Write down Maxwell's equations for the electric field $\mathbf{E}(\mathbf{x}, t)$ and the magnetic field $\mathbf{B}(\mathbf{x}, t)$ in a vacuum. Deduce that both $\mathbf{E}$ and $\mathbf{B}$ satisfy a wave equation, and relate the wave speed $c$ to the physical constants $\epsilon_{0}$ and $\mu_{0}$.

Verify that there exist plane-wave solutions of the form

$$\begin{aligned} &\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \\ &\mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right] \end{aligned}$$

where $\mathbf{e}$ and $\mathbf{b}$ are constant complex vectors, $\mathbf{k}$ is a constant real vector and $\omega$ is a real constant. Derive the dispersion relation that relates the angular frequency $\omega$ of the wave to the wavevector $\mathbf{k}$, and give the algebraic relations between the vectors $\mathbf{e}, \mathbf{b}$ and $\mathbf{k}$ implied by Maxwell's equations.

Let $\mathbf{n}$ be a constant real unit vector. Suppose that a perfect conductor occupies the region $\mathbf{n} \cdot \mathbf{x}<0$ with a plane boundary $\mathbf{n} \cdot \mathbf{x}=0$. In the vacuum region $\mathbf{n} \cdot \mathbf{x}>0$, a plane electromagnetic wave of the above form, with $\mathbf{k} \cdot \mathbf{n}<0$, is incident on the plane boundary. Write down the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface of the conductor. Show that Maxwell's equations and the boundary conditions are satisfied if the solution in the vacuum region is the sum of the incident wave given above and a reflected wave of the form

$$\begin{aligned} &\mathbf{E}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{e}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \\ &\mathbf{B}^{\prime}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{b}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{x}-\omega t\right)}\right] \end{aligned}$$

where

$$\begin{aligned} &\mathbf{e}^{\prime}=-\mathbf{e}+2(\mathbf{n} \cdot \mathbf{e}) \mathbf{n} \\ &\mathbf{b}^{\prime}=\mathbf{b}-2(\mathbf{n} \cdot \mathbf{b}) \mathbf{n} \\ &\mathbf{k}^{\prime}=\mathbf{k}-2(\mathbf{n} \cdot \mathbf{k}) \mathbf{n} \end{aligned}$$