(a) Starting from Maxwell's equations, show that in a vacuum,

$$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=\mathbf{0} \quad \text { and } \quad \boldsymbol{\nabla} \cdot \mathbf{E}=0 \quad \text { where } \quad c=\sqrt{\frac{1}{\epsilon_{0} \mu_{0}}} .$$

(b) Suppose that $\mathbf{E}=\frac{E_{0}}{\sqrt{2}}(1,1,0) \cos (k z-\omega t)$ where $E_{0}, k$ and $\omega$ are real constants.

(i) What are the wavevector and the polarisation? How is $\omega$ related to $k$ ?

(ii) Find the magnetic field $\mathbf{B}$.

(iii) Compute and interpret the time-averaged value of the Poynting vector, $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$.