 Solution

In SI units, Maxwell's equations areDotting the last equation with $\mathbf{E}/{\mu }_0$, and using the curl equation for $\mathbf{E}$, givesIntegrate over the fixed region and apply the divergence theorem. This proves Poynting's theorem: the rate of increase of electromagnetic energy stored inside, plus the outward Poynting flux, plus the rate of work on charges, is zero. The last term need not be positive for arbitrary externally driven currents; when positive it is power transferred from the field to matter.

For the travelling wave put $\phi =kz-\omega t$. A purely oscillatory magnetic field satisfying Faraday's law isBoth divergences vanish. The vacuum Ampère-Maxwell law then requires $k^2={\mu }_0{ϵ}_0{\omega }^2$, so with the positive constants in the question,A superposed static background is not part of the pure plane-wave solution. The electric and magnetic energy contributions are equal, givingConsequently $\boxed{|\mathbf{P}|=c\mathcal{U}}$ pointwise, including the nodes where both sides vanish.