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{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 6}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, February 17, 2003} \\ 
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\centerline{\bf PROBLEM 16}
\centerline{\small{(Problem 9.9 Griffiths)}}

Write down the (real) electric and magnetic fields for a 
monochromatic plane wave of amplitude $E_{0}$, frequency 
$\omega$, and linear polarization $\hat{\bf n}$ that is:

\begin{description}
\item{a)} Traveling in the negative $x$ direction and
          polarized in the $z$ direction.

\item{b)} Traveling in the direction from the origin 
          to the point $(1,1,1)$ with polarization 
	  parallel to the $xz$ plane.
\end{description}

In each case sketch the wave and give the explicit
Cartesian components of $\hat{\bf k}$~and~$\hat{\bf n}$.
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\bigskip
\centerline{\bf PROBLEM 17 and 18}

Consider a magnetic field given by the following expression:
\begin{eqnarray}
  &&  B_{x}(x,y,z,t)=-\imath B_{0}
     \sin\left(\frac{\pi x}{a}\right) e^{i(kz-\omega t)}\;,\cr
  &&  B_{y}(x,y,z,t)\equiv 0\;, \cr
  &&  B_{z}(x,y,z,t)=B_{0}
     \cos\left(\frac{\pi x}{a}\right) e^{i(kz-\omega t)}\;. 
     \nonumber
\end{eqnarray}

\begin{description}
\item{a)} Use $\nabla\cdot{\bf B}\!=\!0$ to express the
          wavenumber $k$ in terms of $a$ (and perhaps
	  other well-known constants).

\item{b)} Use the Ampere-Maxwell law (with ${\bf J}=0$)
          to compute the electric field ${\bf E}$. Hint:
	  Assume that ${\bf E}({\bf r},t)={\bf E}({\bf r})
	  e^{-i\omega t}$.

\item{c)} Verify that the electric field ${\bf E}$
          satisfied Gauss' law (with $\rho=0$).

\item{d)} Use Faraday's law to express the frequency
          $\omega$ in terms of the wavenumber $k$.
\end{description}
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