(i) Write down the general solution of Poisson's equation. Derive from Maxwell's equations the Biot-Savart law for the magnetic field of a steady localised current distribution.

(ii) A plane rectangular loop with sides of length $a$ and $b$ lies in the plane $z=0$ and is centred on the origin. Show that when $r=|\mathbf{r}| \gg a, b$, the vector potential $\mathbf{A}(\mathbf{r})$ is given approximately by

$$\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \frac{\mathbf{m} \wedge \mathbf{r}}{r^{3}}$$

where $\mathbf{m}=I a b \hat{\mathbf{z}}$ is the magnetic moment of the loop.

Hence show that the magnetic field $\mathbf{B}(\mathbf{r})$ at a great distance from an arbitrary small plane loop of area $A$, situated in the $x y$-plane near the origin and carrying a current $I$, is given by

$$\mathbf{B}(\mathbf{r})=\frac{\mu_{0} I A}{4 \pi r^{5}}\left(3 x z, 3 y z, 2 r^{2}-3 x^{2}-3 y^{2}\right)$$