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\centerline{\bf Electrodynamics A (PHY 5346) Fall 2016 
Final Decenber 14.} \bigskip 

\begin{enumerate}

\item {\bf Electron-positron annhilation (40\%).} 
\smallskip

An electron and a positron, each (for $c=1$) with mass equal to 
$0.511\,[MeV]$, annihilate at rest into two photons. 
\begin{enumerate}
\item Working in the rest frame of the positronium, so that one of 
the photons moves in positive $x=x^1$ direction, find the momentum 
four-vector for each photon. 
\item Suppose that the annihilation took place in a galaxy that 
is receding from us at $\beta = 3/5$. What is the energy of the photon 
that we observe?
\end{enumerate} \medskip


\item {\bf Coaxial cable (30\%).} \smallskip

Consider a long, straight coaxial cable with a circular cross section. 
A constant current flows from a source to a target through the inner 
conductor $(0\le\rho\le\rho_1)$ and back through the outer conductor 
$(\rho_2\le\rho\le\rho_3)$. Assume vacuum in the in--between region 
$(\rho_1<\rho<\rho_2)$ and use Amp\'ere's law to calculate the magnetic 
field everywhere.
\bigskip


\item {\bf Electric potential from distinct BCs on half-spheres (30\%).} 
\smallskip

\begin{enumerate}
\item Consider a constant potential $\Phi_0$ on the surface of the upper
half of a hollow sphere of radius $R$ and $\Phi =0$ on the surface 
of the lower half of this sphere with  no charges inside. Choose the 
center of the sphere at $x=y=z=0$, its upper half above and its lower 
half below the $x-y$ plane. Use the Green function of a sphere to 
calculate the potential on the $z$ axis inside this sphere.
\item Take $R=1$ and $\Phi_0=1$. Evaluate $\Phi(z)$ for the values 
$z=R$, $z=R/2$, $z=0$, $z=-R/2$ and $z=-R$. Sketch $\Phi(z)$ in the 
range $-1\le z\le 1$.
\end{enumerate} \medskip


\end{enumerate}
\end{document}

