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\begin{document}

\centerline{\bf Electrodynamics B (PHY 5347): Test on Homework
April 20, 2017.} \bigskip

% \begin{figure}[th] % From EMscript/figures_plt/minkowski_space_ex.
% \vspace{5pc}       % 3mink-prob.plt
% \centerline{\hbox{ \psfig{figure=figures/e3mink.eps,width=10cm} }}
% \caption{Minkowski space in which B travels with speed $4/5$ away 
% from A and then back to A.  \label{fig_exmink3} } \end{figure}

\begin{enumerate} % \vfil\eject

\item {\bf Magnetism in matter (25\%).}
\smallskip

\begin{figure}[tb] % Scan from COMPS 1999.  \vspace{5pc}
\centerline{\hbox{ \psfig{figure=magsphere,width=10cm} }}
\caption{Magnetism in matter: Concentric spheres in a magnetic 
field $\vec{B}_0=B_0\,\hat{z}$. \label{fig_magsphere} } \end{figure}

Consider two concentric spheres with radii $a<b$ placed in a (formerly) 
constant magnetic field $\vec{B}_0=B_0\,\hat{z}$. The inner and outer 
regions are vacuum, while the region between the spheres is made of 
a material of permeability $\mu$. See figure~\ref{fig_magsphere}.

\begin{enumerate}

\item Write down the Maxwell equations and the boundary conditions of 
relevance for this problem.

\item Introduce a magnetic scalar potential and show the differential
equations and boundary conditions that it satisfies.

\end{enumerate}
\bigskip

\item{\bf Covariant retarded Green function (25\%).}
\smallskip

Show that the explicitly covariant expression
$$ G_r (x-y)={1\over 2\pi}\,\theta(x^0-y^0)\,\delta[\,(x-y)^2\,]$$
agrees with the retarded Green function 
$$ G_r(x-y) = {\delta (\tau-\xi) \over 4\pi\,\xi} = {\delta(x^0-y^0
   -|\vec{x}-\vec{y}|)\over 4\pi\,|\vec{x}-\vec{y}| }\ .$$
\bigskip


\item{\bf Charge conjugation, parity and time reversal in 
     electrodynamics (25\%).} \smallskip

\begin{enumerate}
 \item State which of the following electrodynamic quantities are even 
 $(+)$ and which are odd $(-)$ under operations of charge conjugation 
 $(C)$, parity $(P)$, and time reversal $(T)$: 1.~The potential 
 component $A^0$, 2.~The current density $\vec{J}$, 3.~The potential 
 components $\vec{A}$, 4.~The electric field $\vec{E}$, 5.~The magnetic 
 field $\vec{B}$. State the results as $(C,\, P,\, T)$. Start from the 
 assumption that the charge density $\rho$ has $(C,P,T)=(-,+,+)$.
\item Express the following Lorentz scalars in terms of $\vec{E}$
 and $\vec{B}$:
 $$F^{\alpha\beta}\, F_{\alpha\beta}~~~{\rm and}~~~ 
   ^*F^{\alpha\beta}\, F_{\alpha\beta}\ . $$
 State their transformation behavior under $(C,P,T)$.
\end{enumerate}
\bigskip


\item{\bf TM waves in a rectangular wave guide (25\%).}
\smallskip

Consider the propagation of TM waves in a rectangular wave guide with its 
cross section in the $x$-$y$-plane and of inner dimensions $a,b$. Assume
perfectly conducting surfaces, $e^{+ikz}$ for the axial dependence, and 
$e^{-i\omega t}$ for the time dependence.
\begin{enumerate}
\item Calculate  $E^z$.
\item Write down the lowest cut-off frequency.
\end{enumerate}
\vfil

\end{enumerate}

\end{document}
