Low-frequency and high-frequency approximations; the skin depth and the plasma
frequency
Our result, Eq. (32.42), for the index of refraction for metals predicts quite different characteristics for wave propagation at different frequencies. Let’s first see what happens at very low frequencies. If $\omega$ is small enough, we can approximate Eq. (32.42) by \begin{equation} \label{Eq:II:32:44} n^2=-i\,\frac{\sigma}{\epsO\omega}. \end{equation}  Now, as you can check by taking the square,2 \begin{equation} \sqrt{-i}=\frac{1-i}{\sqrt{2}};\notag \end{equation}  so for low frequencies, \begin{equation} \label{Eq:II:32:45} n=\sqrt{\sigma/2\epsO\omega}\,(1-i). \end{equation}  The real and imaginary parts of $n$ have the same magnitude. With such a large imaginary part to $n$, the wave is rapidly attenuated in the metal. Referring to Eq. (32.36), the amplitude of a wave going in the z-direction decreases as \begin{equation} \label{Eq:II:32:46} \exp\bigl[-\sqrt{\sigma\omega/2\epsO c^2}\,z\bigr]. \end{equation}  Let’s write this as \begin{equation} \label{Eq:II:32:47} e^{-z/\delta}, \end{equation}  where $\delta$ is then the distance in which the wave amplitude decreases by the factor $e^{-1}=1/2.72$—or roughly one-third. The amplitude of such a wave as a function of $z$ is shown in Fig. 32–3. Since electromagnetic waves will penetrate into a metal only this distance, $\delta$ is called the skin depth. It is given by \begin{equation} \label{Eq:II:32:48} \delta=\sqrt{2\epsO c^2/\sigma\omega}. \end{equation}  Now what do we mean by “low” frequencies? Looking at Eq. (32.42), we see that it can be approximated by Eq. (32.44) only if $\omega\tau$ is much less than one and if $\omega\epsO/\sigma$ is also much less than one—that is, our low-frequency approximation applies when \begin{equation} \omega\ll\frac{1}{\tau}\notag \end{equation}  and \begin{equation} \label{Eq:II:32:49} \omega\ll\frac{\sigma}{\epsO}. \end{equation}  Let’s see what frequencies these correspond to for a typical metal like copper. We compute $\tau$ by using Eq. (32.43), and $\sigma/\epsO$, by using the measured conductivity. We take the following data from a handbook:  \begin{align*}  &\sigma=5.76\times10^7\text{ (ohm$\cdot$meter)}^{-1},\\[2pt]  &\text{atomic weight${}=63.5$ grams},\\[2pt]  &\text{density${}=8.9$ grams$\cdot$cm$^{-3}$},\\[2pt]  &\text{Avogadro’s number${}=6.02\times10^{23}$ (gram atomic weight)$^{-1}$}. \end{align*} 
 \begin{align*}  &\sigma=5.76\times10^7\text{ (ohm$\cdot$meter)}^{-1},\\[2pt]  &\text{atomic weight${}=63.5$ grams},\\[2pt]  &\text{density${}=8.9$ grams$\cdot$cm$^{-3}$},\\[2pt]  &\text{Avogadro’s number${}=$}\\[2pt]  &\phantom{Av}\text{$6.02\times 10^{23}$ (gram atomic weight)$^{-1}$}. \end{align*}  If we assume that there is one free electron per atom, then the number of electrons per cubic meter is \begin{equation*} N=8.5\times10^{28}\text{ meter}^{-3}. \end{equation*}  Using \begin{align*} q_e &=1.6\times10^{-19}\text{ coulomb},\\[2pt] \epsO &=8.85\times10^{-12}\text{ farad}\cdot\text{meter}^{-1},\\[2pt] m &=9.11\times10^{-31}\text{ kg}, \end{align*}  we get \begin{align*} \tau &=2.4\times10^{-14}\text{ sec},\\[5pt] \frac{1}{\tau} &=4.1\times10^{13}\text{ sec}^{-1},\\[5pt] \frac{\sigma}{\epsO} &=6.5\times10^{18}\text{ sec}^{-1}. \end{align*}  So for frequencies less than about $10^{12}$ cycles per second, copper will have the “low-frequency” behavior we describe (that means for waves whose free-space wavelength is longer than $0.3$ millimeters—very short radio waves!). For these waves, the skin depth in copper is \begin{equation*} \delta=\sqrt{\frac{0.028\:\text{m}^2\cdot\text{sec}^{-1}}{\omega}} \end{equation*}  For microwaves of $10{,}000$ megacycles per second ($3$-cm waves) \begin{equation*} \delta=6.7\times10^{-5}\text{ cm}. \end{equation*}  The wave penetrates a very small distance. We can see from this why in studying cavities (or waveguides) we needed to worry only about the fields inside the cavity, and not in the metal or outside the cavity. Also, we see why the losses in a cavity are reduced by a thin plating of silver or gold. The losses come from the current, which are appreciable only in a thin layer equal to the skin depth. Suppose we look now at the index of a metal like copper at high frequencies. For very high frequencies $\omega\tau$ is much greater than one, and Eq. (32.42) is well approximated by \begin{equation} \label{Eq:II:32:50} n^2=1-\frac{\sigma}{\epsO\omega^2\tau}. \end{equation}  For waves of high frequencies the index of a metal becomes real—and less than one! This is also evident from Eq. (32.38) if the dissipation term with $\gamma$ is neglected, as can be done for very large $\omega$. Equation (32.38) gives \begin{equation} \label{Eq:II:32:51} n^2=1-\frac{Nq_e^2}{m\epsO\omega^2}, \end{equation}  which is, of course, the same as Eq. (32.50). We have seen before the quantity $Nq_e^2/m\epsO$, which we called the square of the plasma frequency (Section 7–3): \begin{equation*} \omega_p^2=\frac{Nq_e^2}{\epsO m}, \end{equation*}  so we can write Eq. (32.50) or Eq. (32.51) as \begin{equation*} n^2=1-\biggl(\frac{\omega_p}{\omega}\biggr)^2. \end{equation*}  The plasma frequency is a kind of “critical” frequency. For $\omega<\omega_p$ the index of a metal has an imaginary part, and waves are attenuated; but for $\omega\gg\omega_p$ the index is real, and the metal becomes transparent. You know, of course, that metals are reasonably transparent to x-rays. But some metals are even transparent in the ultraviolet. In Table 32–3 we give for several metals the experimental observed wavelength at which they begin to become transparent. In the second column we give the calculated critical wavelength $\lambda_p=2\pi c/\omega_p$. Considering that the experimental wavelength is not too well defined, the fit of the theory is fairly good. You may wonder why the plasma frequency $\omega_p$ should have anything to do with the propagation of electromagnetic waves in metals. The plasma frequency came up in Chapter 7 as the natural frequency of density oscillations of the free electrons. (A clump of electrons is repelled by electric forces, and the inertia of the electrons leads to an oscillation of density.) So longitudinal plasma waves are resonant at $\omega_p$. But we are now talking about transverse electromagnetic waves, and we have found that transverse waves are absorbed for frequencies below $\omega_p$. (It’s an interesting and not accidental coincidence.) Although we have been talking about wave propagation in metals, you appreciate by this time the universality of the phenomena of physics—that it doesn’t make any difference whether the free electrons are in a metal or whether they are in the plasma of the ionosphere of the earth, or in the atmosphere of a star. To understand radio propagation in the ionosphere, we can use the same expressions—using, of course, the proper values for $N$ and $\tau$. We can see now why long radio waves are absorbed or reflected by the ionosphere, whereas short waves go right through. (Short waves must be used for communication with satellites.)  We have talked about the high- and low-frequency extremes for wave propagation in metals. For the in-between frequencies the full-blown formula of Eq. (32.42) must be used. In general, the index will have real and imaginary parts; the wave is attenuated as it propagates into the metal. For very thin layers, metals are somewhat transparent even at optical frequencies. As an example, special goggles for people who work around high-temperature furnaces are made by evaporating a thin layer of gold on glass. The visible light is transmitted fairly well—with a strong green tinge—but the infrared is strongly absorbed. Finally, it cannot have escaped the reader that many of these formulas resemble in some ways those for the dielectric constant $\kappa$ discussed in Chapter 10. The dielectric constant $\kappa$ measures the response of the material to a constant field, that is, for $\omega=0$. If you look carefully at the definition of $n$ and $\kappa$ you see that $\kappa$ is simply the limit of $n^2$ as $\omega\to0$. Indeed, placing $\omega=0$ and $n^2=\kappa$ in equations of this chapter will reproduce the equations of the theory of the dielectric constant of Chapter 11.