% BB Nov 11 2016. 21 participants.
%\documentclass[aps,twocolumn]{revtex4}
%\documentclass[aps,preprint]{revtex4}
\documentclass[aps,12pt]{revtex4}
\usepackage{epsfig}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\tq}{\tilde{q}}
                                                                                
\begin{document}

\centerline{\bf Electrodynamics A (PHY 5346) Fall 2016 
Test on Homework December~1.} \bigskip 

\begin{figure}[tb]
\centerline{\hbox{ \psfig{figure=figures/1mink,width=10cm} }}
\caption{Minkowski space in which observer~A is at rest and
flashes a light signal at observer~B, who moves with speed $4c/5$
and flashes the signal back.  \label{fig_exmink1} }
\end{figure}

\begin{enumerate}

\item{\bf Time in Minkowski space (30\%).}

In figure~\ref{fig_exmink1} the Minkowski space is parametrized by the 
coordinates of the rest frame of an observer~A. While observer A stays 
at rest, observer~B moves with speed $4c/5$ along the positive $x$ axis. 
At their common origin both, A and B, have set their clocks to zero. 
After 15~s observer~A emits at position A$_1$ a light signal which 
reaches observer~B at position B$_0$, who flashes it back at observer~A, 
who receives it at position A$_2$  as drawn in the figure.

\begin{enumerate}
\item What are the coordinates of the position B$_0$?
\item What is the time on the clock of observer~B at position $B_0$?
\item What is the time on the clock of observer~A when the back signal
is received at position A$_2$?
\end{enumerate}

\item{\bf Potential in a rectangular box (30\%).}

Consider a hollow, rectangular box with
$0\le x\le a>0$, $0\le y\le b>0$ and $0\le z\le c>0$.
Assume that the potential is zero on five sides and on the
sixth side $(z=c)$ given by 
$$ \Phi (x,y;c)=\sin\left({\pi x\over a}\right)\, 
\sin\left({2\pi y\over b}\right) + \sin\left({3\pi x\over a}\right)\,
\sin\left({\pi y\over b}\right).$$
Calculate the potential $\Phi (x,y,z)$ inside the box.
\smallskip
% Solution: Solutions appendix. EM95 exercises, p.11 (\# 18).


\item{\bf Dielectric sphere (40\%).}

Consider a dielectric sphere of radius $R$ with dielectric 
constant $\epsilon_1$ in a dielectric medium with dielectric constant 
$\epsilon_2$. At large distances from the sphere a homogeneous, constant
electric field $\vec E = E_0\,{\hat z}$ is found. 

\noindent (a)~Expand the Potential $\Phi(\vec x)$ into spherical
harmonics and state the boundary conditions at $r=R$.

\noindent (b)~Calculate the Potential $\Phi (\vec x)$ and the electric 
field $E(\vec x)$ for all $\vec x$. 

\noindent (c)~Calculate the surface charge density.
\smallskip
% Solution: EM94 exercises, p.20 (\# 21).
\medskip

% \begin{figure}[th] % From EMscript/figures_plt/minkowski_space_ex.
% \vspace{5pc}       % 3mink-prob.plt
% \centerline{\hbox{ \psfig{figure=e3mink,width=10cm} }}
% \caption{Minkowski \label{fig_exmink3} } \end{figure}

\end{enumerate}
\end{document}
