Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 4 / 7D / b / iii / Solution

Since $\mathbf{E}\cdot {\mathbf{e}}_z=0$, the vector triple product yields $\mathbf{E}\times ({\mathbf{e}}_z\times \mathbf{E})=|\mathbf{E}{|}^2{\mathbf{e}}_z$. Consequently the Poynting vector isIts average over a full period uses $⟨{\cos }^2⟩=1/2$:This is the mean electromagnetic energy flux per unit area. For positive frequency it points in the propagation direction and has magnitude $E_0^2/(2{\mu }_{0}c)={ϵ}_{0}c{E}_0^2/2$.