Give an expression for the force $\mathbf{F}$ on a charge $q$ moving at velocity $\mathbf{v}$ in electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$. Consider a stationary electric circuit $C$, and let $S$ be a stationary surface bounded by $C$. Derive from Maxwell's equations the result

$$\mathcal{E}=-\frac{d \Phi}{d t}$$

where the electromotive force $\mathcal{E}=\oint_{C} q^{-1} \mathbf{F} \cdot d \mathbf{r}$ and the flux $\Phi=\int_{S} \mathbf{B} \cdot d \mathbf{S}$.

Now assume that $(*)$ also holds for a moving circuit. A circular loop of wire of radius $a$ and total resistance $R$, whose normal is in the $z$-direction, moves at constant speed $v$ in the $x$-direction in the presence of a magnetic field $\mathbf{B}=\left(0,0, B_{0} x / d\right)$. Find the current in the wire.