The field a ring of wires lets through

The inside of a conductor is free of static field because the charges on a conductor’s surface move until they cancel any field inside it, and the potential is where the wanderers stop gave the same fact a second proof: a random walker started in a closed metal box can only end on the box, which is all at one potential, so the potential inside is that potential everywhere. Both arguments need the box to be closed. Every real shield has holes in it, and many shields are mostly holes — a cage of bars, a mesh, a braid of thin wires round a cable.

The folk account, and much of the textbook one, is that holes do not matter much. Michael Faraday sat inside a wire-framed cube covered in metal foil in 1836 while sparks played over its outside, and the cage that bears his name has been built since then out of bars and mesh as often as out of sheets. Richard Feynman’s Lectures give the reason a mesh works: the field of a row of charged wires, evaluated a distance away from the row, varies from point to point by an amount that dies exponentially with distance, with a decay length of the spacing over $2\pi$. A few spacings away, the row looks like a smooth sheet.

The argument is attractive because it is a real calculation rather than a handwave, and because it predicts a decay so steep that a mesh a few spacings thick should be indistinguishable from foil. It is repeated in engineering texts and lecture notes, and the claim that a static field “cannot get through” a fine wire cage is the standard explanation of why a car, a cage or an aircraft protects the people inside from the field of a thundercloud. Nobody seems to have checked it against a direct computation of a cage of wires held at a common potential, perhaps because the answer looked obvious, until a numerical analyst went looking for the formula and found that it did not exist.

That is correct as a statement about the row, and in 2015 Jonathan Chapman, David Hewett and Nick Trefethen showed that it is the wrong statement about a cage. Their analysis, redone in the figures below for a ring of parallel wires, finds that shielding by wires is only as good as a power of the spacing, with the thickness of the wires entering through a logarithm. Twelve thin wires let a quarter of the field through.

A cage in a uniform field #

Put a ring of parallel wires in a uniform field running across them, and join the wires so that they all sit at one potential and carry no net charge between them. Each wire takes up whatever charge it needs to be at the common potential, and the arrangement of those charges is what shields the inside.

In two dimensions each wire is a line of charge, its potential a logarithm of distance, and the condition that every wire be at the same potential is a set of linear equations, one per wire, with the total charge fixed at zero. Solving them gives every wire’s charge, and from the charges the field everywhere.

Outside, the cage looks very like a solid cylinder: the equipotentials crowd against its sides and bend round it. Near the wires the lines wriggle round each one. Inside, a few wire spacings from the ring, the wriggles have gone and the lines run straight and evenly spaced. That is the textbook picture so far — the inside is smooth. But the lines inside are there. They run across the cage at a quarter of the density they have outside, and a quarter of the density of equipotentials is a quarter of the field.

What the wires carry #

A solid conducting cylinder in a uniform field cancels it inside exactly. It does so with a surface charge that varies round the cylinder as the cosine of the angle — positive on the side the field points to, negative on the other — and of exactly the size that makes the field of that charge, inside the cylinder, equal and opposite to the field applied. Every conductor arranges its charge this way, and the surface charge is the whole of the shielding.

The wires try to do the same thing, and they get the pattern right. The charge they carry follows the cosine round the ring. It falls short in size, at 74 per cent of what a solid shell would hold on the same arc, and the shortfall is exactly the leak: the field inside is the applied field minus the field of the charges, and charges at 74 per cent of what is needed cancel 74 per cent of it.

The wires fall short for a reason that has nothing to do with the gaps letting the field “through”. A thin wire is a poor container of charge. Its potential, relative to its neighbours, is dominated by its own charge, and the potential of a line of charge diverges logarithmically as the wire gets thinner. A wire that must sit at the same potential as all the others can therefore hold only a limited charge before its own potential runs away, and a thin wire hits that limit sooner than a thick one. The shield is limited by how much charge each wire can take on, which is a question about how much charge a shape will hold, and the answer for a thin wire is: not much.

The walkers’ view of a thin wire #

There is a way to see why thin wires are poor at this without solving any equations, and it uses the random walkers that turn a potential into an average. Start a walker at the centre of the cage and let it wander. If it touches a wire it collects the cage’s potential. If it slips out through a gap and wanders away, it collects whatever the potential is far outside, where the applied field sets it. The potential at the centre is the average of those payments over many walkers, so the fraction of walkers that escape is a measure of how much the outside gets in.

In two dimensions a walker is very bad at finding a thin target. It will return arbitrarily close to its starting point eventually, but the chance that it actually touches a disc of radius $r$ before wandering a distance $s$ away falls only as one over the logarithm of $s/r$ — slowly, but towards zero. A cage of thin wires is a ring of targets that a walker from the centre mostly misses, and the walkers that miss carry the outside potential back in. Thickening the wires tenfold raises the chance of a hit by a fixed amount, which is the logarithm again, now as a probability. And adding more wires lowers the chance of slipping through a gap in proportion to the gap, which is the power law.

The same picture shows why the inside is smooth. Walkers that start well inside the ring have wandered far before they reach it, and by then they have forgotten which gap or wire they were heading for: the potential they average is the same whether they started a little to the left or a little to the right. Forgetting is exponential in the distance travelled, measured in spacings, and that is the ripple’s decay. The average they bring back, though, is set by the odds of escape, and those odds do not depend on where inside the cage the walk began.

Thicker, but only logarithmically #

Because the wire’s capacity for charge depends on the logarithm of its radius, so does the cage’s shielding. A tenfold thicker wire removes about the same fraction of the leak as the previous tenfold thickening did. Going from a wire one thousandth of the spacing in radius to one hundredth takes the leak from 46 per cent to 32; going from a hundredth to a tenth takes it to 7. Only as the wires grow so fat that the gaps between them close does the leak go to zero, and that is no longer a cage.

The logarithm is familiar from elsewhere. The capacitance of a coaxial cable, the inductance of a long wire and the resistance of a thin electrode in a large tank all depend on the logarithm of a ratio of radii, and all for the same reason: the field of a line of charge, or of a line of current, falls as one over the distance, and integrating one over the distance gives a logarithm. In each case the practical consequence is that the thickness of a thin conductor matters, but weakly. Doubling a cable’s inner conductor changes its capacitance by a few per cent. Doubling a cage’s wires changes its leak by about as much.

More wires, a power law #

The obvious way to improve a cage is to add wires. The textbook exponential suggests that this should help enormously. If the field inside fell off as $e^{-2\pi d/s}$ from the ring, and the centre is a distance $R$ in from a ring of $N$ wires spaced $s = 2\pi R/N$ apart, the field at the centre would be $e^{-N}$ of the field outside: a millionth for twelve wires, $10^{-21}$ for forty-eight.

The computed cage does nothing of the kind. Doubling the number of wires, with each wire kept at the same fraction of the spacing, roughly halves the leak: 41 per cent for six wires, 26 for twelve, 15 for twenty-four, 8 for forty-eight. The leak falls as about $N^{-0.86}$, a power law with the logarithm of the wire radius folded in, and the exponential it was supposed to follow is off the bottom of the graph by twenty decades. Chapman, Hewett and Trefethen put the result in a phrase: the shielding of a Faraday cage is linear, not exponential, in the spacing.

The part that does die exponentially #

The textbook argument is not wrong. It answers a different question, and the last figure shows which.

Subtract the uniform field at the centre from the field everywhere else inside the cage, and what is left — the ripple, the part that knows about individual wires — does die exponentially inward, at exactly the rate the textbook argument gives. Measured from the computed field, its decay rate near the ring is 11.9 per radius, where $2\pi/s$ for twelve wires is 12.0. Deeper in, it falls faster still, as the $N$th power of the distance from the centre, because a pattern with twelve-fold symmetry inside a circle can only be built from the twelfth and higher harmonics.

The exponential is a fact about Laplace’s equation. A potential that repeats with period $s$ along a line, in a region with no charge, must change with distance from the line as $e^{\pm 2\pi d/s}$, and the bounded solution dies. Feynman’s grid argument applies that to the periodic part of the field of a row of wires, correctly. What it does not cover is the part of the field that is not periodic — the average across one spacing, the zeroth Fourier component, which has wavelength infinity and no reason to decay at all. In a field applied from outside, the average is the field, and how much of it is cancelled depends entirely on how much charge the wires can take up. The ripple is the cage’s texture; the average is its job.

Where this leaves real shields #

A real Faraday cage is usually better than this ring of wires, and the reasons are worth separating from the result.

Most cages are not built from parallel wires but from a mesh, with wires in two directions joined at every crossing, or from perforated sheet. A mesh can carry currents and charges along paths a set of parallel wires cannot, and its effective “wires” are not isolated thin conductors but the edges of a connected surface with holes in it. The two-dimensional analysis here does not describe it, and three-dimensional cages are harder to compute; what carries over is the lesson that thin conductors take up charge only logarithmically, and that a shield’s performance against a uniform field is set by that, not by the exponential decay of its texture.

The more important difference is frequency. Almost every practical shield is meant to stop waves, not static fields — the microwave oven’s door, the screen round a sensitive circuit, the braid round a coaxial cable. For waves, currents in the shield do the work, a closed loop of wire can carry them, and what matters is the size of the holes compared with the wavelength. That is how far a field gets into metal, and there a mesh with holes much smaller than the wavelength genuinely does behave like a solid sheet. The static case is the one in which the folk account fails, and it fails because at zero frequency a cage of separate wires cannot circulate a current and has only its charges to work with.

The static case is not academic. A cage of rods round a high-voltage experiment, a ring of grounded bars round an electron beam or an electron lens, and the grid of a vacuum tube are all arrangements of separate conductors meant to fix a potential, and each of them leaks the uniform part of whatever field is applied from outside at a level set by its spacing and the logarithm of its wire size. Designers of electron optics learned to make such grids fine and their wires not too thin long before the general result was stated.

What the pictures cannot show #

Every figure is computed for a ring of infinitely long parallel wires in two dimensions, with each wire treated as a line of charge at its centre. That treatment is accurate while the wires are thin compared with their spacing, which covers every case drawn except the thickest wires in the radius figure, where the charge on each wire would shift towards its neighbours and the leak would be slightly overestimated. A finite cage, with ends, behaves differently near its ends; the drawn numbers apply to a long cage far from them.

The figures also show one orientation of the applied field, across the wires. A field along the wires is not shielded at all by parallel wires, since it never meets them; it needs wires running the other way. And they show a cage with no net charge. A grounded cage, held at the potential of distant objects, would leak in the same proportion.

The domain of the argument is electrostatics: fields that do not change, or change so slowly that the wires’ charges keep up. There the cage’s shielding is set by how much charge its wires can carry, and it improves as a power of the spacing and as a logarithm of the wire radius. At frequencies where currents can circulate in the shield, a different argument applies and the cage does better.

Still open: what a three-dimensional cage does #

The two-dimensional problem is now understood, with explicit formulas for the leak in terms of the spacing and the radius, but the cage Faraday sat in was three-dimensional, built of wires running in several directions and joined at the corners. Whether the same linear-with-a-logarithm law holds for a cubic mesh of wires, how the joints change it, and how it passes over into the behaviour of a perforated sheet as the wires fatten into a surface with holes, are questions that have been studied numerically and are not settled in general. They matter for any shield made of thin conductors that must fix a static potential, from particle-detector grids to the electrostatic screens on spacecraft.

The textbook exponential survives in its proper place. A ring of joined wires in a uniform field carries the cosine charge pattern a solid shell would, but only at the strength its thin wires can take up — 74 per cent for twelve wires at 2 per cent of the spacing — so a quarter of the field goes through; the leak falls about as 1/N with more wires and only logarithmically with thicker ones, while the exponential e^(−2πd/s) belongs to the ripple, which is the cage’s texture rather than its job.