What is the relationship between the electric field $\mathbf{E}$ and the charge per unit area $\sigma$ on the surface of a perfect conductor?

Consider a charge distribution $\rho(\mathbf{r})$ distributed with potential $\phi(\mathbf{r})$ over a finite volume $V$ within which there is a set of perfect conductors with charges $Q_{i}$, each at a potential $\phi_{i}$ (normalised such that the potential at infinity is zero). Using Maxwell's equations and the divergence theorem, derive a relationship between the electrostatic energy $W$ and a volume integral of an explicit function of the electric field $\mathbf{E}$, where

$$W=\frac{1}{2} \int_{V} \rho \phi d \tau+\frac{1}{2} \sum_{i} Q_{i} \phi_{i}$$

Consider $N$ concentric perfectly conducting spherical shells. Shell $n$ has radius $r_{n}$ (where $r_{n}>r_{n-1}$ ) and charge $q$ for $n=1$, and charge $2(-1)^{(n+1)} q$ for $n>1$. Show that

$$W \propto \frac{1}{r_{1}},$$

and determine the constant of proportionality.