Three sides of a closed rectangular circuit $C$ are fixed and one is moving. The circuit lies in the plane $z=0$ and the sides are $x=0, y=0, x=a(t), y=b$, where $a(t)$ is a given function of time. A magnetic field $\mathbf{B}=\left(0,0, \frac{\partial f}{\partial x}\right)$ is applied, where $f(x, t)$ is a given function of $x$ and $t$ only. Find the magnetic flux $\Phi$ of $\mathbf{B}$ through the surface $S$ bounded by $C$.

Find an electric field $\mathbf{E}_{\mathbf{0}}$ that satisfies the Maxwell equation

$$\boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

and then write down the most general solution $\mathbf{E}$ in terms of $\mathbf{E}_{0}$ and an undetermined scalar function independent of $f$.

Verify that

$$\oint_{C}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t},$$

where $\mathbf{v}$ is the velocity of the relevant side of $C$. Interpret the left hand side of this equation.

If a unit current flows round $C$, what is the rate of work required to maintain the motion of the moving side of the rectangle? You should ignore any electromagnetic fields produced by the current.