Waves in Conductors and Skin Depth

The Physical Picture

Imagine sending a radio wave into a metal block. In a perfect insulator, the wave would travel through indefinitely, losing no energy. In a conductor, however, the electric field drives free electrons into motion, creating a current. This current dissipates energy as heat via Joule heating. Consequently, the wave loses amplitude as it penetrates the material. The distance over which the field amplitude drops by a factor of \( 1/e \) is called the skin depth. For good conductors like copper, this depth is very small at radio frequencies, meaning the wave is effectively absorbed in a thin layer near the surface. Furthermore, the magnetic field does not stay in phase with the electric field; it lags by a specific angle, a direct consequence of the energy dissipation mechanism.

Complex Wavevector and Propagation

Starting from Maxwell’s equations in a linear, homogeneous medium with conductivity \( \sigma \), we substitute the free current density \( \vec{J}_f = \sigma \vec{E} \) into Ampère’s law. For a monochromatic plane wave, this leads to a dispersion relation where the square of the complex wavevector \( \tilde{k} \) is given by:

Here, \( \mu \) is the permeability, \( \varepsilon \) is the permittivity, and \( \omega \) is the angular frequency. The imaginary term \( i\,\mu\sigma\omega \) arises from the conduction current. We define the complex wavevector as \( \tilde{k} = k + i\kappa \), where \( k \) is the real part governing the phase oscillation and \( \kappa \) is the imaginary part governing the exponential decay. The electric field can be written in phasor form as \( \vec{E} = \vec{E}_0 e^{i(kz - \omega t)} e^{-\kappa z} \). The term \( e^{-\kappa z} \) represents the attenuation of the wave amplitude as it travels in the \( z \)-direction.

The Good Conductor Approximation

For most metals at optical and radio frequencies, the conduction current dominates over the displacement current. This condition is expressed as \( \sigma \gg \varepsilon\omega \). Under this assumption, the term \( \mu\varepsilon\omega^2 \) in the dispersion relation becomes negligible compared to \( i\,\mu\sigma\omega \). The equation simplifies to \( \tilde{k}^2 \approx i\,\mu\sigma\omega \). To solve for \( \tilde{k} \), we take the square root of \( i \). Since \( i = e^{i\pi/2} \), the square root is \( e^{i\pi/4} = \frac{1}{\sqrt{2}}(1 + i) \). Thus, \( \tilde{k} \approx \sqrt{\frac{\mu\sigma\omega}{2}} (1 + i) \). This implies that the real and imaginary parts are equal: \( k = \kappa \). We define the skin depth \( \delta \) as the inverse of this common value:

The electric field then takes the form \( \vec{E} = E_0 e^{-z/\delta} \cos\left(\frac{z}{\delta} - \omega t\right)\hat{\mathbf{x}} \). The field oscillates with a wavelength \( \lambda = 2\pi\delta \) and decays with a characteristic length \( \delta \). Note that \( \delta \) depends on the square root of the frequency; doubling the frequency reduces the skin depth by a factor of \( \sqrt{2} \).

Field Phase and Magnitude

In a lossless dielectric, the electric and magnetic fields are in phase, and their magnitudes are related by the intrinsic impedance of the medium. In a good conductor, the situation is different. Using Faraday’s law \( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \), we can determine the magnetic field. For a wave propagating in the \( z \)-direction with \( \vec{E} \) in the \( x \)-direction, the magnetic field \( \vec{B} \) is in the \( y \)-direction. The complex amplitude ratio is derived from the wavevector. In the good conductor limit, the magnitude ratio is:

More importantly, the phase relationship changes. The complex wavevector \( \tilde{k} \) has an argument of \( \pi/4 \). This results in the magnetic field lagging the electric field by \( \pi/4 \) radians (45 degrees). This phase shift is a direct signature of the dissipative nature of the medium. The energy flow, described by the Poynting vector, is no longer purely real; it has an imaginary component representing reactive power, but the real part indicates the net energy dissipated in the conductor.