Home exercise 7.3.3: Fields and wall currents at high power

$f_{co,10}=2.079\mathrm{GHz}$, $s=\sqrt{1-(2.079/3{)}^2}=0.721$, $Z_{TE}=377/0.721=523.0\Omega$:

$4.6$ times below breakdown.

$|H_x{|}_{\max }=E_0/Z_{TE}=1248A/m$ at $x=a/2$, and with $\omega {\mu }_{0}a=2\pi \times 3\times 10^9\times 4\pi \times 10^{-7}\times 0.07214=1709\Omega$,

On the broad walls $|J_s{|}^2=|H_x{|}^2+|H_z{|}^2$ with $H_x\propto \sin (\pi x/a)$ and $H_z\propto \cos (\pi x/a)$, and the largest value, $1248A/m$ along $z$, is on the centre line. On the narrow walls the only tangential component is $H_z$: $1200A/m$, vertical.

${\rho }_s={\epsilon }_0{E}_0=\frac{10^{-9}}{36\pi }\times 0.653\times 10^6=5.77\mu C/{\mathrm{m}}^2$, on the broad walls at $x=a/2$, of opposite signs on top and bottom.

$|H_z{|}_{\max }/|H_x{|}_{\max }=(f_c/f)/s=0.693/0.721=0.962$, which is $\tan \theta$ of the two plane waves of TE${}_{10}$: their $H$ is split between $x$ and $z$ by the bounce angle.