\item Draw conclusions concerning optical applications.
\end{enumerate}
-\section{\ldots}
-
-\ldots
+\section{Dielectric function of the free electron gas}
\begin{enumerate}
- \item \ldots
- \item \ldots
+ \item Derive an expression for the dieletric function $\epsilon(\omega)$
+ of the free electron gas.
+ {\bf Hint:} The equation of motion for a free electron
+ (position vector $x$) in an electric field $E$ is given by
+ $m\frac{d^2x}{dt^2}=-eE$.
+ For an electric field which has a
+ $e^{-i\omega t}$ dependance on time
+ the ansatz $x=x_0 e^{-i\omega t}$ is suitable
+ to solve the equation of motion.
+ What is the dipole moment of that electron?
+ Now write down the polarization $P$ which is defined as
+ the dipole moment of all electrons per volume.
+ As known from electro statics the polarization is connected
+ to the dielectric constant by
+ $\epsilon\epsilon_0E=\epsilon_0E+P$.
+ \item Rewrite $\epsilon(\omega)$ using the plasma frequency $\omega_p$
+ defined as $\omega_p^2=\frac{ne^2}{\epsilon_0m}$
+ ($n$: electron density, $e$: electron charge,
+ $\epsilon_0$: vacuum premitivity, $m$: electron mass).
+ Sketch $\epsilon(\omega)$ against $\frac{\omega}{\omega_p}$.
+ Explain what is happening to electromagnetic waves in the regions
+ $\frac{\omega}{\omega_p}<1$ and $\frac{\omega}{\omega_p}>1$.
\end{enumerate}
\end{document}