The charge that follows no local rule

The inside of a conductor explains why the field in a piece of metal at rest is exactly zero: any field would move the free electrons, and they keep moving until the field they set up cancels whatever was there. They end up on the surface, and the surface becomes an equipotential. The same essay notes that the charge does not spread itself evenly. It crowds where the surface curves most sharply, which is why a point concentrates a field and why every high-voltage terminal is fitted with a smooth toroidal ring.

That statement is true and close to empty, because “curves most sharply” admits many measures. A surface has two principal curvatures at each point; their mean is one measure and their product, the Gaussian curvature, another. One can take any power of either. A rule worth the name would say which. The answer is that no local rule exists — the charge density at a point is not a function of the shape at that point at all — and the clearest way to see it is to look at the one shape for which the density is known exactly and then at two shapes for which it is known just as exactly and follows something else.

A conducting egg #

A conducting ellipsoid, $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$, carrying a charge $Q$, has a surface density known in closed form since the early nineteenth century:

$$\sigma = \frac{Q}{4\pi abc}\,\frac{1}{\sqrt{x^2/a^4 + y^2/b^4 + z^2/c^4}}.$$

The square root in the denominator has a geometric meaning: its reciprocal is the distance from the centre of the ellipsoid to the plane tangent at the point. So the charge density on a conducting egg is proportional to how far the tangent plane lies from the centre — large at the ends, where the tangent plane is far out, and small at the waist.

The reason this works is older than electricity. Newton showed that a thin uniform spherical shell exerts no gravitational force on anything inside it, and the result was later extended to a shell bounded by two similar, concentric ellipsoids — a homoeoid, a shell whose thickness varies exactly in proportion to the distance from the centre to the tangent plane. A homoeoid of charge, like one of mass, produces no field anywhere in its interior. That is precisely the condition for the charge on a conductor: no field inside. So the conductor’s surface charge is a homoeoid of vanishing thickness, and its density is the homoeoid’s thickness. Newton’s theorem about why the inside of a hollow planet would be weightless is the reason charge collects at the ends of an egg.

The figure draws the charge as a band of thickness proportional to the local density, and the distribution can be checked without trusting the formula. The potential produced at points inside by rings of charge with that density — summed with the elliptic integrals that give a ring’s potential away from its axis — comes out the same, to better than a part in a thousand, at every interior point tried: centre, along the axis, off it. An arrangement that left any interior field would fail that test.

The curvature it actually follows #

Writing the Gaussian curvature of the ellipsoid in the same coordinates gives $K = 1/[a^2 b^2 c^2 (x^2/a^4+y^2/b^4+z^2/c^4)^2]$, and comparing with the density shows at once that

$$\sigma = \frac{Q}{4\pi\sqrt{abc}}\,K^{1/4}.$$

The charge on a conducting ellipsoid is proportional to the fourth root of the Gaussian curvature, exactly.

For a spheroid five times as long as it is wide the density at the tip is five times the density at the waist. The Gaussian curvature at the tip is 625 times that at the waist, and its fourth root is five. The mean curvature rises 9.6 times and the square root of the Gaussian curvature 25 times, both far above the charge. Between the waist and the tip the density and $K^{1/4}$ lie exactly on top of one another, and neither of the other measures comes close.

It is tempting to take this as the rule. It is not; it is a fact about ellipsoids. Nothing in the derivation said “curvature”. It said “homoeoid”, which is a property of a family of nested similar surfaces, and the fourth root appears because the ellipsoid’s curvature and the distance to its tangent plane happen to be related by that power. For a shape that is not an ellipsoid there is no reason to expect it.

Two shapes, two rules #

The simplest counter-example is a pair of conducting spheres, of radii $R_1$ and $R_2$, joined by a long thin wire so that they share a potential. Far apart, each sphere’s potential is its own charge divided by $4\pi\varepsilon_0$ times its radius, so equal potentials require charges in proportion to the radii. Spread over areas in proportion to the squares of the radii, the densities go as $1/R$. The Gaussian curvature of a sphere is $1/R^2$, so here the density goes as the square root of the curvature.

Plotted on logarithmic axes against the ratio of curvatures, the two families lie on two different straight lines, of slopes one quarter and one half. At the same contrast in curvature, a factor of ten thousand, the spheroid’s tip is ten times as densely charged as its waist and the small sphere a hundred times as densely as the large one. If charge density were any function of the local curvature, the two families would fall on one line. They do not, because they are different global problems: in the joined spheres each surface is shielded from the other by distance, and the only link is the wire; in the spheroid every part of the surface is near every other and each part’s charge affects the potential everywhere.

That is the general statement. The charge density on a conductor is the solution of a problem posed on the whole surface at once: find the distribution that makes the potential the same everywhere on it. One number for every point is the essay about what that condition means, and it is a global condition in the plainest sense. Moving a bump on one side of a conductor changes the charge on the other side. A local rule would mean that the charge at a point could be predicted from a small patch around it, and the potential at that point is set by charges everywhere.

Where a random walker lands #

There is a second way to see that the problem is global, and it supplies the most surprising description of the answer. The potential is where the wanderers stop shows that the solution of Laplace’s equation can be written as an average over random walks. In the same language, the charge distribution on an isolated conductor has a striking meaning: release a random walker from very far away, let it wander until it first touches the conductor, and record where. The probability density of the landing point, over the surface, is proportional to the surface charge density. Mathematicians call it the harmonic measure from infinity.

That explains the crowding at once and explains why it is not local. A walker approaching a long egg from a random direction is much more likely to strike an end than the waist, because the ends stick out and intercept walkers that would otherwise wander into the region beside the waist and find their way to it only rarely. A walker approaching a disc almost never reaches the middle of a face before touching the rim. Re-entrant corners, pockets and the insides of cavities are almost never reached at all, which is why, in Faraday’s ice-pail and in every hollow conductor, the inside surface carries no charge. Whether a point is reached depends on what surrounds it at every scale, not on how the surface curves there.

The same picture gives the field just outside a conductor, which is the charge density divided by $\varepsilon_0$, and so it also gives the rate at which diffusing particles from far away reach each part of a surface — the reason the tips of growing crystals and the ends of a branching deposit grow fastest, a rule that the cluster that grows where the walkers arrive follows to its fractal conclusion.

How much a point multiplies a field #

The engineering consequence of charge crowding is that sharp conductors break air down. Air ionises at about three million volts a metre; a cloud above a field sets up a field at ground level of tens of kilovolts a metre. Anything that multiplies that field by a hundred at its tip will discharge.

A grounded spheroid standing upright on a conducting plane in a uniform field — the half of a spheroid above the plane, which is the same problem by symmetry — has a tip field that can be found exactly. For a sphere, a hemisphere on the ground, it is three times the applied field. For a spheroid ten times as long as it is wide, it is 49 times; for one a hundred times as long, 2,326 times. The growth is close to the square of the slenderness, divided by a slowly growing logarithm.

The square is the interesting part. The radius of curvature at the tip of a spheroid is $b^2/a$, so for fixed width a longer spheroid has a sharper point; but the enhancement is not set by that radius alone. A needle a hundred times as long as it is wide has its tip radius a hundredth of its width and multiplies the field 2,326 times; a short blunt spike with the same tip radius multiplies it far less. The length gathers the field from a larger region, and the tip concentrates what has been gathered. That is why a lightning conductor works by being tall as well as pointed, and why the faint glow a thin wire wears before it sparks appears first on long thin conductors at high voltage. It is also why Benjamin Franklin’s debate with the advocates of blunt lightning rods in the 1770s, about whether a point invited strikes or quietly discharged the cloud, could not be settled by reasoning about the tip.

The tip’s charge density also exerts an outward pull on the surface, half the field times the charge density, which the pressure a charge puts on its own metal shows is what makes a charged drop draw itself into a cone. A conducting liquid with a slightly elongated shape, charged, concentrates its charge at the ends, which pulls the ends out further, which concentrates more charge: the instability that ends in an electrospray jet is the spheroid’s tip enhancement feeding itself.

A flat plate and its rim #

The most extreme case is a shape with no curvature over most of its surface. Flatten an oblate spheroid until it becomes a disc, and the density on each face takes a limiting form.

On the faces of a disc the curvature is zero everywhere, and if charge density were any power of curvature it would be zero there too. Instead it is $\sigma \propto 1/\sqrt{1-\rho^2}$, finite at the centre and rising without limit at the rim, where it diverges as the inverse square root of the distance from the edge. Half the charge lies in the outer thirteen per cent of the radius, a quarter of the disc’s area. The spheroids of finite thickness approach this distribution smoothly; at a hundredth of the radius in thickness the density at nine-tenths of the way out is already indistinguishable from the disc’s.

The inverse square root is the universal behaviour near any sharp edge with a right angle of opening, and it is what makes a real parallel-plate capacitor depart from the idealised one in how much charge a shape will hold: the fringing field at the edges carries a charge that the uniform-field calculation omits, and for a capacitor whose plates are not much wider than their separation, the edge charge is a large fraction of the whole. A sharp edge has no curvature at all in the sense of a smooth surface; what it has is an opening angle, and the exponent of the divergence depends on that angle, steeper for a knife-edge and gentler for a blunt corner. A re-entrant corner, opening into the metal, has an exponent that sends the density to zero rather than to infinity, which is the random walker’s pocket again.

Why no rule could have worked #

There is a simple argument that no local rule can exist, independent of any example. Take a conductor and attach to it, far away on the other side, a long thin wire leading to a large sphere. The charge on the original conductor’s surface changes everywhere — some of it flows to the sphere — but the local geometry of the surface has not changed at any point. So the charge density at a point cannot be a function of the local geometry. Even its ratio between two points can change: a second conductor brought near one side of the first draws charge to that side by induction, as the charge that has to be somewhere else shows with an image charge, without altering the curvature anywhere.

What the local geometry does determine is the behaviour near singular points: the exponent of the divergence at an edge or a corner, which depends only on the angle. That is a genuinely local statement, and it is the part of the folk rule that survives. Sharper features carry more charge in a precisely calculable way as their scale shrinks. How much charge they carry in absolute terms depends on everything else.

Ideal metal in ideal shapes #

The figures treat ideal conductors in electrostatic equilibrium, with no surface roughness, no work-function differences between faces of a crystal, and no space charge in the air around them. Real conductors have microscopic asperities whose own tip enhancements multiply the macroscopic one, which is why measured breakdown fields from polished metal fall well short of the ideal. The joined-sphere result assumes the spheres far apart compared with their radii; close together, each sphere’s charge is drawn towards the other. The tip enhancement is for a spheroid in a uniform field; a real lightning rod stands in the non-uniform field under a cloud, near other structures, and the field at its tip is modified by the space charge it emits once corona begins. The disc’s divergent density is integrable — the charge near the rim is finite — but the field at the rim is genuinely infinite in the ideal, and any real edge, with a finite radius, caps it.

Still open: the best shape for a given purpose #

Since charge density is a global property, so is the problem of shaping a conductor to keep its maximum field below a given value — the design of high-voltage electrodes, of the terminals of electrostatic generators and of the electrodes of particle accelerators, where breakdown limits the voltage achievable. For a given volume or a given surface area, the shape minimising the maximum surface field is a problem in shape optimisation that has been solved numerically for many configurations but has no general closed-form answer, and the electrodes in use are rounded according to experience and computation rather than theory. Whether a sphere is the unique optimum among all bodies of a given capacitance — so that no other shape could hold the same charge at the same voltage with a lower peak field — is a question related to old isoperimetric inequalities for capacitance, proved for some measures and conjectured for others.

The habit worth carrying away is to ask whether a statement about a field is local or global. On a conducting ellipsoid the charge goes exactly as the fourth root of the Gaussian curvature, because its charge is a homoeoid, Newton’s shell that exerts no force inside; on joined spheres it goes as the square root, and on a disc it diverges at an edge with no curvature at all. Charge density is where random walkers from infinity first touch the surface, and that is decided by the whole shape.