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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 03:40:34 1997}
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\section{Electromagnetic Waves in Matter\label%
{Electromagnetic Waves in Matter}}

The wave equations in matter, with $c=(\varepsilon \mu )^{-1/2}$ are: 
\begin{eqnarray*}
\left( \nabla ^{2}-\varepsilon \mu \frac{\partial ^{2}}{\partial t^{2}}-%
\frac{\mu }{\rho }\frac{\partial }{\partial t}\right) \vec{E} &=&0~ \\
\left( \nabla ^{2}-\varepsilon \mu \frac{\partial ^{2}}{\partial t^{2}}-%
\frac{\mu }{\rho }\frac{\partial }{\partial t}\right) \vec{B} &=&0
\end{eqnarray*}
These give, after substitution of monochromatic plane waves: 
\[
\vec{E}=E\exp (i(\vec{k}\cdot \vec{r}-\omega t)) 
\]
and 
\[
\vec{B}=B\exp (i(\vec{k}\cdot \vec{r}-\omega t)) 
\]
the \textit{dispersion relation}: 
\[
k^{2}=\varepsilon \mu \omega ^{2}+\frac{i\mu \omega }{\rho } 
\]
The first term arises from the displacement current, the second from the
conductance current. If $k$ is written as $k:=k^{\prime }+ik^{\prime \prime
} $ we have: 
\[
k^{\prime }=\omega \sqrt{\frac{1}{2}\varepsilon \mu }\sqrt{1+\sqrt{1+\frac{1%
}{(\rho \varepsilon \omega )^{2}}}}~~~\text{and}~~~k^{\prime \prime }=\omega 
\sqrt{\frac{1}{2}\varepsilon \mu }\sqrt{-1+\sqrt{1+\frac{1}{(\rho
\varepsilon \omega )^{2}}}} 
\]
This results in a damped wave: 
\[
\vec{E}=E\exp (-k^{\prime \prime }\vec{n}\cdot \vec{r})\exp (i(k^{\prime }%
\vec{n}\cdot \vec{r}-\omega t)) 
\]
If the material is a good conductor, the wave vanishes after approximately
one wavelength, 
\[
k=(1+i)\sqrt{\frac{\mu \omega }{2\rho }} 
\]

\subsection{\ Related Reference Topics}

{\small \hyperref{Calculus}{}{}{../Mathematics/CAL_contents.tex}}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Matrix Algebra}{}{}{../Mathematics/MA_contents.tex}}

{\small \hyperref{Table of Integrals}{}{}{%
../Mathematics/TAB_integrals.tex#table of integrals}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Matrix Algebra}{}{}{../../Help/DM8.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

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