
+      "[('3.6.3', 'CD')]\n",
+      "[('d-Block', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('Transition', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
       "[('Group', 'NNP')]\n",
-      "[('1', 'CD')]\n",
-      "[('alkali', 'JJ'), ('metals', 'NNS')]\n",
-      "[('Group', 'NNP')]\n",
-      "[('2', 'CD')]\n",
-      "[('alkaline', 'NN'), ('earth', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
-      "[('ns1', 'NN')]\n",
-      "[('ns2', 'NNS'), ('outermost', 'VBD')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('belong', 'NN')]\n",
-      "[('s-Block', 'JJ'), ('Elements', 'NNS')]\n",
-      "[('reactive', 'JJ'), ('metals', 'NNS')]\n",
-      "[('low', 'JJ'), ('ionization', 'NN')]\n",
-      "[('enthalpies', 'NNS')]\n",
-      "[('outermost', 'NN')]\n",
-      "[('electron', 'NN')]\n",
-      "[('1+', 'CD')]\n",
-      "[('ion', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('alkali', 'JJ'), ('metals', 'NNS')]\n",
-      "[('2+', 'CD')]\n",
-      "[('ion', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('alkaline', 'JJ'), ('earth', 'NN'), ('metals', 'NNS')]\n",
-      "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('reactivity', 'NN')]\n",
-      "[('increase', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('compounds', 'NNS')]\n",
-      "[('s-block', 'JJ'), ('elements', 'NNS')]\n",
-      "[('exception', 'NN')]\n",
-      "[('lithium', 'NN')]\n",
-      "[('beryllium', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
+      "[('3', 'CD')]\n",
+      "[('12', 'CD')]\n",
+      "[('centre', 'NN')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
       "[('coloured', 'VBN'), ('ions', 'NNS')]\n",
       "[('exhibit', 'NN')]\n",
       "[('variable', 'JJ'), ('valence', 'NN')]\n",
       "[('oxidation', 'NN'), ('states', 'NNS')]\n",
       "[('paramagnetism', 'NN')]\n",
       "[('catalysts', 'NNS')]\n",
-      "[('Zn', 'NN')]\n",
-      "[('Cd', 'NN')]\n",
-      "[('Hg', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
-      "[('d10ns2', 'NN')]\n",
-      "[('properties', 'NNS')]\n",
-      "[('transition', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('way', 'NN')]\n",
-      "[('transition', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
-      "[('bridge', 'NN')]\n",
-      "[('active', 'JJ'), ('metals', 'NNS')]\n",
-      "[('s-block', 'JJ'), ('elements', 'NNS')]\n",
-      "[('active', 'JJ'), ('elements', 'NNS')]\n",
-      "[('Groups', 'NNS')]\n",
-      "[('13', 'CD')]\n",
-      "[('14', 'CD')]\n",
-      "[('familiar', 'JJ'), ('name', 'NN')]\n",
-      "[('“', 'NN')]\n",
-      "[('Transition', 'NN')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('”', 'NN')]\n",
-      "[('Problem', 'NN')]\n",
-      "[('3.4', 'CD')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
-      "[('position', 'NN')]\n",
-      "[('periodic', 'NN')]\n",
-      "[('table', 'NN')]\n",
-      "[('following', 'VBG'), ('elements', 'NNS')]\n",
-      "[('order', 'NN')]\n",
-      "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('Si', 'NN')]\n",
-      "[('Be', 'VB')]\n",
-      "[('Mg', 'NN')]\n",
-      "[('Na', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('order', 'NN')]\n",
-      "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('Si', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('Be', 'VB')]\n",
-      "[('<', 'NN')]\n",
-      "[('Mg', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('Na', 'NN')]\n",
-      "[('3.7', 'CD')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('TRENDS', 'NNS')]\n",
-      "[('IN', 'NN')]\n",
-      "[('PROPERTIES', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('ELEMENTS', 'NNS')]\n",
-      "[('many', 'JJ'), ('observable', 'JJ'), ('patterns', 'NNS')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('early', 'JJ'), ('actinoids', 'NNS')]\n",
+      "[('corresponding', 'VBG'), ('lanthanoids', 'NNS')]\n",
+      "[('large', 'JJ'), ('number', 'NN')]\n",
+      "[('oxidation', 'NN')]\n",
+      "[('states', 'NNS')]\n",
+      "[('actinoid', 'JJ'), ('elements', 'NNS')]\n",
+      "[('3.6.5', 'CD')]\n",
+      "[('Metals', 'NNS')]\n",
+      "[('Non-metals', 'NNS')]\n",
+      "[('Metalloids', 'NNS')]\n",
+      "[('addition', 'NN')]\n",
+      "[('classification', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('group', 'NN')]\n",
-      "[('period', 'NN')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('example', 'NN')]\n",
-      "[('period', 'NN')]\n",
-      "[('chemical', 'NN'), ('reactivity', 'NN')]\n",
-      "[('Group', 'NNP')]\n",
-      "[('1', 'CD')]\n",
-      "[('metals', 'NNS')]\n",
+      "[('f-blocks', 'NNS')]\n",
+      "[('broad', 'JJ'), ('classification', 'NN')]\n",
       "[('elements', 'NNS')]\n",
-      "[('middle', 'NN')]\n",
-      "[('table', 'NN')]\n",
-      "[('maximum', 'NN')]\n",
-      "[('Group', 'NNP')]\n",
-      "[('17', 'CD')]\n",
-      "[('non-metals', 'NNS')]\n",
-      "[('section', 'NN')]\n",
+      "[('properties', 'NNS')]\n",
       "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('chemical', 'NN'), ('properties', 'NNS')]\n",
+      "[('respect', 'NN')]\n",
+      "[('ionic', 'JJ'), ('radii', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('Atomic', 'NNP')]\n",
+      "[('Radius', 'NN')]\n",
+      "[('size', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('lot', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('ball', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('cloud', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('sharp', 'JJ'), ('boundary', 'JJ')]\n",
+      "[('determination', 'NN')]\n",
+      "[('atomic', 'JJ'), ('size', 'NN')]\n",
+      "[('trends', 'NNS')]\n",
       "[('terms', 'NNS')]\n",
-      "[('number', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('energy', 'NN')]\n",
-      "[('levels', 'NNS')]\n",
-      "[('3.7.1', 'CD')]\n",
-      "[('Trends', 'NNS')]\n",
-      "[('Physical', 'JJ'), ('Properties', 'NNS')]\n",
-      "[('numerous', 'JJ'), ('physical', 'JJ'), ('properties', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('points', 'NNS')]\n",
-      "[('heats', 'NNS')]\n",
-      "[('fusion', 'NN')]\n",
-      "[('vaporization', 'NN')]\n",
+      "[('nuclear', 'JJ'), ('charge', 'NN')]\n",
       "[('energy', 'NN')]\n",
-      "[('atomization', 'NN')]\n",
-      "[('periodic', 'JJ'), ('variations', 'NNS')]\n",
-      "[('other', 'JJ'), ('words', 'NNS')]\n",
-      "[('practical', 'JJ'), ('way', 'NN')]\n",
-      "[('size', 'NN')]\n",
-      "[('individual', 'JJ'), ('atom', 'NN')]\n",
-      "[('estimate', 'NN')]\n",
+      "[('level', 'NN')]\n",
       "[('atomic', 'JJ'), ('size', 'NN')]\n",
-      "[('distance', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('combined', 'VBN'), ('state', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('second', 'JJ'), ('period', 'NN')]\n",
       "[('family', 'NN')]\n",
       "[('vertical', 'JJ'), ('column', 'NN')]\n",
       "[('periodic', 'NNS'), ('table', 'NN')]\n",
@@ -1702,68 +1732,106 @@
       "[('increases', 'NNS')]\n",
       "[('atomic', 'JJ'), ('number', 'NN')]\n",
       "[('b', 'NN')]\n",
-      "[('size', 'NN')]\n",
-      "[('anion', 'NN')]\n",
-      "[('parent', 'NN')]\n",
-      "[('atom', 'NN')]\n",
-      "[('addition', 'NN')]\n",
-      "[('one', 'CD')]\n",
-      "[('more', 'RBR'), ('electrons', 'NNS')]\n",
-      "[('increased', 'VBN'), ('repulsion', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('decrease', 'NN')]\n",
-      "[('effective', 'JJ'), ('nuclear', 'JJ'), ('charge', 'NN')]\n",
-      "[('example', 'NN')]\n",
-      "[('ionic', 'JJ'), ('radius', 'NN')]\n",
-      "[('fluoride', 'JJ'), ('ion', 'NN')]\n",
-      "[('F–', 'NN')]\n",
-      "[('136', 'CD')]\n",
-      "[('pm', 'NN')]\n",
-      "[('atomic', 'JJ'), ('radius', 'NN')]\n",
-      "[('fluorine', 'NN')]\n",
-      "[('64', 'CD')]\n",
-      "[('pm', 'NN')]\n",
-      "[('*', 'CD'), ('Two', 'CD')]\n",
-      "[('more', 'JJR'), ('species', 'NNS')]\n",
-      "[('same', 'JJ'), ('number', 'NN')]\n",
-      "[('atoms', 'NNS')]\n",
-      "[('same', 'JJ'), ('number', 'NN')]\n",
-      "[('valence', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('same', 'JJ'), ('structure', 'NN')]\n",
-      "[('nature', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('first', 'JJ'), ('ionization', 'NN')]\n",
-      "[('enthalpies', 'NNS')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
-      "[('60', 'CD')]\n",
-      "[('situation', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('alkali', 'NN')]\n",
-      "[('metals', 'NNS')]\n",
-      "[('ns-electron', 'NN')]\n",
-      "[('noble', 'JJ'), ('gas', 'NN')]\n",
-      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
+      "[('fact', 'NN')]\n",
+      "[('radii', 'NN')]\n",
+      "[('noble', 'JJ'), ('gases', 'NNS')]\n",
+      "[('covalent', 'NN')]\n",
+      "[('radii', 'NN')]\n",
+      "[('van', 'NN')]\n",
+      "[('der', 'NN')]\n",
+      "[('Waals', 'NNS')]\n",
+      "[('radii', 'NN')]\n",
+      "[('other', 'JJ'), ('elements', 'NNS')]\n",
       "[('case', 'NN')]\n",
-      "[('increase', 'NN')]\n",
+      "[('net', 'JJ'), ('repulsion', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
       "[('nuclear', 'JJ'), ('charge', 'NN')]\n",
-      "[('removal', 'NN')]\n",
-      "[('outermost', 'NN')]\n",
+      "[('ion', 'NN')]\n",
+      "[('size', 'NN')]\n",
+      "[('largest', 'JJS'), ('species', 'NNS')]\n",
+      "[('Mg', 'NN')]\n",
+      "[('smallest', 'JJS'), ('one', 'CD')]\n",
+      "[('Al3+', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('Ionization', 'NN')]\n",
+      "[('Enthalpy', 'NN')]\n",
+      "[('A', 'DT')]\n",
+      "[('quantitative', 'JJ'), ('measure', 'NN')]\n",
+      "[('tendency', 'NN')]\n",
+      "[('element', 'NN')]\n",
       "[('electron', 'NN')]\n",
-      "[('less', 'RBR'), ('energy', 'NN')]\n",
-      "[('group', 'NN')]\n",
+      "[('Ionization', 'NN')]\n",
+      "[('Enthalpy', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('isolated', 'VBN'), ('gaseous', 'JJ'), ('atom', 'NN')]\n",
+      "[('X', 'NN')]\n",
+      "[('ground', 'NN')]\n",
+      "[('state', 'NN')]\n",
+      "[('X', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('→', 'NN')]\n",
+      "[('X+', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('+', 'NN')]\n",
+      "[('e–', 'NN')]\n",
+      "[('3.1', 'CD')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('units', 'NNS')]\n",
+      "[('kJ', 'NN')]\n",
+      "[('mol–1', 'NN')]\n",
+      "[('X+', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('→', 'NN')]\n",
+      "[('X2+', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('+', 'NN')]\n",
+      "[('e–', 'NN')]\n",
+      "[('3.2', 'CD')]\n",
+      "[('Energy', 'NNP')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('atom', 'NN')]\n",
+      "[('hence', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpies', 'NNS')]\n",
+      "[('first', 'JJ'), ('ionization', 'NN')]\n",
+      "[('enthalpies', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('atomic', 'JJ'), ('numbers', 'NNS')]\n",
+      "[('60', 'CD')]\n",
+      "[('periodicity', 'NN')]\n",
+      "[('graph', 'NN')]\n",
+      "[('addition', 'NN')]\n",
+      "[('two', 'CD')]\n",
       "[('first', 'JJ'), ('ionization', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('boron', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('5', 'CD')]\n",
-      "[('beryllium', 'NN')]\n",
-      "[('Z', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('greater', 'JJR'), ('nuclear', 'JJ'), ('charge', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('decreases', 'NNS')]\n",
+      "[('group', 'NN')]\n",
+      "[('effective', 'JJ'), ('nuclear', 'JJ'), ('charge', 'NN')]\n",
+      "[('valence', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('actual', 'JJ'), ('charge', 'NN')]\n",
+      "[('nucleus', 'NN')]\n",
+      "[('“', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('valence', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('nucleus', 'NN')]\n",
+      "[('intervening', 'VBG')]\n",
+      "[('core', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('example', 'NN')]\n",
+      "[('2s', 'CD')]\n",
+      "[('electron', 'NN')]\n",
+      "[('lithium', 'NN')]\n",
+      "[('nucleus', 'NN')]\n",
+      "[('inner', 'NN'), ('core', 'NN')]\n",
+      "[('1s', 'CD')]\n",
+      "[('electrons', 'NNS')]\n",
       "[('same', 'JJ'), ('principal', 'JJ'), ('quantum', 'NN')]\n",
       "[('level', 'NN')]\n",
       "[('s-electron', 'NN')]\n",
@@ -1777,135 +1845,138 @@
       "[('ionization', 'NN')]\n",
       "[('boron', 'NN')]\n",
       "[('p-electron', 'NN')]\n",
-      "[('Predict', 'NN')]\n",
-      "[('first', 'RB'), ('∆i', 'NN')]\n",
-      "[('H', 'NN')]\n",
-      "[('value', 'NN')]\n",
-      "[('Al', 'NN')]\n",
-      "[('575', 'CD')]\n",
-      "[('760', 'CD')]\n",
-      "[('kJ', 'NN')]\n",
+      "[('2p-electron', 'JJ')]\n",
+      "[('boron', 'NN')]\n",
+      "[('removal', 'NN')]\n",
+      "[('2s-', 'JJ'), ('electron', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('boron', 'NN')]\n",
+      "[('smaller', 'JJR'), ('first', 'JJ'), ('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('Problem', 'NN')]\n",
+      "[('3.7', 'CD')]\n",
+      "[('Which', 'WDT')]\n",
+      "[('following', 'VBG')]\n",
+      "[('negative', 'JJ'), ('electron', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('P', 'NN')]\n",
+      "[('S', 'NN')]\n",
+      "[('Cl', 'NN')]\n",
+      "[('F', 'NN')]\n",
       "[('answer', 'NN')]\n",
-      "[('many', 'JJ'), ('elements', 'NNS')]\n",
-      "[('energy', 'NN')]\n",
+      "[('Solution', 'NN')]\n",
+      "[('Electron', 'NN')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('period', 'NN')]\n",
       "[('electron', 'NN')]\n",
-      "[('atom', 'NN')]\n",
+      "[('2p-orbital', 'JJ'), ('leads', 'NNS')]\n",
+      "[('greater', 'JJR'), ('repulsion', 'NN')]\n",
       "[('electron', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('negative', 'JJ'), ('electron', 'NN')]\n",
       "[('gain', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('electron', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('least', 'RBS'), ('negative', 'JJ'), ('electron', 'NN')]\n",
       "[('gain', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('size', 'NN')]\n",
+      "[('e', 'NN')]\n",
+      "[('Electronegativity', 'NN')]\n",
+      "[('A', 'DT')]\n",
+      "[('qualitative', 'JJ'), ('measure', 'NN')]\n",
+      "[('ability', 'NN')]\n",
       "[('atom', 'NN')]\n",
-      "[('increases', 'NNS')]\n",
-      "[('added', 'VBD'), ('electron', 'NN')]\n",
-      "[('nucleus', 'NN')]\n",
-      "[('case', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('3.7', 'CD')]\n",
+      "[('chemical', 'NN')]\n",
+      "[('compound', 'NN')]\n",
+      "[('shared', 'VBN'), ('electrons', 'NNS')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electron', 'NN')]\n",
       "[('gain', 'NN')]\n",
       "[('enthalpy', 'NN')]\n",
-      "[('O', 'NN')]\n",
-      "[('F', 'NN')]\n",
-      "[('succeeding', 'VBG')]\n",
-      "[('element', 'NN')]\n",
+      "[('measureable', 'JJ'), ('quantity', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('scale', 'NN')]\n",
+      "[('energy', 'NN')]\n",
       "[('electron', 'NN')]\n",
-      "[('O', 'NN')]\n",
-      "[('F', 'NN')]\n",
-      "[('added', 'VBD'), ('electron', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('quantum', 'NNS'), ('level', 'NN')]\n",
-      "[('suffers', 'NNS')]\n",
-      "[('significant', 'JJ'), ('repulsion', 'NN')]\n",
-      "[('other', 'JJ'), ('electrons', 'NNS')]\n",
-      "[('level', 'NN')]\n",
-      "[('n', 'NNS'), ('=', 'VBP')]\n",
-      "[('3', 'CD')]\n",
-      "[('quantum', 'NN')]\n",
-      "[('level', 'NN')]\n",
-      "[('S', 'NN')]\n",
-      "[('Cl', 'NN')]\n",
-      "[('added', 'VBD'), ('electron', 'NN')]\n",
-      "[('larger', 'JJR'), ('region', 'NN')]\n",
-      "[('space', 'NN')]\n",
-      "[('electron-electron', 'JJ'), ('repulsion', 'NN')]\n",
-      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('figure', 'NN')]\n",
-      "[('3.7', 'CD')]\n",
-      "[('Periodicity', 'NN')]\n",
-      "[('Valence', 'NN')]\n",
-      "[('Oxidation', 'NN')]\n",
-      "[('States', 'NNS')]\n",
-      "[('valence', 'NN')]\n",
-      "[('characteristic', 'JJ'), ('property', 'NN')]\n",
-      "[('elements', 'NNS')]\n",
-      "[('terms', 'NNS')]\n",
-      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
-      "[('valence', 'NN')]\n",
-      "[('representative', 'JJ'), ('elements', 'NNS')]\n",
-      "[('number', 'NN')]\n",
-      "[('electrons', 'NNS')]\n",
-      "[('outermost', 'NN')]\n",
-      "[('orbitals', 'NNS')]\n",
-      "[('/', 'NN')]\n",
-      "[('eight', 'CD')]\n",
-      "[('minus', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('outermost', 'NN'), ('electrons', 'NNS')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('Silicon', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('14', 'CD')]\n",
-      "[('element', 'NN')]\n",
-      "[('valence', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('bromine', 'NN'), ('belongs', 'NNS')]\n",
-      "[('halogen', 'NN')]\n",
-      "[('family', 'NN')]\n",
-      "[('valence', 'NN')]\n",
-      "[('1', 'CD')]\n",
+      "[('atom', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('affinity', 'NN')]\n",
+      "[('convention', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('affinity', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('negative', 'JJ'), ('sign', 'NN')]\n",
+      "[('electron', 'NN'), ('affinity', 'NN')]\n",
+      "[('absolute', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('other', 'JJ'), ('temperature', 'NN')]\n",
+      "[('T', 'NN')]\n",
+      "[('heat', 'NN')]\n",
+      "[('capacities', 'NNS')]\n",
+      "[('reactants', 'NNS')]\n",
+      "[('products', 'NNS')]\n",
+      "[('account', 'NN')]\n",
+      "[('∆egH', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('–Ae', 'NN')]\n",
+      "[('5/2', 'CD')]\n",
+      "[('RT', 'NN')]\n",
+      "[('trend', 'NN')]\n",
+      "[('ionization', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('Table', 'JJ'), ('3.8', 'CD')]\n",
+      "[('Electronegativity', 'NN')]\n",
+      "[('Values', 'NNS')]\n",
+      "[('Pauling', 'VBG')]\n",
+      "[('scale', 'NN')]\n",
+      "[('Periods', 'NNS')]\n",
+      "[('Table', 'JJ'), ('3.8', 'CD')]\n",
       "[('b', 'NN')]\n",
-      "[('Aluminium', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('13', 'CD')]\n",
-      "[('valence', 'NN')]\n",
-      "[('3', 'CD')]\n",
-      "[('sulphur', 'NN'), ('belongs', 'NNS')]\n",
-      "[('group', 'NN')]\n",
-      "[('16', 'CD')]\n",
+      "[('Electronegativity', 'NN')]\n",
+      "[('Values', 'NNS')]\n",
+      "[('Pauling', 'VBG')]\n",
+      "[('scale', 'NN')]\n",
+      "[('3.7.2', 'CD')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Trends', 'NNS')]\n",
+      "[('Chemical', 'NNP')]\n",
+      "[('Properties', 'NNS')]\n",
+      "[('Most', 'JJS')]\n",
+      "[('trends', 'NNS')]\n",
+      "[('chemical', 'NN')]\n",
+      "[('properties', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
+      "[('diagonal', 'JJ'), ('relationships', 'NNS')]\n",
+      "[('inert', 'NN'), ('pair', 'NN')]\n",
+      "[('effect', 'NN')]\n",
+      "[('effects', 'NNS')]\n",
+      "[('lanthanoid', 'JJ'), ('contraction', 'NN')]\n",
+      "[('etc', 'NN')]\n",
+      "[('term', 'NN')]\n",
+      "[('oxidation', 'NN')]\n",
+      "[('state', 'NN')]\n",
       "[('valence', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('Hence', 'NN')]\n",
-      "[('formula', 'NN')]\n",
-      "[('compound', 'NN')]\n",
-      "[('Al2S3', 'NN')]\n",
-      "[('periodic', 'JJ'), ('trends', 'NNS')]\n",
-      "[('valence', 'NN')]\n",
+      "[('Problem', 'NN')]\n",
+      "[('3.8', 'CD')]\n",
+      "[('Periodic', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('formulas', 'NNS')]\n",
+      "[('compounds', 'NNS')]\n",
+      "[('following', 'VBG'), ('pairs', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
-      "[('hydrides', 'NNS')]\n",
-      "[('oxides', 'NNS')]\n",
-      "[('Table', 'JJ'), ('3.9', 'CD')]\n",
+      "[('silicon', 'NN')]\n",
+      "[('bromine', 'NN')]\n",
       "[('b', 'NN')]\n",
-      "[('Anomalous', 'JJ')]\n",
-      "[('Properties', 'NNS')]\n",
-      "[('Second', 'JJ')]\n",
-      "[('Period', 'NN')]\n",
-      "[('Elements', 'NNS')]\n",
-      "[('first', 'RB'), ('element', 'NN')]\n",
-      "[('groups', 'NNS')]\n",
-      "[('1', 'CD')]\n",
-      "[('lithium', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('beryllium', 'NN')]\n",
-      "[('groups', 'NNS')]\n",
-      "[('boron', 'NN')]\n",
-      "[('differs', 'NNS')]\n",
-      "[('many', 'JJ'), ('respects', 'NNS')]\n",
-      "[('other', 'JJ'), ('members', 'NNS')]\n",
-      "[('respective', 'JJ'), ('group', 'NN')]\n",
+      "[('aluminium', 'NN')]\n",
+      "[('sulphur', 'NN')]\n",
       "[('example', 'NN')]\n",
       "[('lithium', 'NN')]\n",
       "[('other', 'JJ'), ('alkali', 'JJ'), ('metals', 'NNS')]\n",
@@ -1917,138 +1988,222 @@
       "[('other', 'JJ'), ('members', 'NNS')]\n",
       "[('groups', 'NNS')]\n",
       "[('ionic', 'JJ'), ('compounds', 'NNS')]\n",
-      "[('Problem', 'NN')]\n",
-      "[('3.9', 'CD')]\n",
-      "[('Are', 'NN')]\n",
-      "[('oxidation', 'NN')]\n",
-      "[('state', 'NN')]\n",
-      "[('covalency', 'NN')]\n",
-      "[('Al', 'NN')]\n",
-      "[('[', 'NN')]\n",
-      "[('AlCl', 'NN')]\n",
-      "[('H2O', 'NN')]\n",
-      "[('5', 'CD')]\n",
-      "[('2+', 'CD')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('No', 'DT')]\n",
-      "[('Oxides', 'NNS')]\n",
+      "[('fact', 'NN')]\n",
+      "[('behaviour', 'NN')]\n",
+      "[('lithium', 'NN')]\n",
+      "[('beryllium', 'NN')]\n",
+      "[('second', 'JJ'), ('element', 'NN')]\n",
+      "[('group', 'NN')]\n",
+      "[('magnesium', 'NN')]\n",
+      "[('aluminium', 'NN')]\n",
+      "[('chemical', 'NN')]\n",
+      "[('physical', 'JJ'), ('properties', 'NNS')]\n",
+      "[('manifestation', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configuration', 'NN')]\n",
       "[('elements', 'NNS')]\n",
+      "[('results', 'NNS')]\n",
+      "[('high', 'JJ'), ('chemical', 'NN'), ('reactivity', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('extremes', 'NNS')]\n",
       "[('centre', 'NN')]\n",
-      "[('Al2O3', 'NN')]\n",
-      "[('As2O3', 'NN')]\n",
-      "[('CO', 'NN')]\n",
-      "[('NO', 'DT')]\n",
-      "[('N2O', 'NN')]\n",
-      "[('Solution', 'NN')]\n",
-      "[('Na2O', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('forms', 'NNS')]\n",
-      "[('strong', 'JJ'), ('base', 'NN')]\n",
-      "[('whereas', 'NNS')]\n",
-      "[('Cl2O7', 'NN')]\n",
-      "[('forms', 'NNS')]\n",
-      "[('strong', 'JJ'), ('acid', 'NN')]\n",
-      "[('Na2O', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('H2O', 'NN')]\n",
-      "[('2NaOH', 'CD')]\n",
-      "[('Cl2O7', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('H2O', 'NN')]\n",
-      "[('2HClO4', 'CD')]\n",
-      "[('Their', 'PRP$')]\n",
-      "[('basic', 'JJ')]\n",
-      "[('acidic', 'JJ'), ('nature', 'NN')]\n",
-      "[('litmus', 'JJ'), ('paper', 'NN')]\n",
+      "[('property', 'NN')]\n",
+      "[('reducing', 'VBG')]\n",
+      "[('behaviour', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
       "[('metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('increases', 'NNS')]\n",
-      "[('group', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('left', 'NN'), ('decreases', 'NNS')]\n",
       "[('non-metallic', 'JJ'), ('character', 'NN')]\n",
-      "[('decreases', 'NNS')]\n",
-      "[('Non-metals', 'NNS')]\n",
-      "[('top', 'NN')]\n",
-      "[('periodic', 'NNS'), ('table', 'NN')]\n",
-      "[('twenty', 'NN')]\n",
-      "[('number', 'NN')]\n",
-      "[('atomic', 'JJ'), ('radii', 'NN')]\n",
-      "[('decrease', 'NN')]\n",
+      "[('increases', 'NNS')]\n",
       "[('period', 'NN')]\n",
-      "[('increase', 'NN')]\n",
-      "[('atomic', 'JJ'), ('number', 'NN')]\n",
+      "[('consequence', 'NN')]\n",
       "[('group', 'NN')]\n",
+      "[('1', 'CD')]\n",
+      "[('2', 'CD')]\n",
+      "[('metals', 'NNS')]\n",
+      "[('Four', 'CD')]\n",
+      "[('types', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('periodic', 'NNS'), ('table', 'NN')]\n",
+      "[('basis', 'NN')]\n",
+      "[('electronic', 'JJ'), ('configurations', 'NNS')]\n",
+      "[('s-block', 'NN')]\n",
+      "[('p-block', 'NN')]\n",
+      "[('d-block', 'NN')]\n",
+      "[('f-block', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('Hydrogen', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('electron', 'NN')]\n",
+      "[('1s', 'CD')]\n",
+      "[('orbital', 'JJ'), ('occupies', 'NNS')]\n",
+      "[('unique', 'JJ'), ('position', 'NN')]\n",
+      "[('periodic', 'NN')]\n",
+      "[('table', 'NN')]\n",
+      "[('Oxides', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('left', 'NN')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('right', 'NN')]\n",
+      "[('nature', 'NN')]\n",
       "[('Oxides', 'NNS')]\n",
       "[('elements', 'NNS')]\n",
       "[('centre', 'NN')]\n",
+      "[('3.15', 'CD')]\n",
+      "[('Energy', 'NNP')]\n",
+      "[('electron', 'NN')]\n",
+      "[('ground', 'NN')]\n",
+      "[('state', 'NN')]\n",
+      "[('hydrogen', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('3.22', 'CD')]\n",
+      "[('basic', 'JJ'), ('difference', 'NN')]\n",
+      "[('terms', 'NNS')]\n",
+      "[('gain', 'NN')]\n",
+      "[('enthalpy', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('3.23', 'CD')]\n",
+      "[('statement', 'NN')]\n",
+      "[('electronegativity', 'NN')]\n",
+      "[('N', 'NN')]\n",
+      "[('Pauling', 'VBG')]\n",
+      "[('scale', 'NN')]\n",
+      "[('3.0', 'CD')]\n",
+      "[('nitrogen', 'NN')]\n",
+      "[('compounds', 'NNS')]\n",
+      "[('3.24', 'CD')]\n",
+      "[('Describe', 'NN')]\n",
+      "[('theory', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('atom', 'NN')]\n",
+      "[('gains', 'NNS')]\n",
+      "[('electron', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('electron', 'NN')]\n",
+      "[('3.25', 'CD')]\n",
+      "[('Would', 'MD')]\n",
+      "[('first', 'JJ'), ('ionization', 'NN')]\n",
+      "[('enthalpies', 'NNS')]\n",
+      "[('two', 'CD')]\n",
+      "[('isotopes', 'NNS')]\n",
+      "[('same', 'JJ'), ('element', 'NN')]\n",
+      "[('answer', 'NN')]\n",
       "[('b', 'NN')]\n",
       "[('element', 'NN')]\n",
       "[('two', 'CD')]\n",
       "[('electrons', 'NNS')]\n",
+      "[('above', 'IN'), ('elements', 'NNS')]\n",
+      "[('least', 'RBS'), ('reactive', 'JJ'), ('element', 'NN')]\n",
       "[('c', 'NNS')]\n",
-      "[('element', 'NN')]\n",
-      "[('two', 'CD')]\n",
+      "[('c', 'NNS')]\n",
+      "[('block', 'NN')]\n",
+      "[('number', 'NN')]\n",
+      "[('number', 'NN')]\n",
       "[('electrons', 'NNS')]\n",
-      "[('d', 'NN')]\n",
-      "[('group', 'NN')]\n",
-      "[('metal', 'NN')]\n",
-      "[('liquid', 'NN')]\n",
-      "[('gas', 'NN')]\n",
-      "[('room', 'NN')]\n",
-      "[('temperature', 'NN')]\n",
-      "[('Table', 'NN')]\n",
-      "[('Contents', 'NNS')]\n",
-      "[('Unit', 'NN')]\n",
-      "[('3', 'CD')]\n",
-      "[('3.2', 'CD')]\n",
-      "[('GENESIS', 'NN')]\n",
-      "[('OF', 'IN')]\n",
-      "[('PERIODIC', 'NN')]\n",
-      "[('CLASSIFICATION', 'NN')]\n",
-      "[('3.7.2', 'CD')]\n",
-      "[('Periodic', 'NN')]\n",
-      "[('Trends', 'NNS')]\n",
-      "[('Chemical', 'NNP')]\n",
+      "[('subshell', 'NN')]\n",
+      "[('3.35', 'CD')]\n",
+      "[('Anything', 'NN')]\n",
+      "[('valence', 'NN')]\n",
+      "[('electrons', 'NNS')]\n",
+      "[('chemistry', 'NN')]\n",
+      "[('element', 'NN')]\n",
+      "[('Classification', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('Periodicity', 'NN')]\n",
       "[('Properties', 'NNS')]\n",
-      "[('3.7.3', 'CD')]\n",
-      "[('Periodic', 'NN')]\n",
+      "[('3.4', 'CD')]\n",
+      "[('NOMENCLATURE', 'NN')]\n",
+      "[('ELEMENTS', 'NNS')]\n",
+      "[('WITH', 'NN')]\n",
+      "[('ATOMIC', 'NN')]\n",
+      "[('NUMBERS', 'NNS')]\n",
+      "[('100', 'CD')]\n",
+      "[('3.6.1', 'CD')]\n",
+      "[('s-Block', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('3.6.2', 'CD')]\n",
+      "[('p-Block', 'NN')]\n",
+      "[('Elements', 'NNS')]\n",
+      "[('3.7.1', 'CD')]\n",
       "[('Trends', 'NNS')]\n",
-      "[('Chemical', 'NNP')]\n",
-      "[('Reactivity', 'NN')]\n",
-      "[('thousand', 'NN')]\n",
+      "[('Physical', 'JJ')]\n",
+      "[('Properties', 'NNS')]\n",
+      "[('SUMMARY', 'NN')]\n",
+      "[('Exercises', 'NNS')]\n",
+      "[('8.1', 'CD')]\n",
+      "[('Introduction', 'NN')]\n",
+      "[('lives', 'NNS')]\n",
+      "[('tendency', 'NN')]\n",
+      "[('material', 'NN'), ('objects', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('falls', 'NNS')]\n",
+      "[('towards', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('downhill', 'NN')]\n",
+      "[('clouds', 'NNS')]\n",
+      "[('fall', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('many', 'JJ'), ('other', 'JJ'), ('such', 'JJ'), ('phenomena', 'NNS')]\n",
+      "[('Italian', 'JJ'), ('Physicist', 'NN')]\n",
+      "[('Galileo', 'NN')]\n",
+      "[('fact', 'NN')]\n",
+      "[('bodies', 'NNS')]\n",
+      "[('irrespective', 'JJ')]\n",
+      "[('masses', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('constant', 'JJ'), ('acceleration', 'NN')]\n",
+      "[('public', 'JJ'), ('demonstration', 'NN')]\n",
+      "[('fact', 'NN')]\n",
+      "[('only', 'RB'), ('motion', 'NN')]\n",
+      "[('celestial', 'JJ'), ('objects', 'NNS')]\n",
+      "[('motion', 'NN')]\n",
+      "[('circle', 'NN')]\n",
+      "[('Similar', 'JJ'), ('theories', 'NNS')]\n",
+      "[('Indian', 'JJ'), ('astronomers', 'NNS')]\n",
+      "[('400', 'CD')]\n",
       "[('years', 'NNS')]\n",
-      "[('Polish', 'JJ'), ('monk', 'NN')]\n",
-      "[('Nicolas', 'NNS')]\n",
-      "[('Copernicus', 'NN')]\n",
-      "[('definitive', 'JJ'), ('model', 'NN')]\n",
-      "[('planets', 'NNS')]\n",
-      "[('circles', 'NNS')]\n",
-      "[('fixed', 'JJ'), ('central', 'JJ'), ('sun', 'NN')]\n"
+      "[('data', 'NNS')]\n",
+      "[('assistant', 'NN')]\n",
+      "[('Johannes', 'NNS')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('1571-1640', 'JJ')]\n",
+      "[('length', 'NN')]\n",
+      "[('string', 'NN')]\n",
+      "[('ends', 'NNS')]\n",
+      "[('F1', 'NN')]\n",
+      "[('F2', 'NN')]\n",
+      "[('pins', 'NNS')]\n",
+      "[('circle', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('focii', 'NN')]\n",
+      "[('merge', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('semi-major', 'JJ'), ('axis', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('circle', 'NN')]\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[('theory', 'NN')]\n",
-      "[('church', 'NN')]\n",
-      "[('notable', 'JJ'), ('amongst', 'NN')]\n",
-      "[('supporters', 'NNS')]\n",
-      "[('Galileo', 'NN')]\n",
-      "[('prosecution', 'NN')]\n",
-      "[('state', 'NN')]\n",
-      "[('beliefs', 'NNS')]\n",
-      "[('ellipse', 'NN')]\n",
-      "[('circle', 'NN')]\n",
-      "[('special', 'JJ'), ('case', 'NN')]\n",
-      "[('closed', 'VBD'), ('curve', 'NN')]\n",
-      "[('points', 'NNS')]\n",
-      "[('F1', 'NN')]\n",
-      "[('F2', 'NN')]\n",
-      "[('line', 'NN')]\n",
-      "[('ellipse', 'NN')]\n",
-      "[('points', 'NNS')]\n",
-      "[('P', 'NN')]\n",
-      "[('A', 'DT')]\n",
-      "[('b', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('observations', 'NNS')]\n",
+      "[('sun', 'NN')]\n",
+      "[('Table', 'JJ'), ('8.1', 'CD')]\n",
+      "[('Data', 'NNS')]\n",
+      "[('measurement', 'NN')]\n",
+      "[('planetary', 'JJ'), ('motions', 'NNS')]\n",
+      "[('confirm', 'NN')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('Law', 'NN')]\n",
+      "[('Periods', 'NNS')]\n",
+      "[('≡', 'NN')]\n",
+      "[('Semi-major', 'JJ'), ('axis', 'NN')]\n",
+      "[('units', 'NNS')]\n",
+      "[('1010', 'CD')]\n",
+      "[('m', 'NN')]\n",
       "[('T', 'NN')]\n",
       "[('≡', 'JJ'), ('Time', 'NNP')]\n",
       "[('period', 'NN')]\n",
@@ -2056,96 +2211,108 @@
       "[('planet', 'NN')]\n",
       "[('years', 'NNS')]\n",
       "[('y', 'NN')]\n",
+      "[('central', 'JJ'), ('force', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('planet', 'NN')]\n",
+      "[('vector', 'NN')]\n",
       "[('Sun', 'NNP')]\n",
-      "[('origin', 'NN')]\n",
-      "[('position', 'NN')]\n",
-      "[('momentum', 'NN')]\n",
       "[('planet', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('p', 'NN')]\n",
       "[('law', 'NN')]\n",
       "[('areas', 'NNS')]\n",
-      "[('8.1', 'CD')]\n",
-      "[('speed', 'NN')]\n",
+      "[('three', 'CD')]\n",
+      "[('laws', 'NNS')]\n",
+      "[('planetary', 'JJ'), ('motion', 'NN')]\n",
+      "[('painstaking', 'VBG'), ('observations', 'NNS')]\n",
+      "[('Tycho', 'NN')]\n",
+      "[('Brahe', 'NN')]\n",
+      "[('coworkers', 'NNS')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('assistant', 'NN')]\n",
+      "[('Brahe', 'NN')]\n",
+      "[('years', 'NNS')]\n",
+      "[('three', 'CD')]\n",
+      "[('planetary', 'JJ'), ('laws', 'NNS')]\n",
       "[('planet', 'NN')]\n",
-      "[('perihelion', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('Sun-planet', 'NN')]\n",
-      "[('distance', 'NN')]\n",
-      "[('SP', 'NN')]\n",
-      "[('LA', 'NNP')]\n",
-      "[('=', 'NN')]\n",
-      "[('mp', 'NN')]\n",
-      "[('rA', 'NN')]\n",
-      "[('vA', 'NN')]\n",
-      "[('two', 'CD')]\n",
-      "[('results', 'NNS')]\n",
+      "[('longer', 'JJR'), ('time', 'NN')]\n",
+      "[('BAC', 'NN')]\n",
+      "[('CPB', 'NN')]\n",
+      "[('time', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('T', 'NN')]\n",
+      "[('27.3', 'CD')]\n",
+      "[('days', 'NNS')]\n",
+      "[('Rm', 'NN')]\n",
+      "[('3.84', 'CD')]\n",
+      "[('×', 'NN')]\n",
+      "[('10\\xad8m', 'CD')]\n",
+      "[('angular', 'JJ'), ('momentum', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('torquedue', 'JJ')]\n",
+      "[('force', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('vanishes', 'NNS')]\n",
+      "[('Central', 'JJ'), ('forces', 'NNS')]\n",
+      "[('condition', 'NN')]\n",
+      "[('Two', 'CD')]\n",
+      "[('important', 'JJ'), ('results', 'NNS')]\n",
       "[('1', 'CD')]\n",
-      "[('2', 'CD')]\n",
       "[('motion', 'NN')]\n",
-      "[('planet', 'NN')]\n",
-      "[('fact', 'NN')]\n",
-      "[('result', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('well-known', 'JJ'), ('second', 'JJ'), ('law', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('Tr', 'NN')]\n",
-      "[('trejectory', 'NN')]\n",
       "[('particle', 'NN')]\n",
       "[('central', 'JJ'), ('force', 'NN')]\n",
-      "[('position', 'NN')]\n",
-      "[('P', 'NN')]\n",
-      "[('force', 'NN')]\n",
-      "[('OP', 'NN')]\n",
-      "[('O', 'NN')]\n",
-      "[('centre', 'NN')]\n",
-      "[('force', 'NN')]\n",
-      "[('origin', 'NN')]\n",
-      "[('area', 'NN')]\n",
-      "[('∆t', 'NN')]\n",
-      "[('area', 'NN')]\n",
-      "[('sector', 'NN')]\n",
-      "[('POP′', 'NN')]\n",
-      "[('≈', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('sin', 'NN')]\n",
-      "[('α', 'NN')]\n",
-      "[('PP′/2', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('v', 'NN')]\n",
-      "[('∆t/2', 'NN')]\n",
-      "[('8.3', 'CD')]\n",
+      "[('plane', 'NN')]\n",
+      "[('Mathematically', 'RB')]\n",
+      "[('Newton', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('gravitation', 'NN')]\n",
+      "[('law', 'NN')]\n",
       "[('force', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('m2', 'NN')]\n",
+      "[('point', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('2m', 'CD')]\n",
-      "[('centroid', 'NN'), ('G', 'NNP')]\n",
-      "[('triangle', 'NN')]\n",
+      "[('m1', 'NN')]\n",
+      "[('magnitude', 'NN')]\n",
+      "[('8.5', 'CD')]\n",
+      "[('one', 'CD')]\n",
+      "[('basis', 'NN')]\n",
+      "[('symmetry', 'NN')]\n",
+      "[('resultant', 'JJ'), ('force', 'NN')]\n",
+      "[('zero', 'NN')]\n",
       "[('b', 'NN')]\n",
-      "[('force', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('vertex', 'NN')]\n",
-      "[('A', 'DT')]\n",
-      "[('Take', 'VB')]\n",
-      "[('AG', 'NNP')]\n",
-      "[('=', 'NN')]\n",
-      "[('BG', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('CG', 'NN')]\n",
+      "[('calculus', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('special', 'JJ'), ('cases', 'NNS')]\n",
+      "[('simple', 'NN'), ('law', 'NN')]\n",
+      "[('results', 'NNS')]\n",
       "[('1', 'CD')]\n",
-      "[('m', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('attraction', 'NN')]\n",
+      "[('hollow', 'JJ'), ('spherical', 'NN'), ('shell', 'NN')]\n",
+      "[('uniform', 'JJ'), ('density', 'NN')]\n",
       "[('point', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('extended', 'VBN'), ('object', 'NN')]\n",
-      "[('force', 'NN')]\n",
+      "[('entire', 'JJ'), ('mass', 'NN')]\n",
+      "[('shell', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('shell', 'NN')]\n",
+      "[('Gravitational', 'JJ'), ('forces', 'NNS')]\n",
+      "[('various', 'JJ'), ('regions', 'NNS')]\n",
+      "[('shell', 'NN')]\n",
+      "[('components', 'NNS')]\n",
+      "[('line', 'NN')]\n",
       "[('point', 'NN')]\n",
       "[('mass', 'NN')]\n",
-      "[('force', 'NN')]\n",
-      "[('same', 'JJ'), ('direction', 'NN')]\n",
-      "[('calculus', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('third', 'JJ'), ('law', 'NN')]\n",
-      "[('1619', 'CD')]\n",
+      "[('centre', 'NN')]\n",
+      "[('direction', 'NN')]\n",
+      "[('prependicular', 'NN')]\n",
+      "[('line', 'NN')]\n",
+      "[('forces', 'NNS')]\n",
+      "[('point', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('shells', 'NNS')]\n",
       "[('total', 'JJ'), ('mass', 'NN')]\n",
       "[('shells', 'NNS')]\n",
       "[('mass', 'NN')]\n",
@@ -2158,31 +2325,77 @@
       "[('RE', 'NN')]\n",
       "[('gravitational', 'JJ'), ('force', 'NN')]\n",
       "[('Eq', 'NN')]\n",
-      "[('8.10', 'CD')]\n",
-      "[('8.11', 'CD')]\n",
-      "[('8.6', 'CD')]\n",
-      "[('Acceleration', 'NN')]\n",
-      "[('gravity', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('earth', 'NN')]\n",
-      "[('point', 'NN')]\n",
-      "[('mass', 'NN')]\n",
-      "[('m', 'NN')]\n",
-      "[('height', 'NN')]\n",
-      "[('h', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('earth', 'NN')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
       "[('radius', 'NN')]\n",
-      "[('earth', 'NN')]\n",
       "[('RE', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
       "[('point', 'NN')]\n",
-      "[('earth', 'NN')]\n",
-      "[('distance', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('result', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
+      "[('entire', 'JJ'), ('mass', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
       "[('centre', 'NN')]\n",
-      "[('earth', 'NN')]\n",
+      "[('Ms', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
+      "[('Ms/ME', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('RE', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('3', 'CD')]\n",
+      "[('/', 'NN'), ('RE3', 'NNP')]\n",
+      "[('8.16', 'CD')]\n",
+      "[('mass', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('cube', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('depth', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('case', 'NN')]\n",
+      "[('smaller', 'JJR'), ('sphere', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('RE–d', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('G', 'NN')]\n",
+      "[('Ms', 'NN')]\n",
+      "[('m', 'NN')]\n",
+      "[('/', 'NN')]\n",
+      "[('RE', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('8.17', 'CD')]\n",
+      "[('Ms', 'NN')]\n",
+      "[('F', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('G', 'NN')]\n",
+      "[('ME', 'NN')]\n",
+      "[('m', 'NN')]\n",
       "[('RE', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('h', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('3', 'CD')]\n",
+      "[('8.18', 'CD')]\n",
+      "[('acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('depth', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('8.19', 'CD')]\n",
       "[('earth', 'NN'), ('’', 'NNS')]\n",
       "[('s', 'NN')]\n",
       "[('surface', 'NN')]\n",
@@ -2195,14 +2408,6 @@
       "[('’', 'NN')]\n",
       "[('s', 'NN'), ('gravity', 'NN')]\n",
       "[('surface', 'NN')]\n",
-      "[('8.7', 'CD')]\n",
-      "[('Gravitational', 'JJ')]\n",
-      "[('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('notion', 'NN')]\n",
-      "[('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('energy', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('position', 'NN')]\n",
       "[('position', 'NN')]\n",
       "[('particle', 'NN')]\n",
       "[('changes', 'NNS')]\n",
@@ -2214,131 +2419,279 @@
       "[('work', 'NN')]\n",
       "[('body', 'NN')]\n",
       "[('force', 'NN')]\n",
-      "[('course', 'NN')]\n",
-      "[('machines', 'NNS')]\n",
-      "[('object', 'NN')]\n",
-      "[('much', 'JJ'), ('greater', 'JJR'), ('speeds', 'NNS')]\n",
-      "[('greater', 'JJR'), ('initial', 'JJ'), ('speed', 'NN')]\n",
-      "[('object', 'NN')]\n",
-      "[('higher', 'JJR'), ('heights', 'NNS')]\n",
-      "[('reason', 'NN')]\n",
-      "[('moon', 'NN')]\n",
-      "[('atmosphere', 'RB')]\n",
-      "[('greater', 'JJR'), ('gravitational', 'JJ'), ('pull', 'NN')]\n",
-      "[('4M', 'CD')]\n",
-      "[('mechanical', 'JJ'), ('energy', 'NN')]\n",
+      "[('forces', 'NNS')]\n",
+      "[('work', 'NN')]\n",
+      "[('path', 'NN')]\n",
+      "[('conservative', 'JJ'), ('forces', 'NNS')]\n",
+      "[('force', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('conservative', 'JJ'), ('force', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('points', 'NNS')]\n",
       "[('surface', 'NN')]\n",
-      "[('M', 'NN')]\n",
-      "[('neutral', 'JJ'), ('point', 'NN')]\n",
-      "[('N', 'NN')]\n",
-      "[('speed', 'NN')]\n",
-      "[('zero', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('distances', 'NNS')]\n",
+      "[('surface', 'NN')]\n",
+      "[('radius', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('height', 'NN'), ('h1', 'NN')]\n",
+      "[('surface', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('height', 'NN'), ('h2', 'NN')]\n",
+      "[('surface', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('m', 'NN')]\n",
+      "[('second', 'JJ'), ('position', 'NN')]\n",
+      "[('W12', 'NN')]\n",
+      "[('W12', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('Force', 'NN')]\n",
+      "[('×', 'NN')]\n",
+      "[('displacement', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('mg', 'NN')]\n",
+      "[('h2', 'NN'), ('–', 'NN')]\n",
+      "[('h1', 'NN')]\n",
+      "[('8.20', 'CD')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('height', 'NN')]\n",
+      "[('h', 'NN')]\n",
+      "[('surface', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h', 'NN')]\n",
+      "[('mgh', 'NN'), ('+', 'NN')]\n",
+      "[('Wo', 'NN')]\n",
+      "[('8.21', 'CD')]\n",
+      "[('Wo', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('constant', 'JJ')]\n",
+      "[('W12', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h2', 'NN')]\n",
+      "[('–', 'NN')]\n",
+      "[('W', 'NN')]\n",
+      "[('h1', 'NN')]\n",
+      "[('8.22', 'CD')]\n",
+      "[('work', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('difference', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('initial', 'JJ'), ('positions.Observe', 'NN')]\n",
+      "[('constant', 'JJ'), ('Wo', 'NNP')]\n",
+      "[('cancels', 'NNS')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('r', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('r1', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('r2', 'NN')]\n",
+      "[('r2', 'NN'), ('>', 'NN')]\n",
+      "[('r1', 'NN')]\n",
+      "[('vertical', 'JJ'), ('path', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.20', 'CD')]\n",
+      "[('8.24', 'CD')]\n",
+      "[('8.22', 'CD')]\n",
+      "[('8.24', 'CD')]\n",
+      "[('example', 'NN')]\n",
+      "[('application', 'NN')]\n",
+      "[('superposition', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('Answer', 'NN')]\n",
+      "[('Consider', 'VB')]\n",
+      "[('four', 'CD')]\n",
+      "[('mass', 'NN')]\n",
+      "[('m', 'NN')]\n",
+      "[('corners', 'NNS')]\n",
+      "[('square', 'NN')]\n",
+      "[('side', 'NN')]\n",
+      "[('l', 'NN')]\n",
+      "[('8.8', 'CD')]\n",
+      "[('Escape', 'NN')]\n",
+      "[('Speed', 'NN')]\n",
+      "[('stone', 'NN')]\n",
+      "[('hand', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('natural', 'JJ'), ('query', 'NN')]\n",
+      "[('mind', 'NN')]\n",
+      "[('‘', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('such', 'JJ'), ('high', 'JJ'), ('initial', 'JJ'), ('speeds', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
       "[('principle', 'NN')]\n",
       "[('conservation', 'NN')]\n",
-      "[('mechanical', 'JJ'), ('energy', 'NN')]\n",
-      "[('point', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('question', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('energy', 'NN')]\n",
+      "[('conservation', 'NN')]\n",
+      "[('Eqs', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('Vi', 'NN')]\n",
+      "[('8.29', 'CD')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('Moon', 'NNP')]\n",
+      "[('only', 'RB'), ('natural', 'JJ'), ('satellite', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('near', 'IN'), ('circular', 'JJ'), ('orbit', 'NN')]\n",
+      "[('time', 'NN')]\n",
+      "[('period', 'NN')]\n",
+      "[('27.3', 'CD')]\n",
+      "[('days', 'NNS')]\n",
+      "[('rotational', 'JJ'), ('period', 'NN')]\n",
+      "[('moon', 'NN')]\n",
+      "[('own', 'JJ'), ('axis', 'NN')]\n",
+      "[('R.H.S', 'NN')]\n",
+      "[('Eqs', 'NN')]\n",
       "[('speed', 'NN')]\n",
-      "[('projectile', 'NN')]\n",
-      "[('zero', 'NN')]\n",
-      "[('N', 'NN')]\n",
-      "[('heavier', 'JJR'), ('sphere', 'RB'), ('4', 'CD')]\n",
-      "[('M', 'NN')]\n",
-      "[('8.35', 'CD')]\n",
-      "[('8.37', 'CD')]\n",
-      "[('T', 'CD'), ('2', 'CD')]\n",
-      "[('=', 'NN'), ('k', 'NN')]\n",
-      "[('RE', 'NN')]\n",
-      "[('+', 'NN')]\n",
+      "[('V', 'NN')]\n",
       "[('h', 'NN')]\n",
-      "[('3', 'CD')]\n",
-      "[('k', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('π2', 'NN')]\n",
-      "[('/', 'NN')]\n",
-      "[('GME', 'NN')]\n",
-      "[('8.38', 'CD')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('’', 'NN')]\n",
-      "[('law', 'NN')]\n",
-      "[('periods', 'NNS')]\n",
-      "[('motion', 'NN')]\n",
-      "[('satellites', 'NNS')]\n",
+      "[('=', '$'), ('0', 'CD')]\n",
+      "[('8.36', 'CD')]\n",
+      "[('relation', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('sides', 'NNS')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('surface', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('numerical', 'JJ'), ('values', 'NNS')]\n",
-      "[('9.8', 'CD')]\n",
+      "[('h', 'NN')]\n",
+      "[('comparison', 'NN')]\n",
       "[('RE', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('6400', 'CD')]\n",
-      "[('km', 'NN'), ('.', '.')]\n",
-      "[('85', 'CD')]\n",
-      "[('minutes', 'NNS')]\n",
-      "[('8.5', 'CD')]\n",
-      "[('planet', 'NN')]\n",
-      "[('Mars', 'NNS')]\n",
-      "[('two', 'CD')]\n",
-      "[('moons', 'NNS')]\n",
-      "[('phobos', 'NNS')]\n",
-      "[('delmos', 'NN')]\n",
-      "[('phobos', 'NNS')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[('5.97×', 'CD')]\n",
+      "[('1024', 'CD')]\n",
+      "[('kg', 'NN')]\n",
+      "[('moon', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('Earth', 'NN')]\n",
+      "[('8.7', 'CD')]\n",
+      "[('Express', 'NN')]\n",
+      "[('constant', 'JJ'), ('k', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[('value', 'NN')]\n",
+      "[('k', 'NN')]\n",
+      "[('time', 'NN')]\n",
       "[('period', 'NN')]\n",
-      "[('7', 'CD')]\n",
-      "[('hours', 'NNS')]\n",
-      "[('39', 'CD')]\n",
-      "[('minutes', 'NNS')]\n",
-      "[('orbital', 'JJ'), ('radius', 'NN')]\n",
-      "[('9.4', 'CD')]\n",
-      "[('×103', 'NN')]\n",
-      "[('km', 'NN')]\n",
-      "[('length', 'NN')]\n",
-      "[('martian', 'JJ'), ('year', 'NN')]\n",
-      "[('days', 'NNS')]\n",
-      "[('∴', 'NN'), ('TM', 'NNP')]\n",
+      "[('moon', 'NN')]\n",
+      "[('T2', 'NN')]\n",
       "[('=', 'NN')]\n",
-      "[('1.52', 'CD')]\n",
-      "[('3/2', 'CD')]\n",
-      "[('365', 'CD')]\n",
-      "[('=', '$'), ('684', 'CD')]\n",
-      "[('days', 'NNS')]\n",
-      "[('orbits', 'NNS')]\n",
-      "[('planets', 'NNS')]\n",
-      "[('Mercury', 'NN')]\n",
-      "[('Mars', 'NNS')]\n",
-      "[('Pluto*', 'NN')]\n",
-      "[('derivation', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('’', 'NN')]\n",
-      "[('third', 'JJ'), ('law', 'NN')]\n",
-      "[('[', 'NN')]\n",
+      "[('1.33', 'CD')]\n",
+      "[('×', 'NN')]\n",
+      "[('10-14', 'JJ')]\n",
+      "[('3.84', 'CD')]\n",
+      "[('105', 'CD')]\n",
+      "[('3', 'CD')]\n",
+      "[('T', 'NN')]\n",
+      "[('27.3', 'CD')]\n",
+      "[('d', 'NN'), ('Note', 'NN')]\n",
       "[('Eq', 'NN')]\n",
-      "[('moon', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[('elliptical', 'JJ'), ('orbits', 'NNS')]\n",
+      "[('RE+h', 'NN')]\n",
+      "[('semi-major', 'JJ'), ('axis', 'NN')]\n",
+      "[('ellipse', 'NN')]\n",
+      "[('8.40', 'CD')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
       "[('distance', 'NN')]\n",
-      "[('3.84', 'CD')]\n",
-      "[('×', '$'), ('105', 'CD')]\n",
-      "[('km', 'NN')]\n",
+      "[('RE+h', 'NN')]\n",
+      "[('centre', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('8.40', 'CD')]\n",
-      "[('magnitude', 'NN')]\n",
+      "[('8.41', 'CD')]\n",
+      "[('K.E', 'NNP')]\n",
+      "[('P.E', 'NN')]\n",
+      "[('orbit', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
       "[('K.E', 'NNP')]\n",
-      "[('much', 'JJ'), ('energy', 'NN')]\n",
-      "[('circular', 'JJ'), ('orbit', 'NN')]\n",
-      "[('radius', 'NN')]\n",
-      "[('4RE', 'CD')]\n",
-      "[('changes', 'NNS')]\n",
-      "[('potential', 'JJ'), ('energies', 'NNS')]\n",
-      "[('8.11', 'CD')]\n",
-      "[('Geostationary', 'JJ')]\n",
-      "[('Polar', 'NN')]\n",
       "[('Satellites', 'NNS')]\n",
-      "[('interesting', 'VBG'), ('phenomenon', 'NN')]\n",
-      "[('arises', 'NNS')]\n",
-      "[('value', 'NN')]\n",
-      "[('RE+', 'NN')]\n",
+      "[('finite', 'JJ'), ('distance', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('energies', 'NNS')]\n",
+      "[('zero', 'NN')]\n",
+      "[('8.37', 'CD')]\n",
+      "[('24', 'CD')]\n",
+      "[('hours', 'NNS')]\n",
+      "[('T', 'NN')]\n",
+      "[('=', 'NN')]\n",
+      "[('24', 'CD')]\n",
+      "[('hours', 'NNS')]\n",
       "[('h', 'NN')]\n",
+      "[('35800', 'CD')]\n",
+      "[('km', 'NN')]\n",
+      "[('Satellites', 'NNS')]\n",
+      "[('circular', 'JJ'), ('orbits', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('equatorial', 'JJ'), ('plane', 'NN')]\n",
       "[('T', 'NN')]\n",
-      "[('Eq', 'NN')]\n",
-      "[('8.37', 'CD')]\n",
+      "[('=', 'NN')]\n",
       "[('24', 'CD')]\n",
       "[('hours', 'NNS')]\n",
+      "[('Geostationery', 'NN')]\n",
+      "[('Satellites', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('same', 'JJ'), ('period', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('point', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('powerful', 'JJ'), ('rockets', 'NNS')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('such', 'JJ'), ('large', 'JJ'), ('heights', 'NNS')]\n",
+      "[('earth', 'NN')]\n",
+      "[('view', 'NN')]\n",
+      "[('several', 'JJ'), ('benefits', 'NNS')]\n",
+      "[('many', 'JJ'), ('practical', 'JJ'), ('applications', 'NNS')]\n",
+      "[('India', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('s', 'NN')]\n",
+      "[('Leap', 'VB')]\n",
+      "[('Space', 'NN')]\n",
+      "[('India', 'NNP')]\n",
+      "[('space', 'NN')]\n",
+      "[('programme', 'NN')]\n",
+      "[('1962', 'CD')]\n",
+      "[('Indian', 'JJ'), ('National', 'NNP')]\n",
+      "[('Committee', 'NNP')]\n",
+      "[('Space', 'NN')]\n",
+      "[('Research', 'NN')]\n",
+      "[('Government', 'NN')]\n",
+      "[('India', 'NNP')]\n",
+      "[('Indian', 'JJ'), ('Space', 'NNP')]\n",
+      "[('Research', 'NN')]\n",
+      "[('Organisation', 'NN')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('1969', 'CD')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('role', 'NN')]\n",
+      "[('importance', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('technology', 'NN')]\n",
+      "[('nation', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('s', 'NN')]\n",
+      "[('development', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('service', 'NN')]\n",
+      "[('common', 'JJ'), ('man', 'NN')]\n",
       "[('India', 'NNP')]\n",
       "[('first', 'RB'), ('low', 'JJ'), ('orbit', 'NN')]\n",
       "[('satellite', 'NN')]\n",
@@ -2349,26 +2702,58 @@
       "[('erstwhile', 'NN')]\n",
       "[('Soviet', 'JJ')]\n",
       "[('Union', 'NNP')]\n",
-      "[('Various', 'JJ'), ('research', 'NN')]\n",
-      "[('centers', 'NNS')]\n",
-      "[('autonomous', 'JJ'), ('institutions', 'NNS')]\n",
-      "[('remote', 'NN'), ('sensing', 'VBG')]\n",
-      "[('astronomy', 'NN')]\n",
-      "[('astrophysics', 'NNS')]\n",
-      "[('atmospheric', 'JJ'), ('sciences', 'NNS')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('indigenous', 'JJ'), ('launching', 'VBG')]\n",
+      "[('vehicle', 'NN')]\n",
+      "[('1979', 'CD')]\n",
+      "[('Rohini', 'NN')]\n",
+      "[('series', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
       "[('space', 'NN')]\n",
-      "[('research', 'NN')]\n",
-      "[('aegis', 'NN')]\n",
-      "[('Department', 'NNP')]\n",
+      "[('main', 'JJ'), ('launch', 'NN'), ('site', 'NN')]\n",
+      "[('Satish', 'JJ')]\n",
+      "[('Dhawan', 'NNP')]\n",
       "[('Space', 'NN')]\n",
-      "[('Government', 'NN')]\n",
+      "[('Center', 'NNP')]\n",
+      "[('Sriharikota', 'NN')]\n",
+      "[('Andhra', 'NN')]\n",
+      "[('Pradesh', 'NN')]\n",
+      "[('tremendous', 'JJ'), ('progress', 'NN')]\n",
       "[('India', 'NNP')]\n",
-      "[('Geostationery', 'NN')]\n",
-      "[('satellite', 'NN')]\n",
-      "[('station', 'NN')]\n",
-      "[('signals', 'NNS')]\n",
-      "[('wide', 'JJ'), ('area', 'NN')]\n",
-      "[('earth', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('space', 'NN')]\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[('programme', 'NN')]\n",
+      "[('ISRO', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('six', 'CD')]\n",
+      "[('largest', 'JJS'), ('space', 'NN')]\n",
+      "[('agencies', 'NNS')]\n",
+      "[('world', 'NN')]\n",
+      "[('order', 'NN')]\n",
+      "[('complete', 'JJ'), ('self-reliance', 'NN')]\n",
+      "[('applications', 'NNS')]\n",
+      "[('cost', 'NN')]\n",
+      "[('reliable', 'JJ'), ('Polar', 'NN')]\n",
+      "[('Satellite', 'NN')]\n",
+      "[('Launch', 'NN')]\n",
+      "[('Vehicle', 'NN')]\n",
+      "[('PSLV', 'NN')]\n",
+      "[('1990s', 'CD')]\n",
+      "[('PSLV', 'NN')]\n",
+      "[('favoured', 'VBN'), ('carrier', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
+      "[('various', 'JJ'), ('countries', 'NNS')]\n",
+      "[('unprecedented', 'JJ'), ('international', 'JJ'), ('collaboration', 'NN')]\n",
+      "[('class', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
+      "[('Polar', 'NN')]\n",
+      "[('satellites', 'NNS')]\n",
       "[('low', 'JJ'), ('altitude', 'NN')]\n",
       "[('500', 'CD')]\n",
       "[('800', 'CD')]\n",
@@ -2380,117 +2765,122 @@
       "[('earth', 'NN')]\n",
       "[('axis', 'NN')]\n",
       "[('east-west', 'JJ'), ('direction', 'NN')]\n",
-      "[('Information', 'NN')]\n",
-      "[('such', 'JJ'), ('satellites', 'NNS')]\n",
-      "[('remote', 'NN'), ('sensing', 'VBG')]\n",
-      "[('meterology', 'NN')]\n",
-      "[('environmental', 'JJ'), ('studies', 'NNS')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('Weightlessness', 'NN')]\n",
+      "[('Weight', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('force', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('spring', 'NN')]\n",
-      "[('gravitational', 'JJ'), ('pull', 'NN')]\n",
+      "[('same', 'JJ'), ('principle', 'NN')]\n",
+      "[('weight', 'NN')]\n",
       "[('object', 'NN')]\n",
-      "[('turn', 'NN')]\n",
       "[('spring', 'NN')]\n",
-      "[('force', 'NN')]\n",
+      "[('balance', 'NN')]\n",
+      "[('hung', 'NN')]\n",
+      "[('fixed', 'VBN'), ('point', 'NN')]\n",
+      "[('e.g', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('exerts', 'NNS')]\n",
       "[('object', 'NN')]\n",
-      "[('upwards', 'NNS')]\n",
       "[('imagine', 'NN')]\n",
       "[('top', 'JJ'), ('end', 'NN')]\n",
       "[('balance', 'NN')]\n",
       "[('top', 'JJ'), ('ceiling', 'NN')]\n",
       "[('room', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('upward', 'RB'), ('force', 'NN')]\n",
       "[('object', 'NN')]\n",
-      "[('free', 'JJ'), ('fall', 'NN')]\n",
-      "[('phenomenon', 'NN')]\n",
-      "[('phenomenon', 'NN')]\n",
-      "[('weightlessness', 'NN')]\n",
-      "[('resultant', 'JJ'), ('force', 'NN')]\n",
-      "[('FR', 'NN')]\n",
-      "[('vector', 'NN')]\n",
-      "[('addition', 'NN')]\n",
-      "[('FR', 'NN')]\n",
-      "[('=', 'NN')]\n",
+      "[('acceleration', 'NN')]\n",
+      "[('g', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('reading', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('balance', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('spring', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('height', 'NN')]\n",
+      "[('Summary', 'JJ'), ('1', 'CD')]\n",
+      "[('Newton', 'NN')]\n",
+      "[('’', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('universal', 'JJ'), ('gravitation', 'NN')]\n",
+      "[('states', 'NNS')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('attraction', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('particles', 'NNS')]\n",
+      "[('masses', 'NNS')]\n",
+      "[('m1', 'NN')]\n",
+      "[('m2', 'NN')]\n",
+      "[('distance', 'NN')]\n",
+      "[('r', 'NN')]\n",
+      "[('magnitude', 'NN')]\n",
+      "[('G', 'NN')]\n",
+      "[('universal', 'JJ'), ('gravitational', 'JJ'), ('constant', 'NN')]\n",
+      "[('value', 'NN')]\n",
+      "[('6.672', 'CD')]\n",
+      "[('×10–11', 'NN')]\n",
+      "[('N', 'NN')]\n",
+      "[('m2', 'NN')]\n",
+      "[('kg–2', 'NN')]\n",
       "[('F1', 'NN')]\n",
-      "[('+', 'NN')]\n",
       "[('F2', 'NN')]\n",
-      "[('+', 'NN')]\n",
-      "[('……+', 'NN')]\n",
-      "[('Fn', 'NN')]\n",
-      "[('=', 'NN')]\n",
-      "[('symbol', 'NN')]\n",
-      "[('‘', 'NN')]\n",
-      "[('Σ', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('summation', 'NN')]\n",
-      "[('Kepler', 'NNP')]\n",
-      "[('’', 'NN')]\n",
-      "[('laws', 'NNS')]\n",
-      "[('planetary', 'JJ'), ('motion', 'NN')]\n",
-      "[('state', 'NN')]\n",
-      "[('planets', 'NNS')]\n",
-      "[('elliptical', 'JJ'), ('orbits', 'NNS')]\n",
-      "[('Sun', 'NNP')]\n",
-      "[('one', 'CD')]\n",
-      "[('focal', 'JJ'), ('points', 'NNS')]\n",
-      "[('b', 'NN')]\n",
-      "[('radius', 'NN')]\n",
-      "[('vector', 'NN')]\n",
-      "[('drawn', 'NN')]\n",
+      "[('individual', 'JJ'), ('forces', 'NNS')]\n",
+      "[('M1', 'NN')]\n",
+      "[('M2', 'NN')]\n",
+      "[('….Mn', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('gravitation', 'NN')]\n",
+      "[('principle', 'NN')]\n",
+      "[('superposition', 'NN')]\n",
+      "[('force', 'NN')]\n",
+      "[('other', 'JJ'), ('bodies', 'NNS')]\n",
+      "[('Most', 'JJS'), ('planets', 'NNS')]\n",
+      "[('circular', 'JJ'), ('orbits', 'NNS')]\n",
       "[('Sun', 'NNP')]\n",
-      "[('planet', 'NN')]\n",
-      "[('equal', 'JJ'), ('areas', 'NNS')]\n",
-      "[('equal', 'JJ'), ('time', 'NN')]\n",
-      "[('intervals', 'NNS')]\n",
       "[('elliptical', 'JJ'), ('orbits', 'NNS')]\n",
       "[('above', 'IN'), ('equation', 'NN')]\n",
       "[('R', 'NN')]\n",
       "[('semi-major', 'JJ'), ('axis', 'NN')]\n",
-      "[('4', 'CD')]\n",
-      "[('acceleration', 'NN')]\n",
-      "[('gravity', 'NN')]\n",
-      "[('height', 'NN')]\n",
-      "[('h', 'NN')]\n",
-      "[('earth', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN')]\n",
+      "[('total', 'JJ'), ('energy', 'NN')]\n",
+      "[('bound', 'NN')]\n",
+      "[('system', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('orbit', 'NN')]\n",
+      "[('elliptical', 'JJ'), ('orbit', 'NN')]\n",
+      "[('potential', 'JJ'), ('energies', 'NNS')]\n",
+      "[('escape', 'NN')]\n",
+      "[('speed', 'NN')]\n",
       "[('surface', 'NN')]\n",
-      "[('h', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('<', 'NN')]\n",
-      "[('RE', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('depth', 'NN')]\n",
-      "[('d', 'NN')]\n",
       "[('earth', 'NN')]\n",
-      "[('’', 'NN')]\n",
-      "[('s', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('5', 'CD')]\n",
+      "[('value', 'NN')]\n",
+      "[('11.2', 'CD')]\n",
+      "[('km', 'NN')]\n",
+      "[('s–1', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('uniform', 'JJ'), ('spherical', 'NN'), ('shell', 'NN')]\n",
+      "[('solid', 'NNS'), ('sphere', 'RB')]\n",
+      "[('symmetric', 'JJ'), ('internal', 'JJ'), ('mass', 'NN')]\n",
+      "[('distribution', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('shell', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('10', 'CD')]\n",
+      "[('particle', 'NN')]\n",
+      "[('uniform', 'JJ'), ('spherical', 'NN'), ('shell', 'NN')]\n",
       "[('gravitational', 'JJ'), ('force', 'NN')]\n",
-      "[('conservative', 'JJ'), ('force', 'NN')]\n",
-      "[('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('function', 'NN')]\n",
-      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
-      "[('two', 'CD')]\n",
-      "[('particles', 'NNS')]\n",
-      "[('distance', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('V', 'NN')]\n",
+      "[('particle', 'NN')]\n",
       "[('zero', 'NN')]\n",
-      "[('r', 'NN')]\n",
-      "[('→', 'NN')]\n",
-      "[('∞', 'NN')]\n",
-      "[('total', 'JJ'), ('potential', 'JJ'), ('energy', 'NN')]\n",
-      "[('system', 'NN')]\n",
-      "[('particles', 'NNS')]\n",
-      "[('sum', 'NN')]\n",
-      "[('energies', 'NNS')]\n",
-      "[('pairs', 'NNS')]\n",
-      "[('particles', 'NNS')]\n",
-      "[('pair', 'NN')]\n",
-      "[('term', 'NN')]\n",
-      "[('form', 'NN')]\n",
-      "[('equation', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('homogeneous', 'JJ'), ('solid', 'NNS')]\n",
+      "[('force', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
       "[('geosynchronous', 'JJ'), ('communication', 'NN')]\n",
       "[('satellite', 'NN')]\n",
       "[('moves', 'NNS')]\n",
@@ -2504,18 +2894,106 @@
       "[('’', 'NN')]\n",
       "[('s', 'NN')]\n",
       "[('centre', 'NN')]\n",
+      "[('Points', 'NNS')]\n",
+      "[('1', 'CD')]\n",
+      "[('motion', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('influence', 'NN')]\n",
+      "[('following', 'VBG'), ('quantities', 'NNS')]\n",
+      "[('Angular', 'JJ')]\n",
+      "[('momentum', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('Total', 'JJ')]\n",
+      "[('mechanical', 'JJ'), ('energy', 'NN')]\n",
+      "[('Linear', 'JJ')]\n",
+      "[('momentum', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('Angular', 'JJ'), ('momentum', 'NN')]\n",
+      "[('conservation', 'NN')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('second', 'JJ'), ('law', 'NN')]\n",
+      "[('inverse', 'NN')]\n",
+      "[('square', 'NN')]\n",
+      "[('law', 'NN')]\n",
+      "[('gravitation', 'NN')]\n",
+      "[('central', 'JJ'), ('force', 'NN')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('third', 'JJ'), ('law', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
       "[('8.1', 'CD')]\n",
       "[('T2', 'NN')]\n",
       "[('=', 'NN')]\n",
       "[('KS', 'NNP')]\n",
       "[('R3', 'NN')]\n",
-      "[('Exercises', 'NNS')]\n",
-      "[('8.1', 'CD')]\n",
-      "[('Answer', 'NN')]\n",
-      "[('following', 'VBG')]\n",
-      "[('charge', 'NN')]\n",
-      "[('electrical', 'JJ'), ('forces', 'NNS')]\n",
-      "[('hollow', 'NN'), ('conductor', 'NN')]\n",
+      "[('constant', 'JJ'), ('KS', 'NNP')]\n",
+      "[('planets', 'NNS')]\n",
+      "[('circular', 'JJ'), ('orbits', 'NNS')]\n",
+      "[('satellites', 'NNS')]\n",
+      "[('Earth', 'NN')]\n",
+      "[('[', 'NN')]\n",
+      "[('Eq', 'NN')]\n",
+      "[('8.38', 'CD')]\n",
+      "[(']', 'NN')]\n",
+      "[('4', 'CD')]\n",
+      "[('astronaut', 'NN'), ('experiences', 'NNS')]\n",
+      "[('weightlessness', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('location', 'NN')]\n",
+      "[('space', 'NN')]\n",
+      "[('astronaut', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('“', 'NN')]\n",
+      "[('free', 'JJ'), ('fall', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('Earth', 'NN')]\n",
+      "[('5', 'CD')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('particles', 'NNS')]\n",
+      "[('distance', 'NN')]\n",
+      "[('r', 'NN')]\n",
+      "[('constant', 'JJ')]\n",
+      "[('value', 'NN')]\n",
+      "[('choice', 'NN')]\n",
+      "[('choice', 'NN')]\n",
+      "[('V', 'NN')]\n",
+      "[('0', 'CD')]\n",
+      "[('r', 'NN')]\n",
+      "[('→', 'NN')]\n",
+      "[('∞', 'NN')]\n",
+      "[('Note', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('choice', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('i.e', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('infinity', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('potential', 'NN'), ('energy', 'NN')]\n",
+      "[('object', 'NN')]\n",
+      "[('total', 'JJ'), ('energy', 'NN')]\n",
+      "[('satellite', 'NN')]\n",
+      "[('8', 'CD')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('particles', 'NNS')]\n",
+      "[('force', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('finite', 'JJ'), ('rigid', 'JJ'), ('bodies', 'NNS')]\n",
+      "[('line', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('gravitational', 'JJ'), ('force', 'NN')]\n",
+      "[('particle', 'NN')]\n",
+      "[('spherical', 'JJ'), ('shell', 'NN')]\n",
+      "[('zero', 'NN')]\n",
+      "[('Gravitational', 'JJ')]\n",
+      "[('shielding', 'VBG')]\n",
       "[('8.2', 'CD')]\n",
       "[('Choose', 'VB')]\n",
       "[('correct', 'JJ'), ('alternative', 'NN')]\n",
@@ -2523,13 +3001,58 @@
       "[('gravity', 'NN')]\n",
       "[('increases/decreases', 'NNS')]\n",
       "[('altitude', 'NN')]\n",
-      "[('8.5', 'CD')]\n",
-      "[('Let', 'VB')]\n",
-      "[('galaxy', 'NN')]\n",
-      "[('2.5', 'CD')]\n",
-      "[('×', 'NN'), ('1011', 'CD')]\n",
-      "[('one', 'CD')]\n",
-      "[('solar', 'JJ'), ('mass', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('Acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('increases/decreases', 'NNS')]\n",
+      "[('depth', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('sphere', 'RB')]\n",
+      "[('uniform', 'JJ'), ('density', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('Acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('earth/mass', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('formula', 'NN')]\n",
+      "[('–G', 'NN')]\n",
+      "[('Mm', 'NN')]\n",
+      "[('1/r2', 'CD')]\n",
+      "[('1/r1', 'CD')]\n",
+      "[('more/less', 'NN'), ('accurate', 'NN')]\n",
+      "[('formulamg', 'NN')]\n",
+      "[('r2', 'NN'), ('–', 'NN')]\n",
+      "[('r1', 'NN')]\n",
+      "[('difference', 'NN')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('points', 'NNS')]\n",
+      "[('r1', 'NN'), ('distance', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('8.7', 'CD')]\n",
+      "[('escape', 'NN')]\n",
+      "[('speed', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('earth', 'NN')]\n",
+      "[('mass', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('b', 'NN')]\n",
+      "[('location', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('direction', 'NN')]\n",
+      "[('projection', 'NN')]\n",
+      "[('d', 'NN')]\n",
+      "[('height', 'NN')]\n",
+      "[('location', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('8', 'CD')]\n",
+      "[('A', 'DT')]\n",
+      "[('comet', 'NN')]\n",
+      "[('sun', 'NN')]\n",
+      "[('elliptical', 'JJ'), ('orbit', 'NN')]\n",
       "[('8.9', 'CD')]\n",
       "[('Which', 'WDT')]\n",
       "[('following', 'VBG'), ('symptoms', 'NNS')]\n",
@@ -2544,6 +3067,29 @@
       "[('d', 'NN')]\n",
       "[('orientational', 'JJ')]\n",
       "[('problem', 'NN')]\n",
+      "[('8.10', 'CD')]\n",
+      "[('following', 'VBG'), ('two', 'CD')]\n",
+      "[('exercises', 'NNS')]\n",
+      "[('correct', 'NN')]\n",
+      "[('answer', 'NN')]\n",
+      "[('ones', 'NNS')]\n",
+      "[('gravitational', 'JJ'), ('intensity', 'NN')]\n",
+      "[('centre', 'NN')]\n",
+      "[('hemispherical', 'JJ'), ('shell', 'NN')]\n",
+      "[('uniform', 'JJ'), ('mass', 'NN')]\n",
+      "[('density', 'NN')]\n",
+      "[('direction', 'NN')]\n",
+      "[('arrow', 'NN')]\n",
+      "[('Fig', 'NN')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('b', 'NN')]\n",
+      "[('c', 'NNS')]\n",
+      "[('iv', 'NN')]\n",
+      "[('0', 'CD')]\n",
+      "[('8.1', 'CD')]\n",
+      "[('LIST', 'NN')]\n",
+      "[('INDIAN', 'NN')]\n",
+      "[('SATELLITES', 'NNS')]\n",
       "[('India', 'NNP')]\n",
       "[('239', 'CD')]\n",
       "[('foreign', 'JJ'), ('satellites', 'NNS')]\n",
@@ -2559,51 +3105,137 @@
       "[('26', 'CD')]\n",
       "[('1999', 'CD')]\n",
       "[('02', 'CD')]\n",
-      "[('15', 'CD')]\n",
-      "[('2017', 'CD')]\n",
-      "[('101', 'CD')]\n",
-      "[('world', 'NN')]\n",
-      "[('record', 'NN')]\n",
-      "[('June', 'NNP')]\n",
       "[('23', 'CD')]\n",
-      "[('2017', 'CD')]\n",
-      "[('29', 'CD')]\n",
+      "[('2007', 'CD')]\n",
+      "[('01', 'CD')]\n",
+      "[('Jan', 'NN')]\n",
+      "[('20', 'CD')]\n",
+      "[('2011', 'CD')]\n",
+      "[('01', 'CD')]\n",
+      "[('Sep', 'NN')]\n",
+      "[('09', 'CD')]\n",
+      "[('2012', 'CD')]\n",
+      "[('02', 'CD')]\n",
+      "[('Feb', 'NN')]\n",
+      "[('25', 'CD')]\n",
+      "[('2013', 'CD')]\n",
+      "[('06', 'CD')]\n",
+      "[('June', 'NNP')]\n",
+      "[('30', 'CD')]\n",
+      "[('2014', 'CD')]\n",
+      "[('05', 'CD')]\n",
+      "[('July', 'NNP')]\n",
+      "[('10', 'CD')]\n",
+      "[('2015', 'CD')]\n",
+      "[('05', 'CD')]\n",
+      "[('Sep', 'NN')]\n",
+      "[('26', 'CD')]\n",
+      "[('2016', 'CD')]\n",
+      "[('05', 'CD')]\n",
+      "[('Feb', 'NN')]\n",
+      "[('Jan', 'NN')]\n",
+      "[('12', 'CD')]\n",
+      "[('2018', 'CD')]\n",
+      "[('28', 'CD')]\n",
+      "[('Sep', 'NN')]\n",
       "[('Table', 'NN')]\n",
       "[('Contents', 'NNS')]\n",
+      "[('Chapter', 'NN')]\n",
+      "[('Eight', 'CD')]\n",
+      "[('Gravitation', 'NN')]\n",
+      "[('8.1', 'CD')]\n",
+      "[('Introduction', 'NN')]\n",
+      "[('8.2', 'CD')]\n",
+      "[('Kepler', 'NNP')]\n",
+      "[('’', 'NN')]\n",
+      "[('laws', 'NNS')]\n",
+      "[('8.4', 'CD')]\n",
+      "[('Gravitational', 'JJ')]\n",
+      "[('Constant', 'JJ')]\n",
+      "[('8.5', 'CD')]\n",
+      "[('Acceleration', 'NN')]\n",
+      "[('gravity', 'NN')]\n",
+      "[('earth', 'NN')]\n",
       "[('8.6', 'CD')]\n",
       "[('Acceleration', 'NN')]\n",
       "[('gravity', 'NN')]\n",
       "[('surface', 'NN')]\n",
       "[('earth', 'NN')]\n",
+      "[('8.7', 'CD')]\n",
+      "[('Gravitational', 'JJ')]\n",
+      "[('potential', 'JJ'), ('energy', 'NN')]\n",
+      "[('8.8', 'CD')]\n",
+      "[('Escape', 'NN')]\n",
+      "[('Speed', 'NN')]\n",
       "[('8.9', 'CD')]\n",
       "[('Earth', 'NN')]\n",
       "[('Satellites', 'NNS')]\n",
+      "[('8.10', 'CD')]\n",
+      "[('Energy', 'NNP')]\n",
+      "[('Satellite', 'NN')]\n",
       "[('8.11', 'CD')]\n",
       "[('Geostationary', 'JJ')]\n",
       "[('Polar', 'NN')]\n",
       "[('Satellites', 'NNS')]\n",
-      "[('example', 'NN')]\n",
-      "[('Amoeba', 'NN')]\n",
+      "[('8.12', 'CD')]\n",
+      "[('Weightlessness', 'NN')]\n",
+      "[('Cover', 'NN')]\n",
+      "[('last', 'JJ'), ('chapter', 'NN')]\n",
+      "[('living', 'NN')]\n",
+      "[('organisms', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('unicellular', 'JJ'), ('organisms', 'NNS')]\n",
       "[('single', 'JJ'), ('cell', 'NN')]\n",
-      "[('movement', 'NN')]\n",
-      "[('intake', 'NN')]\n",
-      "[('food', 'NN')]\n",
-      "[('gaseous', 'JJ'), ('exchange', 'NN')]\n",
-      "[('excretion', 'NN')]\n",
-      "[('specialised', 'VBN'), ('function', 'NN')]\n",
-      "[('different', 'JJ'), ('group', 'NN')]\n",
+      "[('performs', 'NNS')]\n",
+      "[('basic', 'JJ'), ('functions', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
-      "[('plants', 'NNS')]\n",
-      "[('vascular', 'JJ'), ('tissues', 'NNS')]\n",
-      "[('food', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('one', 'CD')]\n",
-      "[('part', 'NN')]\n",
-      "[('plant', 'NN')]\n",
-      "[('other', 'JJ'), ('parts', 'NNS')]\n",
-      "[('6.2', 'CD')]\n",
-      "[('Plant', 'NN')]\n",
+      "[('specific', 'JJ'), ('functions', 'NNS')]\n",
+      "[('particular', 'JJ'), ('function', 'NN')]\n",
+      "[('cluster', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('definite', 'JJ'), ('place', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('group', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('structure', 'NN')]\n",
+      "[('and/or', 'NN')]\n",
+      "[('work', 'NN')]\n",
+      "[('particular', 'JJ'), ('function', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('6.1', 'CD')]\n",
+      "[('Are', 'NN'), ('Plants', 'NNS')]\n",
+      "[('Animals', 'NNS')]\n",
+      "[('Made', 'VBN')]\n",
+      "[('Same', 'NN')]\n",
+      "[('Types', 'NNS')]\n",
       "[('Tissues', 'NNS')]\n",
+      "[('structure', 'NN')]\n",
+      "[('functions', 'NNS')]\n",
+      "[('plants', 'NNS')]\n",
+      "[('animals', 'NNS')]\n",
+      "[('same', 'JJ'), ('structure', 'NN')]\n",
+      "[('similar', 'JJ'), ('functions', 'NNS')]\n",
+      "[('noticeable', 'JJ'), ('differences', 'NNS')]\n",
+      "[('two', 'CD')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('certain', 'JJ'), ('regions', 'NNS')]\n",
+      "[('capacity', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('various', 'JJ'), ('plant', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
+      "[('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Cell', 'NN')]\n",
+      "[('growth', 'NN')]\n",
+      "[('animals', 'NNS')]\n",
+      "[('fundamental', 'JJ'), ('difference', 'NN')]\n",
+      "[('different', 'JJ'), ('modes', 'NNS')]\n",
+      "[('life', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('major', 'JJ'), ('groups', 'NNS')]\n",
+      "[('organisms', 'NNS')]\n",
+      "[('different', 'JJ'), ('feeding', 'NN')]\n",
+      "[('methods', 'NNS')]\n",
       "[('Jar', 'NN')]\n",
       "[('1', 'CD')]\n",
       "[('Jar', 'NN')]\n",
@@ -2612,23 +3244,125 @@
       "[('roots', 'NNS')]\n",
       "[('onion', 'NN')]\n",
       "[('bulbs', 'NN')]\n",
-      "[('girth', 'NN')]\n"
+      "[('two', 'CD')]\n",
+      "[('onion', 'NN')]\n",
+      "[('bulbs', 'NN')]\n",
+      "[('place', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('jar', 'NN')]\n"
      ]
     },
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[('stem', 'NN')]\n",
-      "[('root', 'NN')]\n",
-      "[('increases', 'NNS')]\n",
-      "[('lateral', 'JJ'), ('meristem', 'NN')]\n",
-      "[('cambium', 'NN')]\n",
-      "[('Activity', 'NN')]\n",
-      "[('6.3', 'CD')]\n",
       "[('•', 'NN')]\n",
-      "[('leaf', 'NN')]\n",
-      "[('Rhoeo', 'NN')]\n",
+      "[('Observe', 'NN')]\n",
+      "[('growth', 'NN')]\n",
+      "[('roots', 'NNS')]\n",
+      "[('bulbs', 'NN')]\n",
+      "[('few', 'JJ'), ('days', 'NNS')]\n",
+      "[('growth', 'NN')]\n",
+      "[('roots', 'NNS')]\n",
+      "[('jars', 'NNS')]\n",
+      "[('lengths', 'NNS')]\n",
+      "[('day', 'NN')]\n",
+      "[('five', 'CD')]\n",
+      "[('more', 'RBR'), ('days', 'NNS')]\n",
+      "[('record', 'NN')]\n",
+      "[('observations', 'NNS')]\n",
+      "[('tables', 'NNS')]\n",
+      "[('table', 'NN')]\n",
+      "[('Length', 'NN')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('1', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('2', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('3', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('4', 'CD')]\n",
+      "[('Day', 'NNP')]\n",
+      "[('5', 'CD')]\n",
+      "[('Jar', 'NN')]\n",
+      "[('1', 'CD')]\n",
+      "[('Jar', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('growth', 'NN')]\n",
+      "[('plants', 'NNS')]\n",
+      "[('certain', 'JJ'), ('specific', 'JJ'), ('regions', 'NNS')]\n",
+      "[('dividing', 'VBG')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
+      "[('points', 'NNS')]\n",
+      "[('region', 'NN')]\n",
+      "[('meristematic', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('New', 'NNP')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('meristem', 'NN')]\n",
+      "[('meristem', 'NN')]\n",
+      "[('characteristics', 'NNS')]\n",
+      "[('components', 'NNS')]\n",
+      "[('other', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('6.2.2', 'CD')]\n",
+      "[('Permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
+      "[('specific', 'JJ'), ('role', 'NN')]\n",
+      "[('ability', 'NN')]\n",
+      "[('process', 'NN')]\n",
+      "[('permanent', 'JJ'), ('shape', 'NN')]\n",
+      "[('size', 'NN')]\n",
+      "[('function', 'NN')]\n",
+      "[('differentiation', 'NN')]\n",
+      "[('Place', 'NN')]\n",
+      "[('one', 'CD')]\n",
+      "[('section', 'NN')]\n",
+      "[('slide', 'NN')]\n",
+      "[('drop', 'NN')]\n",
+      "[('glycerine', 'NN')]\n",
+      "[('sections', 'NNS')]\n",
+      "[('root', 'NN')]\n",
+      "[('stem', 'NN')]\n",
+      "[('different', 'JJ'), ('plants', 'NNS')]\n",
+      "[('6.2.2', 'CD')]\n",
+      "[('Simple', 'JJ'), ('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('A', 'DT')]\n",
+      "[('few', 'JJ'), ('layers', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('simple', 'JJ'), ('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('unspecialised', 'JJ'), ('cells', 'NNS')]\n",
+      "[('thin', 'NN'), ('cell', 'NN')]\n",
+      "[('walls', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('large', 'JJ'), ('spaces', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('intercellular', 'JJ'), ('spaces', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('stores', 'NNS')]\n",
+      "[('food', 'NN')]\n",
+      "[('situations', 'NNS')]\n",
+      "[('performs', 'NNS'), ('photosynthesis', 'NN')]\n",
+      "[('chlorenchyma', 'NN')]\n",
+      "[('parenchyma', 'NN')]\n",
+      "[('type', 'NN')]\n",
+      "[('aerenchyma', 'NN')]\n",
+      "[('bending', 'NN')]\n",
+      "[('various', 'JJ'), ('parts', 'NNS')]\n",
+      "[('plant', 'NN')]\n",
+      "[('tendrils', 'NNS')]\n",
+      "[('stems', 'NNS')]\n",
+      "[('climbers', 'NNS')]\n",
+      "[('mechanical', 'JJ'), ('support', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('leaf', 'NN'), ('stalks', 'NNS')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('corners', 'NNS')]\n",
+      "[('little', 'JJ'), ('intercellular', 'JJ'), ('space', 'NN')]\n",
       "[('•', 'NNS'), ('Stretch', 'VBP')]\n",
       "[('pressure', 'NN')]\n",
       "[('•', 'NN')]\n",
@@ -2636,71 +3370,185 @@
       "[('skin', 'NN')]\n",
       "[('projects', 'NNS')]\n",
       "[('cut', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('Guard', 'NN')]\n",
-      "[('cells', 'NNS')]\n",
-      "[('epidermal', 'JJ'), ('cells', 'NNS')]\n",
-      "[('lateral', 'JJ'), ('view', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('surface', 'NN')]\n",
-      "[('view', 'NN')]\n",
-      "[('pores', 'NNS')]\n",
-      "[('stomata', 'NNS')]\n",
-      "[('Transpiration', 'NN')]\n",
-      "[('loss', 'NN')]\n",
+      "[('Remove', 'VB')]\n",
+      "[('peel', 'NN')]\n",
+      "[('petri', 'NNS'), ('dish', 'NN')]\n",
       "[('water', 'NN')]\n",
-      "[('form', 'NN')]\n",
-      "[('vapour', 'NN')]\n",
+      "[('•', 'NN')]\n",
+      "[('Add', 'VB')]\n",
+      "[('few', 'JJ'), ('drops', 'NNS')]\n",
+      "[('safranin', 'NN')]\n",
+      "[('•', 'NNS'), ('Wait', 'VBP')]\n",
+      "[('couple', 'NN')]\n",
+      "[('minutes', 'NNS')]\n",
+      "[('slide', 'NN')]\n",
       "[('place', 'NN')]\n",
-      "[('stomata', 'NNS')]\n",
+      "[('cover', 'NN')]\n",
+      "[('slip', 'NN')]\n",
+      "[('•', 'NNS'), ('Observe', 'VBP')]\n",
+      "[('microscope', 'NN')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('single', 'JJ'), ('layer', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('entire', 'JJ'), ('surface', 'NN')]\n",
+      "[('plant', 'NN')]\n",
+      "[('outer', 'NN')]\n",
+      "[('epidermis', 'NN')]\n",
+      "[('gases', 'NNS')]\n",
+      "[('atmosphere', 'RB')]\n",
       "[('role', 'NN')]\n",
       "[('transpiration', 'NN')]\n",
       "[('plants', 'NNS')]\n",
-      "[('Epidermal', 'JJ')]\n",
+      "[('reason', 'NN')]\n",
+      "[('outer', 'NN'), ('layer', 'NN')]\n",
+      "[('branch', 'NN')]\n",
+      "[('outer', 'NN')]\n",
+      "[('layer', 'NN')]\n",
+      "[('young', 'JJ'), ('stem', 'NN')]\n",
+      "[('plants', 'NNS')]\n",
+      "[('outer', 'NN')]\n",
+      "[('protective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('undergoes', 'NNS')]\n",
+      "[('certain', 'JJ'), ('changes', 'NNS')]\n",
+      "[('strip', 'NN')]\n",
+      "[('secondary', 'JJ'), ('meristem', 'NN')]\n",
+      "[('cortex', 'NN')]\n",
+      "[('layers', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
-      "[('roots', 'NNS')]\n",
-      "[('function', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('absorption', 'NN')]\n",
-      "[('hair-like', 'NN'), ('parts', 'NNS')]\n",
-      "[('total', 'JJ'), ('absorptive', 'JJ'), ('surface', 'NN')]\n",
-      "[('area', 'NN')]\n",
-      "[('water', 'NN')]\n",
-      "[('minerals', 'NNS')]\n",
+      "[('cork', 'NN')]\n",
+      "[('Cells', 'NNS')]\n",
+      "[('cork', 'NN')]\n",
+      "[('intercellular', 'JJ'), ('spaces', 'NNS')]\n",
+      "[('Such', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('simple', 'JJ'), ('permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Complex', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('one', 'CD')]\n",
+      "[('type', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('common', 'JJ'), ('function', 'NN')]\n",
+      "[('Xylem', 'NN')]\n",
+      "[('phloem', 'NN')]\n",
+      "[('examples', 'NNS')]\n",
+      "[('such', 'JJ'), ('complex', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('vascular', 'JJ'), ('bundle', 'NN')]\n",
+      "[('Vascular', 'JJ'), ('tissue', 'NN')]\n",
+      "[('distinctive', 'JJ'), ('feature', 'NN')]\n",
+      "[('complex', 'JJ'), ('plants', 'NNS')]\n",
+      "[('one', 'CD')]\n",
+      "[('survival', 'NN')]\n",
+      "[('terrestrial', 'JJ'), ('environment', 'NN')]\n",
+      "[('section', 'NN')]\n",
+      "[('stem', 'NN')]\n",
+      "[('different', 'JJ'), ('types', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('vascular', 'NN')]\n",
+      "[('bundle', 'NN')]\n",
+      "[('Xylem', 'NN')]\n",
+      "[('tracheids', 'NNS')]\n",
+      "[('vessels', 'NNS')]\n",
+      "[('xylem', 'NN')]\n",
       "[('parenchyma', 'NN')]\n",
-      "[('stores', 'NNS')]\n",
-      "[('food', 'NN')]\n",
-      "[('Sieve', 'NN')]\n",
-      "[('tubes', 'NNS')]\n",
-      "[('tubular', 'JJ'), ('cells', 'NNS')]\n",
-      "[('perforated', 'VBN'), ('walls', 'NNS')]\n",
-      "[('Phloem', 'NN')]\n",
-      "[('transports', 'NNS')]\n",
+      "[('xylem', 'NN')]\n",
+      "[('fibres', 'NNS')]\n",
+      "[('Questions', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('oxygen', 'NN')]\n",
+      "[('functions', 'NNS')]\n",
+      "[('mitochondria', 'NNS')]\n",
+      "[('clue', 'NN')]\n",
+      "[('question', 'NN')]\n",
+      "[('example', 'NN')]\n",
+      "[('oxygen', 'NN')]\n",
       "[('food', 'NN')]\n",
-      "[('leaves', 'NNS')]\n",
-      "[('other', 'JJ'), ('parts', 'NNS')]\n",
-      "[('plant', 'NN')]\n",
-      "[('1', 'CD')]\n",
-      "[('Name', 'NN')]\n",
-      "[('types', 'NNS')]\n",
-      "[('simple', 'NN'), ('tissues', 'NNS')]\n",
-      "[('movement', 'NN')]\n",
-      "[('chest', 'NN')]\n",
-      "[('small', 'JJ'), ('amount', 'NN')]\n",
-      "[('material', 'NN')]\n",
-      "[('intercellular', 'JJ'), ('spaces', 'NNS')]\n",
-      "[('anything', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('6.3.1', 'CD')]\n",
+      "[('Epithelial', 'JJ'), ('tissue', 'NN')]\n",
+      "[('covering', 'VBG')]\n",
+      "[('protective', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('animal', 'NN')]\n",
       "[('body', 'NN')]\n",
-      "[('least', 'JJS'), ('one', 'CD')]\n",
-      "[('layer', 'NN')]\n",
+      "[('epithelial', 'JJ'), ('tissues', 'NNS')]\n",
+      "[('Epithelium', 'NN')]\n",
+      "[('organs', 'NNS')]\n",
+      "[('cavities', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('result', 'NN')]\n",
+      "[('permeability', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('various', 'JJ'), ('epithelia', 'NNS')]\n",
+      "[('important', 'JJ'), ('role', 'NN')]\n",
+      "[('exchange', 'NN')]\n",
+      "[('materials', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('external', 'JJ'), ('environment', 'NN')]\n",
+      "[('different', 'JJ'), ('parts', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('oesophagus', 'NN')]\n",
+      "[('lining', 'VBG')]\n",
+      "[('mouth', 'NN')]\n",
+      "[('squamous', 'JJ'), ('epithelium', 'NN')]\n",
+      "[('skin', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('squamous', 'JJ'), ('epithelium', 'NN')]\n",
+      "[('columnar', 'NN')]\n",
+      "[('pillar-like', 'JJ'), ('’', 'NN')]\n",
       "[('epithelium', 'NN')]\n",
-      "[('Different', 'NN')]\n",
-      "[('show', 'NN')]\n",
-      "[('structures', 'NNS')]\n",
-      "[('unique', 'JJ'), ('functions', 'NNS')]\n",
+      "[('movement', 'NN')]\n",
+      "[('epithelial', 'JJ'), ('barrier', 'NN')]\n",
+      "[('glandular', 'JJ'), ('epithelium', 'NN')]\n",
+      "[('framework', 'NN')]\n",
+      "[('body', 'NN')]\n",
+      "[('Bone', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('hard', 'JJ'), ('matrix', 'NN')]\n",
+      "[('calcium', 'NN')]\n",
+      "[('phosphorus', 'NN')]\n",
+      "[('compounds', 'NNS')]\n",
+      "[('Two', 'CD')]\n",
+      "[('bones', 'NNS')]\n",
+      "[('type', 'NN')]\n",
+      "[('connective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('ligament', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('considerable', 'JJ'), ('strength', 'NN')]\n",
+      "[('Tendons', 'NNS')]\n",
+      "[('muscles', 'NNS')]\n",
+      "[('bones', 'NNS')]\n",
+      "[('type', 'NN')]\n",
+      "[('connective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Tendons', 'NNS')]\n",
+      "[('fibrous', 'JJ'), ('tissue', 'NN')]\n",
+      "[('great', 'JJ'), ('strength', 'NN')]\n",
+      "[('limited', 'JJ'), ('flexibility', 'NN')]\n",
+      "[('type', 'NN')]\n",
+      "[('connective', 'JJ'), ('tissue', 'NN')]\n",
+      "[('cartilage', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('solid', 'JJ'), ('matrix', 'NN')]\n",
+      "[('proteins', 'NNS')]\n",
+      "[('sugars', 'NNS')]\n",
+      "[('space', 'NN')]\n",
+      "[('organs', 'NNS')]\n",
+      "[('internal', 'JJ'), ('organs', 'NNS')]\n",
+      "[('repair', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('fats', 'NNS')]\n",
+      "[('body', 'NN')]\n",
+      "[('Fat-storing', 'VBG'), ('adipose', 'JJ'), ('tissue', 'NN')]\n",
+      "[('skin', 'NN')]\n",
+      "[('internal', 'JJ'), ('organs', 'NNS')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('fat', 'JJ'), ('globules', 'NNS')]\n",
+      "[('Storage', 'NN')]\n",
+      "[('fats', 'NNS')]\n",
+      "[('insulator', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('fat', 'JJ'), ('globules', 'NNS')]\n",
       "[('muscles', 'NNS')]\n",
-      "[('Muscles', 'NNS')]\n",
-      "[('limbs', 'NNS')]\n",
       "[('Such', 'JJ'), ('muscles', 'NNS')]\n",
       "[('voluntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('muscles', 'NNS')]\n",
@@ -2709,17 +3557,20 @@
       "[('help', 'NN')]\n",
       "[('body', 'NN')]\n",
       "[('movement', 'NN')]\n",
-      "[('microscope', 'NN')]\n",
-      "[('muscles', 'NNS')]\n",
-      "[('alternate', 'NN'), ('light', 'NN')]\n",
-      "[('dark', 'JJ'), ('bands', 'NNS')]\n",
-      "[('striations', 'NNS')]\n",
       "[('result', 'NN')]\n",
       "[('muscles', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
       "[('tissue', 'NN')]\n",
       "[('multinucleate', 'NN')]\n",
       "[('many', 'JJ'), ('nuclei', 'NNS')]\n",
+      "[('movement', 'NN')]\n",
+      "[('food', 'NN')]\n",
+      "[('alimentary', 'JJ'), ('canal', 'NN')]\n",
+      "[('contraction', 'NN')]\n",
+      "[('relaxation', 'NN')]\n",
+      "[('blood', 'NN')]\n",
+      "[('vessels', 'NNS')]\n",
+      "[('involuntary', 'JJ'), ('movements', 'NNS')]\n",
       "[('Smooth', 'NNP')]\n",
       "[('involuntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('control', 'NN')]\n",
@@ -2732,73 +3583,22 @@
       "[('cells', 'NNS')]\n",
       "[('pointed', 'VBN'), ('ends', 'NNS')]\n",
       "[('single', 'JJ'), ('nucleus', 'NN')]\n",
-      "[('unstriated', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('Muscular', 'JJ'), ('tissue', 'NN')]\n",
-      "[('elongated', 'VBN'), ('cells', 'NNS')]\n",
-      "[('muscle', 'NN')]\n",
-      "[('fibres', 'NNS')]\n",
-      "[('tissue', 'NN')]\n",
-      "[('movement', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('Muscles', 'NNS')]\n",
-      "[('special', 'JJ'), ('proteins', 'NNS')]\n",
-      "[('contractile', 'NN'), ('proteins', 'NNS')]\n",
-      "[('contract', 'NN')]\n",
-      "[('relax', 'NN')]\n",
-      "[('movement', 'NN')]\n",
       "[('muscles', 'NNS')]\n",
+      "[('Muscles', 'NNS')]\n",
+      "[('limbs', 'NNS')]\n",
       "[('Such', 'JJ'), ('muscles', 'NNS')]\n",
       "[('voluntary', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('microscope', 'NN')]\n",
-      "[('muscles', 'NNS')]\n",
-      "[('alternate', 'NN'), ('light', 'NN')]\n",
-      "[('dark', 'JJ'), ('bands', 'NNS')]\n",
-      "[('striations', 'NNS')]\n",
       "[('result', 'NN')]\n",
       "[('muscles', 'NNS')]\n",
-      "[('cells', 'NNS')]\n",
-      "[('tissue', 'NN')]\n",
-      "[('multinucleate', 'NN')]\n",
-      "[('many', 'JJ'), ('nuclei', 'NNS')]\n",
-      "[('movement', 'NN')]\n",
-      "[('food', 'NN')]\n",
-      "[('alimentary', 'JJ'), ('canal', 'NN')]\n",
-      "[('contraction', 'NN')]\n",
-      "[('relaxation', 'NN')]\n",
-      "[('blood', 'NN')]\n",
-      "[('vessels', 'NNS')]\n",
-      "[('involuntary', 'JJ'), ('movements', 'NNS')]\n",
       "[('Smooth', 'NNP')]\n",
       "[('involuntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('control', 'NN')]\n",
       "[('such', 'JJ'), ('movements', 'NNS')]\n",
-      "[('iris', 'NN')]\n",
-      "[('eye', 'NN')]\n",
-      "[('ureters', 'NNS')]\n",
-      "[('bronchi', 'NN')]\n",
-      "[('lungs', 'NNS')]\n",
       "[('cells', 'NNS')]\n",
       "[('pointed', 'VBN'), ('ends', 'NNS')]\n",
       "[('single', 'JJ'), ('nucleus', 'NN')]\n",
-      "[('unstriated', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('muscles', 'NNS')]\n",
-      "[('heart', 'NN')]\n",
-      "[('show', 'NN')]\n",
-      "[('rhythmic', 'JJ'), ('contraction', 'NN')]\n",
-      "[('relaxation', 'NN')]\n",
-      "[('life', 'NN')]\n",
       "[('involuntary', 'JJ'), ('muscles', 'NNS')]\n",
       "[('cardiac', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('Heart', 'NN')]\n",
-      "[('muscle', 'NN')]\n",
-      "[('cells', 'NNS')]\n",
-      "[('Activity', 'NN')]\n",
-      "[('6.5', 'CD')]\n",
-      "[('Compare', 'NN')]\n",
-      "[('structures', 'NNS')]\n",
-      "[('different', 'JJ'), ('types', 'NNS')]\n",
-      "[('muscular', 'NN')]\n",
-      "[('tissues', 'NNS')]\n",
       "[('shape', 'NN')]\n",
       "[('number', 'NN')]\n",
       "[('nuclei', 'NN')]\n",
@@ -2808,54 +3608,39 @@
       "[('Table', 'NN')]\n",
       "[('6.1', 'CD')]\n",
       "[('cells', 'NNS')]\n",
-      "[('nervous', 'JJ'), ('tissue', 'NN')]\n",
-      "[('one', 'CD')]\n",
-      "[('place', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('brain', 'NN')]\n",
-      "[('spinal', 'JJ'), ('cord', 'NN')]\n",
-      "[('nerves', 'NNS')]\n",
-      "[('nervous', 'JJ'), ('tissue', 'NN')]\n",
-      "[('Questions', 'NNS')]\n",
-      "[('1', 'CD')]\n",
       "[('tissue', 'NN')]\n",
-      "[('movement', 'NN')]\n",
-      "[('body', 'NN')]\n",
-      "[('2', 'CD')]\n",
-      "[('neuron', 'NNS'), ('look', 'VBP')]\n",
-      "[('3', 'CD')]\n",
-      "[('three', 'CD')]\n",
-      "[('features', 'NNS')]\n",
-      "[('cardiac', 'JJ'), ('muscles', 'NNS')]\n",
-      "[('•', 'JJ'), ('Permanent', 'NNP')]\n",
+      "[('nerve', 'NN')]\n",
+      "[('cells', 'NNS')]\n",
+      "[('neurons', 'NNS')]\n",
+      "[('4', 'CD')]\n",
+      "[('functions', 'NNS')]\n",
+      "[('areolar', 'JJ'), ('tissue', 'NN')]\n",
+      "[('•', 'NN')]\n",
+      "[('Plant', 'NN')]\n",
       "[('tissues', 'NNS')]\n",
-      "[('meristematic', 'JJ'), ('tissue', 'NN')]\n",
-      "[('ability', 'NN')]\n",
+      "[('two', 'CD')]\n",
+      "[('main', 'JJ'), ('types', 'NNS')]\n",
+      "[('Xylem', 'NN')]\n",
+      "[('phloem', 'NN')]\n",
+      "[('types', 'NNS')]\n",
       "[('complex', 'JJ'), ('tissues', 'NNS')]\n",
       "[('•', 'NN')]\n",
-      "[('Parenchyma', 'NN')]\n",
-      "[('collenchyma', 'NN')]\n",
-      "[('sclerenchyma', 'NN')]\n",
+      "[('Striated', 'VBN')]\n",
       "[('three', 'CD')]\n",
       "[('types', 'NNS')]\n",
-      "[('simple', 'NN'), ('tissues', 'NNS')]\n",
-      "[('shape', 'NN')]\n",
-      "[('function', 'NN')]\n",
-      "[('epithelial', 'JJ'), ('tissue', 'NN')]\n",
-      "[('columnar', 'NN')]\n",
-      "[('different', 'JJ'), ('types', 'NNS')]\n",
-      "[('connective', 'JJ'), ('tissues', 'NNS')]\n",
-      "[('body', 'NN')]\n",
-      "[('areolar', 'JJ'), ('tissue', 'NN')]\n",
-      "[('adipose', 'JJ'), ('tissue', 'NN')]\n",
-      "[('bone', 'NN')]\n",
-      "[('tendon', 'NN')]\n",
-      "[('ligament', 'NN')]\n",
-      "[('cartilage', 'NN')]\n",
-      "[('blood', 'NN')]\n",
-      "[('Nervous', 'JJ'), ('tissue', 'NN')]\n",
-      "[('neurons', 'NNS')]\n",
-      "[('impulses', 'NNS')]\n",
+      "[('muscle', 'NN')]\n",
+      "[('tissues', 'NNS')]\n",
+      "[('Exercises', 'NNS')]\n",
+      "[('1', 'CD')]\n",
+      "[('term', 'NN')]\n",
+      "[('“', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('”', 'NN')]\n",
+      "[('2', 'CD')]\n",
+      "[('many', 'JJ'), ('types', 'NNS')]\n",
+      "[('elements', 'NNS')]\n",
+      "[('xylem', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
       "[('simple', 'NN'), ('tissues', 'NNS')]\n",
       "[('complex', 'JJ'), ('tissues', 'NNS')]\n",
       "[('plants', 'NNS')]\n",
@@ -2877,48 +3662,44 @@
       "[('types', 'NNS')]\n",
       "[('muscle', 'NN')]\n",
       "[('fibres', 'NNS')]\n",
-      "[('10', 'CD')]\n",
-      "[('following', 'VBG')]\n",
       "[('Tissue', 'NN')]\n",
       "[('inner', 'NN'), ('lining', 'NN')]\n",
       "[('mouth', 'NN')]\n",
-      "[('b', 'NN')]\n",
-      "[('Tissue', 'NN')]\n",
-      "[('muscle', 'NN')]\n",
-      "[('humans', 'NNS')]\n",
-      "[('c', 'NNS')]\n",
-      "[('Tissue', 'NN')]\n",
-      "[('food', 'NN')]\n",
-      "[('plants', 'NNS')]\n",
-      "[('d', 'NN')]\n",
+      "[('e', 'NN')]\n",
+      "[('Connective', 'JJ')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('fluid', 'NN')]\n",
+      "[('matrix', 'NN')]\n",
+      "[('f', 'NN')]\n",
       "[('Tissue', 'NN')]\n",
-      "[('stores', 'NNS')]\n",
-      "[('body', 'NN')]\n",
-      "[('12', 'CD')]\n",
-      "[('regions', 'NNS')]\n",
-      "[('parenchyma', 'NN'), ('tissue', 'NN')]\n",
-      "[('13', 'CD')]\n",
-      "[('role', 'NN')]\n",
-      "[('epidermis', 'NN')]\n",
-      "[('plants', 'NNS')]\n",
+      "[('present', 'NN')]\n",
+      "[('brain', 'NN')]\n",
+      "[('11', 'CD')]\n",
+      "[('type', 'NN')]\n",
+      "[('tissue', 'NN')]\n",
+      "[('following', 'VBG')]\n",
+      "[('skin', 'NN')]\n",
+      "[('bark', 'NN')]\n",
+      "[('tree', 'NN')]\n",
+      "[('bone', 'NN')]\n",
+      "[('lining', 'VBG')]\n",
+      "[('kidney', 'NN')]\n",
+      "[('tubule', 'NN')]\n",
+      "[('vascular', 'JJ'), ('bundle', 'NN')]\n",
       "[('14', 'CD')]\n",
       "[('cork', 'NN')]\n",
       "[('act', 'NN')]\n",
       "[('protective', 'JJ'), ('tissue', 'NN')]\n",
       "[('15', 'CD')]\n",
       "[('following', 'VBG'), ('chart', 'NN')]\n",
-      "[('Chapter', 'NN')]\n",
-      "[('6', 'CD')]\n",
-      "[('6.2', 'CD')]\n",
-      "[('Plant', 'NN')]\n",
-      "[('Tissues', 'NNS')]\n",
-      "[('6.3', 'CD')]\n",
-      "[('Animal', 'NN')]\n",
+      "[('Table', 'NN')]\n",
+      "[('Contents', 'NNS')]\n",
       "[('Tissues', 'NNS')]\n",
-      "[('6.3.2', 'CD')]\n",
-      "[('Connective', 'JJ'), ('tissue', 'NN')]\n",
-      "[('6.3.4', 'CD')]\n",
-      "[('Nervous', 'JJ'), ('tissue', 'NN')]\n",
+      "[('6.2.2', 'CD')]\n",
+      "[('Permanent', 'JJ'), ('tissue', 'NN')]\n",
+      "[('6.3.1', 'CD')]\n",
+      "[('Epithelial', 'JJ'), ('tissue', 'NN')]\n",
+      "[('Landmarks', 'NNS')]\n",
       "[('Cover', 'NN')]\n",
       "[('Chapter', 'NN')]\n",
       "[('6Tissues', 'NNS')]\n"
@@ -2943,7 +3724,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 103,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -2956,14 +3737,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 104,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "124\n"
+      "39\n"
      ]
     }
    ],
diff --git a/sentences-test.txt b/sentences-test.txt
new file mode 100644
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--- /dev/null
+++ b/sentences-test.txt
@@ -0,0 +1,2468 @@
+Unit 1
+DIVERSITY IN THE LIVING WORLD
+Chapter 1 The Living World Chapter 2 Biological Classification Chapter 3 Plant Kingdom Chapter 4 Animal Kingdom
+Biology is the science of life forms and living processes
+The living world comprises an amazing diversity of living organisms
+Early man could easily perceive the difference between inanimate matter and living organisms
+Early man deified some of the inanimate matter (wind, sea, fire etc.) and some among the animals and plants
+A common feature of all such forms of inanimate and animate objects was the sense of awe or fear that they evoked
+The description of living organisms including human beings began much later in human history
+Societies which indulged in anthropocentric view of biology could register limited progress in biological knowledge
+Systematic and monumental description of life forms brought in, out of necessity, detailed systems of identification, nomenclature and classification
+The biggest spin off of such studies was the recognition of the sharing of similarities among living organisms both horizontally and vertically
+That all present day living organisms are related to each other and also to all organisms that ever lived on this earth, was a revelation which humbled man and led to cultural movements for conservation of biodiversity
+In the following chapters of this unit, you will get a description, including classification, of animals and plants from a taxonomist’s perspective.
+Ernst Mayr (1904 – 2004) Born on 5 July 1904, in Kempten, Germany, Ernst Mayr, the Harvard University evolutionary biologist who has been called ‘The Darwin of the 20th century’, was one of the 100 greatest scientists of all time
+Mayr joined Harvard’s Faculty of Arts and Sciences in 1953 and retired in 1975, assuming the title Alexander Agassiz Professor of Zoology Emeritus
+Throughout his nearly 80-year career, his research spanned ornithology, taxonomy, zoogeography, evolution, systematics, and the history and philosophy of biology
+He almost single-handedly made the origin of species diversity the central question of evolutionary biology that it is today
+He also pioneered the currently accepted definition of a biological species
+Mayr was awarded the three prizes widely regarded as the triple crown of biology: the Balzan Prize in 1983, the International Prize for Biology in 1994, and the Crafoord Prize in 1999
+Mayr died at the age of 100 in the year 2004.
+Chapter 1
+The Living World
+1.1 What is ‘Living’? 1.2 Diversity in the Living World 1.3 Taxonomic Categories 1.4 Taxonomical Aids
+How wonderful is the living world ! The wide range of living types is amazing
+The extraordinary habitats in which we find living organisms, be it cold mountains, deciduous forests, oceans, fresh water lakes, deserts or hot springs, leave us speechless
+The beauty of a galloping horse, of the migrating birds, the valley of flowers or the attacking shark evokes awe and a deep sense of wonder
+The ecological conflict and cooperation among members of a population and among populations of a community or even the molecular traffic inside a cell make us deeply reflect on – what indeed is life? This question has two implicit questions within it
+The first is a technical one and seeks answer to what living is as opposed to the non-living, and the second is a philosophical one, and seeks answer to what the purpose of life is
+As scientists, we shall not attempt answering the second question
+We will try to reflect on – what is living?
+1.1 What is ‘Living’? When we try to define ‘living’, we conventionally look for distinctive characteristics exhibited by living organisms
+Growth, reproduction, ability to sense environment and mount a suitable response come to our mind immediately as unique features of living organisms
+One can add a few more features like metabolism, ability to self-replicate, self-organise, interact and emergence to this list
+Let us try to understand each of these
+All living organisms grow
+Increase in mass and increase in number of individuals are twin characteristics of growth
+A multicellular organism grows by cell division
+In plants, this growth by cell division occurs continuously throughout their life span
+In animals, this growth is seen only up to a certain age
+However, cell division occurs in certain tissues to replace lost cells
+Unicellular organisms grow by cell division
+One can easily observe this in in vitro cultures by simply counting the number of cells under the microscope
+In majority of higher animals and plants, growth and reproduction are mutually exclusive events
+One must remember that increase in body mass is considered as growth
+Non-living objects also grow if we take increase in body mass as a criterion for growth
+Mountains, boulders and sand mounds do grow
+However, this kind of growth exhibited by non-living objects is by accumulation of material on the surface
+In living organisms, growth is from inside
+Growth, therefore, cannot be taken as a defining property of living organisms
+Conditions under which it can be observed in all living organisms have to be explained and then we understand that it is a characteristic of living systems
+A dead organism does not grow
+Reproduction, likewise, is a characteristic of living organisms
+In multicellular organisms, reproduction refers to the production of progeny possessing features more or less similar to those of parents
+Invariably and implicitly we refer to sexual reproduction
+Organisms reproduce by asexual means also
+Fungi multiply and spread easily due to the millions of asexual spores they produce
+In lower organisms like yeast and hydra, we observe budding
+In Planaria (flat worms), we observe true regeneration, i.e., a fragmented organism regenerates the lost part of its body and becomes, a new organism
+The fungi, the filamentous algae, the protonema of mosses, all easily multiply by fragmentation
+When it comes to unicellular organisms like bacteria, unicellular algae or Amoeba, reproduction is synonymous with growth, i.e., increase in number of cells
+We have already defined growth as equivalent to increase in cell number or mass
+Hence, we notice that in single-celled organisms, we are not very clear about the usage of these two terms – growth and reproduction
+Further, there are many organisms which do not reproduce (mules, sterile worker bees, infertile human couples, etc)
+Hence, reproduction also cannot be an all-inclusive defining characteristic of living organisms
+Of course, no non-living object is capable of reproducing or replicating by itself
+Another characteristic of life is metabolism
+All living organisms are made of chemicals
+These chemicals, small and big, belonging to various classes, sizes, functions, etc., are constantly being made and changed into some other biomolecules
+These conversions are chemical reactions or metabolic reactions
+There are thousands of metabolic reactions occurring simultaneously inside all living organisms, be they unicellular or multicellular
+All plants, animals, fungi and microbes exhibit metabolism
+The sum total of all the chemical reactions occurring in our body is metabolism
+No non­-living object exhibits metabolism
+Metabolic reactions can be demonstrated outside the body in cell-free systems
+An isolated metabolic reaction(s) outside the body of an organism, performed in a test tube is neither living nor non-living
+Hence, while metabolism is a defining feature of all living organisms without exception, isolated metabolic reactions in vitro are not living things but surely living reactions
+Hence, cellular organisation of the body is the defining feature of life forms
+Perhaps, the most obvious and technically complicated feature of all living organisms is this ability to sense their surroundings or environment and respond to these environmental stimuli which could be physical, chemical or biological
+We sense our environment through our sense organs
+Plants respond to external factors like light, water, temperature, other organisms, pollutants, etc
+All organisms, from the prokaryotes to the most complex eukaryotes can sense and respond to environmental cues
+Photoperiod affects reproduction in seasonal breeders, both plants and animals
+All organisms handle chemicals entering their bodies
+All organisms therefore, are ‘aware’ of their surroundings
+Human being is the only organism who is aware of himself, i.e., has self-consciousness
+Consciousness therefore, becomes the defining property of living organisms
+When it comes to human beings, it is all the more difficult to define the living state
+We observe patients lying in coma in hospitals virtually supported by machines which replace heart and lungs
+The patient is otherwise brain-dead
+The patient has no self-­consciousness
+Are such patients who never come back to normal life, living or non-living? In higher classes, you will come to know that all living phenomena are due to underlying interactions
+Properties of tissues are not present in the constituent cells but arise as a result of interactions among the constituent cells
+Similarly, properties of cellular organelles are not present in the molecular constituents of the organelle but arise as a result of interactions among the molecular components comprising the organelle
+These interactions result in emergent properties at a higher level of organisation
+This phenomenon is true in the hierarchy of organisational complexity at all levels
+Therefore, we can say that living organisms are self-replicating, evolving and self-regulating interactive systems capable of responding to external stimuli
+Biology is the story of life on earth
+Biology is the story of evolution of living organisms on earth
+All living organisms – present, past and future, are linked to one another by the sharing of the common genetic material, but to varying degrees.
+1.2 Diversity in the Living World If you look around you will see a large variety of living organisms, be it potted plants, insects, birds, your pets or other animals and plants
+There are also several organisms that you cannot see with your naked eye but they are all around you
+If you were to increase the area that you make observations in, the range and variety of organisms that you see would increase
+Obviously, if you were to visit a dense forest, you would probably see a much greater number and kinds of living organisms in it
+Each different kind of plant, animal or organism that you see, represents a species
+The number of species that are known and described range between 1.7-1.8 million
+This refers to biodiversity or the number and types of organisms present on earth
+We should remember here that as we explore new areas, and even old ones, new organisms are continuously being identified
+As stated earlier, there are millions of plants and animals in the world; we know the plants and animals in our own area by their local names
+These local names would vary from place to place, even within a country
+Probably you would recognise the confusion that would be created if we did not find ways and means to talk to each other, to refer to organisms we are talking about
+Hence, there is a need to standardise the naming of living organisms such that a particular organism is known by the same name all over the world
+This process is called nomenclature
+Obviously, nomenclature or naming is only possible when the organism is described correctly and we know to what organism the name is attached to
+This is identification
+In order to facilitate the study, number of scientists have established procedures to assign a scientific name to each known organism
+This is acceptable to biologists all over the world
+For plants, scientific names are based on agreed principles and criteria, which are provided in International Code for Botanical Nomenclature (ICBN)
+You may ask, how are animals named? Animal taxonomists have evolved International Code of Zoological Nomenclature (ICZN)
+The scientific names ensure that each organism has only one name
+Description of any organism should enable the people (in any part of the world) to arrive at the same name
+They also ensure that such a name has not been used for any other known organism
+Biologists follow universally accepted principles to provide scientific names to known organisms
+Each name has two components – the Generic name and the specific epithet
+This system of providing a name with two components is called Binomial nomenclature
+This naming system given by Carolus Linnaeus is being practised by biologists all over the world
+This naming system using a two word format was found convenient
+Let us take the example of mango to understand the way of providing scientific names better
+The scientific name of mango is written as Mangifera indica
+Let us see how it is a binomial name
+In this name Mangifera represents the genus while indica, is a particular species, or a specific epithet
+Other universal rules of nomenclature are as follows: 1
+Biological names are generally in Latin and written in italics
+They are Latinised or derived from Latin irrespective of their origin
+2
+The first word in a biological name represents the genus while the second component denotes the specific epithet
+3
+Both the words in a biological name, when handwritten, are separately underlined, or printed in italics to indicate their Latin origin
+4
+The first word denoting the genus starts with a capital letter while the specific epithet starts with a small letter
+It can be illustrated with the example of Mangifera indica
+Name of the author appears after the specific epithet, i.e., at the end of the biological name and is written in an abbreviated form, e.g., Mangifera indica Linn
+It indicates that this species was first described by Linnaeus
+Since it is nearly impossible to study all the living organisms, it is necessary to devise some means to make this possible
+This process is classification
+Classification is the process by which anything is grouped into convenient categories based on some easily observable characters
+For example, we easily recognise groups such as plants or animals or dogs, cats or insects
+The moment we use any of these terms, we associate certain characters with the organism in that group
+What image do you see when you think of a dog ? Obviously, each one of us will see ‘dogs’ and not ‘cats’
+Now, if we were to think of ‘Alsatians’ we know what we are talking about
+Similarly, suppose we were to say ‘mammals’, you would, of course, think of animals with external ears and body hair
+Likewise, in plants, if we try to talk of ‘Wheat’, the picture in each of our minds will be of wheat plants, not of rice or any other plant
+Hence, all these - ‘Dogs’, ‘Cats’, ‘Mammals’, ‘Wheat’, ‘Rice’, ‘Plants’, ‘Animals’, etc., are convenient categories we use to study organisms
+The scientific term for these categories is taxa
+Here you must recognise that taxa can indicate categories at very different levels
+‘Plants’ – also form a taxa
+‘Wheat’ is also a taxa
+Similarly, ‘animals’, ‘mammals’, ‘dogs’ are all taxa – but you know that a dog is a mammal and mammals are animals
+Therefore, ‘animals’, ‘mammals’ and ‘dogs’ represent taxa at different levels
+Hence, based on characteristics, all living organisms can be classified into different taxa
+This process of classification is taxonomy
+External and internal structure, along with the structure of cell, development process and ecological information of organisms are essential and form the basis of modern taxonomic studies
+Hence, characterisation, identification, classification and nomenclature are the processes that are basic to taxonomy
+Taxonomy is not something new
+Human beings have always been interested in knowing more and more about the various kinds of organisms, particularly with reference to their own use
+In early days, human beings needed to find sources for their basic needs of food, clothing and shelter
+Hence, the earliest classifications were based on the ‘uses’ of various organisms
+Human beings were, since long, not only interested in knowing more about different kinds of organisms and their diversities, but also the relationships among them
+This branch of study was referred to as systematics
+The word systematics is derived from the Latin word ‘systema’ which means systematic arrangement of organisms
+Linnaeus used Systema Naturae as the title of his publication
+The scope of systematics was later enlarged to include identification, nomenclature and classification
+Systematics takes into account evolutionary relationships between organisms.
+1.3 Taxonomic Categories Classification is not a single step process but involves hierarchy of steps in which each step represents a rank or category
+Since the category is a part of overall taxonomic arrangement, it is called the taxonomic category and all categories together constitute the taxonomic hierarchy
+Each category, referred to as a unit of classification, in fact, represents a rank and is commonly termed as taxon (pl.: taxa)
+Taxonomic categories and hierarchy can be illustrated by an example
+Insects represent a group of organisms sharing common features like three pairs of jointed legs
+It means insects are recognisable concrete objects which can be classified, and thus were given a rank or category
+Can you name other such groups of organisms? Remember, groups represent category
+Category further denotes rank
+Each rank or taxon, in fact, represents a unit of classification
+These taxonomic groups/categories are distinct biological entities and not merely morphological aggregates
+Taxonomical studies of all known organisms have led to the development of common categories such as kingdom, phylum or division (for plants), class, order, family, genus and species
+All organisms, including those in the plant and animal kingdoms have species as the lowest category
+Now the question you may ask is, how to place an organism in various categories? The basic requirement is the knowledge of characters of an individual or group of organisms
+This helps in identifying similarities and dissimilarities among the individuals of the same kind of organisms as well as of other kinds of organisms.
+1.3.1 Species Taxonomic studies consider a group of individual organisms with fundamental similarities as a species
+One should be able to distinguish one species from the other closely related species based on the distinct morphological differences
+Let us consider Mangifera indica, Solanum tuberosum (potato) and Panthera leo (lion)
+All the three names, indica, tuberosum and leo, represent the specific epithets, while the first words Mangifera, Solanum and Panthera are genera and represents another higher level of taxon or category
+Each genus may have one or more than one specific epithets representing different organisms, but having morphological similarities
+For example, Panthera has another specific epithet called tigris and Solanum includes species like nigrum and melongena
+Human beings belong to the species sapiens which is grouped in the genus Homo
+The scientific name thus, for human being, is written as Homo sapiens.
+1.3.2 Genus Genus comprises a group of related species which has more characters in common in comparison to species of other genera
+We can say that genera are aggregates of closely related species
+For example, potato and brinjal are two different species but both belong to the genus Solanum
+Lion (Panthera leo), leopard (P
+pardus) and tiger (P
+tigris) with several common features, are all species of the genus Panthera
+This genus differs from another genus Felis which includes cats.
+1.3.3 Family The next category, Family, has a group of related genera with still less number of similarities as compared to genus and species
+Families are characterised on the basis of both vegetative and reproductive features of plant species
+Among plants for example, three different genera Solanum, Petunia and Datura are placed in the family Solanaceae
+Among animals for example, genus Panthera, comprising lion, tiger, leopard is put along with genus, Felis (cats) in the family Felidae
+Similarly, if you observe the features of a cat and a dog, you will find some similarities and some differences as well
+They are separated into two different families – Felidae and Canidae, respectively.
+1.3.4 Order You have seen earlier that categories like species, genus and families are based on a number of similar characters
+Generally, order and other higher taxonomic categories are identified based on the aggregates of characters
+Order being a higher category, is the assemblage of families which exhibit a few similar characters
+The similar characters are less in number as compared to different genera included in a family
+Plant families like Convolvulaceae, Solanaceae are included in the order Polymoniales mainly based on the floral characters
+The animal order, Carnivora, includes families like Felidae and Canidae.
+Figure 1.1 Taxonomic categories showing hierarchial arrangement in ascending order
+1.3.5 Class This category includes related orders
+For example, order Primata comprising monkey, gorilla and gibbon is placed in class Mammalia along with order Carnivora that includes animals like tiger, cat and dog
+Class Mammalia has other orders also.
+1.3.6 Phylum Classes comprising animals like fishes, amphibians, reptiles, birds along with mammals constitute the next higher category called Phylum
+All these, based on the common features like presence of notochord and dorsal hollow neural system, are included in phylum Chordata
+In case of plants, classes with a few similar characters are assigned to a higher category called Division.
+1.3.7 Kingdom All animals belonging to various phyla are assigned to the highest category called Kingdom Animalia in the classification system of animals
+The Kingdom Plantae, on the other hand, is distinct, and comprises all plants from various divisions
+Henceforth, we will refer to these two groups as animal and plant kingdoms
+The taxonomic categories from species to kingdom have been shown in ascending order starting with species in Figure 1.1
+These are broad categories
+However, taxonomists have also developed sub-categories in this hierarchy to facilitate more sound and scientific placement of various taxa
+Look at the hierarchy in Figure 1.1
+Can you recall the basis of arrangement? Say, for example, as we go higher from species to kingdom, the number of common characteristics goes on decreasing
+Lower the taxa, more are the characteristics that the members within the taxon share
+Higher the category, greater is the difficulty of determining the relationship to other taxa at the same level
+Hence, the problem of classification becomes more complex.
+Table 1.1 indicates the taxonomic categories to which some common organisms like housefly, man, mango and wheat belong.
+TABLE 1.1 Organisms with their Taxonomic Categories
+1.4 Taxonomical Aids
+Taxonomic studies of various species of plants, animals and other organisms are useful in agriculture, forestry, industry and in general in knowing our bio-resources and their diversity
+These studies would require correct classification and identification of organisms
+Identification of organisms requires intensive laboratory and field studies
+The collection of actual specimens of plant and animal species is essential and is the prime source of taxonomic studies
+These are also fundamental to studies and essential for training in systematics
+It is used for classification of an organism, and the information gathered is also stored along with the specimens
+In some cases the specimen is preserved for future studies
+Biologists have established certain procedures and techniques to store and preserve the information as well as the specimens
+Some of these are explained to help you understand the usage of these aids.
+1.4.1 Herbarium Herbarium is a store house of collected plant specimens that are dried, pressed and preserved on sheets
+Further, these sheets are arranged according to a universally accepted system of classification
+These specimens, along with their descriptions on herbarium sheets, become a store house or repository for future use
+The herbarium sheets also carry a label providing information about date and place of collection, English, local and botanical names, family, collector’s name, etc
+Herbaria also serve as quick referral systems in taxonomical studies.
+Figure 1.2 Herbarium showing stored specimens
+1.4.2 Botanical Gardens These specialised gardens have collections of living plants for reference
+Plant species in these gardens are grown for identification purposes and each plant is labelled indicating its botanical/scientific name and its family
+The famous botanical gardens are at Kew (England), Indian Botanical Garden, Howrah (India) and at National Botanical Research Institute, Lucknow (India).
+1.4.3 Museum Biological museums are generally set up in educational institutes such as schools and colleges
+Museums have collections of preserved plant and animal specimens for study and reference
+Specimens are preserved in the containers or jars in preservative solutions
+Plant and animal specimens may also be preserved as dry specimens
+Insects are preserved in insect boxes after collecting, killing and pinning
+Larger animals like birds and mammals are usually stuffed and preserved
+Museums often have collections of skeletons of animals too.
+1.4.4 Zoological Parks These are the places where wild animals are kept in protected environments under human care and which enable us to learn about their food habits and behaviour
+All animals in a zoo are provided, as far as possible, the conditions similar to their natural habitats
+Children love visiting these parks, commonly called Zoos .
+Figure 1.3 Pictures showing animals in different zoological parks of India
+1.4.5 Key Key is another taxonomical aid used for identification of plants and animals based on the similarities and dissimilarities
+The keys are based on the contrasting characters generally in a pair called couplet
+It represents the choice made between two opposite options
+This results in acceptance of only one and rejection of the other
+Each statement in the key is called a lead
+Separate taxonomic keys are required for each taxonomic category such as family, genus and species for identification purposes
+Keys are generally analytical in nature
+Flora, manuals, monographs and catalogues are some other means of recording descriptions
+They also help in correct identification
+Flora contains the actual account of habitat and distribution of plants of a given area
+These provide the index to the plant species found in a particular area
+Manuals are useful in providing information for identification of names of species found in an area
+Monographs contain information on any one taxon.
+Summary The living world is rich in variety
+Millions of plants and animals have been identified and described but a large number still remains unknown
+The very range of organisms in terms of size, colour, habitat, physiological and morphological features make us seek the defining characteristics of living organisms
+In order to facilitate the study of kinds and diversity of organisms, biologists have evolved certain rules and principles for identification, nomenclature and classification of organisms
+The branch of knowledge dealing with these aspects is referred to as taxonomy
+The taxonomic studies of various species of plants and animals are useful in agriculture, forestry, industry and in general for knowing our bio-resources and their diversity
+The basics of taxonomy like identification, naming and classification of organisms are universally evolved under international codes
+Based on the resemblances and distinct differences, each organism is identified and assigned a correct scientific/biological name comprising two words as per the binomial system of nomenclature
+An organism represents/occupies a place or position in the system of classification
+There are many categories/ranks and are generally referred to as taxonomic categories or taxa
+All the categories constitute a taxonomic hierarchy
+Taxonomists have developed a variety of taxonomic aids to facilitate identification, naming and classification of organisms
+These studies are carried out from the actual specimens which are collected from the field and preserved as referrals in the form of herbaria, museums and in botanical gardens and zoological parks
+It requires special techniques for collection and preservation of specimens in herbaria and museums
+Live specimens, on the other hand, of plants and animals, are found in botanical gardens or in zoological parks
+Taxonomists also prepare and disseminate information through manuals and monographs for further taxonomic studies
+Taxonomic keys are tools that help in identification based on characteristics.
+Exercises 1
+Why are living organisms classified? 2
+Why are the classification systems changing every now and then? 3
+What different criteria would you choose to classify people that you meet often? 4
+What do we learn from identification of individuals and populations? 5
+Given below is the scientific name of Mango
+Identify the correctly written name
+Mangifera Indica Mangifera indica 6
+Define a taxon
+Give some examples of taxa at different hierarchical levels
+7
+Can you identify the correct sequence of taxonomical categories? (a) Species Order Phylum Kingdom (b) Genus Species Order Kingdom (c) Species Genus Order Phylum 8
+Try to collect all the currently accepted meanings for the word ‘species’
+Discuss with your teacher the meaning of species in case of higher plants and animals on one hand, and bacteria on the other hand
+9
+Define and understand the following terms: Phylum Class Family (iv) Order (v) Genus 10
+How is a key helpful in the identification and classification of an organism? 11
+Illustrate the taxonomical hierarchy with suitable examples of a plant and an animal.
+The Living World
+Table of Contents
+Unit 1
+DIVERSITY IN THE LIVING WORLD
+Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 1
+The Living World
+1.1 What is ‘Living’? 1.2 Diversity in the Living World 1.3 Taxonomic Categories 1.4 Taxonomical Aids 1.1 What is ‘Living’? 1.2 Diversity in the Living World 1.3 Taxonomic Categories 1.3.1 Species 1.3.2 Genus 1.3.3 Family 1.3.4 Order 1.3.5 Class 1.3.6 Phylum 1.3.7 Kingdom 1.4 Taxonomical Aids 1.4.1 Herbarium 1.4.2 Botanical Gardens 1.4.3 Museum 1.4.4 Zoological Parks 1.4.5 Key
+Summary Exercises
+Landmarks
+Table of Contents
+Unit 12 Aldehydes, Ketones and Carboxylic Acids
+Objectives
+After studying this Unit, you will be able to • write the common and IUPAC names of aldehydes, ketones and carboxylic acids; • write the structures of the compounds containing functional groups namely carbonyl and carboxyl groups; • describe the important methods of preparation and reactions of these classes of compounds; • correlate physical properties and chemical reactions of aldehydes, ketones and carboxylic acids, with their structures; • explain the mechanism of a few selected reactions of aldehydes and ketones; • understand various factors affecting the acidity of carboxylic acids and their reactions; • describe the uses of aldehydes, ketones and carboxylic acids.
+Carbonyl compounds are of utmost importance to organic chemistry
+They are constituents of fabrics, flavourings, plastics and drugs.
+In the previous Unit, you have studied organic compounds with functional groups containing carbon-oxygen single bond
+In this Unit, we will study about the organic compounds containing carbon-oxygen double bond (>C=O) called carbonyl group, which is one of the most important functional groups in organic chemistry
+In aldehydes, the carbonyl group is bonded to a carbon and hydrogen while in the ketones, it is bonded to two carbon atoms
+The carbonyl compounds in which carbon of carbonyl group is bonded to carbon or hydrogen and oxygen of hydroxyl moiety (-OH) are known as carboxylic acids, while in compounds where carbon is attached to carbon or hydrogen and nitrogen of -NH2 moiety or to halogens are called amides and acyl halides respectively
+Esters and anhydrides are derivatives of carboxylic acids
+The general formulas of these classes of compounds are given below:
+Aldehydes, ketones and carboxylic acids are widespread in plants and animal kingdom
+They play an important role in biochemical processes of life
+They add fragrance and flavour to nature, for example, vanillin (from vanilla beans), salicylaldehyde (from meadow sweet) and cinnamaldehyde (from cinnamon) have very pleasant fragrances.
+They are used in many food products and pharmaceuticals to add flavours
+Some of these families are manufactured for use as solvents (i.e., acetone) and for preparing materials like adhesives, paints, resins, perfumes, plastics, fabrics, etc.
+12.1 Nomenclature and Structure of Carbonyl Group
+12.1.1 Nomenclature
+I
+Aldehydes and ketones Aldehydes and ketones are the simplest and most important carbonyl compounds
+There are two systems of nomenclature of aldehydes and ketones
+(a) Common names Aldehydes and ketones are often called by their common names instead of IUPAC names
+The common names of most aldehydes are derived from the common names of the corresponding carboxylic acids [Section 12.6.1] by replacing the ending –ic of acid with aldehyde
+At the same time, the names reflect the Latin or Greek term for the original source of the acid or aldehyde
+The location of the substituent in the carbon chain is indicated by Greek letters α, β, γ, δ, etc
+The α-carbon being the one directly linked to the aldehyde group, β-carbon the next, and so on
+For example
+The common names of ketones are derived by naming two alkyl or aryl groups bonded to the carbonyl group
+The locations of substituents are indicated by Greek letters, α α′, β β′ and so on beginning with the carbon atoms next to the carbonyl group, indicated as αα′
+Some ketones have historical common names, the simplest dimethyl ketone is called acetone
+Alkyl phenyl ketones are usually named by adding the name of acyl group as prefix to the word phenone
+For example
+(b) IUPAC names The IUPAC names of open chain aliphatic aldehydes and ketones are derived from the names of the corresponding alkanes by replacing the ending –e with –al and –one respectively
+In case of aldehydes the longest carbon chain is numbered starting from the carbon of the aldehyde group while in case of ketones the numbering begins from the end nearer to the carbonyl group
+The substituents are prefixed in alphabetical order along with numerals indicating their positions in the carbon chain
+The same applies to cyclic ketones, where the carbonyl carbon is numbered one
+When the aldehyde group is attached to a ring, the suffix carbaldehyde is added after the full name of the cycloalkane
+The numbering of the ring carbon atoms start from the carbon atom attached to the aldehyde group
+The name of the simplest aromatic aldehyde carrying the aldehyde group on a benzene ring is benzenecarbaldehyde
+However, the common name benzaldehyde is also accepted by IUPAC
+Other aromatic aldehydes are hence named as substituted benzaldehydes.
+The common and IUPAC names of some aldehydes and ketones are given in Table 12.1.
+Table 12.1: Common and IUPAC Names of Some Aldehydes and Ketones
+12.1.2 Structure of the Carbonyl Group
+The carbonyl carbon atom is sp2-hybridised and forms three sigma (σ) bonds
+The fourth valence electron of carbon remains in its p-orbital and forms a π-bond with oxygen by overlap with p-orbital of an oxygen
+In addition, the oxygen atom also has two non bonding electron pairs
+Thus, the carbonyl carbon and the three atoms attached to it lie in the same plane and the π-electron cloud is above and below this plane
+The bond angles are approximately 120° as expected of a trigonal coplanar structure .
+The carbon-oxygen double bond is polarised due to higher electronegativity of oxygen relative to carbon
+Hence, the carbonyl carbon is an electrophilic (Lewis acid), and carbonyl oxygen, a nucleophilic (Lewis base) centre
+Carbonyl compounds have substantial dipole moments and are polar than ethers
+The high polarity of the carbonyl group is explained on the basis of resonance involving a neutral (A) and a dipolar (B) structures as shown.
+Intext Questions 12.1 Write the structures of the following compounds
+α-Methoxypropionaldehyde 3-Hydroxybutanal 2-Hydroxycyclopentane carbaldehyde (iv) 4-Oxopentanal (v) Di-sec
+butyl ketone (vi) 4-Fluoroacetophenone
+12.2 Preparation of Aldehydes and Ketones Some important methods for the preparation of aldehydes and ketones are as follows:
+12.2.1 Preparation of Aldehydes and Ketones 1
+By oxidation of alcohols
+Aldehydes and ketones are generally prepared by oxidation of primary and secondary alcohols, respectively (Unit 11, Class XII).
+By dehydrogenation of alcohols This method is suitable for volatile alcohols and is of industrial application
+In this method alcohol vapours are passed over heavy metal catalysts (Ag or Cu)
+Primary and secondary alcohols give aldehydes and ketones, respectively (Unit 11, Class XII)
+3
+From hydrocarbons By ozonolysis of alkenes: As we know, ozonolysis of alkenes followed by reaction with zinc dust and water gives aldehydes, ketones or a mixture of both depending on the substitution pattern of the alkene (Unit 13, Class XI)
+By hydration of alkynes: Addition of water to ethyne in the presence of H2SO4 and HgSO4 gives acetaldehyde
+All other alkynes give ketones in this reaction (Unit 13, Class XI).
+12.2.2 Preparation of Aldehydes 1
+From acyl chloride (acid chloride) Acyl chloride (acid chloride) is hydrogenated over catalyst, palladium on barium sulphate
+This reaction is called Rosenmund reduction.
+From nitriles and esters Nitriles are reduced to corresponding imine with stannous chloride in the presence of hydrochloric acid, which on hydrolysis give corresponding aldehyde.
+This reaction is called Stephen reaction
+Alternatively, nitriles are selectively reduced by diisobutylaluminium hydride, (DIBAL-H) to imines followed by hydrolysis to aldehydes:
+Similarly, esters are also reduced to aldehydes with DIBAL-H.
+From hydrocarbons
+Aromatic aldehydes (benzaldehyde and its derivatives) are prepared from aromatic hydrocarbons by the following methods:
+By oxidation of methylbenzene Strong oxidising agents oxidise toluene and its derivatives to benzoic acids
+However, it is possible to stop the oxidation at the aldehyde stage with suitable reagents that convert the methyl group to an intermediate that is difficult to oxidise further
+The following methods are used for this purpose
+(a) Use of chromyl chloride (CrO2Cl2): Chromyl chloride oxidises methyl group to a chromium complex, which on hydrolysis gives corresponding benzaldehyde.
+This reaction is called Etard reaction
+(b) Use of chromic oxide (CrO3): Toluene or substituted toluene is converted to benzylidene diacetate on treating with chromic oxide in acetic anhydride
+The benzylidene diacetate can be hydrolysed to corresponding benzaldehyde with aqueous acid.
+By side chain chlorination followed by hydrolysis Side chain chlorination of toluene gives benzal chloride, which on hydrolysis gives benzaldehyde
+This is a commercial method of manufacture of benzaldehyde.
+By Gatterman – Koch reaction
+When benzene or its derivative is treated with carbon monoxide and hydrogen chloride in the presence of anhydrous aluminium chloride or cuprous chloride, it gives benzaldehyde or substituted benzaldehyde.
+This reaction is known as Gatterman-Koch reaction
+12.2.3 Preparation of Ketones 1
+From acyl chlorides
+Treatment of acyl chlorides with dialkylcadmium, prepared by the reaction of cadmium chloride with Grignard reagent, gives ketones.
+From nitriles
+Treating a nitrile with Grignard reagent followed by hydrolysis yields a ketone.
+3
+From benzene or substituted benzenes When benzene or substituted benzene is treated with acid chloride in the presence of anhydrous aluminium chloride, it affords the corresponding ketone
+This reaction is known as Friedel-Crafts acylation reaction.
+Example 12.1 Give names of the reagents to bring about the following transformations: Hexan-1-ol to hexanal Cyclohexanol to cyclohexanone p-Fluorotoluene to p-fluorobenzaldehyde (iv) Ethanenitrile to ethanal
+(v) Allyl alcohol to propenal (vi) But-2-ene to ethanal Solution
+C5H5NH+CrO3Cl-(PCC) Anhydrous CrO3 CrO3 in the presence of acetic anhydride/1
+CrO2Cl2 2
+HOH (iv) (Diisobutyl)aluminiumhydride (DIBAL-H) (v) PCC (vi) O3/H2O-Zn dust
+Intext Question
+12.2 Write the structures of products of the following reactions;
+(iv)
+12.3 Physical Properties The physical properties of aldehydes and ketones are described as follows
+Methanal is a gas at room temperature
+Ethanal is a volatile liquid
+Other aldehydes and ketones are liquid or solid at room temperature
+The boiling points of aldehydes and ketones are higher than hydrocarbons and ethers of comparable molecular masses
+It is due to weak molecular association in aldehydes and ketones arising out of the dipole-dipole interactions
+Also, their boiling points are lower than those of alcohols of similar molecular masses due to absence of intermolecular hydrogen bonding
+The following compounds of molecular masses 58 and 60 are ranked in order of increasing boiling points.
+The lower members of aldehydes and ketones such as methanal, ethanal and propanone are miscible with water in all proportions, because they form hydrogen bond with water.
+However, the solubility of aldehydes and ketones decreases rapidly on increasing the length of alkyl chain
+All aldehydes and ketones are fairly soluble in organic solvents like benzene, ether, methanol, chloroform, etc
+The lower aldehydes have sharp pungent odours
+As the size of the molecule increases, the odour becomes less pungent and more fragrant
+In fact, many naturally occurring aldehydes and ketones are used in the blending of perfumes and flavouring agents.
+Example 12.2
+Arrange the following compounds in the increasing order of their boiling points:
+CH3CH2CH2CHO, CH3CH2CH2CH2OH, H5C2-O-C2H5, CH3CH2CH2CH3 Solution The molecular masses of these compounds are in the range of 72 to 74
+Since only butan-1-ol molecules are associated due to extensive intermolecular hydrogen bonding, therefore, the boiling point of butan-1-ol would be the highest
+Butanal is more polar than ethoxyethane
+Therefore, the intermolecular dipole-dipole attraction is stronger in the former
+n-Pentane molecules have only weak van der Waals forces
+Hence increasing order of boiling points of the given compounds is as follows: CH3CH2CH2CH3 < H5C2-O-C2H5 < CH3CH2CH2CHO < CH3CH2CH2CH2OH
+Intext Question 12.3 Arrange the following compounds in increasing order of their boiling points
+CH3CHO, CH3CH2OH, CH3OCH3, CH3CH2CH3
+12.4 Chemical Reactions
+Since aldehydes and ketones both possess the carbonyl functional group, they undergo similar chemical reactions
+1
+Nucleophilic addition reactions Contrary to electrophilic addition reactions observed in alkenes (refer Unit 13, Class XI), the aldehydes and ketones undergo nucleophilic addition reactions.
+Mechanism of nucleophilic addition reactions A nucleophile attacks the electrophilic carbon atom of the polar carbonyl group from a direction approximately perpendicular to the plane of sp2 hybridised orbitals of carbonyl carbon
+The hybridisation of carbon changes from sp2 to sp3 in this process, and a tetrahedral alkoxide intermediate is produced
+This intermediate captures a proton from the reaction medium to give the electrically neutral product
+The net result is addition of Nu– and H+ across the carbon oxygen double bond as shown in Fig
+12.2.
+Reactivity Aldehydes are generally more reactive than ketones in nucleophilic addition reactions due to steric and electronic reasons
+Sterically, the presence of two relatively large substituents in ketones hinders the approach of nucleophile to carbonyl carbon than in aldehydes having only one such substituent
+Electronically, aldehydes are more reactive than ketones because two alkyl groups reduce the electrophilicity of the carbonyl carbon more effectively than in former.
+Example 12.3
+Would you expect benzaldehyde to be more reactive or less reactive in nucleophilic addition reactions than propanal? Explain your answer.
+Solution The carbon atom of the carbonyl group of benzaldehyde is less electrophilic than carbon atom of the carbonyl group present in propanal
+The polarity of the carbonyl group is reduced in benzaldehyde due to resonance as shown below and hence it is less reactive than propanal.
+Some important examples of nucleophilic addition and nucleophilic addition-elimination reactions: (a) Addition of hydrogen cyanide (HCN): Aldehydes and ketones react with hydrogen cyanide (HCN) to yield cyanohydrins
+This reaction occurs very slowly with pure HCN
+Therefore, it is catalysed by a base and the generated cyanide ion (CN-) being a stronger nucleophile readily adds to carbonyl compounds to yield corresponding cyanohydrin
+Cyanohydrins are useful synthetic intermediates
+(b) Addition of sodium hydrogensulphite: Sodium hydrogensulphite adds to aldehydes and ketones to form the addition products.
+The position of the equilibrium lies largely to the right hand side for most aldehydes and to the left for most ketones due to steric reasons
+The hydrogensulphite addition compound is water soluble and can be converted back to the original carbonyl compound by treating it with dilute mineral acid or alkali
+Therefore, these are useful for separation and purification of aldehydes.
+(c) Addition of Grignard reagents: (refer Unit 11, Class XII)
+(d) Addition of alcohols: Aldehydes react with one equivalent of monohydric alcohol in the presence of dry hydrogen chloride to yield alkoxyalcohol intermediate, known as hemiacetals, which further react with one more molecule of alcohol to give a gem-dialkoxy compound known as acetal as shown in the reaction.
+Ketones react with ethylene glycol under similar conditions to form cyclic products known as ethylene glycol ketals
+Dry hydrogen chloride protonates the oxygen of the carbonyl compounds and therefore, increases the electrophilicity of the carbonyl carbon facilitating the nucleophilic attack of ethylene glycol
+Acetals and ketals are hydrolysed with aqueous mineral acids to yield corresponding aldehydes and ketones respectively
+(e) Addition of ammonia and its derivatives: Nucleophiles, such as ammonia and its derivatives H2N-Z add to the carbonyl group of aldehydes and ketones
+The reaction is reversible and catalysed by acid
+The equilibrium favours the product formation due to rapid dehydration of the intermediate to form >C=N-Z.
+Z = Alkyl, aryl, OH, NH2, C6H5NH, NHCONH2, etc.
+Table 12.2: Some N-Substituted Derivatives of Aldehydes and Ketones (>C=N-Z)
+* 2,4-DNP-derivatives are yellow, orange or red solids, useful for characterisation of aldehydes and ketones.
+2
+Reduction
+Reduction to alcohols: Aldehydes and ketones are reduced to primary and secondary alcohols respectively by sodium borohydride (NaBH4) or lithium aluminium hydride (LiAlH4) as well as by catalytic hydrogenation (Unit 11, Class XII)
+Reduction to hydrocarbons: The carbonyl group of aldehydes and ketones is reduced to CH2 group on treatment with zinc-amalgam and concentrated hydrochloric acid [Clemmensen reduction] or with hydrazine followed by heating with sodium or potassium hydroxide in high boiling solvent such as ethylene glycol (Wolff-Kishner reduction).
+Bernhard Tollens (1841-1918) was a Professor of Chemistry at the University of Gottingen, Germany.
+3
+Oxidation Aldehydes differ from ketones in their oxidation reactions
+Aldehydes are easily oxidised to carboxylic acids on treatment with common oxidising agents like nitric acid, potassium permanganate, potassium dichromate, etc
+Even mild oxidising agents, mainly Tollens’ reagent and Fehlings’ reagent also oxidise aldehydes.
+Ketones are generally oxidised under vigorous conditions, i.e., strong oxidising agents and at elevated temperatures
+Their oxidation involves carbon-carbon bond cleavage to afford a mixture of carboxylic acids having lesser number of carbon atoms than the parent ketone.
+The mild oxidising agents given below are used to distinguish aldehydes from ketones: Tollens’ test: On warming an aldehyde with freshly prepared ammoniacal silver nitrate solution (Tollens’ reagent), a bright silver mirror is produced due to the formation of silver metal
+The aldehydes are oxidised to corresponding carboxylate anion
+The reaction occurs in alkaline medium.
+Fehling’s test: Fehling reagent comprises of two solutions, Fehling solution A and Fehling solution B
+Fehling solution A is aqueous copper sulphate and Fehling solution B is alkaline sodium potassium tartarate (Rochelle salt)
+These two solutions are mixed in equal amounts before test
+On heating an aldehyde with Fehling’s reagent, a reddish brown precipitate is obtained
+Aldehydes are oxidised to corresponding carboxylate anion
+Aromatic aldehydes do not respond to this test.
+Oxidation of methyl ketones by haloform reaction: Aldehydes and ketones having at least one methyl group linked to the carbonyl carbon atom (methyl ketones) are oxidised by sodium hypohalite to sodium salts of corresponding carboxylic acids having one carbon atom less than that of carbonyl compound
+The methyl group is converted to haloform
+This oxidation does not affect a carbon-carbon double bond, if present in the molecule.
+Iodoform reaction with sodium hypoiodite is also used for detection of CH3CO group or CH3CH(OH) group which produces CH3CO group on oxidation.
+Example 12.4
+An organic compound (A) with molecular formula C8H8O forms an orange-red precipitate with 2,4-DNP reagent and gives yellow precipitate on heating with iodine in the presence of sodium hydroxide
+It neither reduces Tollens’ or Fehlings’ reagent, nor does it decolourise bromine water or Baeyer’s reagent
+On drastic oxidation with chromic acid, it gives a carboxylic acid (B) having molecular formula C7H6O2
+Identify the compounds (A) and (B) and explain the reactions involved.
+Solution (A) forms 2,4-DNP derivative
+Therefore, it is an aldehyde or a ketone
+Since it does not reduce Tollens’ or Fehling reagent, (A) must be a ketone
+(A) responds to iodoform test
+Therefore, it should be a methyl ketone
+The molecular formula of (A) indicates high degree of unsaturation, yet it does not decolourise bromine water or Baeyer’s reagent
+This indicates the presence of unsaturation due to an aromatic ring
+Compound (B), being an oxidation product of a ketone should be a carboxylic acid
+The molecular formula of (B) indicates that it should be benzoic acid and compound (A) should, therefore, be a monosubstituted aromatic methyl ketone
+The molecular formula of (A) indicates that it should be phenyl methyl ketone (acetophenone)
+Reactions are as follows:
+Reactions due to a-hydrogen Acidity of α-hydrogens of aldehydes and ketones: The aldehydes and ketones undergo a number of reactions due to the acidic nature of α-hydrogen. The acidity of α-hydrogen atoms of carbonyl compounds is due to the strong electron withdrawing effect of the carbonyl group and resonance stabilisation of the conjugate base.
+Aldol condensation: Aldehydes and ketones having at least one α-hydrogen undergo a reaction in the presence of dilute alkali as catalyst to form β-hydroxy aldehydes (aldol) or β-hydroxy ketones (ketol), respectively
+This is known as Aldol reaction.
+The name aldol is derived from the names of the two functional groups, aldehyde and alcohol, present in the products
+The aldol and ketol readily lose water to give α,β-unsaturated carbonyl compounds which are aldol condensation products and the reaction is called Aldol condensation
+Though ketones give ketols (compounds containing a keto and alcohol groups), the general name aldol condensation still applies to the reactions of ketones due to their similarity with aldehydes.
+Cross aldol condensation: When aldol condensation is carried out between two different aldehydes and / or ketones, it is called cross aldol condensation
+If both of them contain α-hydrogen atoms, it gives a mixture of four products
+This is illustrated below by aldol reaction of a mixture of ethanal and propanal.
+Ketones can also be used as one component in the cross aldol reactions.
+Other reactions
+Cannizzaro reaction: Aldehydes which do not have an α-hydrogen atom, undergo self oxidation and reduction (disproportionation) reaction on heating with concentrated alkali
+In this reaction, one molecule of the aldehyde is reduced to alcohol while another is oxidised to carboxylic acid salt.
+Electrophilic substitution reaction: Aromatic aldehydes and ketones undergo electrophilic substitution at the ring in which the carbonyl group acts as a deactivating and meta-directing group.
+Intext Questions 12.4 Arrange the following compounds in increasing order of their reactivity in nucleophilic addition reactions
+Ethanal, Propanal, Propanone, Butanone
+Benzaldehyde, p-Tolualdehyde, p-Nitrobenzaldehyde, Acetophenone
+Hint: Consider steric effect and electronic effect
+12.5 Predict the products of the following reactions:
+(iv)
+12.5 Uses of Aldehydes and Ketones In chemical industry aldehydes and ketones are used as solvents, starting materials and reagents for the synthesis of other products
+Formaldehyde is well known as formalin (40%) solution used to preserve biological specimens and to prepare bakelite (a phenol-formaldehyde resin), urea-formaldehyde glues and other polymeric products
+Acetaldehyde is used primarily as a starting material in the manufacture of acetic acid, ethyl acetate, vinyl acetate, polymers and drugs
+Benzaldehyde is used in perfumery and in dye industries
+Acetone and ethyl methyl ketone are common industrial solvents
+Many aldehydes and ketones, e.g., butyraldehyde, vanillin, acetophenone, camphor, etc
+are well known for their odours and flavours.
+Carboxylic Acids
+Carbon compounds containing a carboxyl functional group, –COOH are called carboxylic acids
+The carboxyl group, consists of a carbonyl group attached to a hydroxyl group, hence its name carboxyl
+Carboxylic acids may be aliphatic (RCOOH) or aromatic (ArCOOH) depending on the group, alkyl or aryl, attached to carboxylic carbon
+Large number of carboxylic acids are found in nature
+Some higher members of aliphatic carboxylic acids (C12 – C18) known as fatty acids, occur in natural fats as esters of glycerol
+Carboxylic acids serve as starting material for several other important organic compounds such as anhydrides, esters, acid chlorides, amides, etc.
+12.6 Nomenclature and Structure of Carboxyl Group
+12.6.1 Nomenclature
+Since carboxylic acids are amongst the earliest organic compounds to be isolated from nature, a large number of them are known by their common names
+The common names end with the suffix –ic acid and have been derived from Latin or Greek names of their natural sources
+For example, formic acid (HCOOH) was first obtained from red ants (Latin: formica means ant), acetic acid (CH3COOH) from vinegar (Latin: acetum, means vinegar), butyric acid (CH3CH2CH2COOH) from rancid butter (Latin: butyrum, means butter).
+In the IUPAC system, aliphatic carboxylic acids are named by replacing the ending –e in the name of the corresponding alkane with – oic acid
+In numbering the carbon chain, the carboxylic carbon is numbered one
+For naming compounds containing more than one carboxyl group, the alkyl chain leaving carboxyl groups is numbered and the number of carboxyl groups is indicated by adding the multiplicative prefix, dicarboxylic acid, tricarboxylic acid, etc
+to the name of parent alkyl chain
+The position of –COOH groups are indicated by the arabic numeral before the multiplicative prefix
+Some of the carboxylic acids along with their common and IUPAC names are listed in Table 12.3.
+Table 12.3 Names and Structures of Some Carboxylic Acids
+12.6.2 Structure of Carboxyl Group
+In carboxylic acids, the bonds to the carboxyl carbon lie in one plane and are separated by about 120°
+The carboxylic carbon is less electrophilic than carbonyl carbon because of the possible resonance structure shown below:
+Intext Question 12.6 Give the IUPAC names of the following compounds: Ph CH2CH2COOH (CH3)2C=CHCOOH (iv)
+12.7 Methods of Preparation of Carboxylic Acids
+Some important methods of preparation of carboxylic acids are as follows
+1
+From primary alcohols and aldehydes Primary alcohols are readily oxidised to carboxylic acids with common oxidising agents such as potassium permanganate (KMnO4) in neutral, acidic or alkaline media or by potassium dichromate (K2Cr2O7) and chromium trioxide (CrO3) in acidic media (Jones reagent).
+Carboxylic acids are also prepared from aldehydes by the use of mild oxidising agents (Section 12.4).
+2
+From alkylbenzenes Aromatic carboxylic acids can be prepared by vigorous oxidation of alkyl benzenes with chromic acid or acidic or alkaline potassium permanganate
+The entire side chain is oxidised to the carboxyl group irrespective of length of the side chain
+Primary and secondary alkyl groups are oxidised in this manner while tertiary group is not affected
+Suitably substituted alkenes are also oxidised to carboxylic acids with these oxidising reagents (refer Unit 13, Class XI).
+3
+From nitriles and amides Nitriles are hydrolysed to amides and then to acids in the presence of H+ or as catalyst
+Mild reaction conditions are used to stop the reaction at the amide stage.
+4
+From Grignard reagents Grignard reagents react with carbon dioxide (dry ice) to form salts of carboxylic acids which in turn give corresponding carboxylic acids after acidification with mineral acid.
+As we know, the Grignard reagents and nitriles can be prepared from alkyl halides (refer Unit 10, Class XII)
+The above methods (3 and 4) are useful for converting alkyl halides into corresponding carboxylic acids having one carbon atom more than that present in alkyl halides (ascending the series).
+5
+From acyl halides and anhydrides Acid chlorides when hydrolysed with water give carboxylic acids or more readily hydrolysed with aqueous base to give carboxylate ions which on acidification provide corresponding carboxylic acids
+Anhydrides on the other hand are hydrolysed to corresponding acid(s) with water.
+6
+From esters
+Acidic hydrolysis of esters gives directly carboxylic acids while basic hydrolysis gives carboxylates, which on acidification give corresponding carboxylic acids.
+Example 12.5 Write chemical reactions to affect the following transformations: Butan-1-ol to butanoic acid Benzyl alcohol to phenylethanoic acid 3-Nitrobromobenzene to 3-nitrobenzoic acid (iv) 4-Methylacetophenone to benzene-1,4-dicarboxylic acid (v) Cyclohexene to hexane-1,6-dioic acid
+(vi) Butanal to butanoic acid
+Solution
+(iv)
+(v)
+(vi)
+Intext Question 12.7 Show how each of the following compounds can be converted to benzoic acid
+Ethylbenzene Acetophenone Bromobenzene (iv) Phenylethene (Styrene)
+12.8 Physical Properties Aliphatic carboxylic acids upto nine carbon atoms are colourless liquids at room temperature with unpleasant odours
+The higher acids are wax like solids and are practically odourless due to their low volatility
+Carboxylic acids are higher boiling liquids than aldehydes, ketones and even alcohols of comparable molecular masses
+This is due to more extensive association of carboxylic acid molecules through intermolecular hydrogen bonding
+The hydrogen bonds are not broken completely even in the vapour phase
+In fact, most carboxylic acids exist as dimer in the vapour phase or in the aprotic solvents.
+In vapour state or in aprotic solvent
+Simple aliphatic carboxylic acids having upto four carbon atoms are miscible in water due to the formation of hydrogen bonds with water
+The solubility decreases with increasing number of carbon atoms
+Higher carboxylic acids are practically insoluble in water due to the increased hydrophobic interaction of hydrocarbon part
+Benzoic acid, the simplest aromatic carboxylic acid is nearly insoluble in cold water
+Carboxylic acids are also soluble in less polar organic solvents like benzene, ether, alcohol, chloroform, etc.
+Hydrogen bonding of RCOOH with H2O
+12.9 Chemical Reactions The reaction of carboxylic acids are classified as follows:
+12.9.1 Reactions Involving Cleavage of O–H Bond
+Acidity
+Reactions with metals and alkalies
+The carboxylic acids like alcohols evolve hydrogen with electropositive metals and form salts with alkalies similar to phenols
+However, unlike phenols they react with weaker bases such as carbonates and hydrogencarbonates to evolve carbon dioxide
+This reaction is used to detect the presence of carboxyl group in an organic compound.
+Carboxylic acids dissociate in water to give resonance stabilised carboxylate anions and hydronium ion.
+For the above reaction:
+where Keq, is equilibrium constant and Ka is the acid dissociation constant
+For convenience, the strength of an acid is generally indicated by its pka value rather than its Ka value
+pKa = – log Ka The pKa of hydrochloric acid is –7.0, where as pKa of trifluoroacetic acid (the strongest carboxylic acid), benzoic acid and acetic acid are 0.23, 4.19 and 4.76, respectively
+Smaller the pKa, the stronger the acid ( the better it is as a proton donor)
+Strong acids have pKa values < 1, the acids with pKa values between 1 and 5 are considered to be moderately strong acids, weak acids have pKa values between 5 and 15, and extremely weak acids have pKa values >15.
+Carboxylic acids are weaker than mineral acids, but they are stronger acids than alcohols and many simple phenols (pKa is ~16 for ethanol and 10 for phenol)
+In fact, carboxylic acids are amongst the most acidic organic compounds you have studied so far
+You already know why phenols are more acidic than alcohols
+The higher acidity of carboxylic acids as compared to phenols can be understood similarly
+The conjugate base of carboxylic acid, a carboxylate ion, is stabilised by two equivalent resonance structures in which the negative charge is at the more electronegative oxygen atom
+The conjugate base of phenol, a phenoxide ion, has non-equivalent resonance structures in which the negative charge is at the less electronegative carbon atom
+Therefore, resonance in phenoxide ion is not as important as it is in carboxylate ion
+Further, the negative charge is delocalised over two electronegative oxygen atoms in carboxylate ion whereas it is less effectively delocalised over one oxygen atom and less electronegative carbon atoms in phenoxide ion (Unit 11, Class XII)
+Thus, the carboxylate ion is more stabilised than phenoxide ion, so carboxylic acids are more acidic than phenols.
+Effect of substituents on the acidity of carboxylic acids: Substituents may affect the stability of the conjugate base and thus, also affect the acidity of the carboxylic acids
+Electron withdrawing groups increase the acidity of carboxylic acids by stabilising the conjugate base through delocalisation of the negative charge by inductive and/or resonance effects
+Conversely, electron donating groups decrease the acidity by destabilising the conjugate base.
+Electron withdrawing group (EWG) stabilises the carboxylate anion and strengthens the acid
+Electron donating group (EDG) destabilises the carboxylate anion and weakens the acid
+The effect of the following groups in increasing acidity order is Ph < I < Br < Cl < F < CN < NO2 < CF3 Thus, the following acids are arranged in order of increasing acidity (based on pKa values): CF3COOH > CCl3COOH > CHCl2COOH > NO2CH2COOH > NC-CH2COOH >
+FCH2COOH > ClCH2COOH > BrCH2COOH > HCOOH > ClCH2CH2COOH > (continue) C6H5COOH > C6H5CH2COOH > CH3COOH > CH3CH2COOH (continue ) Direct attachment of groups such as phenyl or vinyl to the carboxylic acid, increases the acidity of corresponding carboxylic acid, contrary to the decrease expected due to resonance effect shown below:
+This is because of greater electronegativity of sp2 hybridised carbon to which carboxyl carbon is attached
+The presence of electron withdrawing group on the phenyl of aromatic carboxylic acid increases their acidity while electron donating groups decrease their acidity.
+12.9.2 Reactions Involving Cleavage of C–OH Bond 1
+Formation of anhydride Carboxylic acids on heating with mineral acids such as H2SO4 or with P2O5 give corresponding anhydride.
+2
+Esterification Carboxylic acids are esterified with alcohols or phenols in the presence of a mineral acid such as concentrated H2SO4 or HCl gas as a catalyst.
+Mechanism of esterification of carboxylic acids: The esterification of carboxylic acids with alcohols is a kind of nucleophilic acyl substitution
+Protonation of the carbonyl oxygen activates the carbonyl group towards nucleophilic addition of the alcohol
+Proton transfer in the tetrahedral intermediate converts the hydroxyl group into –+OH2 group, which, being a better leaving group, is eliminated as neutral water molecule
+The protonated ester so formed finally loses a proton to give the ester.
+Reactions with PCl5, PCl3 and SOCl2 The hydroxyl group of carboxylic acids, behaves like that of alcohols and is easily replaced by chlorine atom on treating with PCl5, PCl3 or SOCl2
+Thionyl chloride (SOCl2) is preferred because the other two products are gaseous and escape the reaction mixture making the purification of the products easier.
+4
+Reaction with ammonia Carboxylic acids react with ammonia to give ammonium salt which on further heating at high temperature give amides
+For example:
+12.9.3 Reactions Involving –COOH Group 1
+Reduction Carboxylic acids are reduced to primary alcohols by lithium aluminium hydride or better with diborane
+Diborane does not easily reduce functional groups such as ester, nitro, halo, etc
+Sodium borohydride does not reduce the carboxyl group.
+Decarboxylation Carboxylic acids lose carbon dioxide to form hydrocarbons when their sodium salts are heated with sodalime (NaOH and CaO in the ratio of 3 : 1)
+The reaction is known as decarboxylation.
+Alkali metal salts of carboxylic acids also undergo decarboxylation on electrolysis of their aqueous solutions and form hydrocarbons having twice the number of carbon atoms present in the alkyl group of the acid
+The reaction is known as Kolbe electrolysis (Unit 13, Class XI).
+12.9.4
+Substitution Reactions in the Hydrocarbon Part 1
+Halogenation
+Carboxylic acids having an α-hydrogen are halogenated at the α-position on treatment with chlorine or bromine in the presence of small amount of red phosphorus to give α-halocarboxylic acids
+The reaction is known as Hell-Volhard-Zelinsky reaction.
+Ring substitution
+Aromatic carboxylic acids undergo electrophilic substitution reactions in which the carboxyl group acts as a deactivating and meta-directing group
+They however, do not undergo Friedel-Crafts reaction (because the carboxyl group is deactivating and the catalyst aluminium chloride (Lewis acid) gets bonded to the carboxyl group).
+Intext Question 12.8 Which acid of each pair shown here would you expect to be stronger? CH3CO2H or CH2FCO2H CH2FCO2H or CH2ClCO2H CH2FCH2CH2CO2H or CH3CHFCH2CO2H
+(iv)
+12.10 Uses of Carboxylic Acids
+Methanoic acid is used in rubber, textile, dyeing, leather and electroplating industries
+Ethanoic acid is used as solvent and as vinegar in food industry
+Hexanedioic acid is used in the manufacture of nylon-6, 6
+Esters of benzoic acid are used in perfumery
+Sodium benzoate is used as a food preservative
+Higher fatty acids are used for the manufacture of soaps and detergents.
+Summary Aldehydes, ketones and carboxylic acids are some of the important classes of organic compounds containing carbonyl group
+These are highly polar molecules
+Therefore, they boil at higher temperatures than the hydrocarbons and weakly polar compounds such as ethers of comparable molecular masses
+The lower members are more soluble in water because they form hydrogen bonds with water
+The higher members, because of large size of hydrophobic chain of carbon atoms, are insoluble in water but soluble in common organic solvents
+Aldehydes are prepared by dehydrogenation or controlled oxidation of primary alcohols and controlled or selective reduction of acyl halides
+Aromatic aldehydes may also be prepared by oxidation of methylbenzene with chromyl chloride or CrO3 in the presence of acetic anhydride, formylation of arenes with carbon monoxide and hydrochloric acid in the presence of anhydrous aluminium chloride, and cuprous chloride or by hydrolysis of benzal chloride
+Ketones are prepared by oxidation of secondary alcohols and hydration of alkynes
+Ketones are also prepared by reaction of acyl chloride with dialkylcadmium
+A good method for the preparation of aromatic ketones is the Friedel-Crafts acylation of aromatic hydrocarbons with acyl chlorides or anhydrides
+Both aldehydes and ketones can be prepared by ozonolysis of alkenes
+Aldehydes and ketones undergo nucleophilic addition reactions onto the carbonyl group with a number of nucleophiles such as, HCN, NaHSO3, alcohols (or diols), ammonia derivatives, and Grignard reagents
+The α-hydrogens in aldehydes and ketones are acidic
+Therefore, aldehydes and ketones having at least one α-hydrogen, undergo Aldol condensation in the presence of a base to give α-hydroxyaldehydes (aldol) and α-hydroxyketones(ketol), respectively
+Aldehydes having no α-hydrogen undergo Cannizzaro reaction in the presence of concentrated alkali
+Aldehydes and ketones are reduced to alcohols with NaBH4, LiAlH4, or by catalytic hydrogenation
+The carbonyl group of aldehydes and ketones can be reduced to a methylene group by Clemmensen reduction or Wolff-Kishner reduction
+Aldehydes are easily oxidised to carboxylic acids by mild oxidising reagents such as Tollens’ reagent and Fehling’s reagent
+These oxidation reactions are used to distinguish aldehydes from ketones
+Carboxylic acids are prepared by the oxidation of primary alcohols, aldehydes and alkenes by hydrolysis of nitriles, and by treatment of Grignard reagents with carbon dioxide
+Aromatic carboxylic acids are also prepared by side-chain oxidation of alkylbenzenes
+Carboxylic acids are considerably more acidic than alcohols and most of simple phenols
+Carboxylic acids are reduced to primary alcohols with LiAlH4, or better with diborane in ether solution and also undergo α-halogenation with Cl2 and Br2 in the presence of red phosphorus (Hell-Volhard Zelinsky reaction)
+Methanal, ethanal, propanone, benzaldehyde, formic acid, acetic acid and benzoic acid are highly useful compounds in industry.
+Exercises
+12.1 What is meant by the following terms ? Give an example of the reaction in each case
+Cyanohydrin Acetal Semicarbazone (iv) Aldol (v) Hemiacetal (vi) Oxime (vii) Ketal (vii) Imine (ix) 2,4-DNP-derivative (x) Schiff’s base 12.2 Name the following compounds according to IUPAC system of nomenclature: CH3CH(CH3)CH2CH2CHO CH3CH2COCH(C2H5)CH2CH2Cl CH3CH=CHCHO (iv) CH3COCH2COCH3 (v) CH3CH(CH3)CH2C(CH3)2COCH3 (vi) (CH3)3CCH2COOH (vii) OHCC6H4CHO-p 12.3 Draw the structures of the following compounds
+3-Methylbutanal p-Nitropropiophenone p-Methylbenzaldehyde (iv) 4-Methylpent-3-en-2-one (v) 4-Chloropentan-2-one (vi) 3-Bromo-4-phenylpentanoic acid (vii) p,p’-Dihydroxybenzophenone (viii) Hex-2-en-4-ynoic acid 12.4 Write the IUPAC names of the following ketones and aldehydes
+Wherever possible, give also common names
+CH3CO(CH2)4CH3 CH3CH2CHBrCH2CH(CH3)CHO CH3(CH2)5CHO (iv) Ph-CH=CH-CHO (v) (vi) PhCOPh 12.5 Draw structures of the following derivatives
+The 2,4-dinitrophenylhydrazone of benzaldehyde Cyclopropanone oxime Acetaldehydedimethylacetal (iv) The semicarbazone of cyclobutanone (v) The ethylene ketal of hexan-3-one (vi) The methyl hemiacetal of formaldehyde 12.6 Predict the products formed when cyclohexanecarbaldehyde reacts with following reagents
+PhMgBr and then H3O+ Tollens’ reagent Semicarbazide and weak acid (iv) Excess ethanol and acid (v) Zinc amalgam and dilute hydrochloric acid 12.7 Which of the following compounds would undergo aldol condensation, which the Cannizzaro reaction and which neither? Write the structures of the expected products of aldol condensation and Cannizzaro reaction
+Methanal 2-Methylpentanal Benzaldehyde (iv) Benzophenone (v) Cyclohexanone (vi) 1-Phenylpropanone (vii) Phenylacetaldehyde (viii) Butan-1-ol (ix) 2,2-Dimethylbutanal 12.8 How will you convert ethanal into the following compounds? Butane-1,3-diol But-2-enal But-2-enoic acid 12.9 Write structural formulas and names of four possible aldol condensation products from propanal and butanal
+In each case, indicate which aldehyde acts as nucleophile and which as electrophile
+12.10 An organic compound with the molecular formula C9H10O forms 2,4-DNP derivative, reduces Tollens’ reagent and undergoes Cannizzaro reaction
+On vigorous oxidation, it gives 1,2-benzenedicarboxylic acid
+Identify the compound
+12.11 An organic compound (A) (molecular formula C8H16O2) was hydrolysed with dilute sulphuric acid to give a carboxylic acid (B) and an alcohol (C)
+Oxidation of (C) with chromic acid produced (B)
+(C) on dehydration gives but-1-ene
+Write equations for the reactions involved
+12.12 Arrange the following compounds in increasing order of their property as indicated: Acetaldehyde, Acetone, Di-tert-butyl ketone, Methyl tert-butyl ketone (reactivity towards HCN) CH3CH2CH(Br)COOH, CH3CH(Br)CH2COOH, (CH3)2CHCOOH, CH3CH2CH2COOH (acid strength) Benzoic acid, 4-Nitrobenzoic acid, 3,4-Dinitrobenzoic acid, 4-Methoxybenzoic acid (acid strength) 12.13 Give simple chemical tests to distinguish between the following pairs of compounds
+Propanal and Propanone Acetophenone and Benzophenone Phenol and Benzoic acid (iv) Benzoic acid and Ethyl benzoate (v) Pentan-2-one and Pentan-3-one (vi) Benzaldehyde and Acetophenone (vii) Ethanal and Propanal 12.14 How will you prepare the following compounds from benzene? You may use any inorganic reagent and any organic reagent having not more than one carbon atom Methyl benzoate m-Nitrobenzoic acid p-Nitrobenzoic acid (iv) Phenylacetic acid (v) p-Nitrobenzaldehyde
+12.15 How will you bring about the following conversions in not more than two steps? Propanone to Propene Benzoic acid to Benzaldehyde Ethanol to 3-Hydroxybutanal (iv) Benzene to m-Nitroacetophenone (v) Benzaldehyde to Benzophenone (vi) Bromobenzene to 1-Phenylethanol (vii) Benzaldehyde to 3-Phenylpropan-1-ol (viii) Benazaldehyde to α-Hydroxyphenylacetic acid (ix) Benzoic acid to m- Nitrobenzyl alcohol 12.16 Describe the following: Acetylation Cannizzaro reaction Cross aldol condensation (iv) Decarboxylation 12.17 Complete each synthesis by giving missing starting material, reagent or products
+12.18 Give plausible explanation for each of the following: Cyclohexanone forms cyanohydrin in good yield but 2,2,6-trimethylcyclo-hexanone does not
+There are two –NH2 groups in semicarbazide
+However, only one is involved in the formation of semicarbazones
+During the preparation of esters from a carboxylic acid and an alcohol in the presence of an acid catalyst, the water or the ester should be removed as soon as it is formed
+12.19 An organic compound contains 69.77% carbon, 11.63% hydrogen and rest oxygen
+The molecular mass of the compound is 86
+It does not reduce Tollens’ reagent but forms an addition compound with sodium hydrogensulphite and give positive iodoform test
+On vigorous oxidation it gives ethanoic and propanoic acid
+Write the possible structure of the compound
+12.20 Although phenoxide ion has more number of resonating structures than carboxylate ion, carboxylic acid is a stronger acid than phenol
+Why?
+Answers to Some Intext Questions 12.1
+12.2
+12.3 CH3CH2CH3 < CH3OCH3 < CH3CHO < CH3CH2OH
+12.4 Butanone < Propanone < Propanal < Ethanal Acetophenone < p-Tolualdehyde , Benzaldehyde < p-Nitrobenzaldehyde
+12.5
+12.6 3-Phenylpropanoic acid 3-Methylbut-2-enoic acid 2-Methylcyclopentanecarboxylic acid
+(iv) 2,4,6-Trinitrobenzoic acid 12.7
+12.8
+Table of Contents
+Unit 12
+Aldehydes, Ketones and Carboxylic Acids
+12.1 Nomenclature and Structure of Carbonyl Group
+12.1.1 Nomenclature
+12.1.2 Structure of the Carbonyl Group
+12.2 Preparation of Aldehydes and Ketones
+12.2.1 Preparation of Aldehydes and Ketones
+12.2.2 Preparation of Aldehydes
+12.2.3 Preparation of Ketones
+12.3 Physical Properties
+12.4 Chemical Reactions
+12.5 Uses of Aldehydes and Ketones
+12.6 Nomenclature and Structure of Carboxyl Group
+12.6.1 Nomenclature
+12.6.2 Structure of Carboxyl Group
+12.7 Methods of Preparation of Carboxylic Acids
+12.9 Chemical Reactions
+12.9.1 Reactions Involving Cleavage of O–H Bond
+12.9.2 Reactions Involving Cleavage of C–OH Bond
+12.9.3 Reactions Involving –COOH Group
+12.10 Uses of Carboxylic Acids
+Summary
+Exercises
+Answers to Some Intext Questions
+Landmarks
+Cover
+Chapter 12 Aldehydes, Ketones and Carboxylic Acids
+Figure 1.7 Solution Let the original charge on sphere A be q and that on B be q′
+At a distance r between their centres, the magnitude of the electrostatic force on each is given by neglecting the sizes of spheres A and B in comparison to r
+When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2
+Similarly, after D touches B, the redistributed charge on each is q′/2
+Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is
+Thus the electrostatic force on A, due to B, remains unaltered
+7 Forces between Multiple Charges The mutual electric force between two charges is given by Coulomb’s law
+How to calculate the force on a charge where there are not one but several charges around? Consider a system of n stationary charges q1, q2, q3, ..., qn in vacuum
+What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question
+Recall that forces of mechanical origin add according to the parallelogram law of addition
+Is the same true for forces of electrostatic origin?
+Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time
+The individual forces are unaffected due to the presence of other charges
+This is termed as the principle of superposition
+To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in (a)
+The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges
+Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq
+(1.3) even though other charges are present
+Thus, F12 In the same way, the force on q1 due to q3, denoted by F13, is given by
+which again is the Coulomb force on q1 due to q3, even though other charge q2 is present
+Thus the total force F1 on q1 due to the two charges q2 and q3 is given as (1.4) The above calculation of force can be generalised to a system of charges more than three, as shown in (b)
+The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn
+The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n: i.e.,
+Figure 1.8 A system of (a) three charges (b) multiple charges.
+(1.5) The vector sum is obtained as usual by the parallelogram law of addition of vectors
+All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle
+Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l
+What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in ?
+Figure 1.9 Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC, AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l
+By symmatry AO = BO = CO
+Thus, Force F1 on Q due to charge q at A = along AO Force F2 on Q due to charge q at B = along BO Force F3 on Q due to charge q at C = along CO The resultant of forces F2 and F3 is along OA, by the parallelogram law
+Therefore, the total force on Q = = 0, where is the unit vector along OA
+It is clear also by symmetry that the three forces will sum to zero
+Suppose that the resultant force was non-zero but in some direction
+Consider what would happen if the system was rotated through 60° about O
+Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in 0
+What is the force on each charge?
+Figure 1.10 Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in 0
+By the parallelogram law, the total force F1 on the charge q at A is given by F1 = F where is a unit vector along BC
+The force of attraction or repulsion for each pair of charges has the same magnitude The total force F2 on charge q at B is thus F2 = F 2, where 2 is a unit vector along AC
+Similarly the total force on charge –q at C is F3 = F , where is the unit vector along the direction bisecting the ∠BCA
+It is interesting to see that the sum of the forces on the three charges is zero, i.e., F1 + F2 + F3 = 0 The result is not at all surprising
+It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law
+The proof is left to you as an exercise
+1.8 Electric Field Let us consider a point charge Q placed in vacuum, at the origin O
+If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law
+We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P
+In order to answer such questions, the early scientists introduced the concept of field
+According to this, we say that the charge Q produces an electric field everywhere in the surrounding
+When another charge q is brought at some point P, the field there acts on it and produces a force
+The electric field produced by the charge Q at a point r is given as (1.6) where r/r, is a unit vector from the origin to the point r
+Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r
+The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position
+The effect of the charge has been incorporated in the existence of the electric field
+We obtain the force F exerted by a charge Q on a charge q, as (1.7) Note that the charge q also exerts an equal and opposite force on the charge Q
+The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa
+If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q
+Thus, F(r) = q E(r) (1.8) Equation (1.8) defines the SI unit of electric field as N/C*
+Some important remarks may be made here: From Eq
+(1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it
+Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point
+The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge
+Note that the source charge Q must remain at its original location
+However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move
+A way out of this difficulty is to make q negligibly small
+The force F is then negligibly small but the ratio F/q is finite and defines the electric field: (1.9)
+Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.
+A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice
+When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet
+Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q
+This is because F is proportional to q, so the ratio F/q does not depend on q
+The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q
+Thus, the electric field E due to Q is also dependent on the space coordinate r
+For different positions of the charge q all over the space, we get different values of electric field E
+The field exists at every point in three-dimensional space
+For a positive charge, the electric field will be directed radially outwards from the charge
+On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards
+(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r
+Thus at equal distances from the charge Q, the magnitude of its electric field E is same
+The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry
+1.8.1 Electric field due to a system of charges Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O
+Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn
+We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r
+Electric field E1 at r due to q1 at r1 is given by E1 = where is a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P
+In the same manner, electric field E2 at r due to q2 at r2 is E2 = where is a unit vector in the direction from q2 to P and r2P is the distance between q2 and P
+Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn
+By the superposition principle, the electric field E at r due to the system of charges is (as shown in 2) E(r) = E1 (r) + E2 (r) + … + En(r) = E(r) (1.10) E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges
+8.2 Physical significance of electric field You may wonder why the notion of electric field has been introduced here at all
+After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq
+(1.5)]
+Why then introduce this intermediate quantity called the electric field? For electrostatics, the concept of electric field is convenient, but not really necessary
+Electric field is an elegant way of characterising the electrical environment of a system of charges
+Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system)
+Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field
+The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point
+Electric field is a vector field, since force is a vector quantity.
+Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.
+The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena
+Suppose we consider the force between two distant charges q1, q2 in accelerated motion
+Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light
+Thus, the effect of any motion of q1 on q2 cannot arise instantaneously
+There will be some time delay between the effect (force on q2) and the cause (motion of q1)
+It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful
+The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2
+The notion of field elegantly accounts for the time delay
+Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs
+They have an independent dynamics of their own, i.e., they evolve according to laws of their own
+They can also transport energy
+Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy
+The concept of field was first introduced by Faraday and is now among the central concepts in physics
+Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1
+The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance
+Compute the time of fall in each case
+Contrast the situation with that of ‘free fall under gravity’
+Figure 1.13
+Solution In 3(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field
+The acceleration of the electron is ae = eE/me where me is the mass of the electron
+Starting from rest, the time required by the electron to fall through a distance h is given by For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m, te = 2.9 × 10–9s In 3 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE
+The acceleration of the proton is ap = eE/mp where mp is the mass of the proton; mp = 1.67 × 10–27 kg
+The time of fall for the proton is
+Thus, the heavier particle (proton) takes a greater time to fall through the same distance
+This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body
+Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall
+To see if this is justified, let us calculate the acceleration of the proton in the given electric field:
+which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity
+The acceleration of the electron is even greater
+Thus, the effect of acceleration due to gravity can be ignored in this example
+Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart
+Calculate the electric fields at points A, B and C shown in 4
+Figure 1.14 Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude
+Hence the magnitude of the total electric field EA at A is EA = E1A + E2A = 7.2 × 104 N C–1 EA is directed toward the right
+The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude = 4 × 103 N C–1 The magnitude of the total electric field at B is EB = E1B – E2B = 3.2 × 104 N C–1 EB is directed towards the left
+The magnitude of each electric field vector at point C, due to charge q1 and q2 is = 9 × 103 N C–1 The directions in which these two vectors point are indicated in 4
+The resultant of these two vectors is = 9 × 103 N C–1 EC points towards the right
+1.9 Electric Field Lines We have studied electric field in the last section
+It is a vector quantity and can be represented as we represent vectors
+Let us try to represent E due to a point charge pictorially
+Let the point charge be placed at the origin
+Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point
+Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward
+Figure 1.15 shows such a picture
+In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow
+Connect the arrows pointing in one direction and the resulting figure represents a field line
+We thus get many field lines, all pointing outwards from the point charge
+Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No
+Now the magnitude of the field is represented by the density of field lines
+E is strong near the charge, so the density of field lines is more near the charge and the lines are closer
+Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines
+Another person may draw more lines
+But the number of lines is not important
+In fact, an infinite number of lines can be drawn in any region
+It is the relative density of lines in different regions which is important
+We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions
+So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines
+Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge
+We started by saying that the field lines carry information about the direction of electric field at different points in space
+Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points
+The field lines crowd where the field is strong and are spaced apart where it is weak
+Figure 1.16 shows a set of field lines
+We can imagine two equal and small elements of area placed at points R and S normal to the field lines there
+The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points
+The picture shows that the field at R is stronger than at S
+To understand the dependence of the field lines on the area, or rather the solid angle subtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions
+Recall how a (plane) angle is defined in two-dimensions
+Let a small transverse line element ∆l be placed at a distance r from a point O
+Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r
+Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2
+We know that in a given solid angle the number of radial field lines is the same
+In 6, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is ∆Ω at P1 and an element of area ∆Ω at P2, respectively
+The number of lines (say n) cutting these area elements are the same
+The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively
+Since n and ∆Ω are common, the strength of the field clearly has a 1/r2 dependence.
+Figure 1.15 Field of a point charge.
+The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations
+Faraday called them lines of force
+This term is somewhat misleading, especially in case of magnetic fields
+The more appropriate term is field lines (electric or magnetic) that we have adopted in this book
+Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges
+An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point
+An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve
+A field line is a space curve, i.e., a curve in three dimensions.
+Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.
+Figure 1.17 shows the field lines around some simple charge configurations
+As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane
+The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward
+The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges
+The field lines follow some important general properties: Field lines start from positive charges and end at negative charges
+If there is a single charge, they may start or end at infinity
+In a charge-free region, electric field lines can be taken to be continuous curves without any breaks
+Two field lines can never cross each other
+(If they did, the field at the point of intersection will not have a unique direction, which is absurd.)
+Figure 1.17 Field lines due to some simple charge configurations.
+(iv) Electrostatic field lines do not form any closed loops
+This follows from the conservative nature of electric field (Chapter 2)
+10 Electric Flux Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface
+The rate of flow of liquid is given by the volume crossing the area per unit time v dS and represents the flux of liquid flowing across the plane
+If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ
+Therefore, the flux going out of the surface dS is v.dS
+For the case of the electric field, we define an analogous quantity and call it electric flux
+We should, however, note that there is no flow of a physically observable quantity unlike the case ofliquid flow
+In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point
+This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S
+Now suppose we tilt the area element by angle θ
+Clearly, the number of field lines crossing the area element will be smaller
+The projection of the area element normal to E is ∆S cosθ
+Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ
+When θ = 90°, field lines will be parallel to ∆S and will not cross it at all .
+Figure 1.18 Dependence of flux on the inclination θ between E and .
+The orientation of area element and not merely its magnitude is important in many contexts
+For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring
+If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation
+This shows that an area element should be treated as a vector
+It has a magnitude and also a direction
+How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane
+Thus the direction of a planar area vector is along its normal
+How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements
+Each small area element may be treated as planar and a vector associated with it, as explained before
+Notice one ambiguity here
+The direction of an area element is along its normal
+But a normal can point in two directions
+Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context
+For the case of a closed surface, this convention is very simple
+The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal
+This is the convention used in 9
+Thus, the area element vector ∆S at a point on a closed surface equals ∆S where ∆S is the magnitude of the area element and is a unit vector in the direction of outward normal at that point
+We now come to the definition of electric flux
+Electric flux ∆φ through an area element ∆S is defined by ∆φ = E.∆S = E ∆S cosθ (1.11) which, as seen before, is proportional to the number of field lines cutting the area element
+The angle θ here is the angle between E and ∆S
+For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element
+Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element
+The unit of electric flux is N C–1 m2
+The basic definition of electric flux given by Eq
+(1.11) can be used, in principle, to calculate the total flux through any given surface
+All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up
+Thus, the total flux φ through a surface S is φ ~ Σ E.∆S (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element
+This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq
+(1.12) is written as an integral
+11 Electric Dipole An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a
+The line connecting the two charges defines a direction in space
+By convention, the direction from –q to q is said to be the direction of the dipole
+The mid-point of locations of –q and q is called the centre of the dipole.
+Figure 1.19 Convention for defining normal and ∆S.
+The total charge of the electric dipole is obviously zero
+This does not mean that the field of the electric dipole is zero
+Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out
+However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out
+The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q)
+These qualitative ideas are borne out by the explicit calculation as follows: 1.11.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle
+The results are simple for the following two cases: when the point is on the dipole axis, and when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre
+The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+q due to the charge q, by the parallelogram law of vectors
+For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in 0(a)
+Then [1.13(a)] where is the unit vector along the dipole axis (from –q to q)
+Also [1.13(b)] The total field at P is (1.14) For r >> a (r >> a) (1.15) For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by [1.16(a)] [1.16(b)] and are equal
+The directions of E+q and E–q are as shown in 0(b)
+Clearly, the components normal to the dipole axis cancel away
+The components along the dipole axis add up
+The total electric field is opposite to
+We have E = – (E +q + E –q) cosθ (1.17) At large distances (r >> a), this reduces to (1.18) From Eqs
+(1.15) and (1.18), it is clear that the dipole field at large distances does not involve q and a separately; it depends on the product qa
+This suggests the definition of dipole moment
+The dipole moment vector p of an electric dipole is defined by p = q × 2a (1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q
+In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis (r >> a) (1.20)
+Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole
+p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q.
+At a point on the equatorial plane (r >> a) (1.21) Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3
+Further, the magnitude and the direction of the dipole field depends not only on the distance r but also on the angle between the position vector r and the dipole moment p
+We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite
+Such a dipole is referred to as a point dipole
+For a point dipole, Eqs
+(1.20) and (1.21) are exact, true for any r
+1.11.2 Physical significance of dipoles In most molecules, the centres of positive charges and of negative charges* lie at the same place
+Therefore, their dipole moment is zero
+CO2 and CH4 are of this type of molecules
+However, they develop a dipole moment when an electric field is applied
+But in some molecules, the centres of negative charges and of positive charges do not coincide
+Therefore they have a permanent electric dipole moment, even in the absence of an electric field
+Such molecules are called polar molecules
+Water molecules, H2O, is an example of this type
+Various materials give rise to interesting properties and important applications in the presence or absence of electric field
+Example 1.10 Two charges ±10 µC are placed 5.0 mm apart
+Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in 1(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in 1(b)
+fIGURE 1.21 Solution (a) Field at P due to charge +10 µC = = 4.13 × 106 N C–1 along BP Field at P due to charge –10 µC = 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP
+In this example, the ratio OP/OB is quite large (= 60)
+Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole
+For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude (r/a >> 1) where p = 2a q is the magnitude of the dipole moment
+The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q)
+Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, E = = 2.6 × 105 N C–1 along the dipole moment direction AB, which is close to the result obtained earlier
+(b) Field at Q due to charge + 10 µC at B = = 3.99 × 106 N C–1 along BQ Field at Q due to charge –10 µC at A = = 3.99 × 106 N C–1 along QA
+Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA
+Therefore, the resultant electric field at Q due to the two charges at A and B is
+* Centre of a collection of positive point charges is defined much the same way as the centre of mass: .
+= 2 × along BA = 1.33 × 105 N C–1 along BA
+As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole: (r/a >> 1)
+= 1.33 × 105 N C–1
+The direction of electric field in this case is opposite to the direction of the dipole moment vector
+Again, the result agrees with that obtained before
+1.12 Dipole in a Uniform External Field Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in 2
+(By permanent dipole, we mean that p exists irrespective of E; it has not been induced by E.) There is a force qE on q and a force –qE on –q
+The net force on the dipole is zero, since E is uniform
+However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole
+When the net force is zero, the torque (couple) is independent of the origin
+Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces)
+Magnitude of torque = q E × 2 a sinθ = 2 q a E sinθ Its direction is normal to the plane of the paper, coming out of it
+The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it
+Thus, τ = p × E (1.22) This torque will tend to align the dipole with the field E
+When p is aligned with E, the torque is zero
+What happens if the field is not uniform? In that case, the net force will evidently be non-zero
+In addition there will, in general, be a torque on the system as before
+The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E
+In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform
+Figure 1.23 is self-explanatory
+It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field
+When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field
+In general, the force depends on the orientation of p with respect to E
+This brings us to a common observation in frictional electricity
+A comb run through dry hair attracts pieces of paper
+The comb, as we know, acquires charge through friction
+But the paper is not charged
+What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field
+Further, the electric field due to the comb is not uniform
+In this situation, it is easily seen that the paper should move in the direction of the comb! 13 Continuous Charge Distribution We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn
+One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus
+For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions
+For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents
+It is more feasible to consider an area element ∆S on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element
+We then define a surface charge density σ at the area element by
+Figure 1.22 Dipole in a uniform electric field.
+(1.23) We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density
+The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*
+σ represents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically
+The units for σ are C/m2
+Similar considerations apply for a line charge distribution and a volume charge distribution
+The linear charge density λ of a wire is defined by
+Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.
+(1.24) where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element
+The units for λ are C/m
+The volume charge density (sometimes simply called charge density) is defined in a similar manner: (1.25) where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents
+The units for ρ are C/m3
+The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics
+When we refer to the density of a liquid, we are referring to its macroscopic density
+We regard it as a continuous fluid and ignore its discrete molecular constitution.
+Figure 1.24 Definition of linear, surface and volume charge densities
+In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.
+The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq
+(1.10)
+Suppose a continuous charge distribution in space has a charge density ρ
+Choose any convenient origin O and let the position vector of any point in the charge distribution be r
+The charge density ρ may vary from point to point, i.e., it is a function of r
+Divide the charge distribution into small volume elements of size ∆V
+The charge in a volume element ∆V is ρ∆V
+Now, consider any general point P (inside or outside the distribution) with position vector R
+Electric field due to the charge ρ∆V is given by Coulomb’s law: (1.26) where r′ is the distance between the charge element and P, and ′ is a unit vector in the direction from the charge element to P
+By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements: (1.27) Note that ρ, r′, all can vary from point to point
+In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity
+In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous
+1.14 Gauss’s Law As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre
+Divide the sphere into small area elements, as shown in 5
+The flux through an area element ∆S is (1.28) where we have used Coulomb’s law for the electric field due to a single charge q
+The unit vector is along the radius vector from the centre to the area element
+Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S and have the same direction
+Therefore,
+* At the microscopic level, charge distribution is discontinuous, because they are discrete charges separated by intervening space where there is no charge.
+(1.29) since the magnitude of a unit vector is 1
+The total flux through the sphere is obtained by adding up flux through all the different area elements: Since each area element of the sphere is at the same distance r from the charge,
+Now S, the total area of the sphere, equals 4πr2
+Thus, (1.30) Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law
+We state Gauss’s law without proof: Electric flux through a closed surface S
+Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.
+= q/ε0 (1.31) q = total charge enclosed by S
+The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface
+We can see that explicitly in the simple situation of 6
+Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E
+The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface
+Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0
+Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E
+Therefore, φ1 = –E S1, φ2 = +E S2 S1 = S2 = S where S is the area of circular cross-section
+Thus, the total flux is zero, as expected by Gauss’s law
+Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero
+The great significance of Gauss’s law Eq
+(1.31), is that it is true in general, and not only for the simple cases we have considered above
+Let us note some important points regarding this law: Gauss’s law is true for any closed surface, no matter what its shape or size
+The term q on the right side of Gauss’s law, Eq
+(1.31), includes the sum of all charges enclosed by the surface
+The charges may be located anywhere inside the surface
+In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq
+(1.31)] is due to all the charges, both inside and outside S
+The term q on the right side of Gauss’s law, however, represents only the total charge inside S
+Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.
+(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface
+You may choose any Gaussian surface and apply Gauss’s law
+However, take care not to let the Gaussian surface pass through any discrete charge
+This is because electric field due to a system of discrete charges is not well defined at the location of any charge
+(As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution
+(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry
+This is facilitated by the choice of a suitable Gaussian surface
+(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law
+Any violation of Gauss’s law will indicate departure from the inverse square law
+Example 1.11 The electric field components in 7 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2
+Calculate (a) the flux through the cube, and (b) the charge within the cube
+Assume that a = 0.1 m.
+Figure 1.27 Solution (a) Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2
+Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones
+Now the magnitude of the electric field at the left face is EL = αx1/2 = αa1/2 (x = a at the left face)
+The magnitude of electric field at the right face is ER = α x1/2 = α (2a)1/2 (x = 2a at the right face)
+The corresponding fluxes are φL= EL.∆S = =EL ∆S cosθ = –EL ∆S, since θ = 180° = –ELa2 φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0° = ERa2 Net flux through the cube = φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2] = αa5/2 = 800 (0.1)5/2 = 1.05 N m2 C–1 (b) We can use Gauss’s law to find the total charge q inside the cube
+We have φ = q/ε0 or q = φε0
+Therefore, q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C. Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x
+It is given that E = 200 N/C for x > 0 and E = –200 N/C for x < 0
+A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm
+(a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder? Solution (a) We can see from the figure that on the left face E and ∆S are parallel
+Therefore, the outward flux is φL= E.∆S = – 200 = + 200 ∆S, since = – ∆S = + 200 × π (0.05)2 = + 1.57 N m2 C–1 On the right face, E and ∆S are parallel and therefore φR = E.∆S = + 1.57 N m2 C–1
+(b) For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0
+Therefore, the flux out of the side of the cylinder is zero
+(c) Net outward flux through the cylinder φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1
+Figure 1.28 (d) The net charge within the cylinder can be found by using Gauss’s law which gives q = ε0φ = 3.14 × 8.854 × 10–12 C = 2.78 × 10–11 C 1.15 Applications of Gauss’s Law The electric field due to a general charge distribution is, as seen above, given by Eq
+(1.27)
+In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space
+For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law
+This is best understood by some examples
+1.15.1 Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ
+The wire is obviously an axis of symmetry
+Suppose we take the radial vector from O to P and rotate it around the wire
+The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire
+This implies that the electric field must have the same magnitude at these points
+The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0)
+This is clear from 9
+Consider a pair of line elements P1 and P2 of the wire, as shown
+The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel)
+This is true for any such pair and hence the total field at any point P is radial
+Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire
+In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r
+To calculate the field, imagine a cylindrical Gaussian surface, as shown in the 9(b)
+Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero
+At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r
+The surface area of the curved part is 2πrl, where l is the length of the cylinder
+Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E × 2πrl The surface includes charge equal to λ l
+Gauss’s law then gives E × 2πrl = λl/ε0 i.e., E = Vectorially, E at any point is given by (1.32) where is the radial unit vector in the plane normal to the wire passing through the point
+E is directed outward if λ is positive and inward if λ is negative.
+Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density.
+Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A , the scalar A is an algebraic number
+It can be negative or positive
+The direction of A will be the same as that of the unit vector if A > 0 and opposite to if A < 0
+When we want to restrict to non-negative values, we use the symbol and call it the modulus of A
+Thus,
+Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire
+Further, the assumption that the wire is infinitely long is crucial
+Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface
+However, Eq
+(1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored
+15.2 Field due to a uniformly charged infinite plane sheet Let σ be the uniform surface charge density of an infinite plane sheet
+We take the x-axis normal to the given plane
+By symmetry, the electric field will not depend on y and z coordinates and its direction at every point must be parallel to the x-direction
+We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional area A, as shown
+(A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux
+The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction
+Therefore, flux E.∆S through both the surfaces are equal and add up
+Therefore the net flux through the Gaussian surface is 2 EA
+The charge enclosed by the closed surface is σA
+Therefore by Gauss’s law, 2 EA = σA/ε0 or, E = σ/2ε0 Vectorically, (1.33) where is a unit vector normal to the plane and going away from it
+E is directed away from the plate if σ is positive and toward the plate if σ is negative
+Note that the above application of the Gauss’ law has brought out an additional fact: E is independent of x also
+For a finite large planar sheet, Eq
+(1.33) is approximately true in the middle regions of the planar sheet, away from the ends
+15.3 Field due to a uniformly charged thin spherical shell Let σ be the uniform surface charge density of a thin spherical shell of radius R
+The situation has obvious spherical symmetry
+The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector)
+Field outside the shell: Consider a point P outside the shell with radius vector r
+To calculate E at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P
+All points on this sphere are equivalent relative to the given charged configuration
+(That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point
+Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S
+Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2
+The charge enclosed is σ × 4 π R2
+By Gauss’s law E × 4 π r2 = Or, where q = 4 π R2 σ is the total charge on the spherical shell
+Vectorially, (1.34) The electric field is directed outward if q > 0 and inward if q < 0
+This, however, is exactly the field produced by a charge q placed at the centre O
+Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.
+Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.
+Field inside the shell: In 1(b), the point P is inside the shell
+The Gaussian surface is again a sphere through P centred at O
+The flux through the Gaussian surface, calculated as before, is E × 4 π r2
+However, in this case, the Gaussian surface encloses no charge
+Gauss’s law then gives E × 4 π r2 = 0 i.e., E = 0 (r < R ) (1.35) that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*
+This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law
+The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law
+Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R
+The atom as a whole is neutral
+For this model, what is the electric field at a distance r from the nucleus?
+Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R.
+Figure 1.32 Solution The charge distribution for this model of the atom is as shown in 2
+The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral
+This immediately gives us the negative charge density ρ, since we must have or To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law
+Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r
+Its direction is along (or opposite to) the radius vector r from the origin to the point P
+The obvious Gaussian surface is a spherical surface centred at the nucleus
+We consider two situations, namely, r < R and r > R
+r < R : The electric flux φ enclosed by the spherical surface is φ = E (r) × 4 π r2 where E (r) is the magnitude of the electric field at r
+This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface
+The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r, i.e., Substituting for the charge density ρ obtained earlier, we have Gauss’s law then gives, The electric field is directed radially outward
+r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral
+Thus, from Gauss’s law, E (r) × 4 π r2 = 0 or E (r) = 0; r > R At r = R, both cases give the same result: E = 0
+Summary 1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter
+2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract
+By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative
+Conductors allow movement of electric charge through them, insulators do not
+In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile
+4. Electric charge has three basic properties: quantisation, additivity and conservation
+Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, ...
+Proton and electron have charges +e, –e, respectively
+For macroscopic charges for which n is a very large number, quantisation of charge can be ignored
+Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system
+Conservation of electric charges means that the total charge of an isolated system remains unchanged with time
+This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge
+5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21 separating them
+Mathematically, F21 = force on q2 due to where is a unit vector in the direction from q1 to q2 and k = is the constant of proportionality
+In SI units, the unit of charge is coulomb
+The experimental value of the constant ε0 is ε0 = 8.854 × 10–12 C2 N–1 m–2 The approximate value of k is k = 9 × 109 N m2 C–2 6. The ratio of electric force and gravitational force between a proton and an electron is
+* Compare this with a uniform mass shell discussed in Section 8.5 of Class XI Textbook of Physics.
+7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s)
+For an assembly of charges q1, q2, q3, ..., the force on any charge, say q1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on
+For each pair, the force is given by the Coulomb’s law for two charges stated earlier
+8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge
+Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative
+Like Coulomb force, electric field also satisfies superposition principle
+9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point
+The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak
+In regions of constant electric field, the field lines are uniformly spaced parallel straight lines
+On symmetry operations In Physics, we often encounter systems with various symmetries
+Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation
+Consider, for example an infinite uniform sheet of charge (surface charge density σ) along the y-z plane
+This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle
+As the system is unchanged under such symmetry operation, so must its properties be
+In particular, in this example, the electric field E must be unchanged
+Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0)
+Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same
+By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction
+Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate
+It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space
+The direction, however, is opposite of each other on either side ofthe sheet
+Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.
+10. Some of the important properties of field lines are: Field lines are continuous curves without any breaks
+Two field lines cannot cross each other
+Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops
+11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a
+Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q
+12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:
+Dipole electric field on the axis at a distance r from the centre:
+The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2 dependence of electric field due to a point charge
+13. In a uniform electric field E, a dipole experiences a torque given by = p × E but experiences no net force
+. The flux ∆φ of electric field E through a small area element ∆S is given by ∆φ = E.∆S The vector area element ∆S is ∆S = ∆S where ∆S is the magnitude of the area element and is normal to the area element, which can be considered planar for sufficiently small ∆S
+For an area element of a closed surface, is taken to be the direction of outward normal, by convention
+15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S
+The law is especially useful in determining electric field E, when the source distribution has simple symmetry: Thin infinitely long straight wire of uniform linear charge density λ
+where r is the perpendicular distance of the point from the wire and is the radial unit vector in the plane normal to the wire passing through the point
+Infinite thin plane sheet of uniform surface charge density σ
+where is a unit vector normal to the plane, outward on either side
+Thin spherical shell of uniform surface charge density σ
+E = 0 (r < R) where r is the distance of the point from the centre of the shell and R the radius of the shell
+q is the total charge of the shell: q = 4πR2σ
+The electric field outside the shell is as though the total charge is concentrated at the centre
+The same result is true for a solid sphere of uniform volume charge density
+The field is zero at all points inside the shell
+Points to Ponder 1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus
+Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together
+The range of distance where this force is effective is, however, very small ~10-14 m
+This is precisely the size of the nucleus
+Also the electrons are not allowed to sit on top of the protons, i.e
+inside the nucleus, due to the laws of quantum mechanics
+This gives the atoms their structure as they exist in nature
+2. Coulomb force and gravitational force follow the same inverse-square law
+But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces
+This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature
+3. The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law
+In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s)
+In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2 C–2
+4. The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces
+Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects
+The additive property of charge is not an ‘obvious’ property
+It is related to the fact that electric charge has no direction associated with it; charge is a scalar
+6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion
+This is not always true for every scalar
+For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion
+7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6
+Conservation refers to invariance in time in a given frame of reference
+A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision)
+On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system)
+8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass
+9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors
+It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges
+10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges
+For continuous volume charge distribution, it is defined at any point in the distribution
+For a surface charge distribution, electric field is discontinuous across the surface
+11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge
+An electric dipole is the simplest example of this fact
+Exercises
+Physical quantity Symbol Dimensions Unit Remarks Vector area element ∆S [L2] m2 ∆S = ∆S Electric field E [MLT–3A–1] V m–1 Electric flux φ [ML3 T–3A–1] V m ∆φ = E.∆S Dipole moment p [LTA] C m Vector directed from negative to positive charge Charge density: linear λ [L–1 TA] C m–1 Charge/length surface σ [L–2 TA] C m–2 Charge/area volume ρ [L–3 TA] C m–3 Charge/volume
+1.1 What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air? 1.2 The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N
+(a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? 1.3 Check that the ratio ke2/G memp is dimensionless
+Look up a Table of Physical Constants and determine the value of this ratio
+What does the ratio signify? 1.4 (a) Explain the meaning of the statement ‘electric charge of a body is quantised’
+(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? 1.5 When a glass rod is rubbed with a silk cloth, charges appear on both
+A similar phenomenon is observed with many other pairs of bodies
+Explain how this observation is consistent with the law of conservation of charge
+6 Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm
+What is the force on a charge of 1 µC placed at the centre of the square? 1.7 (a) An electrostatic field line is a continuous curve
+That is, a field line cannot have sudden breaks
+Why not? (b) Explain why two field lines never cross each other at any point? 8 Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum
+(a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge? 1.9 A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively
+What are the total charge and electric dipole moment of the system? 1.10 An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1
+Calculate the magnitude of the torque acting on the dipole
+1.11 A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C
+(a) Estimate the number of electrons transferred (from which to which?) (b) Is there a transfer of mass from wool to polythene? 1.12 (a) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm
+What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation
+(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? 1.13 Suppose the spheres A and B in Exercise 1.12 have identical sizes
+A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both
+What is the new force of repulsion between A and B? 1.14 Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field
+Give the signs of the three charges
+Which particle has the highest charge to mass ratio?
+Figure 1.33 1.15 Consider a uniform electric field E = 3 × 103 î N/C
+(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis? 1.16 What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes? 1.17 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C
+(a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? 1.18 A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in 4
+What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)
+Figure 1.34 1.19 A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge
+What is the net electric flux through the surface? 1.20 A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge
+(a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 1.21 A conducting sphere of radius 10 cm has an unknown charge
+If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere? 1.22 A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2
+(a) Find the charge on the sphere
+(b) What is the total electric flux leaving the surface of the sphere? 1.23 An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm
+Calculate the linear charge density
+1.24 Two large, thin metal plates are parallel and close to each other
+On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2
+What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates? Additional Exercises 1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment)
+The density of the oil is 1.26 g cm–3
+Estimate the radius of the drop
+(g = 9.81 m s–2; e = 1.60 × 10–19 C)
+1.26 Which among the curves shown in 5 cannot possibly represent electrostatic field lines? Figure 1.35 1.27 In a certain region of space, electric field is along the z-direction throughout
+The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre
+What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ? 1.28 (a) A conductor A with a cavity as shown in 6(a) is given a charge Q
+Show that the entire charge must appear on the outer surface of the conductor
+(b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A
+Show that the total charge on the outside surface of A is Q + q
+(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment
+Suggest a possible way
+Figure 1.36 1.29 A hollow charged conductor has a tiny hole cut into its surface
+Show that the electric field in the hole is (σ/2ε0), where is the unit vector in the outward normal direction, and σ is the surface charge density near the hole
+30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law
+[Hint: Use Coulomb’s law directly and evaluate the necessary integral.] 1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks
+A proton and a neutron consist of three quarks each
+Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter
+(Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron
+32 (a) Consider an arbitrary electrostatic field configuration
+A small test charge is placed at a null point (i.e., where E = 0) of the configuration
+Show that the equilibrium of the test charge is necessarily unstable
+(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart
+33 A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in 3)
+The length of plate is L and an uniform electric field E is maintained between the plates
+Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2)
+Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics
+1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1
+If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)
+Interactive animation on simple electrostatic experiments: http://demoweb.physics.ucla.edu/content/100-simple-electrostatic-experiments
+Example 1.1
+Interactive animation on charging a two-sphere system by induction: http://www.physicsclassroom.com/mmedia/estatics/itsn.cfm
+Example 1.2
+Example 1.3
+Charles Augustin de Coulomb (1736 –1806)
+Example 1.4
+Interactive animation on Coulomb’s law: http://webphysics.davidson.edu/physlet_resources/bu_semester2/menu_semester2.html
+Example 1.5
+Example 1.5
+Example 1.6
+Example 1.6
+Example 1.7
+Example 1.7
+* An alternate unit V/m will be introduced in the next chapter.
+Example 1.8
+Example 1.8
+Example 1.9
+Example 1.9
+* Solid angle is a measure of a cone
+Consider the intersection of the given cone with a sphere of radius R
+The solid angle ∆Ω of the cone is defined to be equal to ∆S/R2, where ∆S is the area on the sphere cut out by the cone.
+* It will not be proper to say that the number of field lines is equal to E∆S
+The number of field lines is after all, a matter of how many field lines we choose to draw
+What is physically significant is the relative number of field lines crossing a given area at different points.
+Example 1.10
+Example 1.10
+Example 1.10
+Example 1.11
+Example 1.11
+Example 1.12
+Example 1.13
+Example 1.13
+Contents
+Chapter-1
+Landmarks
+Cover
+Chapter One
+ELECTRIC CHARGES AND FIELDS
+1.1 Introduction
+All of us have the experience of seeing a spark or hearing a crackle when we take off our synthetic clothes or sweater, particularly in dry weather
+This is almost inevitable with ladies garments like a polyester saree
+Have you ever tried to find any explanation for this phenomenon? Another common example of electric discharge is the lightning that we see in the sky during thunderstorms
+We also experience a sensation of an electric shock either while opening the door of a car or holding the iron bar of a bus after sliding from our seat
+The reason for these experiences is discharge of electric charges through our body, which were accumulated due to rubbing of insulating surfaces
+You might have also heard that this is due to generation of static electricity
+This is precisely the topic we are going to discuss in this and the next chapter
+Static means anything that does not move or change with time
+Electrostatics deals with the study of forces, fields and potentials arising from static charges
+1.2 Electric Charge Historically the credit of discovery of the fact that amber rubbed with wool or silk cloth attracts light objects goes to Thales of Miletus, Greece, around 600 BC
+The name electricity is coined from the Greek word elektron meaning amber
+Many such pairs of materials were known which on rubbing could attract light objects like straw, pith balls and bits of papers
+You can perform the following activity at home to experience such an effect
+Cut out long thin strips of white paper and lightly iron them
+Take them near a TV screen or computer monitor
+You will see that the strips get attracted to the screen
+In fact they remain stuck to the screen for a while
+Figure 1.1 Rods and pith balls: like charges repel and unlike charges attract each other.
+It was observed that if two glass rods rubbed with wool or silk cloth are brought close to each other, they repel each other
+The two strands of wool or two pieces of silk cloth, with which the rods were rubbed, also repel each other
+However, the glass rod and wool attracted each other
+Similarly, two plastic rods rubbed with cat’s fur repelled each other but attracted the fur
+On the other hand, the plastic rod attracts the glass rod and repel the silk or wool with which the glass rod is rubbed
+The glass rod repels the fur
+If a plastic rod rubbed with fur is made to touch two small pith balls (now-a-days we can use polystyrene balls) suspended by silk or nylon thread, then the balls repel each other and are also repelled by the rod
+A similar effect is found if the pith balls are touched with a glass rod rubbed with silk
+A dramatic observation is that a pith ball touched with glass rod attracts another pith ball touched with plastic rod
+These seemingly simple facts were established from years of efforts and careful experiments and their analyses
+It was concluded, after many careful studies by different scientists, that there were only two kinds of an entity which is called the electric charge
+We say that the bodies like glass or plastic rods, silk, fur and pith balls are electrified
+They acquire an electric charge on rubbing
+The experiments on pith balls suggested that there are two kinds of electrification and we find that like charges repel and unlike charges attract each other
+The experiments also demonstrated that the charges are transferred from the rods to the pith balls on contact
+It is said that the pith balls are electrified or are charged by contact
+The property which differentiates the two kinds of charges is called the polarity of charge
+When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge
+This is true for any pair of objects that are rubbed to be electrified
+Now if the electrified glass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other
+They also do not attract or repel other light objects as they did on being electrified
+Thus, the charges acquired after rubbing are lost when the charged bodies are brought in contact
+What can you conclude from these observations? It just tells us that unlike charges acquired by the objects neutralise or nullify each other’s effect
+Therefore, the charges were named as positive and negative by the American scientist Benjamin Franklin
+We know that when we add a positive number to a negative number of the same magnitude, the sum is zero
+This might have been the philosophy in naming the charges as positive and negative
+By convention, the charge on glass rod or cat’s fur is called positive and that on plastic rod or silk is termed negative
+If an object possesses an electric charge, it is said to be electrified or charged
+When it has no charge it is said to be electrically neutral
+A simple apparatus to detect charge on a body is the gold-leaf electroscope
+It consists of a vertical metal rod housed in a box, with two thin gold leaves attached to its bottom end
+When a charged object touches the metal knob at the top of the rod, charge flows on to the leaves and they diverge
+The degree of divergance is an indicator of the amount of charge
+Unification of electricity and magnetism In olden days, electricity and magnetism were treated as separate subjects
+Electricity dealt with charges on glass rods, cat’s fur, batteries, lightning, etc., while magnetism described interactions of magnets, iron filings, compass needles, etc
+In 1820 Danish scientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle
+Ampere and Faraday supported this observation by saying that electric charges in motion produce magnetic fields and moving magnets generate electricity
+The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Lorentz put forward a theory where they showed the interdependence of these two subjects
+This field is called electromagnetism
+Most of the phenomena occurring around us can be described under electromagnetism
+Virtually every force that we can think of like friction, chemical force between atoms holding the matter together, and even the forces describing processes occurring in cells of living organisms, have its origin in electromagnetic force
+Electromagnetic force is one of the fundamental forces of nature
+Maxwell put forth four equations that play the same role in classical electromagnetism as Newton’s equations of motion and gravitation law play in mechanics
+He also argued that light is electromagnetic in nature and its speed can be found by making purely electric and magnetic measurements
+He claimed that the science of optics is intimately related to that of electricity and magnetism
+The science of electricity and magnetism is the foundation for the modern technological civilisation
+Electric power, telecommunication, radio and television, and a wide variety of the practical appliances used in daily life are based on the principles of this science
+Although charged particles in motion exert both electric and magnetic forces, in the frame of reference where all the charges are at rest, the forces are purely electrical
+You know that gravitational force is a long-range force
+Its effect is felt even when the distance between the interacting particles is very large because the force decreases inversely as the square of the distance between the interacting bodies
+We will learn in this chapter that electric force is also as pervasive and is in fact stronger than the gravitational force by several orders of magnitude (refer to Chapter 1 of Class XI Physics Textbook).
+Students can make a simple electroscope as follows : Take a thin aluminium curtain rod with ball ends fitted for hanging the curtain
+Cut out a piece of length about 20 cm with the ball at one end and flatten the cut end
+Take a large bottle that can hold this rod and a cork which will fit in the opening of the bottle
+Make a hole in the cork sufficient to hold the curtain rod snugly
+Slide the rod through the hole in the cork with the cut end on the lower side and ball end projecting above the cork
+Fold a small, thin aluminium foil (about 6 cm in length) in the middle and attach it to the flattened end of the rod by cellulose tape
+This forms the leaves of your electroscope
+Fit the cork in the bottle with about 5 cm of the ball end projecting above the cork
+A paper scale may be put inside the bottle in advance to measure the separation of leaves
+The separation is a rough measure of the amount of charge on the electroscope.
+Figure 1.2 Electroscopes: (a) The gold leaf electroscope, (b) Schematics of a simple electroscope.
+To understand how the electroscope works, use the white paper strips we used for seeing the attraction of charged bodies
+Fold the strips into half so that you make a mark of fold
+Open the strip and iron it lightly with the mountain fold up, as shown in
+Hold the strip by pinching it at the fold
+You would notice that the two halves move apart
+This shows that the strip has acquired charge on ironing
+When you fold it into half, both the halves have the same charge
+Hence they repel each other
+The same effect is seen in the leaf electroscope
+On charging the curtain rod by touching the ball end with an electrified body, charge is transferred to the curtain rod and the attached aluminium foil
+Both the halves of the foil get similar charge and therefore repel each other
+The divergence in the leaves depends on the amount of charge on them
+Let us first try to understand why material bodies acquire charge
+You know that all matter is made up of atoms and/or molecules
+Although normally the materials are electrically neutral, they do contain charges; but their charges are exactly balanced
+Forces that hold the molecules together, forces that hold atoms together in a solid, the adhesive force of glue, forces associated with surface tension, all are basically electrical in nature, arising from the forces between charged particles
+Thus the electric force is all pervasive and it encompasses almost each and every field associated with our life
+It is therefore essential that we learn more about such a force.
+Figure 1.3 Paper strip experiment.
+To electrify a neutral body, we need to add or remove one kind of charge
+When we say that a body is charged, we always refer to this excess charge or deficit of charge
+In solids, some of the electrons, being less tightly bound in the atom, are the charges which are transferred from one body to the other
+A body can thus be charged positively by losing some of its electrons
+Similarly, a body can be charged negatively by gaining electrons
+When we rub a glass rod with silk, some of the electrons from the rod are transferred to the silk cloth
+Thus the rod gets positively charged and the silk gets negatively charged
+No new charge is created in the process of rubbing
+Also the number of electrons, that are transferred, is a very small fraction of the total number of electrons in the material body
+Also only the less tightly bound electrons in a material body can be transferred from it to another by rubbing
+Therefore, when a body is rubbed with another, the bodies get charged and that is why we have to stick to certain pairs of materials to notice charging on rubbing the bodies
+1.3 Conductors and Insulators A metal rod held in hand and rubbed with wool will not show any sign of being charged
+However, if a metal rod with a wooden or plastic handle is rubbed without touching its metal part, it shows signs of charging
+Suppose we connect one end of a copper wire to a neutral pith ball and the other end to a negatively charged plastic rod
+We will find that the pith ball acquires a negative charge
+If a similar experiment is repeated with a nylon thread or a rubber band, no transfer of charge will take place from the plastic rod to the pith ball
+Why does the transfer of charge not take place from the rod to the ball? Some substances readily allow passage of electricity through them, others do not
+Those which allow electricity to pass through them easily are called conductors
+They have electric charges (electrons) that are comparatively free to move inside the material
+Metals, human and animal bodies and earth are conductors
+Most of the non-metals like glass, porcelain, plastic, nylon, wood offer high resistance to the passage of electricity through them
+They are called insulators
+Most substances fall into one of the two classes stated above*
+When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor
+In contrast, if some charge is put on an insulator, it stays at the same place
+You will learn why this happens in the next chapter
+This property of the materials tells you why a nylon or plastic comb gets electrified on combing dry hair or on rubbing, but a metal article like spoon does not
+The charges on metal leak through our body to the ground as both are conductors of electricity
+When we bring a charged body in contact with the earth, all the excess charge on the body disappears by causing a momentary current to pass to the ground through the connecting conductor (such as our body)
+This process of sharing the charges with the earth is called grounding or earthing
+Earthing provides a safety measure for electrical circuits and appliances
+A thick metal plate is buried deep into the earth and thick wires are drawn from this plate; these are used in buildings for the purpose of earthing near the mains supply
+The electric wiring in our houses has three wires: live, neutral and earth
+The first two carry electric current from the power station and the third is earthed by connecting it to the buried metal plate
+Metallic bodies of the electric appliances such as electric iron, refrigerator, TV are connected to the earth wire
+When any fault occurs or live wire touches the metallic body, the charge flows to the earth without damaging the appliance and without causing any injury to the humans; this would have otherwise been unavoidable since the human body is a conductor of electricity.
+* There is a third category called semiconductors, which offer resistance to the movement of charges which is intermediate between the conductors and insulators.
+1.4 Charging by Induction When we touch a pith ball with an electrified plastic rod, some of the negative charges on the rod are transferred to the pith ball and it also gets charged
+Thus the pith ball is charged by contact
+It is then repelled by the plastic rod but is attracted by a glass rod which is oppositely charged
+However, why a electrified rod attracts light objects, is a question we have still left unanswered
+Let us try to understand what could be happening by performing the following experiment.
+Figure 1.4 Charging by induction.
+Bring two metal spheres, A and B, supported on insulating stands, in contact as shown in (a)
+Bring a positively charged rod near one of the spheres, say A, taking care that it does not touch the sphere
+The free electrons in the spheres are attracted towards the rod
+This leaves an excess of positive charge on the rear surface of sphere B
+Both kinds of charges are bound in the metal spheres and cannot escape
+They, therefore, reside on the surfaces, as shown in (b)
+The left surface of sphere A, has an excess of negative charge and the right surface of sphere B, has an excess of positive charge
+However, not all of the electrons in the spheres have accumulated on the left surface of A
+As the negative charge starts building up at the left surface of A, other electrons are repelled by these
+In a short time, equilibrium is reached under the action of force of attraction of the rod and the force of repulsion due to the accumulated charges
+(b) shows the equilibrium situation
+The process is called induction of charge and happens almost instantly
+The accumulated charges remain on the surface, as shown, till the glass rod is held near the sphere
+If the rod is removed, the charges are not acted by any outside force and they redistribute to their original neutral state
+Separate the spheres by a small distance while the glass rod is still held near sphere A, as shown in (c)
+The two spheres are found to be oppositely charged and attract each other
+(iv) Remove the rod
+The charges on spheres rearrange themselves as shown in (d)
+Now, separate the spheres quite apart
+The charges on them get uniformly distributed over them, as shown in (e)
+In this process, the metal spheres will each be equal and oppositely charged
+This is charging by induction
+The positively charged glass rod does not lose any of its charge, contrary to the process of charging by contact
+When electrified rods are brought near light objects, a similar effect takes place
+The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object
+[This happens even when the light object is not a conductor
+The mechanism for how this happens is explained later in Sections 1.10 and 2.10.] The centres of the two types of charges are slightly separated
+We know that opposite charges attract while similar charges repel
+However, the magnitude of force depends on the distance between the charges and in this case the force of attraction overweighs the force of repulsion
+As a result the particles like bits of paper or pith balls, being light, are pulled towards the rods
+Example 1.1 How can you charge a metal sphere positively without touching it? Solution Figure 1.5(a) shows an uncharged metallic sphere on an insulating metal stand
+Bring a negatively charged rod close to the metallic sphere, as shown in (b)
+As the rod is brought close to the sphere, the free electrons in the sphere move away due to repulsion and start piling up at the farther end
+The near end becomes positively charged due to deficit of electrons
+This process of charge distribution stops when the net force on the free electrons inside the metal is zero
+Connect the sphere to the ground by a conducting wire
+The electrons will flow to the ground while the positive charges at the near end will remain held there due to the attractive force of the negative charges on the rod, as shown in (c)
+Disconnect the sphere from the ground
+The positive charge continues to be held at the near end
+Remove the electrified rod
+The positive charge will spread uniformly over the sphere as shown in (e)
+Figure 1.5 In this experiment, the metal sphere gets charged by the process of induction and the rod does not lose any of its charge.
+Similar steps are involved in charging a metal sphere negatively by induction, by bringing a positively charged rod near it
+In this case the electrons will flow from the ground to the sphere when the sphere is connected to the ground with a wire
+Can you explain why?
+1.5 Basic Properties of Electric Charge We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other
+Here, we shall now describe some other properties of the electric charge
+If the sizes of charged bodies are very small as compared to the distances between them, we treat them as point charges
+All the charge content of the body is assumed to be concentrated at one point in space.
+1.5.1 Additivity of charges We have not as yet given a quantitative definition of a charge; we shall follow it up in the next section
+We shall tentatively assume that this can be done and proceed
+If a system contains two point charges q1 and q2, the total charge of the system is obtained simply by adding algebraically q1 and q2 , i.e., charges add up like real numbers or they are scalars like the mass of a body
+If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn
+Charge has magnitude but no direction, similar to mass
+However, there is one difference between mass and charge
+Mass of a body is always positive whereas a charge can be either positive or negative
+Proper signs have to be used while adding the charges in a system
+For example, the total charge of a system containing five charges +1, +2, –3, +4 and –5, in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the same unit.
+1.5.2 Charge is conserved We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed
+A picture of particles of electric charge enables us to understand the idea of conservation of charge
+When we rub two bodies, what one body gains in charge the other body loses
+Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found that the total charge of the isolated system is always conserved
+Conservation of charge has been established experimentally
+It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed in a process
+Sometimes nature creates charged particles: a neutron turns into a proton and an electron
+The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation
+1.5.3 Quantisation of charge Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted by e
+Thus charge q on a body is always given by q = ne where n is any integer, positive or negative
+This basic unit of charge is the charge that an electron or proton carries
+By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as –e and that on a proton as +e
+The fact that electric charge is always an integral multiple of e is termed as quantisation of charge
+There are a large number of situations in physics where certain physical quantities are quantised
+The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday
+It was experimentally demonstrated by Millikan in 1912
+In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C
+A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter
+In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2 of Class XI, Physics Textbook , Part I)
+In this system, the value of the basic unit of charge is e = 1.602192 × 10–19 C Thus, there are about 6 × 1018 electrons in a charge of –1C
+In electrostatics, charges of this large magnitude are seldom encountered and hence we use smaller units 1 µC (micro coulomb) = 10–6 C or 1 mC (milli coulomb) = 10–3 C
+If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples of e
+Thus, if a body contains n1 electrons and n2 protons, the total amount of charge on the body is n2 × e + n1 × (–e) = (n2 – n1) e
+Since n1 and n2 are integers, their difference is also an integer
+Thus the charge on any body is always an integral multiple of e and can be increased or decreased also in steps of e
+The step size e is, however, very small because at the macroscopic level, we deal with charges of a few µC
+At this scale the fact that charge of a body can increase or decrease in units of e is not visible
+In this respect, the grainy nature of the charge is lost and it appears to be continuous
+This situation can be compared with the geometrical concepts of points and lines
+A dotted line viewed from a distance appears continuous to us but is not continuous in reality
+As many points very close to each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution
+At the macroscopic level, one deals with charges that are enormous compared to the magnitude of charge e
+Since e = 1.6 × 10–19 C, a charge of magnitude, say 1 µC, contains something like 1013 times the electronic charge
+At this scale, the fact that charge can increase or decrease only in units of e is not very different from saying that charge can take continuous values
+Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored
+However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds of e, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored
+It is the magnitude of scale involved that is very important.
+Example 1.2 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 C on the other body? Solution In one second 109 electrons move out of the body
+Therefore the charge given out in one second is 1.6 × 10–19 × 109 C = 1.6 × 10–10 C
+The time required to accumulate a charge of 1 C can then be estimated to be 1 C ÷ (1.6 × 10–10 C/s) = 6.25 × 109 s = 6.25 × 109 ÷ (365 × 24 × 3600) years = 198 years
+Thus to collect a charge of one coulomb, from a body from which 109 electrons move out every second, we will need approximately 200 years
+One coulomb is, therefore, a very large unit for many practical purposes
+It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material
+A cubic piece of copper of side 1 cm contains about 2.5 × 1024 electrons
+Example 1.3 How much positive and negative charge is there in a cup of water? Solution Let us assume that the mass of one cup of water is 250 g
+The molecular mass of water is 18g
+Thus, one mole (= 6.02 × 1023 molecules) of water is 18 g
+Therefore the number of molecules in one cup of water is (250/18) × 6.02 × 1023.
+Each molecule of water contains two hydrogen atoms and one oxygen atom, i.e., 10 electrons and 10 protons
+Hence the total positive and total negative charge has the same magnitude
+It is equal to (250/18) × 6.02 × 1023 × 10 × 1.6 × 10–19 C = 1.34 × 107 C.
+1.6 Coulomb’s Law Coulomb’s law is a quantitative statement about the force between two point charges
+When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges
+Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges
+Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by (1.1) How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallic spheres
+When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges
+However, the charges on the spheres were unknown, to begin with
+How then could he discover a relation like Eq
+(1.1)? Coulomb thought of the following simple way: Suppose the charge on a metallic sphere is q
+If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres
+By symmetry, the charge on each sphere will be q/2*
+Repeating this process, we can get charges q/2, q/4, etc
+Coulomb varied the distance for a fixed pair of charges and measured the force for different separations
+He then varied the charges in pairs, keeping the distance fixed for each pair
+Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation, Eq
+(1.1)
+* A torsion balance is a sensitive device to measure force
+It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.
+Coulomb’s law, a simple mathematical statement, was initially experimentally arrived at in the manner described above
+While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level (r ~ 10–10 m)
+Coulomb discovered his law without knowing the explicit magnitude of the charge
+In fact, it is the other way round: Coulomb’s law can now be employed to furnish a definition for a unit of charge
+In the relation, Eq
+(1.1), k is so far arbitrary
+We can choose any positive value of k
+The choice of k determines the size of the unit of charge
+In SI units, the value of k is about 9 × 109
+The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4
+Putting this value of k in Eq
+(1.1), we see that for q1 = q2 = 1 C, r = 1 m F = 9 × 109 N
+Charles Augustin de Coulomb (1736 – 1806) Coulomb, a French physicist, began his career as a military engineer in the West Indies
+In 1776, he returned to Paris and retired to a small estate to do his scientific research
+He invented a torsion balance to measure the quantity of a force and used it for determination of forces of electric attraction or repulsion between small charged spheres
+He thus arrived in 1785 at the inverse square law relation, now known as Coulomb’s law
+The law had been anticipated by Priestley and also by Cavendish earlier, though Cavendish never published his results
+Coulomb also found the inverse square law of force between unlike and like magnetic poles.
+That is, 1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N
+One coulomb is evidently too big a unit to be used
+In practice, in electrostatics, one uses smaller units like 1 mC or 1 µC
+The constant k in Eq
+(1.1) is usually put as k = 1/4πε0 for later convenience, so that Coulomb’s law is written as (1.2) ε0 is called the permittivity of free space
+The value of ε0 in SI units is = 8.854 × 10–12 C2 N–1m–2
+* Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q.
+Since force is a vector, it is better to write Coulomb’s law in the vector notation
+Let the position vectors of charges q1 and q2 be r1 and r2 respectively [see Fig.1.6(a)]
+We denote force on q1 due to q2 by F12 and force on q2 due to q1 by F21
+The two point charges q1 and q2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21: r21 = r2 – r1 In the same way, the vector leading from 2 to 1 is denoted by r12: r12 = r1 – r2 = – r21 The magnitude of the vectors r21 and r12 is denoted by r21 and r12, respectively (r12 = r21)
+The direction of a vector is specified by a unit vector along the vector
+To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors: , Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as (1.3) Some remarks on Eq
+(1.3) are relevant:
+Figure 1.6 (a) Geometry and (b) Forces between charges
+• Equation (1.3) is valid for any sign of q1 and q2 whether positive or negative
+If q1 and q2 are of the same sign (either both positive or both negative), F21 is along 21, which denotes repulsion, as it should be for like charges
+If q1 and q2 are of opposite signs, F21 is along –21(=12), which denotes attraction, as expected for unlike charges
+Thus, we do not have to write separate equations for the cases of like and unlike charges
+Equation (1.3) takes care of both cases correctly
+• The force F12 on charge q1 due to charge q2, is obtained from Eq
+(1.3), by simply interchanging 1 and 2, i.e., Thus, Coulomb’s law agrees with the Newton’s third law
+• Coulomb’s law [Eq
+(1.3)] gives the force between two charges q1 and q2 in vacuum
+If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter
+We shall consider electrostatics in matter in the next chapter.
+Example 1.4 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges and masses respectively.(a) Compare the strength of these forces by determining the ratio of their magnitudes for an electron and a proton and for two protons
+(b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are1 Å (= 10-10 m) apart? (mp = 1.67 × 10–27 kg, me = 9.11 × 10–31 kg) Solution (a) The electric force between an electron and a proton at a distance r apart is:
+where the negative sign indicates that the force is attractive
+The corresponding gravitational force (always attractive) is:
+where mp and me are the masses of a proton and an electron respectively.
+On similar lines, the ratio of the magnitudes of electric force to the gravitational force between two protons at a distance r apart is: 1.3 × 1036 However, it may be mentioned here that the signs of the two forces are different
+For two protons, the gravitational force is attractive in nature and the Coulomb force is repulsive
+The actual values of these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) are Fe ~ 230 N, whereas, FG ~ 1.9 × 10–34 N
+The (dimensionless) ratio of the two forces shows that electrical forces are enormously stronger than the gravitational forces
+(b) The electric force F exerted by a proton on an electron is same in magnitude to the force exerted by an electron on a proton; however, the masses of an electron and a proton are different
+Thus, the magnitude of force is |F| == 8.987 × 109 Nm2/C2 × (1.6 ×10–19C)2 / (10–10m)2 = 2.3 × 10–8 N Using Newton’s second law of motion, F = ma, the acceleration that an electron will undergo is a = 2.3×10–8 N / 9.11 ×10–31 kg = 2.5 × 1022 m/s2 Comparing this with the value of acceleration due to gravity, we can conclude that the effect of gravitational field is negligible on the motion of electron and it undergoes very large accelerations under the action of Coulomb force due to a proton
+The value for acceleration of the proton is
+2.3 × 10–8 N / 1.67 × 10–27 kg = 1.4 × 1019 m/s2 Example 1.5 A charged metallic sphere A is suspended by a nylon thread
+Another charged metallic sphere B held by an insulating handle is brought close to A such that the distance between their centres is 10 cm, as shown in (a)
+The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen)
+Spheres A and B are touched by uncharged spheres C and D respectively, as shown in (b)
+C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in (c).What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes
+Ignore the sizes of A and B in comparison to the separation between their centres.
+Figure 1.7 Solution Let the original charge on sphere A be q and that on B be q′
+At a distance r between their centres, the magnitude of the electrostatic force on each is given by
+neglecting the sizes of spheres A and B in comparison to r
+When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a charge q/2
+Similarly, after D touches B, the redistributed charge on each is q′/2
+Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is
+Thus the electrostatic force on A, due to B, remains unaltered.
+1.7 Forces between Multiple Charges The mutual electric force between two charges is given by Coulomb’s law
+How to calculate the force on a charge where there are not one but several charges around? Consider a system of nstationary charges q1, q2, q3, ..., qn in vacuum. What is the force on q1 due to q2, q3, ..., qn? Coulomb’s law is not enough to answer this question
+Recall that forces of mechanical origin add according to the parallelogram law of addition
+Is the same true for forces of electrostatic origin?
+Figure 1.8 A system of (a) three charges (b) multiple charges.
+Experimentally, it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time
+The individual forces are unaffected due to the presence of other charges
+This is termed as theprinciple of superposition
+To better understand the concept, consider a system of three charges q1, q2 and q3, as shown in (a). The force on one charge, say q1, due to two other charges q2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges
+Thus, if the force on q1 due to q2 is denoted by F12, F12 is given by Eq
+(1.3) even though other charges are present. Thus, F12
+In the same way, the force on q1 due to q3, denoted by F13, is given by
+which again is the Coulomb force on q1 due to q3, even though other charge q2 is present
+Thus the total force F1 on q1 due to the two charges q2 and q3 is given as (1.4) The above calculation of force can be generalised to a system of charges more than three, as shown in (b)
+The principle of superposition says that in a system of charges q1, q2, ..., qn, the force on q1 due to q2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other charges q3, q4, ..., qn
+The total force F1 on the charge q1, due to all other charges, is then given by the vector sum of the forces F12, F13, ..., F1n: i.e.,
+(1.5) The vector sum is obtained as usual by the parallelogram law of addition of vectors
+All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.
+Example 1.6 Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle, as shown in ?
+Figure 1.9 Solution In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC, AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l. By symmatry AO = BO = CO
+Thus, Force F1 on Q due to charge q at A =along AO Force F2 on Q due to charge q at B =along BO Force F3 on Q due to charge q at C =along CO The resultant of forces F2 and F3 isalong OA, by the parallelogram law
+Therefore, the total force on Q == 0, whereis the unit vector along OA.
+It is clear also by symmetry that the three forces will sum to zero
+Suppose that the resultant force was non-zero but in some direction
+Consider what would happen if the system was rotated through 60° about O
+Example 1.7 Consider the charges q, q, and –q placed at the vertices of an equilateral triangle, as shown in 0
+What is the force on each charge?
+Figure 1.10 Solution The forces acting on charge q at A due to charges q at B and –q at C are F12 along BA and F13 along AC respectively, as shown in 0
+By the parallelogram law, the total force F1 on the charge q at A is given by F1 = Fwhereis a unit vector along BC
+The force of attraction or repulsion for each pair of charges has the same magnitude The total force F2 on charge q at B is thus F2 = F2, where2 is a unit vector along AC
+Similarly the total force on charge –q at C is F3 =F, whereis the unit vector along the direction bisecting the ∠BCA
+It is interesting to see that the sum of the forces on the three charges is zero, i.e., F1 + F2 + F3 = 0
+The result is not at all surprising
+It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law
+The proof is left to you as an exercise.
+1.8 Electric Field Let us consider a point charge Q placed in vacuum, at the origin O
+If we place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law
+We may ask the question: If charge q is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the charge q at P
+In order to answer such questions, the early scientists introduced the concept offield
+According to this, we say that the charge Q produces an electric field everywhere in the surrounding
+When another charge q is brought at some point P, the field there acts on it and produces a force
+The electric field produced by the charge Q at a point r is given as (1.6) wherer/r, is a unit vector from the origin to the point r
+Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vector r
+The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position
+The effect of the charge has been incorporated in the existence of the electric field
+We obtain the force F exerted by a charge Q on a charge q, as (1.7) Note that the charge q also exerts an equal and opposite force on the charge Q. The electrostatic force between the charges Q and q can be looked upon as an interaction between charge q and the electric field of Q and vice versa. If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electric field E at the location of q. Thus, F(r) = q E(r) (1.8) Equation (1.8) defines the SI unit of electric field as N/C*
+Some important remarks may be made here: From Eq
+(1.8), we can infer that if q is unity, the electric field due to a charge Q is numerically equal to the force exerted by it
+Thus, the electric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed at that point
+The charge Q, which is producing the electric field, is called a source charge and the charge q, which tests the effect of a source charge, is called a test charge
+Note that the source charge Q must remain at its original location
+However, if a charge q is brought at any point around Q, Q itself is bound to experience an electrical force due to q and will tend to move
+A way out of this difficulty is to make q negligibly small
+The force F is then negligibly small but the ratio F/q is finite and defines the electric field: (1.9)
+Figure 1.11 Electric field (a) due to a charge Q, (b) due to a charge –Q.
+A practical way to get around the problem (of keeping Q undisturbed in the presence of q) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice
+When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet
+Note that the electric field E due to Q, though defined operationally in terms of some test charge q, is independent of q
+This is because F is proportional to q, so the ratio F/q does not depend on q
+The force F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around the charge Q. Thus, the electric field E due to Q is also dependent on the space coordinate r. For different positions of the charge q all over the space, we get different values of electric field E
+The field exists at every point in three-dimensional space
+For a positive charge, the electric field will be directed radially outwards from the charge
+On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards
+(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q, the magnitude of the electric field E will also depend only on the distance r
+Thus at equal distances from the charge Q, the magnitude of its electric field E is same. The magnitude of electric field E due to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry.
+1.8.1 Electric field due to a system of charges Consider a system of charges q1, q2, ..., qn with position vectors r1, r2, ..., rn relative to some origin O
+Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of charges q1, q2, ..., qn
+We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vector r
+Electric field E1 at r due to q1 at r1 is given by E1 = whereis a unit vector in the direction from q1 to P, and r1P is the distance between q1 and P
+In the same manner, electric field E2 at r due to q2 at r2 is E2 =
+whereis a unit vector in the direction from q2 to P and r2P is the distance between q2 and P
+Similar expressions hold good for fields E3, E4, ..., En due to charges q3, q4, ..., qn
+By the superposition principle, the electric field E at r due to the system of charges is (as shown in 2)
+Figure 1.12 Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.
+E(r) = E1 (r) + E2 (r) + … + En(r) = E(r)(1.10) E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges.
+1.8.2 Physical significance of electric field You may wonder why the notion of electric field has been introduced here at all
+After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq
+(1.5)]
+Why then introduce this intermediate quantity called the electric field? For electrostatics, the concept of electric field is convenient, but not really necessary
+Electric field is an elegant way of characterising the electrical environment of a system of charges
+Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system)
+Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field
+The term field in physics generally refers to a quantity that is defined at every point in space and may vary from point to point
+Electric field is a vector field, since force is a vector quantity.
+The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena
+Suppose we consider the force between two distant charges q1, q2 in accelerated motion
+Now the greatest speed with which a signal or information can go from one point to another is c, the speed of light
+Thus, the effect of any motion of q1 on q2 cannot arise instantaneously
+There will be some time delay between the effect (force on q2) and the cause (motion of q1)
+It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful. The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2
+The notion of field elegantly accounts for the time delay
+Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs
+They have an independent dynamics of their own, i.e., they evolve according to laws of their own
+They can also transport energy
+Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy
+The concept of field was first introduced by Faraday and is now among the central concepts in physics.
+Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1
+The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance
+Compute the time of fall in each case
+Contrast the situation with that of ‘free fall under gravity’.
+Figure 1.13
+Solution In 3(a) the field is upward, so the negatively charged electron experiences a downward force of magnitude eE where E is the magnitude of the electric field
+The acceleration of the electron is ae = eE/me where me is the mass of the electron
+Starting from rest, the time required by the electron to fall through a distance h is given by For e = 1.6 × 10–19C, me = 9.11 × 10–31 kg, E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m, te = 2.9 × 10–9s In 3 (b), the field is downward, and the positively charged proton experiences a downward force of magnitude eE
+The acceleration of the proton is ap = eE/mp where mp is the mass of the proton; mp = 1.67 × 10–27 kg
+The time of fall for the proton is
+Thus, the heavier particle (proton) takes a greater time to fall through the same distance
+This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body
+Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall
+To see if this is justified, let us calculate the acceleration of the proton in the given electric field:
+which is enormous compared to the value of g (9.8 m s–2), the acceleration due to gravity
+The acceleration of the electron is even greater
+Thus, the effect of acceleration due to gravity can be ignored in this example
+Example 1.9 Two point charges q1 and q2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart
+Calculate the electric fields at points A, B and C shown in 4.
+Figure 1.14 Solution The electric field vector E1A at A due to the positive charge q1 points towards the right and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2A at A due to the negative charge q2 points towards the right and has the same magnitude
+Hence the magnitude of the total electric field EA at A is EA = E1A + E2A = 7.2 × 104 N C–1 EA is directed toward the right
+The electric field vector E1B at B due to the positive charge q1 points towards the left and has a magnitude = 3.6 × 104 N C–1 The electric field vector E2B at B due to the negative charge q2 points towards the right and has a magnitude = 4 × 103 N C–1 The magnitude of the total electric field at B is EB = E1B – E2B = 3.2 × 104 N C–1 EB is directed towards the left
+The magnitude of each electric field vector at point C, due to charge q1 and q2 is = 9 × 103 N C–1 The directions in which these two vectors point are indicated in 4
+The resultant of these two vectors is = 9 × 103 N C–1
+EC points towards the right.
+1.9 Electric Field Lines We have studied electric field in the last section
+It is a vector quantity and can be represented as we represent vectors
+Let us try to represent E due to a point charge pictorially
+Let the point charge be placed at the origin
+Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point
+Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward
+Figure 1.15 shows such a picture
+In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow
+Connect the arrows pointing in one direction and the resulting figure represents a field line
+We thus get many field lines, all pointing outwards from the point charge
+Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No
+Now the magnitude of the field is represented by the density of field lines. E is strong near the charge, so the density of field lines is more near the charge and the lines are closer
+Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines
+Another person may draw more lines
+But the number of lines is not important
+In fact, an infinite number of lines can be drawn in any region
+It is the relative density of lines in different regions which is important
+We draw the figure on the plane of paper, i.e., in two-dimensions but we live in three-dimensions
+So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines
+Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge
+We started by saying that the field lines carry information about the direction of electric field at different points in space
+Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points
+The field lines crowd where the field is strong and are spaced apart where it is weak
+Figure 1.16 shows a set of field lines
+We can imagine two equal and small elements of area placed at points R and S normal to the field lines there
+The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points
+The picture shows that the field at R is stronger than at S.
+Figure 1.15 Field of a point charge
+To understand the dependence of the field lines on the area, or rather the solid anglesubtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions
+Recall how a (plane) angle is defined in two-dimensions
+Let a small transverse line element ∆l be placed at a distance r from a point O
+Then the angle subtended by ∆l at O can be approximated as ∆θ = ∆l/r
+Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area ∆S, at a distance r, can be written as ∆Ω = ∆S/r2
+We know that in a given solid angle the number of radial field lines is the same
+In 6, for two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is∆Ω at P1 and an element of area
+Figure 1.16 Dependence of electric field strength on the distance and its relation to the number of field lines.
+∆Ω at P2, respectively
+The number of lines (say n) cutting these area elements are the same
+The number of field lines, cutting unit area element is therefore n/(∆Ω) at P1 and n/(∆Ω) at P2, respectively
+Since n and ∆Ω are common, the strength of the field clearly has a 1/r2dependence
+The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations
+Faraday called them lines of force
+This term is somewhat misleading, especially in case of magnetic fields
+The more appropriate term is field lines (electric or magnetic) that we have adopted in this book
+Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges
+An electric field line is, in general, a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point
+An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve
+A field line is a space curve, i.e., a curve in three dimensions.
+Figure 1.17 Field lines due to some simple charge configurations.
+Figure 1.17 shows the field lines around some simple charge configurations
+As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane
+The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward
+The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges
+The field lines follow some important general properties: Field lines start from positive charges and end at negative charges
+If there is a single charge, they may start or end at infinity
+In a charge-free region, electric field lines can be taken to be continuous curves without any breaks
+Two field lines can never cross each other
+(If they did, the field at the point of intersection will not have a unique direction, which is absurd.) (iv) Electrostatic field lines do not form any closed loops
+This follows from the conservative nature of electric field (Chapter 2).
+1.10 Electric Flux Consider flow of a liquid with velocity v, through a small flat surface dS, in a direction normal to the surface
+The rate of flow of liquid is given by the volume crossing the area per unit time v dSand represents the flux of liquid flowing across the plane
+If the normal to the surface is not parallel to the direction of flow of liquid, i.e., to v, but makes an angle θ with it, the projected area in a plane perpendicular to v is v dS cos θ
+Therefore, the flux going out of the surface dSis v.dS. For the case of the electric field, we define an analogous quantity and call it electric flux
+We should, however, note that there is no flow of a physically observable quantity unlike the case of liquid flow
+In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point
+This means that if we place a small planar element of area ∆S normal to E at a point, the number of field lines crossing it is proportional* to E ∆S
+Now suppose we tilt the area element by angle θ
+Clearly, the number of field lines crossing the area element will be smaller
+The projection of the area element normal to E is ∆S cosθ
+Thus, the number of field lines crossing ∆S is proportional to E ∆S cosθ
+When θ = 90°, field lines will be parallel to ∆Sand will not cross it at all .
+Figure 1.18 Dependence of flux on the inclination θ between E and n
+The orientation of area element and not merely its magnitude is important in many contexts
+For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring
+If you hold it normal to the flow, maximum water will flow through it than if you hold it with some other orientation
+This shows that an area element should be treated as a vector
+It has a magnitude and also a direction
+How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane
+Thus the direction of a planar area vector is along its normal
+How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements
+Each small area element may be treated as planar and a vector associated with it, as explained before
+Notice one ambiguity here
+The direction of an area element is along its normal
+But a normal can point in two directions
+Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context
+For the case of a closed surface, this convention is very simple
+The vector associated with every area element of a closed surface is taken to be in the direction of theoutward normal
+This is the convention used in 9
+Thus, the area element vector ∆S at a point on a closed surface equals ∆Sn where ∆S is the magnitude of the area element and n is a unit vector in the direction of outward normal at that point
+We now come to the definition of electric flux
+Electric flux ∆φ through an area element ∆S is defined by ∆φ = E.∆S = E ∆S cosθ (1.11) which, as seen before, is proportional to the number of field lines cutting the area element
+The angle θ here is the angle between E and ∆S
+For a closed surface, with the convention stated already, θ is the angle between E and the outward normal to the area element
+Notice we could look at the expression E ∆S cosθ in two ways: E (∆S cosθ ) i.e., E times the projection of area normal to E, or E⊥ ∆S, i.e., component of E along the normal to the area element times the magnitude of the area element
+The unit of electric flux is N C–1 m2
+The basic definition of electric flux given by Eq
+(1.11) can be used, in principle, to calculate the total flux through any given surface
+All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up
+Thus, the total flux φ through a surface S is φ ~ Σ E.∆S (1.12) The approximation sign is put because the electric field E is taken to be constant over the small area element
+This is mathematically exact only when you take the limit ∆S → 0 and the sum in Eq
+(1.12) is written as an integral.
+Figure 1.19 Convention for defining normal
+1.11 Electric Dipole An electric dipole is a pair of equal and opposite point charges q and –q, separated by a distance 2a
+The line connecting the two charges defines a direction in space
+By convention, the direction from –q to q is said to be the direction of the dipole
+The mid-point of locations of –q and q is called the centre of the dipole
+The total charge of the electric dipole is obviously zero
+This does not mean that the field of the electric dipole is zero
+Since the charge q and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out
+However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due to q and –q nearly cancel out
+The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence on r of the field due to a single charge q)
+These qualitative ideas are borne out by the explicit calculation as follows:
+1.11.1 The field of an electric dipole The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle
+The results are simple for the following two cases: when the point is on the dipole axis, and when it is in the equatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre
+The electric field at any general point P is obtained by adding the electric fields E–q due to the charge –q and E+qdue to the charge q, by the parallelogram law of vectors
+For points on the axis Let the point P be at distance r from the centre of the dipole on the side of the charge q, as shown in 0(a)
+Then [1.13(a)] whereis the unit vector along the dipole axis (from –q to q)
+Also [1.13(b)] The total field at P is
+(1.14) For r >> a (r >> a) (1.15)
+Figure 1.20 Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole. p is the dipole moment vector of magnitude p = q × 2a and directed from –q to q
+For points on the equatorial plane The magnitudes of the electric fields due to the two charges +q and –q are given by [1.16(a)] [1.16(b)] and are equal
+The directions of E+q and E–q are as shown in 0(b)
+Clearly, the components normal to the dipole axis cancel away
+The components along the dipole axis add up
+The total electric field is opposite to
+We have E = – (E +q + E –q) cosθ (1.17) At large distances (r >> a), this reduces to (1.18) From Eqs
+(1.15) and (1.18), it is clear that the dipole field at large distances does not involveq and a separately; it depends on the product qa
+This suggests the definition of dipole moment
+The dipole moment vector p of an electric dipole is defined by p = q × 2a(1.19) that is, it is a vector whose magnitude is charge q times the separation 2a (between the pair of charges q, –q) and the direction is along the line from –q to q
+In terms of p, the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis (r >> a) (1.20)
+At a point on the equatorial plane (r >> a) (1.21) Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3
+Further, the magnitude and the direction of the dipole field depends not only on the distance rbut also on the angle between the position vector r and the dipole moment p
+We can think of the limit when the dipole size 2a approaches zero, the charge q approaches infinity in such a way that the product p = q × 2a is finite
+Such a dipole is referred to as a point dipole
+For a point dipole, Eqs
+(1.20) and (1.21) are exact, true for any r.
+1.11.2 Physical significance of dipoles
+In most molecules, the centres of positive charges and of negative charges* lie at the same place
+Therefore, their dipole moment is zero
+CO2 and CH4 are of this type of molecules
+However, they develop a dipole moment when an electric field is applied
+But in some molecules, the centres of negative charges and of positive charges do not coincide
+Therefore they have a permanent electric dipole moment, even in the absence of an electric field
+Such molecules are called polar molecules
+Water molecules, H2O, is an example of this type
+Various materials give rise to interesting properties and important applications in the presence or absence of electric field.
+Example 1.10 Two charges ±10 µC are placed 5.0 mm apart
+Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in 1(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in 1(b).
+figure 1.21 Solution (a) Field at P due to charge +10 µC = = 4.13 × 106 N C–1 along BP Field at P due to charge –10 µC
+= 3.86 × 106 N C–1 along PA The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP
+In this example, the ratio OP/OB is quite large (= 60)
+Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole
+For a dipole consisting of charges ± q, 2a distance apart, the electric field at a distance r from the centre on the axis of the dipole has a magnitude (r/a >> 1) where p = 2a q is the magnitude of the dipole moment
+The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q to q)
+Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m Therefore, E == 2.6 × 105 N C–1 along the dipole moment direction AB, which is close to the result obtained earlier
+(b) Field at Q due to charge + 10 µC at B = = 3.99 × 106 N C–1 along BQ
+Field at Q due to charge –10 µC at A = = 3.99 × 106 N C–1 along QA.
+Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA
+Therefore, the resultant electric field at Q due to the two charges at A and B is
+.
+= 2 ×along BA = 1.33 × 105 N C–1 along BA
+As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole: (r/a >> 1)
+= 1.33 × 105 N C–1.
+The direction of electric field in this case is opposite to the direction of the dipole moment vector
+Again, the result agrees with that obtained before.
+1.12 Dipole in a Uniform External Field Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in 2
+(By permanent dipole, we mean that p exists irrespective of E; it has not been induced byE.) There is a force qE on q and a force –qE on –q
+The net force on the dipole is zero, since E is uniform
+However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole
+When the net force is zero, the torque (couple) is independent of the origin
+Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).
+Figure 1.22 Dipole in a uniform electric field
+Magnitude of torque = q E × 2 a sinθ = 2 q a E sinθ Its direction is normal to the plane of the paper, coming out of it
+The magnitude of p × E is also p E sinθ and its direction is normal to the paper, coming out of it
+Thus, τ = p × E (1.22) This torque will tend to align the dipole with the field E
+When p is aligned with E, the torque is zero
+What happens if the field is not uniform? In that case, the net force will evidently be non-zero
+In addition there will, in general, be a torque on the system as before
+The general case is involved, so let us consider the simpler situations when p is parallel to E or antiparallel to E
+In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform
+Figure 1.23 is self-explanatory
+It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field
+When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field
+In general, the force depends on the orientation of p with respect to E.
+This brings us to a common observation in frictional electricity
+A comb run through dry hair attracts pieces of paper
+The comb, as we know, acquires charge through friction
+But the paper is not charged
+What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field
+Further, the electric field due to the comb is not uniform
+In this situation, it is easily seen that the paper should move in the direction of the comb!
+1.13 Continuous Charge Distribution We have so far dealt with charge configurations involving discrete charges q1, q2, ..., qn
+One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus
+For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions
+For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents
+It is more feasible to consider an area element ∆S on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ∆Q on that element
+We then define a surface charge density σ at the area element by
+(1.23) We can do this at different points on the conductor and thus arrive at a continuous function σ, called the surface charge density
+The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*. σrepresents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element ∆S which, as said before, is large microscopically but small macroscopically
+The units for σ are C/m2
+Similar considerations apply for a line charge distribution and a volume charge distribution
+The linear charge density λ of a wire is defined by Figure 1.23 Electric force on a dipole: (a) E parallel to p, (b) E antiparallel to p.
+(1.24) where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and ∆Q is the charge contained in that line element
+The units for λ are C/m
+The volume charge density (sometimes simply called charge density) is defined in a similar manner: (1.25) where ∆Q is the charge included in the macroscopically small volume element ∆V that includes a large number of microscopic charged constituents
+The units for ρ are C/m3
+The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics
+When we refer to the density of a liquid, we are referring to its macroscopic density
+We regard it as a continuous fluid and ignore its discrete molecular constitution.
+Figure 1.24 Definition of linear, surface and volume charge densities. In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but contains a very large number of microscopic constituents.
+The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq
+(1.10)
+Suppose a continuous charge distribution in space has a charge density ρ
+Choose any convenient origin O and let the position vector of any point in the charge distribution be r
+The charge density ρ may vary from point to point, i.e., it is a function of r
+Divide the charge distribution into small volume elements of size ∆V
+The charge in a volume element ∆V is ρ∆V
+Now, consider any general point P (inside or outside the distribution) with position vector R
+Electric field due to the charge ρ∆V is given by Coulomb’s law: (1.26) where r′ is the distance between the charge element and P, and′ is a unit vector in the direction from the charge element to P
+By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements: (1.27) Note that ρ, r′,all can vary from point to point
+In a strict mathematical method, we should let ∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity
+In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.
+1.14 Gauss’s Law As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre
+Divide the sphere into small area elements, as shown in 5.
+Figure 1.25 Flux through a sphere enclosing a point charge q at its centre.
+The flux through an area element ∆S is (1.28) where we have used Coulomb’s law for the electric field due to a single charge q
+The unit vectoris along the radius vector from the centre to the area element
+Now, since the normal to a sphere at every point is along the radius vector at that point, the area element ∆S andhave the same direction
+Therefore, (1.29) since the magnitude of a unit vector is 1
+The total flux through the sphere is obtained by adding up flux through all the different area elements:
+Since each area element of the sphere is at the same distance r from the charge,
+Now S, the total area of the sphere, equals 4πr2
+Thus, (1.30)
+Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.
+Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law
+We state Gauss’s law without proof: Electric flux through a closed surface S
+= q/ε0 (1.31) q = total charge enclosed by S
+The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface
+We can see that explicitly in the simple situation of 6
+Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform field E
+The total flux φ through the surface is φ = φ1 + φ2 + φ3, where φ1 and φ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder and φ3 is the flux through the curved cylindrical part of the closed surface
+Now the normal to the surface 3 at every point is perpendicular to E, so by definition of flux, φ3 = 0
+Further, the outward normal to 2 is along E while the outward normal to 1 is opposite to E
+Therefore, φ1 = –E S1, φ2 = +E S2 S1 = S2 = S where S is the area of circular cross-section
+Thus, the total flux is zero, as expected by Gauss’s law
+Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero
+The great significance of Gauss’s law Eq
+(1.31), is that it is true in general, and not only for the simple cases we have considered above
+Let us note some important points regarding this law: Gauss’s law is true for any closed surface, no matter what its shape or size
+The term q on the right side of Gauss’s law, Eq
+(1.31), includes the sum of all charges enclosed by the surface
+The charges may be located anywhere inside the surface
+In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq
+(1.31)] is due to all the charges, both inside and outside S
+The term q on the right side of Gauss’s law, however, represents only the total charge inside S
+(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface
+You may choose any Gaussian surface and apply Gauss’s law
+However, take care not to let the Gaussian surface pass through any discrete charge
+This is because electric field due to a system of discrete charges is not well defined at the location of any charge
+(As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution
+(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface
+(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law
+Any violation of Gauss’s law will indicate departure from the inverse square law.
+Example 1.11 The electric field components in 7 are Ex = αx1/2, Ey = Ez = 0, in which α = 800 N/C m1/2
+Calculate (a) the flux through the cube, and (b) the charge within the cube
+Assume that a = 0.1 m.
+Figure 1.27 Solution (a) Since the electric field has only an x component, for faces perpendicular to x direction, the angle between E and ∆S is ± π/2
+Therefore, the flux φ = E.∆S is separately zero for each face of the cube except the two shaded ones
+Now the magnitude of the electric field at the left face is EL = αx1/2 = αa1/2 (x = a at the left face)
+The magnitude of electric field at the right face is ER = α x1/2 = α (2a)1/2 (x = 2a at the right face)
+The corresponding fluxes are φL= EL.∆S ==EL ∆S cosθ = –EL ∆S, since θ = 180° = –ELa2 φR= ER.∆S = ER ∆S cosθ = ER ∆S, since θ = 0° = ERa2 Net flux through the cube = φR + φL = ERa2 – ELa2 = a2 (ER – EL) = αa2 [(2a)1/2 – a1/2] = αa5/2 = 800 (0.1)5/2
+= 1.05 N m2 C–1 (b) We can use Gauss’s law to find the total charge q inside the cube
+We have φ = q/ε0 or q =φε0
+Therefore,
+q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C
+Example 1.12 An electric field is uniform, and in the positive x direction for positive x, and uniform with the same magnitude but in the negative x direction for negative x
+It is given that E= 200N/C for x > 0 and E = –200N/C for x < 0
+A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = –10 cm
+(a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder? Solution (a) We can see from the figure that on the left face E and ∆S are parallel
+Therefore, the outward flux is φL= E.∆S = – 200 = + 200 ∆S, since= – ∆S = + 200 × π (0.05)2 = + 1.57 N m2 C–1 On the right face, E and ∆S are parallel and therefore φR = E.∆S = + 1.57 N m2 C–1
+(b) For any point on the side of the cylinder E is perpendicular to ∆S and hence E.∆S = 0
+Therefore, the flux out of the side of the cylinder is zero
+(c) Net outward flux through the cylinder φ = 1.57 + 1.57 + 0 = 3.14 N m2 C–1
+Figure 1.28 (d) The net charge within the cylinder can be found by using Gauss’s law which gives q = ε0φ = 3.14 × 8.854 × 10–12 C
+= 2.78 × 10–11 C
+1.15 Applications of Gauss’s Law The electric field due to a general charge distribution is, as seen above, given by Eq
+(1.27)
+In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space
+For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law
+This is best understood by some examples.
+1.15.1 Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ
+The wire is obviously an axis of symmetry
+Suppose we take the radial vector from O to P and rotate it around the wire
+The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire
+This implies that the electric field must have the same magnitude at these points
+The direction of electric field at every point must be radial (outward if λ > 0, inward if λ < 0)
+This is clear from 9.
+Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density
+Consider a pair of line elements P1 and P2 of the wire, as shown
+The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel)
+This is true for any such pair and hence the total field at any point P is radial
+Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire
+In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r
+To calculate the field, imagine a cylindrical Gaussian surface, as shown in the 9(b).Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero
+At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r
+The surface area of the curved part is 2πrl, where l is the length of the cylinder
+Flux through the Gaussian surface = flux through the curved cylindrical part of the surface = E × 2πrl The surface includes charge equal to λ l
+Gauss’s law then gives E × 2πrl = λl/ε0 i.e., E = Vectorially, E at any point is given by (1.32) whereis the radial unit vector in the plane normal to the wire passing through the point. E is directed outward if λ is positive and inward if λ is negative.
+Note that when we write a vector A as a scalar multiplied by a unit vector, i.e., as A = A, the scalar A is an algebraic number
+It can be negative or positive
+The direction of A will be the same as that of the unit vectorif A > 0 and opposite toif A < 0
+When we want to restrict to non-negative values, we use the symboland call it the modulus of A
+Thus,
+Also note that though only the charge enclosed by the surface (λl) was included above, the electric field E is due to the charge on the entire wire
+Further, the assumption that the wire is infinitely long is crucial
+Without this assumption, we cannot take E to be normal to the curved part of the cylindrical Gaussian surface
+However, Eq
+(1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored.
+1.15.2 Field due to a uniformly charged infinite plane sheet Let σ be the uniform surface charge density of an infinite plane sheet
+We take thex-axis normal to the given plane
+By symmetry, the electric field will not depend on y and zcoordinates and its direction at every point must be parallel to the x-direction
+We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional areaA, as shown
+(A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux.
+Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.
+The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction
+Therefore, flux E.∆S through both the surfaces are equal and add up
+Therefore the net flux through the Gaussian surface is 2 EA
+The charge enclosed by the closed surface is σA
+Therefore by Gauss’s law, 2 EA = σA/ε0 or, E = σ/2ε0 Vectorically, (1.33) where n is a unit vector normal to the plane and going away from it
+E is directed away from the plate if σ is positive and toward the plate if σ is negative
+Note that the above application of the Gauss’ law has brought out an additional fact: E is independent ofx also
+For a finite large planar sheet, Eq
+(1.33) is approximately true in the middle regions of the planar sheet, away from the ends.
+1.15.3 Field due to a uniformly charged thin spherical shell Let σ be the uniform surface charge density of a thin spherical shell of radius R
+The situation has obvious spherical symmetry
+The field at any point P, outside or inside, can depend only on r (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector)
+Field outside the shell: Consider a point P outside the shell with radius vector r
+To calculateE at P, we take the Gaussian surface to be a sphere of radius r and with centre O, passing through P
+All points on this sphere are equivalent relative to the given charged configuration
+(That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitude E and is along the radius vector at each point
+Thus, E and ∆S at every point are parallel and the flux through each element is E ∆S. Summing over all ∆S, the flux through the Gaussian surface is E × 4 π r2
+The charge enclosed is σ × 4 π R2
+By Gauss’s law E × 4 π r2 =
+Or, where q = 4 π R2 σ is the total charge on the spherical shell
+Vectorially, (1.34)
+Figure 1.31 Gaussian surfaces for a point with (a) r > R, (b) r < R
+The electric field is directed outward if q > 0 and inward if q < 0. This, however, is exactly the field produced by a charge q placed at the centre O
+Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.
+Field inside the shell: In 1(b), the point P is inside the shell
+The Gaussian surface is again a sphere through P centred at O. The flux through the Gaussian surface, calculated as before, is E × 4 π r2
+However, in this case, the Gaussian surface encloses no charge
+Gauss’s law then gives E × 4 π r2 = 0 i.e., E = 0 (r < R ) (1.35) that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*
+This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law
+The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law.
+Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge up to a radius R
+The atom as a whole is neutral
+For this model, what is the electric field at a distance r from the nucleus?
+Figure 1.32 Solution The charge distribution for this model of the atom is as shown in 2
+The total negative charge in the uniform spherical charge distribution of radius R must be –Z e, since the atom (nucleus of charge Z e + negative charge) is neutral
+This immediately gives us the negative charge density ρ, since we must have or To find the electric field E(r) at a point P which is a distance r away from the nucleus, we use Gauss’s law
+Because of the spherical symmetry of the charge distribution, the magnitude of the electric field E(r) depends only on the radial distance, no matter what the direction of r
+Its direction is along (or opposite to) the radius vector r from the origin to the point P
+The obvious Gaussian surface is a spherical surface centred at the nucleus
+We consider two situations, namely, r < R and r > R
+r < R : The electric flux φ enclosed by the spherical surface is φ = E (r) × 4 π r2 where E (r) is the magnitude of the electric field at r
+This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface
+The charge q enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radius r, i.e., Substituting for the charge density ρ obtained earlier, we have Gauss’s law then gives,
+The electric field is directed radially outward
+r > R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral
+Thus, from Gauss’s law, E (r) × 4 π r2 = 0 or E (r) = 0; r > R
+At r = R, both cases give the same result: E = 0.
+On symmetry operations In Physics, we often encounter systems with various symmetries
+Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation
+Consider, for example an infinite uniform sheet of charge (surface charge densityσ) along the y-z plane
+This system is unchanged if (a) translated parallel to the y-z plane in any direction, (b) rotated about the x-axis through any angle
+As the system is unchanged under such symmetry operation, so must its properties be
+In particular, in this example, the electric field E must be unchanged
+Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0)
+Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same
+By using rotation symmetry around the x-axis, we can conclude that E must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction
+Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate
+It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space
+The direction, however, is opposite of each other on either side ofthe sheet
+Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.
+Summary 1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter
+2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract
+By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative
+3. Conductors allow movement of electric charge through them, insulators do not
+In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile
+4. Electric charge has three basic properties: quantisation, additivity and conservation
+Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q = n e, where n = 0, ±1, ±2, ±3, ...
+Proton and electron have charges +e, –e, respectively
+For macroscopic charges for which n is a very large number, quantisation of charge can be ignored
+Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system
+Conservation of electric charges means that the total charge of an isolated system remains unchanged with time
+This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge
+5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21separating them
+Mathematically, F21 = force on q2 due to whereis a unit vector in the direction from q1 to q2 and k =is the constant of proportionality
+In SI units, the unit of charge is coulomb
+The experimental value of the constant ε0 is ε0 = 8.854 × 10–12 C2 N–1 m–2 The approximate value of k is k = 9 × 109 N m2 C–2
+6. The ratio of electric force and gravitational force between a proton and an electron is
+7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s)
+For an assembly of charges q1, q2, q3, ..., the force on any charge, sayq1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on
+For each pair, the force is given by the Coulomb’s law for two charges stated earlier
+8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge
+Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative
+Like Coulomb force, electric field also satisfies superposition principle
+9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point
+The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak
+In regions of constant electric field, the field lines are uniformly spaced parallel straight lines.
+10. Some of the important properties of field lines are: Field lines are continuous curves without any breaks
+Two field lines cannot cross each other
+Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops
+11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a
+Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q
+12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:
+Dipole electric field on the axis at a distance r from the centre:
+The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2dependence of electric field due to a point charge
+13. In a uniform electric field E, a dipole experiences a torquegiven by = p × E but experiences no net force
+14. The flux ∆φ of electric field E through a small area element ∆S is given by ∆φ = E.∆S The vector area element ∆S is ∆S = ∆S where ∆S is the magnitude of the area element andis normal to the area element, which can be considered planar for sufficiently small ∆S
+For an area element of a closed surface,is taken to be the direction of outward normal, by convention
+15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S
+The law is especially useful in determining electric field E, when the source distribution has simple symmetry: Thin infinitely long straight wire of uniform linear charge density λ
+where r is the perpendicular distance of the point from the wire andis the radial unit vector in the plane normal to the wire passing through the point
+Infinite thin plane sheet of uniform surface charge density σ
+whereis a unit vector normal to the plane, outward on either side
+Thin spherical shell of uniform surface charge density σ
+E = 0 (r < R) where r is the distance of the point from the centre of the shell and R the radius of the shell. q is the total charge of the shell: q = 4πR2σ
+The electric field outside the shell is as though the total charge is concentrated at the centre
+The same result is true for a solid sphere of uniform volume charge density
+The field is zero at all points inside the shell.
+Points to Ponder 1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus
+Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together
+The range of distance where this force is effective is, however, very small ~10-14 m
+This is precisely the size of the nucleus
+Also the electrons are not allowed to sit on top of the protons, i.e
+inside the nucleus, due to the laws of quantum mechanics
+This gives the atoms their structure as they exist in nature
+2. Coulomb force and gravitational force follow the same inverse-square law
+But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces
+This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature
+3. The constant of proportionality k in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law
+In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s)
+In this case, the value of k is no longer arbitrary; it is approximately 9 × 109 N m2C–2
+4. The rather large value of k, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces
+Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects
+5. The additive property of charge is not an ‘obvious’ property
+It is related to the fact that electric charge has no direction associated with it; charge is a scalar
+6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion
+This is not always true for every scalar
+For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion
+7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6
+Conservation refers to invariance in time in a given frame of reference
+A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision)
+On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system)
+8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass
+9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors
+It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges
+10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges
+For continuous volume charge distribution, it is defined at any point in the distribution
+For a surface charge distribution, electric field is discontinuous across the surface.
+11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge
+An electric dipole is the simplest example of this fact.
+Exercises 1.1 What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7C placed 30 cm apart in air?
+1.2 The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in air is 0.2 N
+(a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? 1.3 Check that the ratio ke2/G memp is dimensionless
+Look up a Table of Physical Constants and determine the value of this ratio
+What does the ratio signify? 1.4 (a) Explain the meaning of the statement ‘electric charge of a body is quantised’
+(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? 1.5 When a glass rod is rubbed with a silk cloth, charges appear on both
+A similar phenomenon is observed with many other pairs of bodies
+Explain how this observation is consistent with the law of conservation of charge
+1.6 Four point charges qA = 2 µC, qB = –5 µC, qC = 2 µC, and qD = –5 µC are located at the corners of a square ABCD of side 10 cm
+What is the force on a charge of 1 µC placed at the centre of the square? 1.7 (a) An electrostatic field line is a continuous curve
+That is, a field line cannot have sudden breaks
+Why not? (b) Explain why two field lines never cross each other at any point? 1.8 Two point charges qA = 3 µC and qB = –3 µC are located 20 cm apart in vacuum
+(a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge? 1.9 A system has two charges qA = 2.5 × 10–7 C and qB = –2.5 × 10–7 C located at points A: (0, 0, –15 cm) and B: (0,0, +15 cm), respectively
+What are the total charge and electric dipole moment of the system? 1.10 An electric dipole with dipole moment 4 × 10–9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 104 NC–1
+Calculate the magnitude of the torque acting on the dipole
+1.11 A polythene piece rubbed with wool is found to have a negative charge of 3 × 10–7 C
+(a) Estimate the number of electrons transferred (from which to which?) (b) Is there a transfer of mass from wool to polythene? 1.12 (a) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm
+What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10–7 C? The radii of A and B are negligible compared to the distance of separation
+(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? 1.13 Suppose the spheres A and B in Exercise 1.12 have identical sizes
+A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both
+What is the new force of repulsion between A and B? 1.14 Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field
+Give the signs of the three charges
+Which particle has the highest charge to mass ratio?
+Figure 1.33 1.15 Consider a uniform electric field E = 3 × 103 î N/C
+(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis? 1.16 What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes? 1.17 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C
+(a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? 1.18 A point charge +10 µC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in 4
+What is the magnitude of the electric flux through the square? (Hint:Think of the square as one face of a cube with edge 10 cm.)
+Figure 1.34 1.19 A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge
+What is the net electric flux through the surface? 1.20 A point charge causes an electric flux of –1.0 × 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge
+(a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? 1.21 A conducting sphere of radius 10 cm has an unknown charge
+If the electric field 20 cm from the centre of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere? 1.22 A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m2
+(a) Find the charge on the sphere
+(b) What is the total electric flux leaving the surface of the sphere? 1.23 An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm
+Calculate the linear charge density.
+1.24 Two large, thin metal plates are parallel and close to each other
+On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C/m2
+What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?
+Additional Exercises 1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 104 NC–1 (Millikan’s oil drop experiment)
+The density of the oil is 1.26 g cm–3
+Estimate the radius of the drop. (g = 9.81 m s–2; e = 1.60 × 10–19 C)
+1.26 Which among the curves shown in 5 cannot possibly represent electrostatic field lines?
+Figure 1.35 1.27 In a certain region of space, electric field is along the z-direction throughout
+The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC–1 per metre
+What are the force and torque experienced by a system having a total dipole moment equal to 10–7 Cm in the negative z-direction ? 1.28 (a) A conductor A with a cavity as shown in 6(a) is given a charge Q
+Show that the entire charge must appear on the outer surface of the conductor
+(b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A
+Show that the total charge on the outside surface of A is Q + q
+(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment
+Suggest a possible way.
+Figure 1.36 1.29 A hollow charged conductor has a tiny hole cut into its surface
+Show that the electric field in the hole is (σ/2ε0), whereis the unit vector in the outward normal direction, and σ is the surface charge density near the hole
+1.30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge density E without using Gauss’s law
+[Hint: Use Coulomb’s law directly and evaluate the necessary integral.] 1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks
+A proton and a neutron consist of three quarks each
+Two types of quarks, the so called ‘up’ quark (denoted by u) of charge + (2/3) e, and the ‘down’ quark (denoted by d) of charge (–1/3) e, together with electrons build up ordinary matter
+(Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron
+1.32 (a) Consider an arbitrary electrostatic field configuration
+A small test charge is placed at a null point (i.e., where E = 0) of the configuration
+Show that the equilibrium of the test charge is necessarily unstable
+(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart
+1.33 A particle of mass m and charge (–q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in 3)
+The length of plate is Land an uniform electric field E is maintained between the plates
+Show that the vertical deflection of the particle at the far edge of the plate is qEL2/(2m vx2)
+Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics.
+1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx = 2.0 × 106 m s–1
+If E between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.)
+Chapter 1 ELECTRIC CHARGES AND FIELDS
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