(i) State Maxwell's equations and show that the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ can be expressed in terms of a scalar potential $\phi$ and a vector potential $\mathbf{A}$. Hence derive the inhomogeneous wave equations that are satisfied by $\phi$ and $\mathbf{A}$ respectively.

(ii) The plane $x=0$ separates a vacuum in the half-space $x<0$ from a perfectly conducting medium occupying the half-space $x>0$. Derive the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at $x=0$.

A plane electromagnetic wave with a magnetic field $\mathbf{B}=B(t, x, z) \hat{\mathbf{y}}$, travelling in the $x z$-plane at an angle $\theta$ to the $x$-direction, is incident on the interface at $x=0$. If the wave has frequency $\omega$ show that the total magnetic field is given by

$$\mathbf{B}=B_{0} \cos \left(\frac{\omega x}{c} \cos \theta\right) \exp \left[i\left(\frac{\omega z}{c} \sin \theta-\omega t\right)\right] \hat{\mathbf{y}}$$

where $B_{0}$ is a constant. Hence find the corresponding electric field $\mathbf{E}$, and obtain the surface charge density and the surface current at the interface.