Wave propagation through a conducting medium

In the limit $\omega\rightarrow 0$, there is a significant difference in the response of a dielectric medium, depending on whether the lowest resonant frequency is zero or non-zero. 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 $f_0$ of the electrons are ``free,'' in the sense of having $\omega_0=0$. In this situation, the low frequency dielectric constant takes the form

$$\epsilon(\omega) \equiv n^2(\omega) =n_0^{~2} + {\rm i}\,\fr... ...{\epsilon_0 m} \frac{f_0}{\omega\,(\gamma_0-{\rm i}\,\omega)},$$ (655)

where $n_0$ is the contribution to the refractive index from all of the other resonances, and $\gamma_0=\lim_{\omega_0\rightarrow 0} g_0\,\omega_0$. Note that for a conducting medium the contribution to the refractive index from the free electrons is singular at $\omega= 0$. This singular behaviour can be explained as follows. Consider the Ampère-Maxwell equation

$$\nabla\wedge{\bfm B} =\mu_0\left({\bfm j}_t + \frac{\partial {\bfm D}} {\partial t}\right).$$ (656)

Let us assume that the medium in question obeys Ohm's law, ${\bfm j}_t = \sigma {\bfm E}$, and has a ``normal'' dielectric constant $n_0^{~2}$. Here, $\sigma$ is the conductivity. Assuming an $\exp(-{\rm i}\,\omega t)$ time dependence of all field quantities the above equation yields

$$\frac{\nabla\wedge{\bfm B}}{\mu_0} = -{\rm i}\,\epsilon_0\,\... ... {\rm i}\, \frac{\sigma}{\epsilon_0 \,\omega}\right) {\bfm E}.$$ (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 right-hand side of the above equation. Thus,

$$\epsilon(\omega) \equiv n^2(\omega) = n_0^{~2} + {\rm i}\, \frac{\sigma}{\epsilon_0 \,\omega}.$$ (658)

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

$$\sigma = \frac{f_0 N e^2}{m(\gamma_0 - {\rm i}\,\omega)}.$$ (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 $\epsilon(\omega)$ 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, $\epsilon=n_0^{~2}$, and a real conductivity, $\sigma$. Thus, from Eq. (4.30)

$$n^2(\omega) = \epsilon + {\rm i}\,\frac{\sigma}{\epsilon_0\, \omega}.$$ (660)

For a poor conductor ( $\sigma/\epsilon\,\epsilon_0\,\omega\ll 1$) we find

$$k = n \,\frac{\omega}{c} \simeq \sqrt{\epsilon}\,\frac{\omeg... ... + {\rm i}\, \frac{\sigma}{2\sqrt{\epsilon} \,\epsilon_0\, c}.$$ (661)

In this limit ${\rm Re}(k) \gg {\rm Im}(k)$, and the attenuation of the wave, which is governed by ${\rm Im}(k)$ [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 $\epsilon$, except that the wave attenuates gradually over a distance of very many wavelengths. For a good conductor ( $\sigma/\epsilon\,\epsilon_0\,\omega\gg 1$)

$$k \simeq {\em e}^{{\rm i}\,\pi/4} \sqrt{\mu_0\, \sigma \,\omega}.$$ (662)

It follows from Eq. (4.5) that

$$\frac{c B_0}{E_0} = \frac{kc}{\omega} = {\em e}^{{\rm i}\,\pi/4} \sqrt{\frac{\sigma} {\epsilon_0\,\omega}}.$$ (663)

Thus, the phase of the magnetic field lags that of the electric field by $45^\circ$. Moreover, the magnitude of $c B_0$ is much larger than that of $E_0$ (since $\sigma/\epsilon_0\,\omega\gg \epsilon \stackrel {_{\normalsize >}}{_{\normalsize\sim}}1$). 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 $x$-direction, the amplitudes of the electric and magnetic fields attenuate like $\exp(-x/d)$, where

$$d = \sqrt{\frac{2}{\mu_0\,\sigma\,\omega}}.$$ (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.