Electromagnetic Wave Propagation in Conductors

A so-called Ohmic conductor is a medium that satisfies Ohm's law, which can be written in the form

$$j = \sigma\,E,$$ (619)

where $j$ is the current density (i.e., the current per unit area), $E$ the electric field-strength, and $\sigma$ a constant known as the conductivity of the medium in question. (Of course, the current generally flows in the same direction as the electric field.) The $z$-directed propagation of an electromagnetic wave, polarized in the $x$-direction, through an Ohmic conductor of conductivity $\sigma$ is governed by

$$\frac{\partial E_x}{\partial t} +\frac{\sigma}{\epsilon_0}\,E_x = - \frac{1}{\epsilon_0}\,\frac{\partial H_y}{\partial z},$$ (620)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z},$$ (621)

For a so-called good conductor, which satisfies the inequality $\sigma\gg \epsilon_0\,\omega$, the first term on the left-hand side of Equation (620) is negligible with respect to the second term, and the above two equations can be shown to reduce to

$$\frac{\partial E_x}{\partial t} = \frac{1}{\mu_0\,\sigma}\,\frac{\partial^2 E_x}{\partial z^2},$$ (622)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z}.$$ (623)

These equations can be solved to give (see Exercise 4)

$$E_x(z,t) = E_0\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d),$$ (624)

$$H_y(z,t) = E_0\,Z^{-1}\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d-\pi/4),$$ (625)

where

$$d = \left(\frac{2}{\mu_0\,\sigma\,\omega}\right)^{1/2},$$ (626)

and

$$Z = \left(\frac{\omega\,\mu_0}{\sigma}\right)^{1/2} = \left(\frac{\omega\,\epsilon_0}{\sigma}\right)^{1/2}\,Z_0.$$ (627)

Equations (624) and (625) indicate that the amplitude of an electromagnetic wave propagating through a conductor decays exponentially on a characteristic lengthscale, $d$, which is known as the skin-depth. Consequently, an electromagnetic wave cannot penetrate more than a few skin-depths into a conducting medium. Note that the skin-depth is smaller at higher frequencies. This implies that high frequency waves penetrate a shorter distance into a conductor than low frequency waves.

Consider a typical metallic conductor such as copper, whose electrical conductivity at room temperature is about $6\times 10^{7}\,(\Omega\,{\rm m})^{-1}$. Copper, therefore, acts as a good conductor for all electromagnetic waves of frequency below about $10^{18}\,{\rm Hz}$. The skin-depth in copper for such waves is thus

$$d = \sqrt{\frac{2}{\mu_0\,\sigma\,\omega}} \simeq \frac{6}{\sqrt{f({\rm Hz})}}\,{\rm cm}.$$ (628)

It follows that the skin-depth is about $6\,{\rm cm}$ at 1Hz, but only about 2mm at 1kHz. This gives rise to the so-called skin-effect in copper wires, by which an oscillating electromagnetic signal of increasing frequency, transmitted along such a wire, is confined to an increasingly narrow layer (whose thickness is of order the skin-depth) on the surface of the wire.

The conductivity of sea-water is only about $\sigma\simeq 5\,(\Omega\,{\rm m})^{-1}$. However, this is still sufficiently high for sea-water to act as a good conductor for all radio frequency electromagnetic waves (i.e., $f=\omega/2\pi < 1$GHz). The skin-depth at 1MHz ( $\lambda\sim 300$m) is about $0.2$m, whereas that at 1kHz ( $\lambda\sim 300$km) is still only about 7m. This obviously poses quite severe restrictions for radio communication with submerged submarines. Either the submarines have to come quite close to the surface to communicate (which is dangerous), or the communication must be performed with extremely low frequency (ELF) waves (i.e., $f< 100$Hz). Unfortunately, such waves have very large wavelengths ( $\lambda > 3000\,{\rm km}$), which means that they can only be efficiently generated by gigantic antennas.

According to Equation (625), the phase of the magnetic component of an electromagnetic wave propagating through a good conductor lags that of the electric component by $\pi/4$ radians. It follows that the mean energy flux into the conductor takes the form

$$\langle{\cal I}\rangle = \langle E_x\,H_y\rangle = \vert E_x\vert^2\,Z^{-1}\,\langle \cos(\omega\,t-z/d)\,\cos(\omega\,t-z/d-\pi/4)\rangle$$

$$= \frac{\vert E_x\vert^2}{\sqrt{8}\,Z},$$ (629)

where $\vert E_x\vert = E_0\,{\rm e}^{-z/d}$ is the amplitude of the electric component of the wave. The fact that the mean energy flux is positive indicates that part of the wave energy is absorbed by the conductor. In fact, the absorbed energy corresponds to the energy lost due to Ohmic heating in the conductor (see Exercise 5).

Note, from (627), that the impedance of a good conductor is far less than that of a vacuum (i.e., $Z\ll Z_0$). This implies that the ratio of the magnetic to the electric components of an electromagnetic wave propagating through a good conductor is far larger than that of a wave propagating through a vacuum.

Suppose that the region $z<0$ is a vacuum, and the region $z>0$ is occupied by a good conductor of conductivity $\sigma$. Let the wave electric and magnetic fields in the vacuum region take the form of the incident and reflected waves specified in (607) and (608). The wave electric and magnetic fields in the conductor are written

$$E_x(z,t) = E_t\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d+\phi_t),$$ (630)

$$H_y(z,t) = E_t\,Z_0^{-1}\,\alpha^{-1}\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d-\pi/4+\phi_t),$$ (631)

where $E_t$ is the amplitude of the decaying wave which penetrates into the conductor, $\phi_t$ is the phase of this wave with respect to the incident wave, and

$$\alpha = \frac{Z}{Z_0}=\left(\frac{\epsilon_0\,\omega}{\sigma}\right)^{1/2}\ll 1.$$ (632)

The appropriate matching conditions are the continuity of $E_x$ and $H_y$ at $z=0$: i.e.,

$$E_i\,\cos(\omega\,t) + E_r\,\cos(\omega\,t+\phi_r) = E_t\,\cos(\omega\,t+\phi_t),$$ (633)

$$\alpha\left[E_i\,\cos(\omega\,t) - E_r\,\cos(\omega\,t+\phi_r)\right] = E_t\,\cos(\omega\,t-\pi/4+\phi_t).$$ (634)

Equations (633) and (634), which must be satisfied at all times, can be solved, in the limit $\alpha\ll 1$, to give (see Exercise 6)

$$E_r \simeq -(1-\sqrt{2}\,\alpha)\,E_i,$$ (635)

$$\phi_r \simeq - \sqrt{2}\,\alpha,$$ (636)

$$E_t \simeq 2\,\alpha\,E_i,$$ (637)

$$\phi_t \simeq \pi/4.$$ (638)

Hence, the coefficient of reflection becomes

$$R \simeq\left(\frac{E_r}{E_i}\right)^2\simeq 1-2\,\sqrt{2}\,... ...a =1- \left(\frac{8\,\epsilon_0\,\omega}{\sigma}\right)^{1/2}.$$ (639)

According to the above analysis, a good conductor reflects a normally incident electromagnetic wave with a phase shift of almost $\pi$ radians (i.e., $E_r\simeq -E_i$). The coefficient of reflection is just less than unity, indicating that, whilst most of the incident energy is reflected by the conductor, a small fraction of it is absorbed.

High-quality metallic mirrors are generally coated in silver, whose conductivity is $6.3\times 10^7\,(\Omega\,{\rm m})^{-1}$. It follows, from (639), that at optical frequencies ( $\omega = 4\times 10^{15}\,{\rm rad./s}$) the coefficient of reflection of a silvered mirror is $R\simeq 93.3\%$. This implies that about $7\%$ of the light incident on the mirror is absorbed, rather than being reflected. This rather severe light loss can be problematic in instruments, such as astronomical telescopes, which are used to view faint objects.