Cavity in a conductor

A cavity in a conductor is a void surrounded by an electrical conductor; completely enclosed is the essential qualification in exact electrostatic shielding arguments. At electrostatic equilibrium, the surrounding connected metal is an equipotential. An empty closed cavity in a conductor has constant electric potential and zero electric field, even if the metal is isolated and charged. An occupied closed cavity in a conductor generally has a nonzero electric field; its inner-wall total electric charge cancels the total electric charge of the internal sources. These are distinct statements, illustrated by Faraday's ice pail experiment.^[ref]

A spherical cavity of radius $a$ filled with fixed uniform charge density $\rho$ and surrounded by a grounded conductor has electric field $\rho r\hat{\mathbf{r}}/(3{ϵ}_0)$ inside and zero electric field outside the conductor when no external sources are present. Gauss's law supplies the interior electric field; integration from $V(a)=0$ gives the displayed electric potential. Its stored electrostatic energy is $2\pi {\rho }^2{a}^5/(45{ϵ}_0)$, obtained by integrating ${ϵ}_0|\mathbf{E}{|}^2/2$ over the cavity. Grounding permits the conductor's net electric charge to change.