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{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 11}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Wednesday, April 2, 2003} \\ 
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\centerline{\bf PROBLEM 32}
\centerline{({\small Griffiths Problem 12.59})}
\medskip

A particle of mass $m$ collides elastically with an identical 
particle at rest. {\it Classically}, the outgoing trajectories 
always make an angle of $90^{\circ}$. Calculate this angle 
{\it relativistically}, in terms of $\phi$, the scattering angle, 
and $v$, the speed, in the center-of-momentum frame.
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\centerline{\bf PROBLEM 33}

The electromagnetic field tensor was defined in class as
$$
    F_{\mu\nu}=\partial_{\mu}A_{\nu}-
	       \partial_{\nu}A_{\mu}\;,
$$
and all of its six non-vanishing components were evaluated
in terms of the electric and magnetic fields. Compute all 
the components of the {\bf dual} electromagnetic tensor 
defined by
$$
    \widetilde{F}^{\mu\nu}\equiv G^{\mu\nu}=\frac{1}{2}
    \epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta} \;,
$$
in terms of ${\bf E}$ and ${\bf B}$. Note that 
$\epsilon^{\mu\nu\alpha\beta}$ is the four-dimensional
generalization of the Levi-Civita symbol; that is,
$\epsilon^{\mu\nu\alpha\beta}\!=\!\pm1$ if $\mu\nu\alpha\beta$
is an even(odd) permutation of $0123$, and zero otherwise.
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\centerline{\bf PROBLEM 34}
\centerline{({\small Griffiths Problem 12.50})}

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.

\begin{description}
\item{a)} Compute the three Lorentz invariants from the
	  contraction of the following tensors:
	  $$
            F^{\mu\nu}F_{\mu\nu}\;,
	    G^{\mu\nu}G_{\mu\nu}\;, 
	    \; {\rm and} \;
            F^{\mu\nu}G_{\mu\nu}\;,
	  $$
	  in terms of the electric and magnetic 
	  (${\bf E}$ and ${\bf B}$) fields.

\item{b)} Suppose that in one inertial frame
	  ${\bf B}\!=\!0$ but ${\bf E}\!\ne\!0$ 
	  (at some point $P$). Is it possible to
	  find another system in which the
	  {\it electric field} is zero at $P$?
\end{description}	  
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\centerline{\bf PROBLEM 27}
\medskip

An electron in its own rest frame has an energy equal to 
$E\!=\!m_{0}c^{2}$ and zero linear momentum ${\bf p}\!=\!0$. 
Compute the energy and momentum of the electron as measured 
by an observer in the {\it ``laboratory''} frame. Note that 
in the laboratory frame the electron is moving with constant 
linear velocity $v$ along the x-axis; that is, 
${\bf v}\!=\!v{\hat{\bf x}}$. Show that while $E$ and $p$ 
are different in both frames, the quantity $E^{2}-(pc)^{2}$ 
is invariant.

\medskip
\noindent
{\bf Hint:} Under a Lorentz transformation the energy (momentum) 
	    transforms as the timelike (spacelike) component of 
	    a four-vector.
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\centerline{\bf PROBLEM 28}

Two electrons are moving toward each other. As measured in the
laboratory frame electron 1 has a speed of $v_{1}\!=\!\beta_{1}c$
and electron 2 of $v_{2}\!=\!\beta_{2}c$. What is the speed of
electron 1 in the rest frame of electron 2?

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\centerline{\bf PROBLEM 29}

Cosmic rays are very energetically particles (such as protons)
that occasionally collide with the Earth's atmosphere producing
a copious amount of pions---strongly interacting particles with
a finite lifetime. For example, charged pions, which decay via 
the weak interaction into muons and neutrinos, have a lifetime 
of $\tau\!=\!2.6\times10^{-8}$~s. Compute the total distance 
traveled by a pion moving at 99\% of the speed of light as 
measured by a {\it ``terrestrial''} observer.

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