(i) Show that, in a region where there is no magnetic field and the charge density vanishes, the electric field can be expressed either as minus the gradient of a scalar potential $\phi$ or as the curl of a vector potential A. Verify that the electric field derived from

$$\mathbf{A}=\frac{1}{4 \pi \epsilon_{0}} \frac{\mathbf{p} \wedge \mathbf{r}}{r^{3}}$$

is that of an electrostatic dipole with dipole moment $\mathbf{p}$.

[You may assume the following identities:

$$\begin{gathered} \nabla(\mathbf{a} \cdot \mathbf{b})=\mathbf{a} \wedge(\nabla \wedge \mathbf{b})+\mathbf{b} \wedge(\nabla \wedge \mathbf{a})+(\mathbf{a} \cdot \nabla) \mathbf{b}+(\mathbf{b} \cdot \nabla) \mathbf{a} \\ \nabla \wedge(\mathbf{a} \wedge \mathbf{b})=(\mathbf{b} \cdot \nabla) \mathbf{a}-(\mathbf{a} \cdot \nabla) \mathbf{b}+\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a} .] \end{gathered}$$

(ii) An infinite conducting cylinder of radius $a$ is held at zero potential in the presence of a line charge parallel to the axis of the cylinder at distance $s_{0}>a$, with charge density $q$ per unit length. Show that the electric field outside the cylinder is equivalent to that produced by replacing the cylinder with suitably chosen image charges.