6.2.5 Skin depth and surface resistance

The skin depth $\delta =1/\sqrt{\pi f\mu \sigma }$ falls as $1/\sqrt{f}$. For copper, $\sigma =5.8\times 10^{7}S/m$, it is

Figure 6.2.7. Seawater follows the same $1/\sqrt{f}$ line up to some tens of megahertz and leaves it near $1\mathrm{GHz}$, where it stops being a good conductor and $1/\alpha$ levels off at the low-loss value $1/83.8\mathrm{m}=1.19\mathrm{cm}$.

Slide $f$ below from $50\mathrm{Hz}$ to $10\mathrm{GHz}$ and watch the skin depth of copper shrink as $1/\sqrt{f}$, from $9.35\mathrm{mm}$ to $0.661\mu \mathrm{m}$, while $\alpha =\beta$.

Why the current crowds to the surface

A wave entering a conductor drives a current $J=\sigma E$ which, like the field, decays as $e^{-z/\delta }$. In circuit language this is the skin effect: the eddy currents induced by the changing magnetic field (Faraday's law, Section 5.3) oppose the current in the interior of a conductor and push it to the surface. The faster the field changes, the thinner the layer that carries the current.

Surface resistance

A conductor much thicker than $\delta$ behaves as if a sheet of thickness $\delta$ carried the whole current uniformly. Its resistance per square, a piece of surface as long as it is wide, is the surface resistance.

That $R_s$ is truly the resistance of the wall, and not only a convenient name, is shown in Exercise 6.2.5: the total current per unit width flowing in a thick wall equals the tangential $H_0$ at its surface, and the power entering the wall is $\frac{1}{2}|H_0{|}^2{R}_s$ per square metre. This is the recipe for the losses of every waveguide and cavity of Chapter 7.

⚙Application 6.2.1 · Talking to a submarine

A message to a submerged submarine has to travel through seawater, and seawater ($\sigma \approx 4S/m$, ${\epsilon }_r\approx 81$) has $\omega \epsilon =2\pi f\times 81\times 10^{-9}/36\pi =4.5\times 10^{-9}f$, so that

$$\frac{\sigma }{\omega \epsilon }=\frac{888.9}{f[\mathrm{MHz}]}.\text{\hspace{1em}(6.2.30)}$$

Below about $9\mathrm{MHz}$ this exceeds $100$ and the sea is a good conductor. A wave entering it from the surface decays with the skin depth, and the loss after a depth $d$ is

$$\delta =\frac{1}{\sqrt{\pi f{\mu }_0\sigma }},\text{loss}=8.686\frac{d}{\delta }\mathrm{dB}.\text{\hspace{1em}(6.2.31)}$$

Table 6.2.2 and Figure 6.2.8 show what this means. With a receiver that tolerates $60\mathrm{dB}$ less than the field just below the surface, a $3\mathrm{MHz}$ signal is lost after one metre, a VLF signal at $20\mathrm{kHz}$ reaches $12\mathrm{m}$ and an ELF signal at $76\mathrm{Hz}$ reaches $199\mathrm{m}$.

Table 6.2.2. Seawater ($\sigma =4S/m$) as a good conductor.

$f$	$76\mathrm{Hz}$ (ELF)	$20\mathrm{kHz}$ (VLF)	$3\mathrm{MHz}$ (HF)
$\delta$	$28.9\mathrm{m}$	$1.78\mathrm{m}$	$0.145\mathrm{m}$
dB/m	$0.30$	$4.88$	$59.8$
${\lambda }_{\mathrm{sea}}=2\pi \delta$	$181\mathrm{m}$	$11.2\mathrm{m}$	$0.91\mathrm{m}$
depth for $60\mathrm{dB}$	$199\mathrm{m}$	$12.3\mathrm{m}$	$1.0\mathrm{m}$

The price of depth is bandwidth. Only frequencies far below $1\mathrm{MHz}$ get through, and a carrier of some kilohertz or some tens of hertz can carry very little information: VLF broadcasts carry a few hundred bits per second, tens of characters per second, ELF only a few characters per minute. A submarine therefore listens near periscope depth on a trailing wire antenna (VLF), or receives a short ELF "call me" message at depth and comes up to receive the rest.

Real systems

VLF shore stations work at $15$ to $30\mathrm{kHz}$ with antenna masts several hundred metres high, for example the US Navy's Cutler station in Maine at $24\mathrm{kHz}$. ELF needs antennas that are tens of kilometres long: the US Navy's $76\mathrm{Hz}$ system in Wisconsin and Michigan (closed in 2004) and the Russian ZEVS at $82\mathrm{Hz}$ used ground electrodes tens of kilometres apart. At $76\mathrm{Hz}$ the wavelength in air is $3950\mathrm{km}$, but in the sea it is only $181\mathrm{m}$, and the same $60\mathrm{dB}$ margin reaches $199\mathrm{m}$ instead of $12\mathrm{m}$ at VLF.