How far a field gets into metal

The field inside a conductor is zero, and every step of that argument depends on the electrons having gone where they had to go. The relaxation time for copper is about $10^{-19}$ seconds, which is why the statement is safe for anything static and for a great deal that is not.

It is not safe for a magnetic field, and the reason is that a magnetic field is not screened by rearranging charge. It is screened by currents, the currents are induced by the field’s own change, and a changing field induces currents that oppose it only to the extent that the metal can carry them.

The equation loses a derivative #

Inside a good conductor the conduction current $\sigma\mathbf{E}$ is enormously larger than the displacement current $\varepsilon\,\partial\mathbf{E}/\partial t$ — for copper the two are equal only at about $10^{18}$ hertz — so the displacement term can be dropped. Maxwell’s equations then give

$$\nabla^2\mathbf{E} = \mu\sigma\frac{\partial\mathbf{E}}{\partial t}$$

which is not a wave equation. It is a diffusion equation, of exactly the form that governs heat and the spreading of a concentration. The second time derivative that makes a wave has gone, and with it everything a wave does: there is no fixed speed, no propagation without loss, and no reflection in the ordinary sense.

A field entering metal therefore does what heat entering a wall does. It seeps, it lags, and how far it gets in a given time goes as the square root of that time — which is where the inverse square root of the frequency comes from.

Solving it for a field oscillating at $\omega$ gives an amplitude falling as $e^{-z/\delta}$ with

$$\delta = \sqrt{\frac{2}{\mu\sigma\omega}}$$

and a phase lagging by one radian per skin depth.

The screening currents are ordinary induction, distributed through a solid instead of confined to a wire. Flux through a loop drives a current when it changes, and their strength is set by how fast the flux is changing rather than by how large it is — which is why a static field walks into a conductor unopposed and an alternating one does not. Everything peculiar about the skin effect comes from that derivative.

The numbers, and what they decide #

Copper at mains frequency: 9.2 millimetres. Copper at a gigahertz: 2.1 microns. Six decades of frequency move the scale by a factor of a thousand, and each end of that range decides something.

At mains frequency a busbar of a centimetre or so carries current throughout, and a thicker one does not. That is why heavy conductors in switchgear are flat bars rather than round rods: a bar has more surface for its cross-section, and surface is what carries the current.

At a gigahertz the current runs in a layer thinner than the plating. So the conductivity of the outermost couple of microns is the conductivity of the component, the surface finish matters more than the bulk material, and a silver-plated brass waveguide performs as silver. It is also why a scratch on a microwave surface matters and a void in the middle does not.

Seawater at seventy-six hertz has a skin depth of twenty-nine metres, which is why submarines are signalled at frequencies measured in tens of hertz and receive a few characters a minute. The bandwidth available at such a frequency is essentially nothing, and there is no way round it: raising the frequency to get bandwidth buys skin depth at the inverse square root and loses it far faster than the bandwidth is gained.

The length, read as a time #

The skin depth is usually presented as a distance, and it is more illuminating as a statement about a race.

A field diffusing into a conductor spreads a distance $\sqrt{t/\mu\sigma}$ in time $t$ — the ordinary square-root law of every diffusion. An oscillating field reverses after half a period. So the depth reached is the diffusion distance in half a period, and the skin depth is that distance up to a factor of order one.

Put the other way round, a conductor of thickness $d$ has a magnetic diffusion time $\mu\sigma d^2$: the time a field takes to get through it. For a millimetre of copper that is about eight microseconds. Anything slower than that passes; anything faster is excluded. It is the same number as the skin depth and it is much easier to apply to a transient, where there is no frequency to put in the formula.

That reading also explains why the material properties enter as a product. Conductivity and permeability both slow the diffusion, one by making the induced currents larger and the other by making the flux they have to move larger, and only their product appears.

And it makes the flux-freezing limit a special case rather than a separate idea: a perfect conductor has an infinite diffusion time, so the field never gets in or out, and a real one leaks on the timescale above. The astrophysical version and the laboratory one are the same equation at very different values of $\mu\sigma d^2$.

The resistance that will not stay put #

Confining the current to a surface layer raises the resistance, and the exact calculation is worth doing because the asymptote is so simple.

For a slab of half-thickness $a$ the ratio is $\mathrm{Re}[ka\coth ka]$ with $k = (1+i)/\delta$, which is one at low frequency and $a/\delta$ at high. The resistance therefore climbs as the square root of the frequency without limit: the copper slab drawn here has doubled its resistance by 741 hertz and is ten times worse by 70 kilohertz.

Two pieces of engineering follow directly.

Hollow conductors. A waveguide, a heavy radio-frequency busbar and the outer conductor of a coaxial cable are all tubes, because the metal in the middle carries nothing and costs money and weight.

Litz wire. A bundle of many strands, each thinner than a skin depth, woven so that every strand takes every position in the bundle. The weaving matters: strands merely bundled would still be linked by each other’s fields, and the outer ones would carry more current than the inner ones — the proximity effect, which is the same physics with a neighbouring conductor supplying the changing field instead of the conductor itself. Litz wire holds the low-frequency resistance up to hundreds of kilohertz, where a solid conductor of the same copper would be several times worse.

The shielding that does not work #

The gap between those two numbers is the whole practical story of shielding.

A metal box is an excellent radio shield and a poor magnetic one. At a megahertz a millimetre of copper attenuates by 133 decibels, which is a factor of $10^{13}$ and is why a mobile phone dies inside a biscuit tin. At fifty hertz the same sheet gives 0.9 decibels, which is nothing, and no thickness anybody would build helps — reaching 20 decibels would need two centimetres of copper.

The fix for low-frequency magnetic fields is not conductivity but permeability. Mu-metal has a relative permeability of twenty thousand, so the flux prefers to go through it rather than through the air beside it, and a mu-metal box works by diverting the field around the volume rather than absorbing it. That is a completely different mechanism: it works down to direct current, where absorption gives nothing at all, and it saturates if the field is too strong, which absorption does not.

The two mechanisms show up in the same figure because permeability also raises the conductivity’s effect — mu-metal’s skin depth is small despite its poor conductivity — but the reason mu-metal is used is the diversion, not the absorption.

And the curve is absorption only. A real sheet also reflects, and for an electric field arriving from far away the reflection is the larger term by a long way: a thin foil that absorbs nothing still shields, because the impedance mismatch between free space and metal is enormous. For a magnetic field from a nearby source it is not, which is exactly the case this essay is about, and it is why the two situations are so often confused.

What a hole does #

A shield’s performance is almost never set by its metal, and saying so is the most useful practical remark in this essay.

An aperture in a conducting sheet leaks, and how much depends on the aperture’s size compared with the wavelength. A hole much smaller than a wavelength radiates as a small dipole and passes very little; a slot comparable with half a wavelength is an antenna and passes almost everything. Since a shield of any use is many skin depths thick, the metal contributes 100 decibels or more and the seams contribute perhaps 40, so the seams decide.

That is why a microwave oven’s door is a perforated screen and works: holes of a millimetre against a wavelength of twelve centimetres are a ratio of over a hundred, so they are, to the microwaves, not there — while being entirely there for visible light, whose wavelength is ten thousand times smaller than the holes.

It is also why the long thin gap round a poorly fitted panel is worse than a large round hole of the same area. A slot’s coupling is governed by its longest dimension rather than its area, so a shielded enclosure is assembled with conductive gaskets along every joint, and a single unbonded seam undoes the metal entirely.

What a hole does is worth stating separately, because it is where enclosures actually fail. A winding that is not infinite leaks at its ends; an enclosure that is not closed leaks at its opening. In both cases the object is defined by where it fails to close, and the quality of the material between the failures is almost never what decides the answer. A copper box with a slot in it is a worse shield than a steel box without one, and no amount of thickness repairs the slot.

Where the field goes instead #

Where the field goes instead is into heat. The currents induced in a conducting sheet are the shielding — they are not a by-product of it — and they dissipate against the metal’s resistance. So a cage sitting in a strong alternating field warms up, and the shielding is paid for continuously rather than bought once. That is the practical difference between magnetic screening and the electrostatic kind, and it is why the first has a power budget and the second does not.

The currents that do the screening are the same eddy currents that slow a magnet falling down a copper tube, and they dissipate. A shield in a strong alternating field heats, and induction hobs, induction furnaces and the losses in a transformer’s core are all this effect used or fought.

The transformer case is the one where the design is visibly a skin-depth calculation. A solid iron core at fifty hertz would have a skin depth of about a millimetre — iron’s permeability is high, which makes the depth small — so the flux would not reach the middle of a large core and the eddy losses would be enormous. Laminating the core into sheets thinner than a skin depth, insulated from one another, removes both problems at once. The lamination thickness is chosen by exactly the calculation in the figure above.

The same argument, run in the opposite direction, is how induction heating works. Choose the frequency so that the skin depth is the depth to be hardened, and the heat is deposited there and nowhere else. A gear tooth case-hardened at ten kilohertz is heated over about half a millimetre; the same part at a hundred hertz would be heated throughout and would distort.

The electrostatic mechanism, for contrast, has no timescale in it at all. A cavity inside a conductor is shielded completely by charge rearranging on the surface, and the rearrangement is as fast as the charges can move — which for practical purposes is instantly. The magnetic mechanism has a timescale, that timescale is the whole of this essay, and the difference between the two is why “Faraday cage” is a phrase that promises more than any real box delivers.

The measurement that uses it #

Because the depth depends on the material and the frequency in a known way, sweeping the frequency measures the material — and that is a whole family of instruments.

Eddy-current testing. Drive a coil near a metal surface and measure how its impedance changes. The change depends on the conductivity, the permeability and the distance, and a crack disturbs the induced current pattern. Choosing the frequency chooses the depth interrogated: high frequency for surface cracks, low for subsurface ones, and a sweep for a depth profile. It is how aircraft skins and heat-exchanger tubes are inspected, without contact and without couplant.

Coin discrimination. A vending machine measures the eddy-current response of a coin at two or three frequencies, which samples different depths, and compares the pattern with a stored one. A plated slug of the right size and weight gives the wrong answer because its interior is a different conductor from its surface.

And magnetotellurics. Natural fluctuations of the Earth’s magnetic field induce currents in the ground; measuring the electric and magnetic fields at the surface over a range of periods gives the conductivity against depth, because each period samples its own skin depth. Periods of seconds probe kilometres and periods of days probe hundreds of kilometres, and the whole crust and upper mantle is mapped this way — an instrument whose depth of view is set entirely by the equation at the top of this page.

The same time, computed for a planet #

The magnetic diffusion time $\mu\sigma d^2$ is quadratic in the size, so putting a planet into it gives a number that decides something about the history of the Earth.

The outer core is liquid iron, a fair conductor at some five hundred thousand siemens per metre, and it is about two thousand kilometres thick. Multiplying those together gives a diffusion time of order a hundred thousand years, and the slowest mode a sphere supports decays a further factor of $\pi^2$ faster than the crude estimate — so a magnetic field left in the Earth’s core with nothing maintaining it would be gone in something like ten or twenty thousand years.

The palaeomagnetic record says the field has been there for at least three and a half thousand million years. It is therefore not a leftover; it is being regenerated continuously, and the requirement that something regenerate it faster than this arithmetic destroys it is the whole reason a dynamo has to be invoked at all. The same estimate applied to the Moon, whose core is far smaller, gives a decay time short enough that its ancient field must also have been driven rather than frozen in.

The mantle above the core does the other half of the job described in this essay, as a shield. It is a poor conductor, so its own diffusion time is short — but not zero, and it filters what reaches the surface: the fastest variations of the core field are smoothed away before a magnetometer on the ground can see them, so the observed drift of the field is a low-pass version of whatever the core is doing. Everything known about the deep field’s behaviour has been through that filter, and the filter’s cutoff is the same equation with the mantle’s conductivity in it.

The case that sits on the boundary #

The essay’s approximation — conduction current far larger than displacement current — is stated as a condition and it is worth looking at something that fails it, because the failure is not academic.

Human tissue conducts, weakly: around half a siemens per metre. A magnetic resonance scanner at three tesla drives it with radio-frequency pulses at 128 megahertz, and putting those numbers into the skin depth gives about six centimetres. That is smaller than a torso, so the excitation field is measurably weaker in the middle of a body than at its surface, and the images acquire a shading that has to be corrected for.

But tissue’s permittivity is high — a relative permittivity of order eighty at these frequencies — so the displacement current is comparable with the conduction current rather than negligible. Tissue at scanner frequencies is neither a good conductor nor a good insulator; it is in between, and both terms matter.

The consequence is that a wave propagates as well as diffusing. The wavelength in tissue is about a quarter of a metre, comparable with the object being imaged, so the field forms standing-wave patterns inside the body: bright and dark regions that move if the patient moves, and that get worse at higher field because the frequency rises with it. Managing them is a substantial part of why seven-tesla body imaging is hard, and the fix — driving the transmit coil as several independently phased elements, so the interference pattern can be steered — is an admission that the field inside the subject is a wave rather than a diffusing skin.

Two mechanisms, comparable in size, in a conductor everybody carries around. It is the clearest available illustration of what the first approximation on this page is actually assuming.

What the picture cannot show #

The good-conductor approximation is stated and not drawn. Dropping the displacement current requires $\sigma \gg \varepsilon\omega$, which holds for copper up to the ultraviolet and for seawater only up to a few megahertz. Above that, seawater is a lossy dielectric rather than a conductor and none of these curves applies to it.

The conductivity is treated as a constant. It is not, at high frequency: once the period is shorter than the time between electron collisions the response becomes inertial rather than resistive, and the metal behaves like a plasma with a frequency below which nothing gets in. For copper that crossover is in the infrared, which is why metals are shiny.

And it is not constant at low temperature either. In very pure metal at a few kelvin the electron mean free path can exceed the skin depth, so an electron sampling the field travels through a region where the field is not uniform, and the local relation between current and field breaks down entirely. The anomalous skin effect that results has a depth going as the cube root of the frequency rather than the square root, and it was one of the first probes of the Fermi surface.

Permeability is treated as a number. For mu-metal it is a strong function of the field, falls to nothing at saturation, and depends on the material’s mechanical history — a mu-metal shield that has been dropped has to be annealed before it works again.

The ladder from here #

Later rungs on this anchor: the proximity effect worked out for two conductors, where a neighbour’s field decides the current distribution; the anomalous skin effect and what it measures about a metal’s electrons; the shielding of a real enclosure, where the seams and the holes rather than the metal decide the performance; magnetic shielding by permeability, including the nested shells used to reach nanotesla; and the transient case, where a field switched on rather than oscillated diffuses inward on a timescale set by the same arithmetic.

The neighbouring ladders are the interior of a conductor, which is this argument in the limit of infinite time, and the falling magnet, where the same induced currents produce a force rather than a screen. What a dielectric takes away from a field is the electrostatic counterpart, and it has no timescale in it at all.