(i) Write down the expression for the electrostatic potential $\phi(\mathbf{r})$ due to a distribution of charge $\rho(\mathbf{r})$ contained in a volume $V$. Perform the multipole expansion of $\phi(\mathbf{r})$ taken only as far as the dipole term.

(ii) If the volume $V$ is the sphere $|\mathbf{r}| \leqslant a$ and the charge distribution is given by

$$\rho(\mathbf{r})= \begin{cases}r^{2} \cos \theta & r \leqslant a \\ 0 & r>a\end{cases}$$

where $r, \theta$ are spherical polar coordinates, calculate the charge and dipole moment. Hence deduce $\phi$ as far as the dipole term.

Obtain an exact solution for $r>a$ by solving the boundary value problem using trial solutions of the forms

$$\phi=\frac{A \cos \theta}{r^{2}} \text { for } r>a,$$

and

$$\phi=B r \cos \theta+C r^{4} \cos \theta \text { for } r<a .$$

Show that the solution obtained from the multipole expansion is in fact exact for $r>a$.

[You may use without proof the result

$$\left.\nabla^{2}\left(r^{k} \cos \theta\right)=(k+2)(k-1) r^{k-2} \cos \theta, \quad k \in \mathbb{N} .\right]$$