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\centerline{\bf Electrodynamics B (PHY 5347): Spring 2017 Final,
May 2.} \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 Magnetic moment of a sphere (33\%).}
\smallskip

A sphere with radius $R$ rotates with constant angular velocity 
$\omega$ about the $z$ axis. A uniform charge distribution is fixed 
on the surface. The total charge is $q$. Calculate the magnetic moment 
$\vec{\mu}$.
\bigskip

\item {\bf Principal value integrals and Green functions of the wave 
           equation (33\%).} % Lecture Notes B - pp.105.
\smallskip

Green functions of the Klein-Gordon wave equation are defined by the
equation
$$\left(\square+m^2\right)\,G(x,y)\ =\ \delta^{(4)}(x-y)\,. $$
After Fourier transformation solutions are found in the form
$$ G(x-y)\ =\ \frac{1}{(2\pi)^3} \int d^3p\,I_p\,,~~
   I_p\ =\ \frac{1}{2\pi} \int_{-\infty}^{+\infty} 
   \frac{du\,e^{-i\,u\,\tau}}{\omega_p^2-u^2} $$
with 
$$ \omega_p = +\sqrt{\vec{p}^{\ 2}+m^2}\,,~~p^0=u~~
   {\rm and}~~\tau=x^0-y^0\,. $$
Closing in the complex $w=u+iv$ plane, one obtains Feynman (F), Dyson
(D), advanced (a) and retarded (r) Green functions, depending on the 
convention for the poles in the integrand of $I_p$. For instance,
$$ G_a(x-y)\ =\ \frac{1}{(2\pi)^3} \int d^3p\,I_p^a~~~{\rm and}~~
   G_r(x-y)\ =\ \frac{1}{(2\pi)^3} \int d^3p\,I_p^r\,,~~~      $$
$$ I_p^r\ =\ \cases{ -\frac{i}{2\omega_p}\, \left(e^{+i\omega_p\tau}
   -e^{-i\omega_p\tau}\right)~{\rm for}~\tau>0,\cr 0~{\rm for}~\tau<0;}
   ~~I_p^a\ =\ \cases{0~{\rm for}~\tau>0; \cr \frac{i}{2\omega_p}\, 
   \left(e^{+i\omega_p\tau} -e^{-i\omega_p\tau}\right)~{\rm for}~\tau<0,}
$$
for the retarded and advanced choices.

Calculate the principal value integral
$$ I_p^P\ =\ \frac{1}{2\pi}\,P \int_{-\infty}^{+\infty} 
   \frac{du\,e^{-i\,u\,\tau}}{\omega_p^2-u^2} $$
for $\tau>0$ and $\tau<0$. Relate the $\tau>0$ result to $I_p^r$
and the $\tau<0$ result to $I_p^a$.
\bigskip % \eject


\item {\bf Far field approximation (34\%).}

Use the vector potential 
$$ \vec{A}_{\phi}(r,\theta) = A^0_{\phi}\,\sin(\theta)\,\cos(\theta)\,
                    \frac{e^{i\,k\,r-i\,\omega\,t}}{r}\,\hat{\phi}\ . $$
to calculate the following quantities in the far field approximation:
\begin{enumerate}
\item The magnetic field.
\item The electric field.
\item The time-averaged Poynting vector.
\item The angular intensity distribution.
\item The maxima of the angular intensity distribution.
\item The maxima of the angular intensity distributions divided by
      the average of the angular intensity distribution over the surface 
      of the sphere that contains the maxima.
\end{enumerate}


\end{enumerate}

\end{document}
