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\begin{document}

\centerline{\bf Electrodynamics B (PHY 5347) Winter/Spring 2017 
Midterm, Feb.\ 23.} \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 Bar sliding in magnetic field (33\%).}
\smallskip


A bar slides along two parallel rails separated by a distance $w$,
with no friction and with velocity $v$. The parallel bars, connector,
and sliding bar are all made of a conductor with resistance $r$ per
unit length. The bars are in a uniform magnetic field $B$ which is
perpendicular to the plane of the bars and directed upward. Assume the
bar starts at a distance $l$ from the end.
\begin{figure}[h]
\includegraphics[angle=0,height=2.5in]{ef97_f1}
\end{figure}
\begin{description}
\item{(a)}
Use Gaussian unit and calculate the emf in the loop.
\item{(b)}
Find the current in the bar as a function of time.
\end{description}

\item {\bf Four-potential of a moving point particle (33\%).} 
\smallskip

A pointlike particle with the charge of an electron $q_e$ travels with 
velocity 
$$\beta = \frac{v}{c} = \frac{1}{2}$$ 
on the straight line $y^1=\beta y^0=vt$. Find the electromagnetic 
potentials which are after one picosecond, i.e., $t=1\,[ps]=
10^{-9}\,[s]$, observed at the origin $\vec{x}=0$. 

Take $3\times 10^9\,[dm/s]$ (decimeter $[dm]$) for the speed of light, 
$q_e$ as charge unit and give the results in $[q_e/dm]$. First, draw 
a Minkowski space picture.

You may use
\begin{eqnarray} \nonumber
  \Phi\left(x^0,\vec{x}\right) = \left[\frac{q}{\left(1-
  \vec{\beta}\cdot\hat{n}\right)\,R}\right]_{\rm ret}\!\!,~~~
  \vec{A}\left(x^0,\vec{x}\right) = \left[\frac{q\,\vec\beta}{\left(
  1-\vec{\beta}\cdot\hat{n}\right)\,R}\right]_{\rm ret}\!\! .
\end{eqnarray}
\bigskip


\item {\bf Euler-Lagrange equation (33\%).}
\smallskip

Consider the Lagrangian
$$ {\cal L} = {1\over 2}\, h_{\alpha\beta}\, h^{\alpha\beta}
  + h^{\alpha\beta}\,(\partial_{\beta} A_{\alpha} -
  \partial_{\alpha} A_{\beta}) - 
  {8\pi\over c}\,A_{\alpha}\,J^{\alpha} $$
where $h_{\alpha\beta}$ is a rank two tensor field, $A_{\alpha} (x)$ 
is a vector field and $J_{\alpha}$ an external current. All these fields 
are regarded to be independent, including $h_{\alpha\beta}$. Find the 
equations of motion from the Euler-Lagrange equations of relativistic 
fields.
\bigskip 

\end{enumerate}

\end{document}
