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 Edisc yields

\[ {\bf E} = \frac{\sigma}{\varepsilon_0} \hat{\bf n} \tag{E_sur_cond}\label{E_sur_cond} \]

which in terms of potential reads

\[ \sigma = -\varepsilon_0 \frac{\partial \phi}{\partial n}. \tag{scd_cond}\label{scd_cond} \]

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 (considering the change rate of energy with respect to volume).