If $S$ is a fixed surface enclosing a volume $V$, use Maxwell's equations to show that

$$\frac{d}{d t} \int_{V}\left(\frac{1}{2} \epsilon_{0} E^{2}+\frac{1}{2 \mu_{0}} B^{2}\right) d V+\int_{S} \mathbf{P} \cdot \mathbf{n} d S=-\int_{V} \mathbf{j} \cdot \mathbf{E} d V$$

where $\mathbf{P}=(\mathbf{E} \times \mathbf{B}) / \mu_{0}$. Give a physical interpretation of each term in this equation.

Show that Maxwell's equations for a vacuum permit plane wave solutions with $\mathbf{E}=E_{0}(0,1,0) \cos (k x-\omega t)$ with $E_{0}, k$ and $\omega$ constants, and determine the relationship between $k$ and $\omega$.

Find also the corresponding $\mathbf{B}(\mathbf{x}, t)$ and hence the time average $<\mathbf{P}>$. What does $<\mathbf{P}>$ represent in this case?