= Electrostatic energy of a uniformly charged solid sphere
{title2=$U=\frac{3Q^2}{20\pi\epsilon_0R}$}
A solid sphere of radius $R$ and uniform <charge density> has total <electric charge> $Q$. <Gauss law> gives the radial <electric field> $E_r=Qr/(4\pi\epsilon_0R^3)$ inside and $E_r=Q/(4\pi\epsilon_0r^2)$ outside. The <electric potential> vanishing at infinity is $Q(3-r^2/R^2)/(8\pi\epsilon_0R)$ inside and $Q/(4\pi\epsilon_0r)$ outside. Integrating the <electrostatic energy> density $\epsilon_0E^2/2$ over both regions gives $U=3Q^2/(20\pi\epsilon_0R)$. In a fixed-density family with $Q\propto Z$ and $R^3\propto Z$, the <electrostatic energy> scales as $Z^{5/3}$.
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