Write down Maxwell's equations in vacuo and show that they admit plane wave solutions in which

$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left(\mathbf{E}_{0} e^{i(\omega t-\mathbf{k} \cdot \mathbf{x})}\right), \quad \mathbf{k} \cdot \mathbf{E}_{0}=0,$$

where $\mathbf{E}_{0}$ and $\mathbf{k}$ are constant vectors. Find the corresponding magnetic field $\mathbf{B}(\mathbf{x}, t)$ and the relationship between $\omega$ and $\mathbf{k}$.

Write down the relations giving the discontinuities (if any) in the normal and tangential components of $\mathbf{E}$ and $\mathbf{B}$ across a surface $z=0$ which carries surface charge density $\sigma$ and surface current density $\mathbf{j}$.

Suppose that a perfect conductor occupies the region $z<0$, and that a plane wave with $\mathbf{k}=(0,0,-k), \mathbf{E}_{0}=\left(E_{0}, 0,0\right)$ is incident from the vacuum region $z>0$. Show that the boundary conditions at $z=0$ can be satisfied if a suitable reflected wave is present, and find the induced surface charge and surface current densities.