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%TCIDATA{Created=Sun Sep 01 12:22:08 1996}
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\markright{{\sf Waves in a Conducting Medium}}
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\begin{document}


\section{7.7 \thinspace \thinspace \thinspace \thinspace \thinspace Waves in
a Conducting Medium.}

\subsection{7.7.1}

\quad \thinspace \textbf{a.} In this section, instead of a complex $%
\varepsilon ,$ we use a real conductivity $\sigma .$ We also keep a real $%
\mu \neq 1$ in general. In a typical metal, $\varepsilon ,\sigma $ and $\mu $
are essentially constant up to $\omega =10^{10}$ sec$^{-1}$ at least. A
typical value of $\sigma $ is the conductivity of Cu at room temperature, $%
\sigma \simeq 5\times 10^{17}$ sec$^{-1}\simeq 5\times 10^{7}$ (ohm-m)$%
^{-1}. $ Thus $\sigma /\omega $ can be a very large number in the frequency
range $10-10^{12}$ sec$^{-1}$ which is of primary interest in this section.

\smallskip\ 

\textbf{b. }The relation $k^2=\mu \left( \omega ^2/c^2\right) \left(
\varepsilon +i4\pi \sigma /\omega \right) $ gives

\[
k=\sqrt{\varepsilon \mu }\,\frac \omega c\left[ \sqrt{\frac{P+1}2}+i\sqrt{%
\frac{P-1}2}\,\,\right] ,\quad \quad \text{with}\,\,\,\,P=\sqrt{1+\left( 
\frac{4\pi \sigma }{\omega \varepsilon }\right) ^2}. 
\]
The formulae here, apart from the different notation, are essentially the
same as discussed in 7.5.8, where the ratio $4\pi \sigma /\omega \varepsilon 
$ is called Im$\varepsilon /$Re$\varepsilon .$

\smallskip\ 

\textbf{c. }If $4\pi \sigma /\omega \varepsilon $ $\ll 1,$ it is customary
to set $k=n\left( \omega /c\right) +i\alpha /2.$ One finds $n=\sqrt{%
\varepsilon \mu },$ as usual, and $\alpha =4\pi \sigma n/\varepsilon c.$
This is case (i) of 7.5.8b.

\smallskip\ 

\textbf{d.} The case of main interest here is that of a good conductor, $%
4\pi \sigma /\omega \varepsilon $ $\gg 1.$ This is case (iii) of 7.5.8b. Now 
$k^{2}$ is nearly imaginary and it is customary to put $k=\left( 1+i\right)
/\delta ,$ where $\delta $ is the \textsl{skin depth} or \textsl{penetration
depth}:

\[
\delta =\frac c{\sqrt{2\pi \mu \omega \sigma }}\,\,\,\,\,\,\,\,\,\left[ =%
\sqrt{\frac 2{\mu \omega \sigma }}\,\,\,\,\,\text{in MKS.}\right] 
\]

\smallskip\ 

\textbf{e.} For Cu, $\delta $ decreases from $0.85$ cm at $\omega
=60\,\,\,(2\pi /\sec )$ to $0.71\times 10^{-3}\,$ cm at $\omega =2\pi \times
10^{8}/\sec .$

\newpage

\subsection{7.7.2}

\quad \thinspace \textbf{a}. Another way to characterize the complex wave
number $k$ of 7.7.1b is through its modulus $\left| k\right| $ and phase $%
\phi :$

\[
\left| k\right| =\sqrt{\varepsilon \mu }\,\frac \omega c\left[ 1+\left( 
\frac{4\pi \sigma }{\omega \varepsilon }\right) ^2\right]
^{1/4},\,\,\,\,\,\,\,\tan 2\phi =\frac{4\pi \sigma }{\omega \varepsilon }. 
\]

\smallskip\ 

\textbf{b. }The electric and magnetic fields are related by $\mathbf{H=}$ $%
\left( c/\omega \mu \right) \mathbf{k\times E.}$ Thus the phase of $k,\phi ,$
is also the phase lag of $\mathbf{H}$ with respect to $\mathbf{E}$ and
approaches $45^{\circ }$ for a good conductor.

\smallskip\ 

\textbf{c.} It is still true that the directions of $\mathbf{E,}$ $\mathbf{H}
$ and $\mathbf{k}$ define three mutually orthogonal axes. (It is still true
that $\mathbf{k\cdot E=0}$ and $\mathbf{k\cdot H=0}$). Then 
\[
\frac{\left| \mathbf{H}\right| }{\left| \mathbf{E}\right| }=\frac{c}{\omega
\mu }\left| k\right| =\sqrt{\frac{\varepsilon }{\mu }}\,\,\left[ 1+\left( 
\frac{4\pi \sigma }{\omega \varepsilon }\right) ^{2}\right] ^{1/4}. 
\]
In a good conductor, $\left| \mathbf{H}\right| \gg \left| \mathbf{E}\right| $
and the field energy is almost entirely magnetic.

\smallskip \ 

\textbf{d. }Most of the formulae for reflection and refraction can be
written in such a way that they remain true if $\varepsilon ^{\prime }$ is
complex. The tangential component of the refracted wave vector $\mathbf{k}%
^{\prime }$ must still be real and equal to $k\sin \theta $, while the
perpendicular component $k_{\perp }^{\prime }$ is complex and is determined $%
\left( k\sin \theta \right) ^{2}+\left( k_{\perp }^{\prime }\right)
^{2}=\varepsilon ^{\prime }\mu ^{\prime }\omega ^{2}/c^{2}$. For instance,
the Fresnel formulae, Jackson's (7.39) and (7.41), are

\begin{itemize}
\item  For \textsl{s} polarization ($\mathbf{E}$ perpendicular to the plane
of incidence): 
\begin{equation}
\frac{E_{0}^{\prime }}{E_{0}}=\frac{2k_{\perp }/\mu }{(k_{\perp }/\mu
)+(k_{\perp }^{\prime }/\mu ^{\prime })}
\end{equation}
\begin{equation}
\frac{E_{0}^{\prime \prime }}{E_{0}}=\frac{(k_{\perp }/\mu )-(k_{\perp
}^{\prime }/\mu ^{\prime })}{(k_{\perp }/\mu )+(k_{\perp }^{\prime }/\mu
^{\prime })}
\end{equation}

\item  For \textsl{p} polarization ($\mathbf{B}$ perpendicular to the plane
of incidence): 
\begin{equation}
\frac{H_{0}^{\prime }}{H_{0}}=\frac{2k_{\perp }/\varepsilon }{(k_{\perp
}/\varepsilon )+(k_{\perp }^{\prime }/\varepsilon ^{\prime })}
\end{equation}
\begin{equation}
\frac{H_{0}^{\prime \prime }}{H_{0}}=\frac{(k_{\perp }/\varepsilon
)-(k_{\perp }^{\prime }/\varepsilon ^{\prime })}{(k_{\perp }/\varepsilon
)+(k_{\perp }^{\prime }/\varepsilon ^{\prime })}
\end{equation}
\end{itemize}

We can use $\varepsilon _{b}^{\prime }+4\pi i\sigma ^{\prime }/\omega $
instead of a complex $\varepsilon ^{\prime }$ and set $k^{\prime }=n^{\prime
}\left( \omega /c\right) +i\alpha ^{\prime }/2$. For a good conductor, $%
k^{\prime }\simeq (1+i)/\delta $ with $\delta =c/\sqrt{2\pi \mu ^{\prime
}\omega \sigma ^{\prime }}$ and the skin depth $\delta $ is such that  $%
k\delta \ll 1.$ This also implies that $k_{\perp }^{\prime }\simeq k^{\prime
}\simeq (1+i)/\delta ,$except at near-grazing incidence

\end{document}
