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%TCIDATA{Created=Fri Sep 06 15:16:03 1996}
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\markright{{\sf Fields at a conductor's surface}}
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\begin{document}


\section{8.1\quad Fields at a conductor's surface.}

\subsection{8.1.1}

\quad \ \textbf{a.} In a \textsl{perfect} conductor there can be no electric
fields (in the macroscopic sense) and the only magnetic fields allowed are
time-independent (otherwise they would generate electric fields). In a 
\textsl{good} conductor the fields penetrate (effectively) a distance $%
\delta $, the skin depth; currents flow in the skin layer and dissipation
occurs.

\smallskip\ 

\textbf{b. }In the theory of wave guides and resonant cavities it is
customary to solve the boundary-value problem as if all the conductors were
perfect and then apply the modifications due to the finite penetration.
Obviously this method will give a good approximation only if the field
distortions and especially the energy losses due to the finite penetration
are small. It is of course possible, but much more complicated, to solve the
boundary value problem with real (finite) conductors forming the walls.

\smallskip\ 

\textbf{c. }The skin depth $\delta $ (see 7.7.1d) is $c/\sqrt{2\pi \mu
_{c}\omega \sigma },$ where $\mu _{c}$is the magnetic permeability of the
conductor. The condition of ``small penetration'' is more precisely that $%
\delta \ll \lambda $ with $\lambda =2\pi c/\omega ;$ it is thus equivalent
to $\left( 2\pi \right) ^{3/2}\sqrt{\sigma \mu _{c}}\gg \sqrt{\omega }.$
This implies that the displacement current can be neglected, because the
ratio of $(1/c)\partial \mathbf{E}_{c}/\partial t$ to $(4\pi /c)\mathbf{J=(}%
4\pi \sigma /c)\mathbf{E}_{c}$ is $\omega /4\pi \sigma ,$ which is of the
order ($\delta /\lambda )^{2}$ for $\mu _{c}\sim 1.$

\smallskip\ 

\textbf{d. }Another important simplification is that the surface can be
regarded as flat if the radius of curvature is greater than $\delta .$

\newpage

\subsection{8.1.2}

\textbf{\quad \thinspace a. }We can take over the results of Ch.7 if we
regard the conductor as a medium of dielectric constant $\varepsilon _{b}+$ $%
4\pi i\sigma /\omega ,$ which reduces just to $4\pi i\sigma /\omega $ by
neglecting the displacement current. Then the index of refraction is 
\[
n=\sqrt{\frac{4\pi \sigma \mu _{c}}{\omega }}\frac{(1+i)}{\sqrt{2}} 
\]
and $n\omega /c$ $=(1+i)/\delta .$

\smallskip\ 

\textbf{b. }The wave equation for any (cartesian) component of the fields in
the conductor can be written $\nabla ^2u+2iu/\delta ^2=0.$

\smallskip\ 

\textbf{c.} The variation of the inside fields in the directions parallel to
the surface must match that of the outside fields, which is typically on the
scale of $\lambda =2\pi c/\omega .$ Thus the second derivative with respect
to the normal coordinate $\xi $ is dominant in the wave equation and the
fields inside are well described by 
\[
u(\xi =0)\exp (-\xi /\delta )\exp (i\xi /\delta ). 
\]

\smallskip\ 

\textbf{d. }It follows from \textbf{c}. that all the fields inside are
determined simply from their values at the surface. The magnitudes of these
fields are related by the Maxwell curl equations 
\begin{equation}
\mathbf{E}_c=\frac c{4\pi \sigma }\nabla \times \mathbf{H}_c\quad \quad
\quad \quad \mathbf{H}_c=-\frac{ic}{\mu _c\omega }\nabla \times \mathbf{E}_c
\label{1}
\end{equation}

\smallskip\ 

\textbf{e. }To leading order in $\delta /\lambda ,$ 
\[
\mathbf{\nabla \times \quad }\text{can be replaced by \quad }\dfrac{(1-i)}{%
\delta }\,\mathbf{n\times } 
\]
(Note: $\mathbf{n}$ points along $-\xi ,$ to follow Jackson).

\newpage

\subsection{8.1.3}

\quad \thinspace \textbf{a. }From \textbf{d}. and \textbf{e.} of previous
page it follows that, to leading order in the small parameter $\delta
/\lambda \sim \sqrt{\omega /\sigma }\sim c/\sigma \delta $ (taking $\mu _c$
to be of order 1): 
\begin{equation}
\mathbf{E}_c=\frac c{4\pi \sigma }\,\frac{1-i}\delta \,\mathbf{n\times H}_c
\label{2}
\end{equation}
Thus $\mathbf{E}$ is one order smaller than $\mathbf{H}$ and is tangential
to leading order (provided that $\mathbf{H}_c$ has a tangential component).
Also 
\begin{equation}
\mathbf{H}_c=\frac{-ic}{\mu _c\omega \delta }\,(1-i)\,\mathbf{n\times E}_c
\label{3}
\end{equation}
which is consistent with (\ref{2}) and further indicates that the tangential
component of $\mathbf{H}$ (assuming that it exists) is one order larger than
the normal component. In summary 
\[
\mathbf{H}_{/\!/}\gg \mathbf{H}_{\perp }\sim \mathbf{E}_{/\!/}\gg \mathbf{E}%
_{\perp } 
\]

\smallskip\ 

\textbf{b. }This applies inside the conductor. The matching conditions imply
that $\mathbf{E}_{/\!/}$ and $\mathbf{H}_{/\!/}$ are the same on the
outside, and $\mathbf{H}_{\perp }$ differs only by a factor $\mu _c.$ On the
other hand $\mathbf{E}_{\perp }$ is larger outside by a factor $4\pi i\sigma
/\omega $ $\propto $ $(\lambda /\delta )^2$ and is thus comparable to $%
\mathbf{H}_{/\!/}.$

\smallskip\ 

\textbf{c. }The discontinuity of $\mathbf{E}_{\perp }$ is due, of course, to
a surface charge layer (of atomic thickness). This layer can be regarded as
a polarization charge or as accumulated free charge due to the current $%
\sigma \mathbf{E,}$ depending on the description chosen.

\smallskip\ 

\textbf{d. }A sketch of the fields near the surface is given on the next
page. In summary, to leading order the inside fields are 
\begin{equation}
\mathbf{H}_{c}\simeq \mathbf{H}_{/\!/}(\xi =0)e^{-\xi /\delta }e^{i\xi
/\delta }\,\,;\quad \quad \quad \mathbf{E}_{c}\simeq \sqrt{\frac{\mu
_{c}\omega }{8\pi \sigma }}(1-i)(\mathbf{n\times H}_{c})  \label{4}
\end{equation}

\newpage

\subsection{8.1.4}

\quad \thinspace \textbf{a.} Associated with $\mathbf{E}_{/\!/}$ there is a
current flowing in the skin depth. The integral of $\mathbf{J}_{/\!/}$ gives
the effective surface current 
\begin{equation}
\mathbf{K}_{eff}=\int_0^\infty \mathbf{J}_{/\!/}d\xi =\sigma \int_0^\infty 
\mathbf{E}_{/\!/}(\xi )d\xi .  \label{5}
\end{equation}

\smallskip\ 

\textbf{b. }In a perfect conductor $\delta \rightarrow 0$ and $\mathbf{K}$
is truly a surface current. $\mathbf{E}_{/\!/}$ and $\mathbf{H}_{\perp }$
vanish at the surface in this limit, while $\mathbf{H}_{/\!/}$ as well as $%
\mathbf{E}_{\perp }$ are discontinuous. One must keep in mind, however, that
the surface current, which in this limit is responsible for the
discontinuity of $\mathbf{H}_{\perp }$, is spread over the macroscopic depth 
$\delta ,$ while the surface charge, which is responsible for the
discontinuity of $\mathbf{E}_{\perp },$ is spread over a layer of
microscopic (atomic) depth. The surface current is \textsl{not} due to a
tangential flow of the surface charge.

\newpage

\subsection{8.1.5}

\quad \thinspace \textbf{a. }From (\ref{4}) and (\ref{5}) 
\begin{equation}
\frac{4\pi }c\mathbf{K}_{eff}=\mathbf{n\times H}_{/\!/}(0).  \label{6}
\end{equation}
This relation is quite general (independent of having $\mathbf{J}=\sigma 
\mathbf{E}$ and of all the approximations made above).

It follows simply from 
\[
\nabla \times \mathbf{H}=\frac{4\pi }{c}\mathbf{J} 
\]
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\smallskip \ 

\textbf{b. }All fields inside can be expressed in terms of $\mathbf{K}_{eff}$
instead of $\mathbf{H}_{/\!/}(0).$

\smallskip\ 

\textbf{c. }From the flow of the Poynting vector $-(c/8\pi )\func{Re}(%
\mathbf{n\cdot E\times H}^{*}),$ or from the integral of $(1/2)\mathbf{%
J\cdot E}^{*},$ one can compute the time-averaged rate of energy dissipation
in the skin depth. Expressed in terms of $\mathbf{K}_{eff}$ the result is,
per unit area 
\begin{equation}
\frac{dP}{da}=\frac 1{2\sigma \delta }\left| \mathbf{K}_{eff}\right| ^2.
\label{7}
\end{equation}

\smallskip\ 

\textbf{d. }If we define a surface impedance $Z_s$ by $\mathbf{K}%
_{eff}=Z_s^{-1}\mathbf{E}_{/\!/}(0),$ we see from (\ref{2}) and (\ref{6})
that $Z_s=$ $(1-i)/\sigma \delta ,$ as is appropriate for a layer of depth $%
\delta .$

\smallskip\ 

\textbf{e.} $Z_{s}$ plays in many ways the role of an effective impedance.
For instance, (\ref{7}) can be written 
\[
\frac{dP}{da}=\frac{1}{2}\left( \func{Re}Z_{s}\right) \left| \mathbf{K}%
_{eff}\right| ^{2}=\frac{1}{2}\left( \func{Re}Z_{s}^{-1}\right) \left| 
\mathbf{E}_{/\!/}\right| ^{2}. 
\]
These formulae are valid quite generally, since 
\[
-\func{Re}(\mathbf{n\cdot E\times H}^{*})/8\pi =\func{Re}(\mathbf{E\cdot
n\times H}^{*})/8\pi =(1/2)\func{Re}(\mathbf{E\cdot K}_{eff}^{*}) 
\]
by (\ref{6}).

\subsection{8.1.6}

\textbf{\quad \thinspace a. }We have emphasized in 5.a. and 5.e. that
certain relations hold independently of the validity of Ohm's law, $\mathbf{J%
}=\sigma \mathbf{E.}$ The reason for stressing this point is that the local
relation between $\mathbf{J}$ (or $\mathbf{D)}$ and $\mathbf{E}$ breaks down
when $\delta =c/\sqrt{2\pi \mu _c\omega \sigma }$ is less than the mean free
path $l.$ [One should refer back to sections 7.5.1c(ii) and 7.10.7c].

\smallskip\ 

\textbf{b. }Clearly the breakdown of the local relation $\mathbf{J}=\sigma 
\mathbf{E}$ will occur for very pure materials at low temperature, because
then $l$ is large and at the same time $\delta $ is small (recall that $%
\delta \propto \sigma ^{-1/2}$ and $\sigma \propto l).$ The breakdown occurs
first at high frequency (because $\delta \propto \sigma ^{-1/2}).$ Recall
however that the whole theory is developed for microwaves and
lower-frequency phenomena (typically $\omega \,\widetilde{<}$ $10^{12}$ sec$%
^{-1}).$

\smallskip\ 

\textbf{c. }When the penetration depth is not given by the ``classical''
formula $\delta =c/\sqrt{2\pi \mu _{c}\omega \sigma }$ one has the
``anomalous skin effect''. One can still define a $\QTR{sl}{K}_{eff}$ and a
surface impedance $Z_{s}$ as in 5.a. One can also define an effective ``%
\textsl{anomalous skin depth}''. The value of $Z_{s}$ depends in a detailed
way on the mechanism of electronic conduction in the material. Many
low-temperature experiments on metals are carried out in this ``anomalous''
regime and yield information on the electronic states near the Fermi level.
Further details are left to Solid State courses.

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