Pre-Quantum Electrodynamics

For a surface, Gauss's law states \[ \oint_{\cal S} {\bf E} \cdot d{\bf a} = \frac{Q_{\mbox{enc}}}{\varepsilon_0} = \frac{1}{\varepsilon_0} \sigma A \] where \(A\) is the area of the Gaussian pillbox and \(\sigma\) the surface charge density. The sides contribute nothing if the pillbox is thin. Taking its area very small, 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 \ref{Gr(2.19)} applied to a small loop straddling the surface, \[ {\bf E}^{\parallel}_{\mbox{above}} = {\bf E}^{\parallel}_{\mbox{below}} \label{Gr(2.32)} \]

Put together: \[ {\bf E}_{\mbox{above}} - {\bf E}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n} \label{Gr(2.33)} \] with \(\hat{\bf n}\) a unit vector normal to the surface, pointing 'out'.

The potential is continuous across any boundary: since \[ V_{\mbox{above}} - V_{\mbox{below}} = -\int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l} \] where the path shrinks to zero, \[ V_{\mbox{above}} = V_{\mbox{below}} \label{Gr(2.34)} \] The gradient however inherits the discontinuity of the electrostatic field, since \({\bf E} = -{\boldsymbol \nabla} V\): \[ {\boldsymbol \nabla} V_{\mbox{above}} - {\boldsymbol \nabla} V_{\mbox{below}} = -\frac{\sigma}{\varepsilon_0} \hat{\bf n} \label{Gr(2.35)} \] or \[ \frac{\partial V_{\mbox{above}}}{\partial n} - \frac{\partial V_{\mbox{below}}}{\partial n} = -\frac{\sigma}{\varepsilon_0} \label{Gr(2.36)} \] where \[ \frac{\partial V}{\partial n} = {\boldsymbol \nabla} V \cdot \hat{\bf n} \label{Gr(2.37)} \] 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 \ref{Gr(2.36)} gives the change of the normal derivative of \(V\), not its value. However, if we assume (as in our first case corollary) that there are no charges living outside of our volume \({\cal V}\), we find that \ref{Gr(2.36)} fully specifies the potential's normal derivative if the surface charge is known.