Merging Casey's Thesis Work

Makefile

@@ -20,7 +20,8 @@ include $(FRAMEWORK_DIR)/build.mk
 include $(FRAMEWORK_DIR)/moose.mk
 
 ################################## MODULES ####################################
-ALL_MODULES := no
+ELECTROMAGNETICS := yes
+
 include $(MOOSE_DIR)/modules/modules.mk
 ###############################################################################
 

doc/content/source/auxkernels/Current.md

@@ -4,27 +4,26 @@
 
 ## Overview
 
-`Current` returns the electric current density of a species in logarithmic form. `Current`
-assumes the electrostatic approximation for the electric field.
+`Current` returns the electric current density of a species in logarithmic form.
 
-The electrostatic current density is defined as
+The current density is defined as
 
 \begin{equation}
-J_{j} = q_{j} (\text{sign}_{j} \mu_{j} \left( \text{-} \nabla V\right) n_{j} - D_{j} \nabla (n_{j}))
+J_{j} = q_{j} (\text{sign}_{j} \mu_{j} \vec{E} n_{j} - D_{j} \nabla (n_{j}))
 \end{equation}
 
-Where $J_{j}$ is the current density, $q_{j}$ is the charge of the species, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $V$ is the potential, $n_{j}$ is the density, and $D_{j}$ is the diffusion coefficient. When converting the density to logarithmic form and applying a scaling factor of the mesh, `Current` is defined as
+Where $J_{j}$ is the current density, $q_{j}$ is the charge of the species, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $\vec{E}$ is the electric field, $n_{j}$ is the density, and $D_{j}$ is the diffusion coefficient. When converting the density to logarithmic form and applying a scaling factor of the mesh, `Current` is defined as
 
 \begin{equation}
-J_{j} = q_{j} N_{A} \left(\text{sign}_{j} \mu_{j} \frac{\text{-} \nabla (V)}{l_{c}} \exp(N_{j}) - D_{j} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}} \right)
+J_{j} = q_{j} N_{A} \left(\text{sign}_{j} \mu_{j} \frac{\vec{E}}{l_{c}} \exp(N_{j}) - D_{j} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}} \right)
 \end{equation}
 
 Where $N_{j}$ is the molar density of the specie in logarithmic form, $N_{A}$ is Avogadro's number, $l_{c}$ is the scaling factor of the mesh.
 
 For the case of the where artificial diffusion is introduced to the charge specie flux, an additional term is included in the current density, such that:
 
 \begin{equation}
-J_{j,\text{ Total}} = J_{j} + q_{j} N_{A} \mu_{j} \frac{\text{-}\lVert \nabla (V) \rVert_{2}}{l_{c}} \frac{h_\text{max}}{2} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}}
+J_{j,\text{ Total}} = J_{j} + q_{j} N_{A} \mu_{j} \frac{\text{-}\lVert \vec{E} \rVert_{2}}{l_{c}} \frac{h_\text{max}}{2} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}}
 \end{equation}
 
 Where $h_\text{max}$ is the max length of the current element.

doc/content/source/auxkernels/DriftDiffusionFluxAux.md

@@ -5,16 +5,16 @@
 ## Overview
 
 `DriftDiffusionFluxAux` returns the simplified drift-diffusion flux of a species. `DriftDiffusionFluxAux`
-assumes a mobility and diffusion coefficient of unity, the electrostatic approximation for the electric field, and a non-scaled version of the specie's density.
+assumes a mobility and diffusion coefficient of unity and a non-scaled version of the specie's density.
 
-The electrostatic flux is defined as
+The flux is defined as
 
 \begin{equation}
-\Gamma_{j} =  \text{sign}_{j}  \left( \text{-}\nabla V\right) n_{j} - \nabla (n_{j})
+\Gamma_{j} =  \text{sign}_{j} \vec{E} n_{j} - \nabla (n_{j})
 \end{equation}
 
 Where $\Gamma_{j}$ is the flux assuming drift-diffusion formulation, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species),
-$V$ is the potential, and $n_{j}$ is the density.
+$\vec{E}$ is the electric field, and $n_{j}$ is the density.
 
 !alert note
 When calculating the drift-diffusion flux for scaled densities and non-unity coefficients, please refer to [`TotalFlux`](/auxkernels/TotalFlux.md).

doc/content/source/auxkernels/EFieldAdvAux.md

@@ -4,20 +4,19 @@
 
 ## Overview
 
-`EFieldAdvAux` returns electric field driven advective flux of defined species in logarithmic form. `EFieldAdvAux`
-assumes the electrostatic approximation for the electric field.
+`EFieldAdvAux` returns electric field driven advective flux of defined species in logarithmic form.
 
 The advective flux is defined as
 
 \begin{equation}
-\Gamma_{\text{Advection}}  = \text{sign}_{j} \mu_{j} \left( \text{-} \nabla V\right) n_{j}
+\Gamma_{\text{Advection}}  = \text{sign}_{j} \mu_{j} \vec{E} n_{j}
 \end{equation}
 
-Where $\Gamma_{\text{Advection}}$ is the advective flux, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $V$ is the potential, and $n_{j}$ is the density. When converting the density to logarithmic form and applying a scaling factor of the mesh,
+Where $\Gamma_{\text{Advection}}$ is the advective flux, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $\vec{E}$ is the electric field, and $n_{j}$ is the density. When converting the density to logarithmic form and applying a scaling factor of the mesh,
 `EFieldAdvAux` is defined as
 
 \begin{equation}
-\Gamma_{\text{Advection}}  = N_{A} \text{sign}_{j} \mu_{j} \frac{\text{-} \nabla (V)}{l_{c}} \exp(N_{j})
+\Gamma_{\text{Advection}}  = N_{A} \text{sign}_{j} \mu_{j} \frac{\vec{E}}{l_{c}} \exp(N_{j})
 \end{equation}
 
 Where $N_{j}$ is the molar density of the specie in logarithmic form, $N_{A}$ is Avogadro's

doc/content/source/auxkernels/Efield.md

@@ -4,15 +4,15 @@
 
 ## Overview
 
-`Efield` returns a component of the electrostatic electric field.
+`Efield` returns a component of the electric field.
 
 The formulation of `Efield` is defined as
 
 \begin{equation}
-E_{\text{comp.}} = \frac{\text{-} \nabla_{\text{comp.}} (V) \ V_{c}}{l_{c}}
+    \frac{\vec{E}_{\text{comp.}} \ V_{c}}{l_{c}}
 \end{equation}
 
-Where $E_{\text{comp.}}$ is a component of the electric field, $V$ is the potential, $V_{c}$ is the
+Where $\vec{E}_{\text{comp.}}$ is a component of the electric field, $V_{c}$ is the
 scaling factor of the potential , and $l_{c}$ is the scaling factor of the mesh.
 
 ## Example Input File Syntax

doc/content/source/auxkernels/PowerDep.md

@@ -5,26 +5,25 @@
 ## Overview
 
 `PowerDep` returns the amount of power deposited into a user specified specie by
-Joule Heating. `PowerDep`
-assumes the electrostatic approximation for the electric field.
+Joule Heating.
 
 The power deposited by Joule Heating is defined as
 
 \begin{equation}
-P_{\text{Joule Heating}} = \Gamma_{j} \cdot \text{-} \nabla (V) \\
+P_{\text{Joule Heating}} = \Gamma_{j} \cdot \vec{E} \\
 \\[10pt]
-\Gamma_{j} = q_{j} (\text{sign}_{j} \mu_{j} \left( \text{-} \nabla V\right) n_{j} - D_{j} \nabla (n_{j}))
+\Gamma_{j} = q_{j} (\text{sign}_{j} \mu_{j} \vec{E} n_{j} - D_{j} \nabla (n_{j}))
 \end{equation}
 
 Where $P_{\text{Joule Heating}}$ is the power deposited by Joule heating, $q_{j}$ is the charge of the species, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient,
-$V$ is the potential, $n_{j}$ is the density, and $D_{j}$ is the diffusion coefficient.
+$\vec{E}$ is the electric field, $n_{j}$ is the density, and $D_{j}$ is the diffusion coefficient.
 When converting the density to log form and applying a scaling factor of the mesh / voltage,
 `PowerDep` is defined as
 
 \begin{equation}
-P_{\text{Joule Heating}} = \Gamma_{j}  \cdot \frac{\text{-} \nabla (V) V_{c}}{l_{c}} \\
+P_{\text{Joule Heating}} = \Gamma_{j}  \cdot \frac{\vec{E} V_{c}}{l_{c}} \\
 \\[10pt]
-\Gamma_{j} = q_{j} N_{A} \left( \text{sign}_{j} \mu_{j} \frac{\text{-} \nabla (V)}{l_{c}} \exp(N_{j}) - D_{j} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}} \right)
+\Gamma_{j} = q_{j} N_{A} \left( \text{sign}_{j} \mu_{j} \frac{\vec{E}}{l_{c}} \exp(N_{j}) - D_{j} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}} \right)
 \end{equation}
 
 Where $N_{j}$ is the molar density of the specie in log form, $N_{A}$ is Avogadro's
@@ -34,7 +33,7 @@ of the potential.
 For the case where artificial diffusion is introduced to the charge specie flux, an additional term is included, such that:
 
 \begin{equation}
-\Gamma_{j\text{, Total}} = \Gamma_{j} + q_{j} N_{A} \mu_{j} \frac{\text{-}\lVert \nabla (V) \rVert_{2}}{l_{c}} \frac{h_\text{max}}{2} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}}
+\Gamma_{j\text{, Total}} = \Gamma_{j} + q_{j} N_{A} \mu_{j} \frac{\text{-}\lVert \vec{E} \rVert_{2}}{l_{c}} \frac{h_\text{max}}{2} \exp(N_{j}) \frac{\nabla (N_{j})}{l_{c}}
 \end{equation}
 
 Where $h_\text{max}$ is the max length of the current element.

doc/content/source/auxkernels/ProcRate.md

@@ -4,22 +4,21 @@
 
 ## Overview
 
-`ProcRate` returns the production rate for chemistry reactions determined by Townsend coefficients in units of #/m$^{3}$s. `ProcRate`
-assumes the electrostatic approximation for the current.
+`ProcRate` returns the production rate for chemistry reactions determined by Townsend coefficients in units of #/m$^{3}$s.
 
 The production rate is defined as
 
 \begin{equation}
-S_{\text{Townsend}} = \alpha_{j} (\mu_{e} \left( \nabla -V\right) n_{e} - D_{e} \nabla (n_{e}))
+S_{\text{Townsend}} = \alpha_{j} (\mu_{e} \vec{E} n_{e} - D_{e} \nabla (n_{e}))
 \end{equation}
 
 Where $S_{\text{Townsend}}$ is the production rate determined by Townsend coefficients, $\alpha_{j}$ is the Townsend coefficient for the reaction, $\mu_{e}$ is the mobility coefficient,
-$V$ is the potential, $n_{e}$ is the electron density, and $D_{e}$ is the diffusion coefficient.
+$\vec{E}$ is the electric field, $n_{e}$ is the electron density, and $D_{e}$ is the diffusion coefficient.
 When converting the density to logarithmic form and applying a scaling factor of the mesh,
 `ProcRate` is defined as
 
 \begin{equation}
-S_{\text{Townsend}} = \alpha_{j} N_{A} \left(\mu_{e} \frac{-\nabla (V)}{l_{c}} \exp(N_{e}) - D_{e} \exp(N_{e}) \frac{\nabla (N_{e})}{l_{c}} \right)
+S_{\text{Townsend}} = \alpha_{j} N_{A} \left(\mu_{e} \frac{\vec{E}}{l_{c}} \exp(N_{e}) - D_{e} \exp(N_{e}) \frac{\nabla (N_{e})}{l_{c}} \right)
 \end{equation}
 
 Where $N_{e}$ is the molar density of the electrons in logarithmic form, $N_{A}$ is Avogadro's

doc/content/source/auxkernels/Sigma.md

@@ -4,21 +4,21 @@
 
 ## Overview
 
-`Sigma` calculates a simplifed version of the surface charge on a boundary due to the advection flux of an ion species. `Sigma` assumes a mobility coefficient of unity, the electrostatic approximation for the electric field, and a non-scaled version of the specie's density.
+`Sigma` calculates a simplifed version of the surface charge on a boundary due to the advection flux of an ion species. `Sigma` assumes a mobility coefficient of unity, and a non-scaled version of the specie's density.
 
 The surface charge is defined as
 
 \begin{equation}
 \sigma = \int \Gamma_{i} \cdot \textbf{n} \ \text{d}t \\[10pt]
-\Gamma_{i} = \text{-} \nabla (V) n_{i}
+\Gamma_{i} = \vec{E} n_{i}
 \end{equation}
 
-Where $\sigma$ is the surface charge, $\Gamma_{i}$ is the advective flux of the ions, $\textbf{n}$ is the outward pointing unit normal on the boundary, $V$ is the potential, and $n_{i}$ is the ion density.
+Where $\sigma$ is the surface charge, $\Gamma_{i}$ is the advective flux of the ions, $\textbf{n}$ is the outward pointing unit normal on the boundary, $\vec{E}$ is the electric field, and $n_{i}$ is the ion density.
 
 Using the midpoint method for integration, the surface charge calculation becomes
 
 \begin{equation}
-\sigma_{t} = \sigma_{t-1} - \nabla (V) n_{i}  \cdot \textbf{n} \ \text{d}t
+\sigma_{t} = \sigma_{t-1} + \vec{E} n_{i}  \cdot \textbf{n} \ \text{d}t
 \end{equation}
 
 Where $\sigma_{t}$ is the surface charge of the current time step, $\sigma_{t-1}$ is the surface of the previous time step, and $\text{d}t$ is the difference between time steps.

doc/content/source/auxkernels/TotalFlux.md

@@ -4,19 +4,18 @@
 
 ## Overview
 
-`TotalFlux` returns the total flux of a species in logarithmic form. `TotalFlux`
-assumes the electrostatic approximation for the electric field.
+`TotalFlux` returns the total flux of a species in logarithmic form.
 
-The electrostatic flux is usually defined as
+The flux is usually defined as
 
 \begin{equation}
-\Gamma_{j} = \text{sign}_{j} \mu_{j} \left( \text{-} \nabla V \right) n_{j} - D_{j} \nabla (n_{j})
+\Gamma_{j} = \text{sign}_{j} \mu_{j} \vec{E} n_{j} - D_{j} \nabla (n_{j})
 \end{equation}
 
-Where $\Gamma_{j}$ is the flux assuming drift-diffusion formulation, $\mu_{j}$ is the mobility coefficient, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species, $\text{-}1$ for negatively charged species and $\text{0}$ for neutral species), $V$ is the potential, $n_{j}$ is the density, and $D_{j}$ is the diffusion coefficient. When converting the density to logarithmic form, `TotalFlux` is defined as
+Where $\Gamma_{j}$ is the flux assuming drift-diffusion formulation, $\mu_{j}$ is the mobility coefficient, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species, $\text{-}1$ for negatively charged species and $\text{0}$ for neutral species), $\vec{E}$ is the electric field, $n_{j}$ is the density, and $D_{j}$ is the diffusion coefficient. When converting the density to logarithmic form, `TotalFlux` is defined as
 
 \begin{equation}
-\Gamma_{j} = \text{sign}_{j} \mu_{j} \left(\text{-} \nabla V\right) \exp(N_{j}) - D_{j} \exp(N_{j}) \nabla (N_{j})
+\Gamma_{j} = \text{sign}_{j} \mu_{j} \vec{E} \exp(N_{j}) - D_{j} \exp(N_{j}) \nabla (N_{j})
 \end{equation}
 
 Where $N_{j}$ is the molar density of the specie in logarithmic form.

doc/content/source/bcs/DCIonBC.md

@@ -4,30 +4,30 @@
 
 ## Overview
 
-`DCIonBC` is an electric field driven outflow boundary condition. `DCIonBC` assumes the electrostatic approximation for the electric field.
+`DCIonBC` is an electric field driven outflow boundary condition.
 
-The electrostatic electric field driven outflow is defined as
+The electric field driven outflow is defined as
 
 \begin{equation}
 a =
 \begin{cases}
-1, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} > 0\\
-0, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \leq 0\\
+1, & \text{sign}_{j} \mu_{j} \ \vec{E} \cdot \textbf{n} > 0\\
+0, & \text{sign}_{j} \mu_{j} \ \vec{E} \cdot \textbf{n} \leq 0\\
 \end{cases} \\[10pt]
-\Gamma_{j} \cdot \textbf{n} = a \ \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \ n_{j}
+\Gamma_{j} \cdot \textbf{n} = a \ \text{sign}_{j} \mu_{j} \ \vec{E} \cdot \textbf{n} \ n_{j}
 \end{equation}
 
-Where $\Gamma$ is the outflow normal to the boundary, $\textbf{n}$ is the normal vector of the boundary, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $n_{j}$ is the density, and $V$ is
-the electrostatic potential. $a$ is defined such that the outflow is only defined when the drift velocity is direct towards the wall and zero otherwise. When converting the density to logarithmic form and applying a scaling
+Where $\Gamma$ is the outflow normal to the boundary, $\textbf{n}$ is the normal vector of the boundary, $\text{sign}_{j}$ indicates the advection behavior ($\text{+}1$ for positively charged species and $\text{-}1$ for negatively charged species), $\mu_{j}$ is the mobility coefficient, $n_{j}$ is the density, and $\vec{E}$ is
+the electric field. $a$ is defined such that the outflow is only defined when the drift velocity is direct towards the wall and zero otherwise. When converting the density to logarithmic form and applying a scaling
 factor of the mesh, the strong form for `DCIonBC` is defined as
 
 \begin{equation}
 a =
 \begin{cases}
-1, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} > 0\\
-0, & \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V) \cdot \textbf{n} \leq 0\\
+1, & \text{sign}_{j} \mu_{j} \ \vec{E} \cdot \textbf{n} > 0\\
+0, & \text{sign}_{j} \mu_{j} \ \vec{E} \cdot \textbf{n} \leq 0\\
 \end{cases} \\[10pt]
-\Gamma_{j} \cdot \textbf{n} = a \ \text{sign}_{j} \mu_{j} \ \text{-} \nabla (V / l_{c}) \cdot \textbf{n} \ \exp(N_{j})
+\Gamma_{j} \cdot \textbf{n} = a \ \text{sign}_{j} \mu_{j} \ \left(\vec{E} / l_{c} \right) \cdot \textbf{n} \ \exp(N_{j})
 \end{equation}
 
 Where $N_{j}$ is the molar density of the species in logarithmic form and

doc/content/source/bcs/DielectricBCWithEffEfield.md

@@ -0,0 +1,70 @@
+# DielectricBCWithEffEfield
+
+!syntax description /BCs/DielectricBCWithEffEfield
+
+## Overview
+
+`DielectricBCWithEffEfield` is a type of [`NeumannBC`](/bcs/NeumannBC.md) for the potential on the boundary of a grounded ideal dielectric. The advection term for the ion is determined by an effective electric field.
+
+The potential at the boundary of a grounded ideal dielectric is defined as
+
+\begin{equation}
+\epsilon_{0}\frac{\partial (E \cdot \textbf{n}) }{\partial t} - \frac{\epsilon_{i}}{d_{i}}\frac{\partial V_{i}}{\partial t} = - e \left( \Gamma_{+} \cdot \textbf{n} -\Gamma_{e} \cdot \textbf{n} \right) \\[10pt]
+\Gamma_{e} \cdot \textbf{n}  = \frac{1}{4}\sqrt{\frac{8 k T_{e}}{\pi m_{e}}} \ n_e - \gamma \Gamma_{+} \cdot \textbf{n} \\[10pt]
+\Gamma_{+} \cdot \textbf{n}  = a \ \mu_{+} \ \vec{E}_{\text{Eff.}} \cdot \textbf{n} \ n_{+} \\[10pt]
+a =
+\begin{cases}
+1, & \mu_{+} \ \vec{E}_{\text{Eff.}} \cdot \textbf{n} > 0\\
+0, & \mu_{+} \ \vec{E}_{\text{Eff.}} \cdot \textbf{n} \leq 0\\
+\end{cases}
+\end{equation}
+
+where 
+
+- $\epsilon_{i}$ is the permittivity of the dielectric,
+- $d_{i}$ is the thickness of the dielectric,
+- $V_{i}$ is the voltage on the dielectric,
+- $\textbf{n}$ is the normal to the boundary, 
+- $e$ is the elemental charge, 
+- $\epsilon_{0}$ is the permittivity of free space,
+- $E$ is the electric field normal to the dielectric, 
+- $\Gamma_{e}$ is the electron outflow flux, and 
+- $\Gamma_{+}$ are the ion outflow flux.
+
+!alert note title=Flux Information
+$\Gamma_{e}$ and $\Gamma_{+}$ are defined with the [`SakiyamaElectronDiffusionBC`](/bcs/SakiyamaElectronDiffusionBC.md), [`SakiyamaSecondaryElectronWithEffEfieldBC`](/bcs/SakiyamaSecondaryElectronWithEffEfieldBC.md) and [`SakiyamaIonAdvectionWithEffEfieldBC`](/bcs/SakiyamaIonAdvectionWithEffEfieldBC.md) (please refer to those BC's for more information on the fluxes).
+
+To convert the above equation into the form at a NeumannBC, the time integral is taken such that:
+
+\begin{equation}
+\int{ \epsilon_{0}\frac{\partial (E \cdot \textbf{n}) }{\partial t} } dt - \int{ \frac{\epsilon_{i}}{d_{i}}\frac{\partial V_{i}}{\partial t} } dt = \int{ - e \left( \Gamma_{+} \cdot \textbf{n} -\Gamma_{e} \cdot \textbf{n} \right) } dt
+\end{equation}
+
+Using the trapezoidal rule for the definite integral and rearranging the equation such that the electric field term is on one side results in:
+
+\begin{equation}
+\epsilon_{0} \left( E \cdot \textbf{n} \right) = \epsilon_{0} \left( E_{\text{old}} \cdot \textbf{n} \right) + \frac{\epsilon_{i}}{d_{i}} \left( V - V_
+{old} \right) - 0.5 \left( e \left( \Gamma_{+} \cdot \textbf{n} - \Gamma_{e} \cdot \textbf{n} \right) + e \left( \Gamma_{+} \cdot \textbf{n} - \Gamma_{e} \cdot \textbf{n} \right)_{old} \right) dt
+\end{equation}
+
+where
+
+- The subscript $\text{old}$ is the value of the variable during the previous time step, and
+- $dt$ is the current time step size.
+
+Lastly, the electrostatics approximation is applied to the electric field normal to the dielectric, which results in a NeumannBC for the potential defined as:
+
+\begin{equation}
+\epsilon_{0} \left( -\nabla V \cdot \textbf{n} \right) = \epsilon_{0} \left( -\nabla V_{\text{old}} \cdot \textbf{n} \right) + \frac{\epsilon_{i}}{d_{i}} \left( V - V_
+{old} \right) - 0.5 \left( e \left( \Gamma_{+} \cdot \textbf{n} - \Gamma_{e} \cdot \textbf{n} \right) + e \left( \Gamma_{+} \cdot \textbf{n} - \Gamma_{e} \cdot \textbf{n} \right)_{old} \right) dt
+\end{equation}
+
+## Example Input File Syntax
+
+!listing test/tests/mms/bcs/2D_DielectricBCWithEffEfield.i block=BCs/potential_left_BC
+
+!syntax parameters /BCs/DielectricBCWithEffEfield
+
+!syntax inputs /BCs/DielectricBCWithEffEfield
+
+!syntax children /BCs/DielectricBCWithEffEfield