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Skin depth $1/\sqrt{\pi f\mu\sigma}$ and the skin effect, plasma frequency $8.98\sqrt{n}$ Hz, reflection below cutoff, dispersion, and why metals shine.
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By the end of this lesson you will be able to compute skin depths in conductors, find plasma frequencies and cutoffs, and predict whether a wave is reflected, absorbed or transmitted by a conducting medium.
You know Maxwell's equations in matter, Ohm's law $\vec{J} = \sigma\vec{E}$, eddy currents, and that metals reflect light. You know how waves behave in insulators. This lesson lets the medium conduct: first ordinary metals, where waves die quickly, then plasmas of free electrons, which reflect low frequencies and pass high ones.
| Term | What it means |
|---|---|
| Skin depth | $\delta = 1/\sqrt{\pi f\mu\sigma}$, the distance over which a wave's amplitude in a conductor falls by $e$. |
| Skin effect | The crowding of alternating current into a surface layer about $\delta$ thick. |
| Complex wave number | $k = k_r + ik_i$, with $k_i$ giving exponential decay. |
| Plasma | A gas of free charges, such as the ionosphere or the electrons in a metal. |
| Plasma frequency | $\omega_p = \sqrt{ne^2/\varepsilon_0m}$, the natural oscillation frequency of free electrons. |
| Cutoff | The frequency below which waves cannot propagate in a medium. |
| Dispersion | Dependence of wave speed on frequency. |
In a conductor the wave's electric field drives a current $\sigma\vec{E}$, which adds a damping term to the wave equation. Plane-wave solutions then have a complex wave number. In a good conductor, where $\sigma \gg \omega\varepsilon$, the real and imaginary parts are equal, and the fields decay as $e^{-z/\delta}$ with skin depth
$$\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f\mu\sigma}}.$$
For copper, $\delta = 8.4$ mm at $60$ Hz and $65$ μm at $1$ MHz. Waves hitting a metal therefore penetrate only a skin depth before being absorbed or reflected; alternating currents crowd into a layer about $\delta$ thick at a wire's surface, the skin effect.
In a dilute plasma, such as the ionosphere, electrons move freely without collisions. Their response gives the dispersion relation
$$\omega^2 = \omega_p^2 + c^2k^2, \qquad \omega_p = \sqrt{\frac{ne^2}{\varepsilon_0m_e}},$$
so $f_p = 8.98\sqrt{n}$ Hz. Above the plasma frequency, waves propagate with group velocity $c\sqrt{1 - \omega_p^2/\omega^2}$, slower than light. Below it, $k$ is imaginary: the wave cannot propagate and is reflected. That is why metals, with $f_p$ in the ultraviolet, reflect visible light, and why the ionosphere reflects shortwave radio.
Another way: picture
Picture pushing on a crowd of free electrons with an oscillating field. If you push slowly, below their natural frequency, they have time to rearrange and cancel your field, and the wave bounces off. Push faster than they can respond, and the wave slips through. A metal's electrons are so dense that they cancel everything slower than ultraviolet; the ionosphere's are thin enough that only radio below about $10$ MHz is turned back.
Another way: steps
Checks. Skin depths must shrink with frequency and conductivity. Magnetic metals like iron, with large $\mu$, have much smaller skin depths than copper. A plasma's group velocity must be below $c$ and must go to zero at cutoff. And metals must have plasma frequencies in the ultraviolet, above visible light, or they would be transparent.
An oscillating field entering a metal drives currents, and those currents, by Faraday's and Ampère's laws, produce fields that oppose the original — eddy currents again. The deeper the wave goes, the more it is cancelled. The wave equation with the $\mu\sigma\,\partial E/\partial t$ term is a diffusion equation, and the skin depth is the distance the field diffuses in one cycle.
The energy is not lost mysteriously: it becomes Joule heat in the surface layer. A perfect conductor, $\sigma \to \infty$, has zero skin depth and reflects everything, while a real metal absorbs a small fraction, which is why microwave cavities made of copper lose energy slowly and superconducting cavities, with nearly zero surface resistance, hold it for millions of cycles.
Because alternating current crowds into a layer $\delta$ thick, a wire's AC resistance rises above its DC value once $\delta$ is smaller than the wire's radius. At $60$ Hz copper's skin depth is $8.4$ mm, so household wiring is unaffected; but the thick conductors of high-power transmission lines are made of strands or hollow tubes, since their centers would carry little current anyway.
At radio frequencies the effect dominates. Radio-frequency coils use Litz wire, many thin insulated strands woven together so each strand spends time at the surface. Microwave components are often silver- or gold-plated, since only the surface micrometers carry current, and the plating need be only a few skin depths thick.
A metal enclosure a few skin depths thick blocks electromagnetic waves: each skin depth reduces the amplitude by $e$, so ten skin depths reduce it by over $20{,}000$. At microwave frequencies, skin depths are micrometers, and even thin foil shields effectively. This is the Faraday cage principle extended to changing fields.
Low frequencies are harder. At $60$ Hz copper's skin depth is centimeters, so shielding power-line magnetic fields requires thick plates or high-permeability alloys, whose small skin depth comes from their large $\mu$. MRI rooms are lined with copper to keep out radio interference at the scanner's tens of megahertz, where a thin sheet suffices.
A metal's conduction electrons form a dense plasma, $n \sim 10^{29}$ m⁻³, with plasma frequencies around $10^{15}$ to $10^{16}$ Hz, in the ultraviolet. All visible light is below cutoff and is reflected: metals are shiny. Ultraviolet above the plasma frequency passes through; thin alkali metal films are transparent to ultraviolet, as Robert Wood observed in 1933.
Gold and copper look colored because their $d$-band electrons absorb blue and green light, adding to the free-electron response. Aluminum and silver reflect the whole visible spectrum evenly and look white-silver. Telescope mirrors are coated with aluminum or silver for visible light and gold for infrared.
Sunlight ionizes the upper atmosphere, creating layers with electron densities of $10^{10}$ to $10^{12}$ per cubic meter between about $60$ and $400$ km altitude. Their plasma frequencies range up to about $10$ MHz. Radio waves below that frequency sent straight up are reflected; at slant angles, somewhat higher frequencies reflect too. Shortwave radio bounces between ionosphere and ground to reach around the world.
The layers change with day and night and with the solar cycle, so broadcasters and amateur radio operators change frequencies accordingly. FM, television, cell and GPS signals, at tens of megahertz to gigahertz, pass straight through, which is why they reach satellites and why FM reception is limited to line of sight. GPS receivers must still correct for the ionosphere's slowing of the signal, using two frequencies to measure it.
Above cutoff, a plasma's refractive index $\sqrt{1 - f_p^2/f^2}$ is less than one, so the phase velocity exceeds $c$ while the group velocity, at which signals and energy travel, is less than $c$; their product is exactly $c^2$. Different frequencies travel at different speeds, and a pulse spreads.
Radio astronomers use this to weigh the interstellar medium. A pulsar's radio pulse arrives later at low frequencies than at high ones, because the thin electron plasma between the stars slows low frequencies more. The delay, the dispersion measure, gives the total number of electrons along the line of sight, and it is how fast radio bursts were recognized as coming from other galaxies.
Seawater, with conductivity about $4$ S/m, has a skin depth of about $8$ m at $1$ kHz and only $25$ cm at $1$ MHz. Ordinary radio cannot reach submerged submarines. The U.S. Navy therefore used extremely low frequencies, $76$ Hz, from transmitters in Wisconsin and Michigan, penetrating tens of meters, and still uses very low frequencies around $20$ kHz; the data rates are tiny because the frequencies are so low.
The same physics limits communication and sensing underground and underwater. Ground-penetrating radar works only in dry, poorly conducting soils; wet clay absorbs it within centimeters. Divers and autonomous underwater vehicles communicate acoustically or with blue-green lasers, the frequencies at which seawater is most transparent.
When Guglielmo Marconi sent a radio signal across the Atlantic in 1901, physicists could not understand how it followed Earth's curve. Oliver Heaviside and Arthur Kennelly proposed a conducting layer high in the atmosphere, and in 1924 Edward Appleton measured its height, winning the 1947 Nobel Prize. The layer is a plasma, and it reflects radio waves below its plasma frequency, $8.98\sqrt{n}$ Hz.
Shortwave broadcasters, the Voice of America and amateur radio operators rely on it to reach around the world, choosing lower frequencies at night when the electron density drops. The National Oceanic and Atmospheric Administration's Space Weather Prediction Center monitors the ionosphere, because solar flares can suddenly increase its density and absorption, blacking out high-frequency radio on the sunlit side of the Earth — a concern for airlines on polar routes that depend on it.
Above a few megahertz, copper's skin depth falls below $50$ μm, and current flows only in a thin surface layer. Designers of radio transmitters, cell-phone circuit boards and microwave filters therefore care about surface finish and plating more than about the bulk of their conductors: a rough surface lengthens the current's path, and a thin silver or gold plating a few skin depths thick carries essentially all the current.
At the other extreme, superconducting radio-frequency cavities made of niobium, used in particle accelerators such as those at Fermilab and Jefferson Lab and in the Linac Coherent Light Source at SLAC, have surface resistances millions of times lower than copper's, so their stored microwave energy decays only after tens of billions of cycles. The skin depth and surface resistance of this lesson are the numbers that drive their design.
Direct current spreads evenly across a wire's cross section. Alternating current does not: the changing magnetic field inside the wire induces eddy currents that oppose the current near the center and reinforce it near the surface. At high frequencies nearly all the current flows within a skin depth of the surface, and the wire's effective resistance rises accordingly.
A second misconception is that a phase velocity greater than $c$ in a plasma violates relativity. The phase velocity describes how wave crests move, not how energy or information travels; the group velocity, which carries both, stays below $c$.
Write the skin depth for copper, $\sigma = 5.96 \times 10^{7}$ S/m.
$\delta = \dfrac{1}{\sqrt{\pi f\mu_0\sigma}}$
Copper is nonmagnetic.
Evaluate the constants.
$\pi\mu_0\sigma = \pi \times 4\pi \times 10^{-7} \times 5.96 \times 10^{7} = 235.3\ \text{s/m}^2$
Leaving $f$.
Evaluate at $60$ Hz.
$\delta = \dfrac{1}{\sqrt{235.3 \times 60}} = 8.4 \times 10^{-3}\ \text{m}$
Eight millimeters.
Evaluate at $1$ MHz.
$\delta = \dfrac{1}{\sqrt{235.3 \times 10^{6}}} = 65\ \mu\text{m}$
Much thinner.
Evaluate at $10$ GHz.
$\delta = 0.65\ \mu\text{m}$
A hundredth of a hair's width.
Copper has one free electron per atom, $n = 8.5 \times 10^{28}$ m⁻³. Write the plasma frequency.
$\omega_p = \sqrt{\dfrac{ne^2}{\varepsilon_0m_e}}$
Free-electron model.
Evaluate the numerator.
$ne^2 = 8.5 \times 10^{28} \times (1.60 \times 10^{-19})^2 = 2.18 \times 10^{-9}$
SI units.
Divide by $\varepsilon_0m_e$.
$\omega_p^2 = \dfrac{2.18 \times 10^{-9}}{8.85 \times 10^{-12} \times 9.11 \times 10^{-31}} = 2.70 \times 10^{32}$
In s⁻².
Take the square root.
$\omega_p = 1.64 \times 10^{16}\ \text{rad/s}$
Angular frequency.
Convert to a wavelength.
$\lambda_p = \dfrac{2\pi c}{\omega_p} = 115\ \text{nm}$
Far ultraviolet.
Conclude about visible light.
$400\text{–}700\ \text{nm} > \lambda_p$
Below the plasma frequency: reflected.
A layer has $n = 10^{12}$ m⁻³. Find its plasma frequency.
$f_p = 8.98\sqrt{10^{12}} = 8.98 \times 10^{6}\ \text{Hz}$
About $9$ MHz.
Test a $5$ MHz signal sent straight up.
$5 < 8.98 \quad\Rightarrow\quad \text{reflected}$
Below cutoff.
Test a $100$ MHz FM signal.
$100 > 8.98 \quad\Rightarrow\quad \text{transmitted}$
Passes into space.
Find the FM signal's group velocity in the layer.
$v_g = c\sqrt{1 - \left(\dfrac{8.98}{100}\right)^2} = 0.996c$
Slightly slowed.
Find its phase velocity.
$v_p = \dfrac{c^2}{v_g} = 1.004c$
Faster than light, carrying no information.
Find the delay over $100$ km of layer.
$\Delta t = \dfrac{L}{v_g} - \dfrac{L}{c} \approx \dfrac{L}{c}\cdot\dfrac{f_p^2}{2f^2} = 1.3\ \mu\text{s}$
At GPS frequencies the same layer delays signals by a few nanoseconds, a meter or two of position.
Explain how GPS corrects it.
$\Delta t \propto \dfrac{1}{f^2}$
Comparing arrival times at two frequencies measures and removes the delay.
Find the frequency where reflection begins at a $60°$ slant.
$f = \dfrac{f_p}{\cos 60°} = 18\ \text{MHz}$
Oblique paths reflect higher frequencies: the secant law of long-distance radio.
Write the scaling.
$\delta \propto \dfrac{1}{\sqrt{f}}$
Same metal.
Substitute the ratio.
$\delta = \dfrac{65.2}{\sqrt{4}}$
Four times the frequency.
Evaluate the depth.
At a certain frequency, the skin depth in a metal is $56$ μm. What is it at four times that frequency?
Complete the worked solution: a metal's skin depth is $288$ μm at frequency $f_0$. Find it at $4f_0$, at $100f_0$ and at $f_0/4$, in μm.
Divide by the square root of four.
$\delta(4f_0) =$ a
Half as deep.
Divide by the square root of a hundred.
$\delta(100f_0) =$ b
A tenth.
Multiply by the square root of four.
$\delta(f_0/4) =$ c
Lower frequencies penetrate deeper.
Match each quantity to its expression.
| $1/\sqrt{\pi f\mu\sigma}$ | $\sqrt{ne^2/\varepsilon_0m}$ | $\omega^2 = \omega_p^2 + c^2k^2$ | reflected, decaying inside | |
|---|---|---|---|---|
| the skin depth | ||||
| the plasma frequency | ||||
| the plasma dispersion relation | ||||
| a wave below $\omega_p$ |
Copper's skin depth is $6.52$ mm at $100$ Hz. Fill in the skin depth, in mm, at $100$ Hz, $10$ kHz and $1$ MHz.
| skin depth (mm) | |
|---|---|
| $100$ Hz | |
| $10$ kHz | |
| $1$ MHz |
Copper's skin depth is $65.2$ μm at $1$ MHz. What is it at 0.25 MHz, in μm?
Answer: μm
What is the skin depth of copper ($\sigma = 5.96 \times 10^{7}$ S/m, nonmagnetic) at 100 kHz, in μm? Use $\mu_0 = 4\pi \times 10^{-7}$ T m/A.
Answer: μm
A layer of the ionosphere has 10^{10} free electrons per cubic meter. Radio waves sent straight up are reflected if their frequency is below the layer's plasma frequency. What is that frequency, in MHz?
Answer: MHz
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Complete the worked solution: a metal's skin depth is $148$ μm at frequency $f_0$. Find it at $4f_0$, at $100f_0$ and at $f_0/4$, in μm.
Divide by the square root of four.
$\delta(4f_0) =$ a
Half as deep.
Divide by the square root of a hundred.
$\delta(100f_0) =$ b
A tenth.
Multiply by the square root of four.
$\delta(f_0/4) =$ c
Lower frequencies penetrate deeper.
You can describe waves in conducting media. Explain to someone why the ionosphere reflects shortwave radio but not FM.
19. Your turn: what is copper's skin depth at $4$ MHz, given $65.2$ μm at $1$ MHz?, step 3
$\delta = 32.6\ \mu\text{m}$
Half as deep.