For a static charge density $\rho(\mathbf{x})$ show that the energy may be expressed as

$$E=\frac{1}{2} \int \rho \phi \mathrm{d}^{3} x=\frac{\epsilon_{0}}{2} \int \mathbf{E}^{2} \mathrm{~d}^{3} x,$$

where $\phi(\mathbf{x})$ is the electrostatic potential and $\mathbf{E}(\mathbf{x})$ is the electric field.

Determine the scalar potential and electric field for a sphere of radius $R$ with a constant charge density $\rho$. Also determine the total electrostatic energy.

In a nucleus with $Z$ protons the volume is proportional to $Z$. Show that we may expect the electric contribution to energy to be proportional to $Z^{\frac{5}{3}}$.