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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 14}\\%
\vspace {3mm} (April 13, 2009) \\
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\item{(58)} The electromagnetic field tensor transforms according to
$$ F'^{\, \alpha\beta} = a^{\alpha}_{\ \gamma} 
  a^{\beta}_{\ \delta} F^{\gamma\delta}\, . $$
\begin{enumerate}
\item Choose the particular case of a Lorentz boost in $x^1$-direction,
      $\vec{v}=v\, \hat{e}_1$, and write down the transformation law
      for the electric field $\vec{E}$ and the magnetic induction 
      $\vec{B}$. Due April~15 before class (8~points).
\item Compare with the results you get from (due April~15 before 
      class (4~points).
\begin{eqnarray} \nonumber
  \vec{E}^{\,'}&=&\gamma\,\left(\vec{E}+\vec{\beta}\times\vec{B}\right) 
  - \frac{\gamma^2}{\gamma+1}\vec{\beta}\,\left(\vec{\beta}\,\vec{E}
    \right)\,, \\ \nonumber
  \vec{B}^{\,'}&=&\gamma\,\left(\vec{B}-\vec{\beta}\times\vec{E}\right)- 
  \frac{\gamma^2}{\gamma+1}\vec{\beta}\,\left(\vec{\beta}\,\vec{B}\right)\,.
\end{eqnarray}
\end{enumerate}

\item{(59)} In the same way that the contraction, or relativistic dot 
product, of two four vectors is a Lorentz invariant, the contraction 
of two relativistic tensors is also a Lorentz invariant (Griffiths 
Problem 12.50).
\begin{enumerate}
\item Compute the three Lorentz invariants from the contraction of the 
tensors: 
$$ F^{\mu\nu}F_{\mu\nu}\;,\ ^*F^{\mu\nu}\,^*F_{\mu\nu}\ {\rm 
   and}\ F^{\mu\nu}\,^*F_{\mu\nu}$$
in terms of the electric and magnetic fields $\vec{E}$ and $\vec{B}$. 
Due April~15 in class (3~points).
\item Suppose that in one inertial frame
	  $\vec{B}\!=\!0$ but $\vec{E}\!\ne\!0$ 
	  (at some point $P$). Is it possible to
	  find another system in which the
	  {\it electric field} is zero at $P$?
Due April~15 in class (1~point).
\end{enumerate}

\item{(60)} The non--zero fields $\vec E$ and $\vec B$ are non--parallel 
in inertial frame $K$. Inertial frame $K'$ moves with velocity $\vec v$
with respect to $K$. Find a physical velocity $\vec v$ so that $\vec E'$ 
and $\vec B'$ are parallel. (Hint: Try $\vec E = E_2\,{\hat e}_2$,
$\vec B = B_2\,{\hat e}_2 + B_3\,{\hat e}_3$ and $\vec v = v\,{\hat e}_1$.) 
Due April~17 before class (10~points). \medskip

\item{(61)} Transform 
$$ f^i= \frac{q}{c}\,F^{i\beta}\,U_{\beta} $$
into
$$ \vec{f} = q\,\gamma\,\vec{E} + \frac{q}{c}\,\vec{U}\times\vec{B}\ .$$
Due April~17 in class (4~points).
 
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