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In the limit
, there is a significant difference in
the response of a dielectric medium, depending on whether the lowest resonant
frequency is zero or nonzero. For insulators the lowest
resonant frequency is different from zero. In this case, the low frequency
refractive index is predominately real, and is also greater than unity.
Suppose, however, that some fraction of the electrons are ``free,'' in
the sense of having . In this situation, the low frequency dielectric
constant takes the form

(655) 
where is the contribution to
the refractive index from all of the other resonances, and
. Note that for
a conducting medium the contribution to the refractive index from the free
electrons is singular at . This singular behaviour can be
explained as follows. Consider the AmpèreMaxwell equation

(656) 
Let us assume that the medium in question obeys Ohm's law,
, and has a ``normal''
dielectric constant . Here, is the conductivity.
Assuming an
time dependence of all field quantities
the above equation yields

(657) 
Suppose, however, that we do not explicitly use Ohm's law but, instead, attribute
all of the properties of the medium to the dielectric constant. In this case,
the effective dielectric constant of the medium is equivalent to the term in
round brackets on the righthand side of the above equation. Thus,

(658) 
A comparison of this term with Eq. (4.27) yields the following expression for
the conductivity

(659) 
Thus, at low frequencies conductors possess predominately real conductivities
(i.e., the current remains in phase with the electric field). However, at
higher frequencies the conductivity becomes complex. At these sorts of frequencies
there is little meaningful distinction between a conductor and an insulator, since
the ``conductivity'' contribution to
appears as a resonant
amplitude just like the other contributions. For a good conductor, such as Copper,
the conductivity remains predominately real for all frequencies up to and including
those in the microwave region of the electromagnetic spectrum.
The conventional way in which to represent the complex refractive index of
a conducting medium (in the low frequency limit) is to write it in terms
of a real ``normal'' dielectric constant,
, and a real
conductivity, . Thus, from Eq. (4.30)

(660) 
For a poor conductor (
) we find

(661) 
In this limit
, and the attenuation
of the wave, which is governed by [see Eq. (4.8)], is
independent of the frequency. Thus, for a poor conductor the wave is
basically the same as a wave propagating through a conventional dielectric
with dielectric constant , except that the wave attenuates gradually
over a distance of very many wavelengths.
For a good conductor
(
)

(662) 
It follows from Eq. (4.5) that

(663) 
Thus, the phase of the magnetic field lags that of the electric field by
. Moreover, the magnitude of is much larger than that of
(since
). It follows that the
field energy is almost entirely magnetic in nature. It is clear that an electromagnetic wave propagating through a good conductor has markedly different properties
to a wave propagating through a conventional
dielectric. For a wave propagating
in the direction, the amplitudes of the electric and magnetic fields
attenuate like , where

(664) 
This quantity is known as the skin depth. It is clear that an electromagnetic
wave incident on a conducting medium will not penetrate more than a few skin depths
into that medium.
Next: The high frequency limit
Up: Electromagnetic wave propagation in
Previous: Anomalous dispersion and resonant
Richard Fitzpatrick
20020518