Home exercise 6.1.7: Power through a plane

$H=\frac{1}{\eta }{a}_x\times E$ with $a_x\times a_z=-a_y$, and $4/98.7=40.5mA/m$:

${\mathcal{P}}_{av}=\frac{E_0^2}{2\eta }{a}_x=\frac{4^2}{2\times 98.7}{a}_x=81.1a_{x}mW/{\mathrm{m}}^2$.

The plane $x=0$ has the normal $a_x$, parallel to ${\mathcal{P}}_{av}$: $P=81.1\times 10^{-3}\times 10=0.811\mathrm{W}$.

$E\times H\propto a_z\times (-a_y)=+a_x$, the direction of travel, and $\frac{1}{2}{E}_0{H}_0=\frac{1}{2}(4)(0.0405)=81.1mW/{\mathrm{m}}^2$. A piece of a plane whose normal makes the angle $\psi$ with $a_k$ would collect ${\mathcal{P}}_{av}A\cos \psi$, and any plane containing $a_x$ collects nothing.