Exercise 7.2.1: Charges and currents of the TEM wave

Top plate $a_n=-a_y$, bottom plate $a_n=+a_y$:

$H_x=500/377=1.326A/m$ (times ${\mathrm{e}}^{-\mathrm{j}kz}$):

$k=2\pi f/c=20.94rad/m$ and $\omega =2\pi \times 10^{9}rad/s$. On the top plate $\partial J_{sz}/\partial z=-\mathrm{j}k\times 1.326=-\mathrm{j}27.8$ and $-\mathrm{j}\omega {\rho }_s=-\mathrm{j}2\pi \times 10^9\times 4.42\times 10^{-9}=-\mathrm{j}27.8$ (both times ${\mathrm{e}}^{-\mathrm{j}kz}$), the continuity equation of Section 5.4 for a surface. In symbols $k{E}_0/{\eta }_0=\omega {\epsilon }_0{E}_0$ because $k/{\eta }_0=\omega \sqrt{{\mu }_0{\epsilon }_0}\sqrt{{\epsilon }_0/{\mu }_0}=\omega {\epsilon }_0$.

$V=E_{0}b=5\mathrm{V}$, $I=J_{s}w=66.3\mathrm{mA}$, $Z_0=V/I=75.4\Omega ={\eta }_{0}b/w$. $q^{\prime }={\rho }_{s}w=0.221nC/m$, and

Power: $\frac{1}{2}VI=0.166\mathrm{W}$ and $\frac{b{E}_0^2}{2{\eta }_0}w=\frac{0.01\times 500^2}{754}\times 0.05=0.166\mathrm{W}$. The TEM wave is alone below $f_{co,1}=c/2b=15\mathrm{GHz}$.