(i) A plane electromagnetic wave has electric and magnetic fields

$$\mathbf{E}=\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}, \quad \mathbf{B}=\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}$$

for constant vectors $\mathbf{E}_{0}, \mathbf{B}_{0}$, constant positive angular frequency $\omega$ and constant wavevector $\mathbf{k}$. Write down the vacuum Maxwell equations and show that they imply

$$\mathbf{k} \cdot \mathbf{E}_{0}=0, \quad \mathbf{k} \cdot \mathbf{B}_{0}=0, \quad \omega \mathbf{B}_{0}=\mathbf{k} \times \mathbf{E}_{0}$$

Show also that $|\mathbf{k}|=\omega / c$, where $c$ is the speed of light.

(ii) State the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface $S$ of a perfect conductor. Let $\sigma$ be the surface charge density and s the surface current density on $S$. How are $\sigma$ and $\mathbf{s}$ related to $\mathbf{E}$ and $\mathbf{B}$ ?

A plane electromagnetic wave is incident from the half-space $x<0$ upon the surface $x=0$ of a perfectly conducting medium in $x>0$. Given that the electric and magnetic fields of the incident wave take the form $(*)$ with

$$\mathbf{k}=k(\cos \theta, \sin \theta, 0) \quad(0<\theta<\pi / 2)$$

and

$$\mathbf{E}_{0}=\lambda(-\sin \theta, \cos \theta, 0),$$

find $\mathbf{B}_{0}$.

Reflection of the incident wave at $x=0$ produces a reflected wave with electric field

$$\mathbf{E}_{0}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{r}-\omega t\right)}$$

with

$$\mathbf{k}^{\prime}=k(-\cos \theta, \sin \theta, 0)$$

By considering the boundary conditions at $x=0$ on the total electric field, show that

$$\mathbf{E}_{0}^{\prime}=-\lambda(\sin \theta, \cos \theta, 0)$$

Show further that the electric charge density on the surface $x=0$ takes the form

$$\sigma=\sigma_{0} e^{i k(y \sin \theta-c t)}$$

for a constant $\sigma_{0}$ that you should determine. Find the magnetic field of the reflected wave and hence the surface current density $\mathbf{s}$ on the surface $x=0$.