(i) Show that the work done in assembling a localised charge distribution $\rho(\mathbf{r})$ in a region $V$ with an associated potential $\phi(\mathbf{r})$ is

$$W=\frac{1}{2} \int_{V} \rho(\mathbf{r}) \phi(\mathbf{r}) d \tau$$

and that this can be written as an integral over all space

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

where the electric field $\mathbf{E}=-\nabla \phi$.

(ii) What is the force per unit area on an infinite plane conducting sheet with a charge density $\sigma$ per unit area (a) if it is isolated in space and (b) if the electric field vanishes on one side of the sheet?

An infinite cylindrical capacitor consists of two concentric cylindrical conductors with radii $a, b(a<b)$, carrying charges $\pm q$ per unit length respectively. Calculate the capacitance per unit length and the energy per unit length. Next determine the total force on each conductor, and calculate the rate of change of energy of the inner and outer conductors if they are moved radially inwards and outwards respectively with speed $v$. What is the corresponding rate of change of the capacitance?