Write down the form of Ohm's Law that applies to a conductor if at a point $\mathbf{r}$ it is moving with velocity $\mathbf{v}(\mathbf{r})$.

Use two of Maxwell's equations to prove that

$$\int_{C}(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot d \mathbf{r}=-\frac{d}{d t} \int_{S} \mathbf{B} \cdot d \mathbf{S}$$

where $C(t)$ is a moving closed loop, $\mathbf{v}$ is the velocity at the point $\mathbf{r}$ on $C$, and $S$ is a surface spanning $C$. The time derivative on the right hand side accounts for changes in both $C$ and B. Explain briefly the physical importance of this result.

Find and sketch the magnetic field $\mathbf{B}$ described in the vector potential

$$\mathbf{A}(r, \theta, z)=\left(0, \frac{1}{2} b r z, 0\right)$$

in cylindrical polar coordinates $(r, \theta, z)$, where $b>0$ is constant.

A conducting circular loop of radius $a$ and resistance $R$ lies in the plane $z=h(t)$ with its centre on the $z$-axis.

Find the magnitude and direction of the current induced in the loop as $h(t)$ changes with time, neglecting self-inductance.

At time $t=0$ the loop is at rest at $z=0$. For time $t>0$ the loop moves with constant velocity $d h / d t=v>0$. Ignoring the inertia of the loop, use energy considerations to find the force $F(t)$ necessary to maintain this motion.

[ In cylindrical polar coordinates

$$\left.\operatorname{curl} \mathbf{A}=\left(\frac{1}{r} \frac{\partial A_{z}}{\partial \theta}-\frac{\partial A_{\theta}}{\partial z}, \frac{\partial A_{r}}{\partial z}-\frac{\partial A_{z}}{\partial r}, \frac{1}{r} \frac{\partial}{\partial r}\left(r A_{\theta}\right)-\frac{1}{r} \frac{\partial A_{r}}{\partial \theta}\right)\right]$$

Part II