(i) Develop the theory of electromagnetic waves starting from Maxwell equations in vacuum. You should relate the wave-speed $c$ to $\epsilon_{0}$ and $\mu_{0}$ and establish the existence of plane, plane-polarized waves in which $\mathbf{E}$ takes the form

$$\mathbf{E}=\left(E_{0} \cos (k z-\omega t), 0,0\right) .$$

You should give the form of the magnetic field $\mathbf{B}$ in this case.

(ii) Starting from Maxwell's equation, establish Poynting's theorem.

$$-\mathbf{j} \cdot \mathbf{E}=\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S},$$

where $W=\frac{\epsilon_{0}}{2} \mathbf{E}^{2}+\frac{1}{2 \mu_{0}} \mathbf{B}^{2}$ and $\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \wedge \mathbf{B}$. Give physical interpretations of $W, S$ and the theorem.

Compute the averages over space and time of $W$ and $\mathbf{S}$ for the plane wave described in (i) and relate them. Comment on the result.