The energy stored in a static electric field $\mathbf{E}$ is

$$U=\frac{1}{2} \int \rho \phi d V,$$

where $\phi$ is the associated electric potential, $\mathbf{E}=-\nabla \phi$, and $\rho$ is the volume charge density.

(i) Assuming that the energy is calculated over all space and that $\mathbf{E}$ vanishes at infinity, show that the energy can be written as

$$U=\frac{\epsilon_{0}}{2} \int|\mathbf{E}|^{2} d V$$

(ii) Find the electric field produced by a spherical shell with total charge $Q$ and radius $R$, assuming it to vanish inside the shell. Find the energy stored in the electric field.