##### Electrostatic Boundary Conditions
ems.es.ep.bc

For a surface, Gauss's law gives us that \[ \oint_{\cal S} {\bf E} \cdot d{\bf a} = \frac{Q_{\mbox{enc}}}{\varepsilon_0} = \frac{1}{\varepsilon_0} \sigma A \] where \(A\) is the area of the Gaussian pillbox and \(\sigma\) the surface charge density, which is here assumed to be constant. The sides contribute nothing if the pillbox is made infinitesimally thin, in which case we get

\[ {\bf E}^{\perp}_{\mbox{above}} - {\bf E}^{\perp}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0}, \label{Gr(2.31)} \]

so the normal component of \({\bf E}\) is discontinuous at the boundary by an amount \(\sigma/\varepsilon_0\).

The tangential component is continuous: from the curlless condition curlE0 applied to a small loop straddling the surface:

\[ {\bf E}^{\parallel}_{\mbox{above}} = {\bf E}^{\parallel}_{\mbox{below}} \label{Gr(2.32)} \]

We can unify both boundary conditions into a single equation:

- Gr (2.33)

\[ {\bf E}_{\mbox{above}} - {\bf E}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n} \tag{Edisc}\label{Edisc} \] with \(\hat{\bf n}\) a unit vector normal to the surface, pointing 'out'.

The potential is however continuous across any boundary: since \[ \phi_{\mbox{above}} - \phi_{\mbox{below}} = -\int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l} \] where the path shrinks to zero, we conclude that \[ \phi_{\mbox{above}} = \phi_{\mbox{below}} \label{Gr(2.34)} \] The gradient however inherits the discontinuity of the electrostatic field, since \({\bf E} = -{\boldsymbol \nabla} \phi\): \[ {\boldsymbol \nabla} \phi_{\mbox{above}} - {\boldsymbol \nabla} \phi_{\mbox{below}} = -\frac{\sigma}{\varepsilon_0} \hat{\bf n} \label{Gr(2.35)} \] or

- Gr (2.36)

\[ \frac{\partial \phi_{\mbox{above}}}{\partial n} - \frac{\partial \phi_{\mbox{below}}}{\partial n} = -\frac{\sigma}{\varepsilon_0} \tag{dpdisc}\label{dpdisc} \] where \[ \frac{\partial \phi}{\partial n} = {\boldsymbol \nabla} \phi \cdot \hat{\bf n} \label{Gr(2.37)} \] is the normal derivative of the potential.

This is the kind of boundary condition that we need to fix a unique solution to Poisson's equation 🐟: our only problem is that dpdisc gives the change of the normal derivative of \(V\), not its value. However, if we assume that there are no charges living outside of our volume \({\cal V}\), we find that dpdisc fully specifies the potential's normal derivative if the surface charge is known.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42