\noindent
{\bf Problem (\ref{ex_skin_depth}):} Skin depth.
\smallskip
\bigskip

\noindent
{\bf Problem (\ref{E_TM_boxwg}): TM waves in a rectangular wave guide.}
\smallskip

\noindent (1)
$$ (\nabla_t^2 + \gamma^2)\, E^z = 0\,,\qquad E^z|_S = 0\,, $$
$$ E^z = E^0\, \sin\left(\frac{m\pi x}{a}\right)\,
               \sin\left(\frac{n\pi y}{a}\right)\,
               e^{i\,k\,z}\,e^{-i\,\omega\,t}\,,            $$
$$ \gamma^2_{n\,m} = \pi^2\,\left[\left(\frac{m}{a}\right)^2 +
                            \left(\frac{n}{b}\right)^2 \right]\,,
                   ~~~m=1,2,\dots\,,~~ n=1,2,\dots\,,          $$
(2)
$$ \gamma^2_{1\,1} = \pi^2\,\left(a^{-2}+b^{-2}\right)\,,~~~~~
   \omega_{1\,1} = \frac{c\,\pi}{\sqrt{\mu\,\epsilon}}\
   \sqrt{a^{-2}+b^{-2}}                                        $$ 
(3)
$$ \vec{E}_t = \frac{i\,k}{\gamma^2}\,\nabla_t\,E^z\,,~~~~~~
   {\rm lowest\ mode:} $$
\begin{eqnarray} \nonumber
  E^x &=& E_0\,\frac{i\,k}{\pi\,a\,(a^{-2}+b^{-2})}\,
          \cos\left(\frac{\pi x}{a}\right)\,
          \sin\left(\frac{\pi y}{a}\right) \\ \nonumber
  E^y &=& E_0 \,\frac{i\,k}{\pi\,b\,(a^{-2}+b^{-2})}\,
          \sin\left(\frac{\pi x}{a}\right)\,
          \cos\left(\frac{\pi y}{a}\right)\,.
\end{eqnarray}
(4)
$$ \vec{H}_t = \frac{\epsilon\,\omega}{c\,k}\,\hat{z}\times\vec{E}_t\,,
   ~~~~~~{\rm lowest\ mode:} $$
\begin{eqnarray} \nonumber
  H^x&=&-E_0\,\frac{i}{b\,(a^{-2}+b^{-2})}\,\sqrt{\frac{\epsilon}{\mu}}
  \, \sin\left(\frac{\pi x}{a}\right)\,
     \cos\left(\frac{\pi y}{a}\right) \\ \nonumber
  H^y&=&+E_0\,\frac{i}{a\,(a^{-2}+b^{-2})}\,\sqrt{\frac{\epsilon}{\mu}}
  \, \cos\left(\frac{\pi x}{a}\right)\,
     \sin\left(\frac{\pi y}{a}\right)\,.
\end{eqnarray}
(5)
$$ \vec{K} = \frac{c}{4\pi}\,\hat{n}\times\vec{H} $$
Let us choose the convention that $\hat{n}$ points out of the conductor
(into the wave guide). Then,
$$ {\rm For\ the}\ (x\!-\!z)\!=\!{\rm plane}\ y=0,b:
~~~\hat{n}=\pm\hat{y}\,,~~~\frac{4\pi}{c}\,\vec{K}=\mp\hat{z}\,H^x\,,$$
$$ {\rm For\ the}\ (y\!-\!z)\!=\!{\rm plane}\ x=0,a:
~~~\hat{n}=\pm\hat{x}\,,~~~\frac{4\pi}{c}\,\vec{K}=\pm\hat{z}\,H^y\,.$$
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\noindent
{\bf Problem (\ref{cavity_oscillator}): Cubic cavity oscillator.}
\smallskip

$$ \vec{E}\ =\ {\hat x}\, E_0\, \sin \left( {\pi y\over a} \right)\,
   \sin \left( {\pi z\over a} \right)\, e^{-i\omega t}\, .        $$
(1)
\begin{eqnarray} \nonumber
  \vec{H} &=& -\frac{i\,c}{\omega\,\mu}\, \nabla\times\vec{E}\ =
  \\ \nonumber 
  &-&\frac{i\,c}{\omega\,\mu}\,E_0\,\frac{\pi}{a}\,\left[
  - \hat{z}\,\cos\left(\frac{\pi y}{a}\right)\,
             \sin\left(\frac{\pi z}{a}\right)\,
  + \hat{y}\,\sin\left(\frac{\pi y}{a}\right)\,
             \cos\left(\frac{\pi z}{a}\right)\,\right]\,
    e^{-i\omega t}\,.
\end{eqnarray}
(2)
$$ \nabla\cdot\vec{E} = 0~~~{\rm as}\ \vec{E}\ {\rm does\ not\
   depend\ on}\ x\,, $$
$$ \nabla\cdot\vec{H} \sim \left[ 
 - \hat{z}\,\cos\left(\frac{\pi y}{a}\right)\,
            \cos\left(\frac{\pi z}{a}\right)\,
  + \hat{y}\,\cos\left(\frac{\pi y}{a}\right)\,
             \cos\left(\frac{\pi z}{a}\right)\,\right]\,
    e^{-i\omega t} = 0\,.                                           $$
\begin{eqnarray} \nonumber
  \nabla\times\vec{H} &=& - \frac{i\,c}{\omega\,\mu}\,E_0\,\frac{\pi^2}{a^2}
  \,\left[ \hat{x}\,\sin\left(\frac{\pi y}{a}\right)\,
           \sin\left(\frac{\pi z}{a}\right)\,
  + \hat{x}\,\sin\left(\frac{\pi y}{a}\right)\,
             \sin\left(\frac{\pi z}{a}\right)\,\right]\,e^{-i\omega t}
\\ \nonumber &=& - \frac{i\,c}{\omega\,\mu}\,\frac{2\,\pi^2}{a^2}\,\vec{E}
            \,=\, - \frac{i\,\omega}{c}\,\epsilon\,\vec{E}
\end{eqnarray}
where the result of (3) below was used in the last step. BCs: From
$\sin(\pi a/a)=\sin(\pi)=0$ we see that $\vec{E}|_S=0$ holds on the 
boundaries parallel to $\vec{E}$ and on the boundaries where 
$\vec{H}$ has a perpendicular component $\vec{H}_{\perp}|_S=0$ holds.
\smallskip 

\noindent (3)
$$ 0 = \left( \nabla^2-\frac{\mu\epsilon}{c^2}\,
   \frac{\partial^2~}{\partial t^2}\right)\,\vec{E} = 
   \left( -\frac{\pi^2}{a^2} -\frac{\pi^2}{a^2}
   \frac{\mu\epsilon}{c^2}\,\omega^2 \right)\,\vec{E}\
   \Rightarrow\ \omega = 
   \sqrt{\frac{2}{\mu\epsilon}}\,\frac{c\,\pi}{a}\,. $$
(4) Surface currents:
$$ \vec{K} = \frac{c}{4\pi}\,\hat{n}\times\vec{H}\,. $$
Top, bottom $(x\!-\!y)\!-\!{\rm plane},\ z=0,a$:
$$ \vec{K} = \pm\frac{c}{4\pi}\,\hat{z}\times\vec{H} =
             \pm\hat{x}\,\frac{i\,c^2}{4\omega\mu\epsilon a}\,
             \sin\left(\frac{\pi y}{a}\right)\,, $$
Front, back $(z\!-\!x)\!-\!{\rm plane},\ y=0,a$:
$$ \vec{K} = \pm\frac{c}{4\pi}\,\hat{y}\times\vec{H} =
             \mp\hat{x}\,\frac{i\,c^2}{4\omega\mu\epsilon a}\,
             \sin\left(\frac{\pi y}{a}\right)\,, $$
Left, right $(y\!-\!z)\!-\!{\rm plane},\ x=0,a$:
\begin{eqnarray} \nonumber
 &\vec{K}& =\,\pm\frac{c}{4\pi}\,\hat{x}\times\vec{H}\,=\\ \nonumber
 &\mp & \frac{i\,c^2}{4\omega\mu\epsilon a}\,\left[
   \hat{y}\,\cos\left(\frac{\pi y}{a}\right)\,
            \sin\left(\frac{\pi z}{a}\right) +
   \hat{z}\,\sin\left(\frac{\pi y}{a}\right)\,
            \cos\left(\frac{\pi z}{a}\right)\,
            \right]\, e^{-i\omega t}\,.
\end{eqnarray}
Surface charge density in the $(y\!-\!z)\!-\!{\rm plane}:$
$$ \sigma = \frac{1}{4\pi}\,\frac{2\pi}{\omega^2\mu}\,E_0\,
   \sin\left(\frac{\pi y}{a}\right)\,
   \sin\left(\frac{\pi z}{a}\right)\,e^{-i\omega t}\ .$$
% \bigskip % \hfil\break \bigskip