= Uniform space charge in a grounded spherical cavity
{title2=$V(r)=\rho(a^2-r^2)/(6\epsilon_0)$}
A spherical cavity of radius $a$ filled with fixed uniform <charge density> $\rho$ and surrounded by a grounded <conductor> has <electric field> $\rho r\hat{\mathbf r}/(3\epsilon_0)$ 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 $2\pi\rho^2a^5/(45\epsilon_0)$, obtained by integrating $\epsilon_0|\mathbf E|^2/2$ over the cavity. Grounding permits the conductor's net <electric charge> to change.
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