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\begin{document}

\centerline{\bf Electrodynamics A (PHY 5346) Fall 2016 
Midterm October 11.} \bigskip 

\begin{enumerate}

\item{\bf Spacetrip (40\%).}

(A) Assume that the earth is in an inertial frame. A spaceship leaves 
the earth at time~0. The spaceship is constructed so that it has an 
acceleration $g$ in each of its own instantaneous rest frames. By its 
own clock, it accelerates on a straight-line path for 1~year, decelerates 
at the same rate for 1~year, turns around, accelerates for 1~year,
decelerates for 1~year, and lands on earth. Calculate the time on 
earth at the landing. 

Instructions: Use $g=9.81~[m/s^2]$, one year = $365\times 24\times 
3600~[s]$ and for the speed of light $c=300,000~[km/s]$. State the 
result in units of years and decimal fractions of years to at least 
three significant digits. Hint: Find first $\zeta(\tau)$, where 
$\tau$ is the proper time in the spaceship and $\zeta(\tau)$ its 
rapidity with respect to Earth.

(B) How far away from Earth did the spaceship travel? Express the 
result in light years.
\bigskip


\item{\bf Lorentz invariance of antisymmetry (20\%).}

Show that an antisymmetric tensor $F^{\alpha\beta}$ stays
antisymmetric under Lorentz transformations, i.e.,
$$ F'^{\alpha\beta} = - F'^{\beta\alpha} ~~~{\rm with}~~
   F'^{\alpha\beta} = a^{\alpha}_{~\hat{\alpha}}\,
   a^{\beta}_{~\hat{\beta}}\, F^{\hat{\alpha}\hat{\beta}}\ .$$

\item {\bf Some electromagnetic invariants (40\%).}

Express the invariants $F_{\alpha\beta}\,F^{\alpha\beta}$, 
$F_{\alpha\beta}\,^*\!F^{\alpha\beta}$ and $^*\!F_{\alpha\beta}\,
^*\!F^{\alpha\beta}$ in terms of $\vec{E}$ and $\vec{B}$ fields, 
where $^*\!F^{\alpha\beta}$ is the dual tensor. 
% (Note that the summations are over both indices, i.e., not matrix 
% multiplications. The results are scalars.)
\bigskip

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\end{enumerate}

\end{document}
