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%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 7}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, March 3, 2003} \\ 
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\centerline{\bf PROBLEM 19}
\medskip

Read and summarize briefly, in approximately five lines, the enclosed
article entitled ``{\it Experimental Verification of a Negative Index
of Refraction''} by Shelby, Smith, and Schultz. This article was
selected as part of {\bf Physics News in 2001}, a publication of the
American Physical Society devoted to the most important developments 
in Physics during 2001 (document also enclosed).
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\centerline{\bf PROBLEM 20}
\medskip

Consider an electromagnetic plane wave incident on
an interface that separates two media, as we did
in class. Now, however, let the electric (not the
 magnetic) field be perpendicular to the plane of
incidence.

\begin{description}
\item{a)} Write expressions for the wavenumber
          ${\bf k}$, the electric field ${\bf E}$,
          and the magnetic field ${\bf B}$ for the
          incident, reflected, and transmitted wave.

\item{b)} Write the four boundary conditions
          satisfied by the fields. Show that 
	  only two of them are independent.

\item{c)} By solving the above equations compute
          the reflections and transmission
          intensities. Make sure that they satisfy
          $R\!+\!T\!=\!1$.

\item{d)} Make a plot of the reflection and transmission
          intensities as a function of the incident angle
          using $n_{1}\!=\!1$, $n_{2}\!=\!1.5$, and
          $\mu_{1}\!=\!\mu_{2}\!=\!1$. Is there a
          Brewster's angle for this problem? That is,
          is there an incident angle for which $R$ vanishes
          for this configuration of fields?
\end{description}
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\centerline{\bf PROBLEM 21}
\medskip

Consider a linear medium of dielectric constant $\epsilon$
and permeability $\mu$. Further, assume that the energy
density ({\it i.e.,} the energy per unit volume) stored in
space is given by:
$$
        u = \frac{1}{8\pi}\left({\bf E}\cdot{\bf D}
          + {\bf B}\cdot{\bf H} \right) \;.
$$

\begin{description}
\item{a)} Starting from the above equation, and using
          Maxwell's equations, show that the energy
          satisfies a continuity like equation of the
          form:
          $$
            \frac{\partial u}{\partial t} +
            \nabla\cdot{\bf S}=-{\bf J}\cdot{\bf E}\;.
          $$
          Display an explicit form for the {\it Poynting}
          vector ${\bf S}$ in terms of the electromagnetic
          fields. What is the physical significance of the
          ${\bf J}\cdot{\bf E}$ term?

\item{b)} Compute the {\it time-averaged} Poynting vector
          for an electromagnetic plane wave of wave number
          ${\bf k}$ and frequency $\omega$. Note that the
          definition of the time-averaged Poynting vector
          is given by
          $$
           \langle{\bf S}\rangle_{\rm avg}=\frac{1}{T}
           \int_{0}^{T} {\bf S}({\bf x},t) dt\;, \qquad
           (T=2\pi/\omega) \;.
          $$
\end{description}
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\centerline{\bf PROBLEM 22}
\medskip

As a simple preamble to the {\bf Special Theory of Relativity},
consider an electron moving along the axis of a parallel-plate
capacitor that provides a constant electric field of magnitude
$E_{0}\!=\!10^{6}$V/m. Compute how long will it take for this
electron to reach the speed of light if it obeys Newton's 2nd
law of motion.
\vfill\eject
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