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\begin{center}
\textbf{Phys 743 - EM II}

\textbf{Final exam - 13 Dec. 1996}
\end{center}

\noindent \textbf{1.}

\textbf{(a) }Starting from the Lorentz transformation, obtain the formula
for the addition of velocities.

\textbf{(b) }Using the Lorentz transformation, or otherwise, discuss the
Doppler shift. In particular, obtain explicit formulas for the longitudinal
and transverse Doppler shifts (relative motion parallel and perpendicular to
the line of sight).

\smallskip

\noindent \textbf{2.}

The symmetric form of the electromagnetic energy-momentum tensor is 
\[
\Theta ^{\alpha \beta }=\frac{1}{4\pi }\left( g^{\alpha \mu }F_{\mu \nu
}F^{\nu \beta }+\frac{1}{4}\,g^{\alpha \beta }F_{\mu \nu }F^{\mu \nu
}\right) 
\]

\textbf{(a) }Express $\Theta ^{00}$ in terms $\mathbf{E}$ and $\mathbf{B}$.
What is its physical meaning?

\textbf{(b) }Express $\Theta ^{0a}$ (where $a=1,2,3$) in terms $\mathbf{E}$
and $\mathbf{B}$. What is its physical meaning?

\textbf{(c) }Similarly, find an explicit expression for the trace of this
tensor, $\Theta _{\;\alpha }^{\alpha },$ and discuss its physical meaning.

\smallskip

\noindent \textbf{3.}

The standard relativistic Lagrangian of a particle is 
\[
L=-mc^{2}\sqrt{1-v^{2}/c^{2}}-e\phi +(e/c)\,\mathbf{v\cdot A}
\]

\textbf{(a) }What is the action $\mathcal{A}$?

\textbf{(b) }Is $\mathcal{A}$ Lorentz invariant? Briefly justify your answer.

\textbf{(c) }Is $L$ Lorentz invariant? Briefly justify your answer.

\textbf{(d)} Show that $L$ is not gauge invariant, but $\mathcal{A}$ is
gauge invariant, under appropriate conditions.

\smallskip

\noindent \textbf{4.}

\textbf{(a) }A photon of frequency $\omega $ is incident on a free electron
at rest.\ If the photon's frequency after collision is $\omega ^{\prime },$
and its scattering angle in the laboratory is $\theta $, what is the
relation between $\omega ^{\prime },\,\omega $ and $\theta $ (Compton's
formula)? What are the energy and the momentum of the recoiling electron?

\textbf{(b) }The cosmic background radiation (CBR) provides an abundant
reservoir of low-frequency photons. High energy electrons of cosmic origin
collide with these photons. Find the maximum frequency $\omega ^{\prime }$
of a photon resulting from the collision of an electron of energy $E$ with a
CBR photon of frequency $\omega $ (inverse Compton scattering).

\smallskip

\noindent \textbf{5.}

\textbf{(a) }Discuss the time-averaged power radiated per unit solid angle
in non-relativistic motion by an electron moving perpendicular to a constant
magnetic field $\mathbf{B}.$ What is the total radiated power and what is
the angular distribution? What is the frequency and polarization of the
emitted radiation?

\textbf{(b) }Answer the same questions for a highly relativistic electron.
This is an open-ended question: be as quantitative or descriptive as you can
about the distribution in frequency and angles, the polarization, and the
total radiated power.

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