Consider a parallel plate waveguide with plate separation d as shown in figure. The electric and magnetic fields for the TEM mode are given by
E_{x}=E_{0} e^{-j k z+j \omega t}, H_{y}=\frac{E_{0}}{\eta} e^{-j k z+j \omega t}Where k = T < one.
(a) Determine the surface charge densities \rho_{s} on the plates at x = 0 and x = d.
(b) Determine the surface current densities \vec{J}_{s} on the same plates.
(c) Prove that \rho_{s} \text { and } \vec{J}_{s} satisfy the current continuity condition.
Given that wave is propagating in +2 direction.
\begin{aligned}&E_{x}=E_{0} e^{+j(\omega t-k z)} \\&H_{y}=\frac{E_{0}}{\eta} e^{+j(\omega t-k z)}\end{aligned}(a) \vec{E}=\frac{P_{S}}{E} a_{x} \text { or } E_{x}=\frac{P_{S}}{E}
P_{S}=\in E_{x}=\in E_{0} \cos (\omega t-k z)
(b) J_{S}=\vec{\eta} \times \vec{H}=\text { for surface } x=0,
\begin{aligned}&\vec{J}_{S}=\vec{a}_{x} \times \vec{H}=H_{y} \hat{a}_{z}=\frac{E_{0}}{\eta} \cos (\omega t-k z) \hat{a}_{2} \\&k=\eta \omega E \\&\Rightarrow \frac{1}{\eta}=\frac{\omega E}{K} \\&\vec{J}_{S}=\omega \in \frac{E_{0}}{K} \cos (\omega t-k z) \vec{a}_{2} \\&\text { at, } x=d, \vec{J}_{S}=-\frac{E_{0}}{\eta} \cos (\omega t-k z) a_{2}\end{aligned}(c) From current continuity \operatorname{eg}^{n}. for time varying fields given by
\begin{aligned}&\nabla . \vec{J}=\frac{-\partial}{\partial t} P \Rightarrow \frac{d}{d z} J S J=\frac{-\partial}{\partial t} P . \\&\begin{aligned}\frac{d}{d z} &\left[\frac{\in \omega E_{0}}{k} \cos (\omega t-k z)\right]=\omega \in E_{0} \sin (\omega t-k z)\end{aligned} \\&\begin{aligned}\Rightarrow \frac{-\partial}{\partial t} P &=\frac{\partial}{\partial t}\left(\in E_{0} \cos (\omega t-k z)\right.\\&=\omega \in E_{0} \sin (\omega t-k z),\end{aligned} \\&\text { So, } \frac{d}{d z} J S z=\frac{-\partial}{\partial t} P .\end{aligned}Thus current continuity equation verified.