(i) Using Maxwell's equations as they apply to magnetostatics, show that the magnetic field $\mathbf{B}$ can be described in terms of a vector potential $\mathbf{A}$ on which the condition $\nabla \cdot \mathbf{A}=0$ may be imposed. Hence derive an expression, valid at any point in space, for the vector potential due to a steady current distribution of density $\mathbf{j}$ that is non-zero only within a finite domain.

(ii) Verify that the vector potential $\mathbf{A}$ that you found in Part (i) satisfies $\nabla \cdot \mathbf{A}=0$, and use it to obtain the Biot-Savart law expression for $\mathbf{B}$. What is the corresponding result for a steady surface current distribution of density $\mathbf{s}$ ?

In cylindrical polar coordinates $(\rho, \phi, z)$ (oriented so that $\mathbf{e}_{\rho} \times \mathbf{e}_{\phi}=\mathbf{e}_{z}$ ) a surface current

$$\mathbf{s}=s(\rho) \mathbf{e}_{\phi}$$

flows in the plane $z=0$. Given that

$$s(\rho)= \begin{cases}4 I\left(1+\frac{a^{2}}{\rho^{2}}\right)^{\frac{1}{2}} & a \leqslant \rho \leqslant 3 a \\ 0 & \text { otherwise }\end{cases}$$

show that the magnetic field at the point $\mathbf{r}=a \mathbf{e}_{z}$ has $z$-component

$$B_{z}=\mu_{0} I \log 5 .$$

State, with justification, the full result for $\mathbf{B}$ at the point $\mathbf{r}=a \mathbf{e}_{z}$.