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\newcommand\Title{\centerline{\large\textbf{Lorentz Transformations for Electromagnetism}}}

\Title

\bigskip
\begin{center}
\textit{Working in groups of two or three, do the following problems.}
\end{center}
\medskip

\noindent
The electric field of an infinite metal sheet with charge density $\sigma$
points away from the sheet and has the constant magnitude
\begin{equation}
 |E| = \frac{\sigma}{2\epsilon_0}
\end{equation}
The magnetic field of such a sheet with current density $\vkap$ has constant
magnitude
\begin{equation}
 |B| = \frac{\mu_0}{2} |\vkap|
\end{equation}
and direction determined by the right-hand-rule.



\begin{enumerate}

\item
Consider a capacitor consisting of 2 horizontal
($y=\hbox{constant}$!)\ parallel plates, with equal and opposite charge
densities.  For definiteness, take the charge density on the bottom plate to
be $\sigma_0$, and suppose that the charges are at rest, that is, that the
current density of each plate is zero.  Determine the electric field
$\EE_0$ between the plates.\\
\textit{(What is the electric field elsewhere?)}


\item
Now let the capacitor move to the \textit{left} ($-x$ direction) with speed
$u$, while you remain at rest.  Then the \textit{width} of the plate is
unchanged, but you observe the \textit{length} to be Lorentz contracted.  What
charge density do you observe on each plate?


\item
What current density do you observe on each plate?



\item
What electric and magnetic fields do you observe between the plates?\\
\textit{Express your answers in terms of $E_0=|\EE_0|$ and $\alpha$, where
$u=c\tanh\alpha$.}

\begin{center}
\textbf{Before moving on, compare your results with another group!}
\end{center}

\item
The above discussion gives the electric ($\EE$) and magnetic ($\BB$) fields
due to parallel plates moving to the left \textit{as seen by an observer at
rest}.  What electric ($\vf{E'}$) and magnetic ($\vf{B'}$) fields are seen by
an observer \textit{moving to the right} with speed \hbox{$v=\cc\tanh\beta$}?

\bigskip
\textit{You may wish to recall that}
\begin{eqnarray*}
\sinh(\alpha\pm\beta) &=& \sinh\alpha\cosh\beta\pm\cosh\alpha\sinh\beta \\
\cosh(\alpha\pm\beta) &=& \cosh\alpha\cosh\beta\pm\sinh\alpha\sinh\beta
\end{eqnarray*}



\item
If you have not already done so, express the components $E'_y$ and $B'_z$ in
terms of $E_y$ and $B_z$.  That is, eliminate $E_0$ and $\alpha$ in favor of
$E^y$ and $B^z$. \textit{Recall that $\mu_0\epsilon_0c^2=1$.}

\item
Interpret your results.
\end{enumerate}

\bigskip

\hfill \textit{by Tevian Dray \& Mary Bridget Kustusch}

\hfill \copyright 2012 Tevian Dray

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