Let $\mathbf{E}(\mathbf{r})$ be the electric field due to a continuous static charge distribution $\rho(\mathbf{r})$ for which $|\mathbf{E}| \rightarrow 0$ as $|\mathbf{r}| \rightarrow \infty$. Starting from consideration of a finite system of point charges, deduce that the electrostatic energy of the charge distribution $\rho$ is

$$W=\frac{1}{2} \varepsilon_{0} \int|\mathbf{E}|^{2} d \tau$$

where the volume integral is taken over all space.

A sheet of perfectly conducting material in the form of a surface $S$, with unit normal $\mathbf{n}$, carries a surface charge density $\sigma$. Let $E_{\pm}=\mathbf{n} \cdot \mathbf{E}_{\pm}$denote the normal components of the electric field $\mathbf{E}$ on either side of $S$. Show that

$$\frac{1}{\varepsilon_{0}} \sigma=E_{+}-E_{-} .$$

Three concentric spherical shells of perfectly conducting material have radii $a, b, c$ with $a<b<c$. The innermost and outermost shells are held at zero electric potential. The other shell is held at potential $V$. Find the potentials $\phi_{1}(r)$ in $a<r<b$ and $\phi_{2}(r)$ in $b<r<c$. Compute the surface charge density $\sigma$ on the shell of radius $b$. Use the formula $(*)$ to compute the electrostatic energy of the system.