Pre-Quantum Electrodynamics

For a conductor, we can exploit the fact that electrical fields vanish on the inside to get a proper boundary condition for the potential. Namely, here, boundary condition \ref{Gr(2.33)} yields \[ {\bf E} = \frac{\sigma}{\varepsilon_0} \hat{\bf n} \label{Gr(2.48)} \] which in terms of potential reads \[ \sigma = -\varepsilon_0 \frac{\partial V}{\partial n}. \label{Gr(2.49)} \] The force per unit area on the surface of an object is \[ {\bf f} = \sigma {\bf E}_{\mbox{average}} = \frac{\sigma}{2} ({\bf E}_{\mbox{above}} + {\bf E}_{\mbox{below}}) \label{Gr(2.50)} \] For a conductor, \({\bf E}_{\mbox{below}} = 0\), \({\bf E}_{\mbox{above}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n}\), so \[ {\bf f} = \frac{\sigma^2}{2\varepsilon_0} \hat{\bf n} \label{Gr(2.51)} \] amounting to an outward electrostatic pressure. In terms of the field, \[ P = \frac{\varepsilon_0}{2} E^2 \label{Gr(2.52)} \] which can also be obtained from the principle of virtual work.