Illustrated Physics
physics, drawn
Concept

Surface charge — where it appears

Charge residing on a conductor's boundary, whose density fixes the field just outside and which carries an outward pressure of its own. The field is the density over the permittivity, and it concentrates where the surface curves most sharply — which is why lightning conductors are pointed.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

A conductor in a field, with the surface charge solved for. Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

The inside of a conductor, where the field is exactly nothing

Put a metal object in any electric field and the field inside it is zero. Not small — zero, by an argument that takes one sentence, and with consequences that reach from lightning to the most precise test of Coulomb's law ever made.

electromagnetism · Conductors Two plates 0.20 plate-widths apart, with the field traced. The electric field between two oppositely charged plates separated by 0.20 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 80 per cent of the field at the centre.

How much charge a shape will hold, before anything is charged

Capacitance is decided by geometry alone. Two pieces of metal have a number attached to them, fixed by their shape and their separation, and it is settled before any charge arrives.

electromagnetism · Conductors The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Where the energy of a field actually is

A charged capacitor holds 1.27 µJ, and two entirely different accounts agree on the number: one built from charges and potentials, one built from joules per cubic metre of empty space. They part company at a resistor, where the power arrives sideways through the surface at 1.67 W.

electromagnetism · Field energy The plane deleted, and one charge put in its place. A charge of 1 nC held 20 mm above an earthed conducting plane. The lines are traced through the field of the real charge plus an equal and opposite one at the mirror position, and then cut at the plane, because below it there is metal and no field whatever. Nothing in the tracing knows about the surface: each line follows the local field direction and stops where it arrives. That every one of them arrives perpendicular — the worst departure among the 9 drawn is 2.2° away from square — is the boundary condition showing itself rather than a rule imposed on the drawing. The image charge is drawn faint because it is not there: it is a way of writing a function that happens to satisfy the equation and the boundary values, which by the uniqueness theorem makes it the field and not a model of the field.

The charge that has to be somewhere else

Hold a charge above an earthed metal sheet and the field above it is exactly the field of two charges — the real one and an imaginary partner buried at the mirror position. The partner is not an analogy or an approximation. It is a legal guess, and a legal guess is a proof.

electromagnetism · Conductors The outward pull on a charged surface. Electrostatic pressure against the field at a conductor's surface. The quantity is ½ε₀E², the energy density of the field itself, and it is outward whatever the sign of the charge — like charges repel, and a charged surface is trying to fly apart. The factor of a half is the interesting part and is where a first attempt goes wrong: the field is σ/ε₀ outside and zero inside, and the layer of charge feels neither of those but their mean, because no charge exerts a force on itself. 0.5 MV/m gives 1.1 Pa, 1 MV/m gives 4.4 Pa, 2 MV/m gives 17.7 Pa, 3 MV/m gives 39.8 Pa, 5 MV/m gives 110.7 Pa. Those are small pressures — three megavolts per metre is the breakdown field of air and pulls with about a hundredth of an atmosphere — which is why electrostatic forces shape soap films and dust and not much that is stiffer, and why the same pressure set against surface tension has a definite size of drop at which it wins.

The pressure a charge puts on its own metal

Charge on a conductor sits on the surface and tries to leave. The outward pull is half epsilon-nought E squared, the half is because a charge exerts no force on itself, and setting that pull against surface tension gives the largest a charged drop is allowed to be — a number Rayleigh wrote down in 1882 and an industry now depends on.

electromagnetism · Conductors The potential at a point is where its walkers end up. A square divided into a 24-step grid, with its top edge held at a potential of 1 and the other three edges at 0. From the probe point (0.29, 0.71), 4000 random walkers each step to one of their four neighbours with equal chance until they touch an edge; six of them are drawn, each ending with a dot on the edge it reached. The fraction that end on the held edge is 0.405 ± 0.008, one standard error, after an average of 122 steps. Solving Laplace's equation on the same grid by repeatedly replacing every value with the average of its four neighbours gives 0.408, and the series solution for the continuous square gives 0.408. The walkers were never told the equation: a value that is the average of its neighbours and a probability of ending somewhere are the same arithmetic.

The potential is where the wanderers stop

Start a random walker at a point between charged conductors and let it wander until it touches one of them. The average potential of the surfaces the walkers touch is the potential at the starting point — exactly, with no equation solved — and the charge a conductor keeps at each place on its surface is the chance that a walker arriving from far away touches it there first.

electromagnetism · Potential Three ships, three local times, one clock. The fair-weather field at sea level, in the same model, as recorded by ships at longitudes 120° W, 0°, 120° E, each plotting it against its own local solar time. An ordinary daily effect — sunshine, heating, turbulence — would peak at the same local time everywhere. This one peaks at 11.4, 19.4, 3.4 hours local time in turn, because every ship is reading the same global potential, which follows the world's storms and peaks at one moment in Universal Time. Seeing the maxima line up in Greenwich time rather than local time, over clean ocean air far from any storm, was how the global circuit was demonstrated: the field in fair weather is the far end of a circuit whose generators are thunderstorms thousands of kilometres away.

The field that keeps Greenwich time

Over a calm ocean, a thousand kilometres from any storm, the air carries a downward electric field of about a hundred volts per metre, and it rises and falls by a sixth once a day. It peaks at the same instant on every ship on every ocean — near seven in the evening in Greenwich, whatever the local clock says. Nothing local can do that. The field is the far end of a circuit whose upper plate is a single conductor round the whole planet, and a conductor has one potential, so every patch of fair-weather sky is reading the sum of all the world's thunderstorms at once.

electromagnetism · Potential A cage of wires in a uniform field. Equipotential lines round a cage of 12 parallel wires, each 2 per cent of the spacing in radius, joined so that they share one potential and carry no net charge, in a uniform field running left to right. Outside, the lines bunch and bend round the cage as they would round a solid cylinder. Inside, they do not vanish: they run straight across, carrying a uniform field 25.8 per cent as strong as the one outside (dashed, drawn at a fifth of the outside spacing because the potential differences inside are so much smaller). The wires' charges are computed so that every wire is at the same potential.

The field a ring of wires lets through

A cage made of wires is supposed to work almost as well as a solid box, because the field that leaks through the gaps dies away exponentially within a gap's width. The exponential is real and it is the wrong part of the story. It describes the ripple the separate wires make; underneath it the average field goes straight through, at a level that falls only in proportion to the spacing and depends on the wires' thickness through a logarithm. A cage of twelve thin wires lets a quarter of the field in.

electromagnetism · Potential Where the charge goes on a conducting egg. The surface charge on three conducting spheroids of the same length, a sphere and spheroids 3 and 10 times as long as they are wide, drawn as a band whose thickness is proportional to the charge per unit area at each point. On the sphere it is uniform. On an ellipsoid the density is exactly proportional to the fourth root of the surface's Gaussian curvature, and so it is 3 and 10 times as dense at the tips as at the waist — not the 9 and 100 times by which the tips' curvature exceeds the waist's, and not the ratio of their mean curvatures either. The charge does crowd to the sharp ends, but by how much is set by the whole shape, not by any property of the surface at the point.

The charge that follows no local rule

Charge on a conductor crowds where the surface curves sharply, and every high-voltage engineer rounds the edges for that reason. But curves sharply by what measure? On a conducting egg the density is exactly the fourth root of the surface's Gaussian curvature, so a tip a hundred times more curved than the waist carries only ten times the charge. On two spheres joined by a wire it is the square root, and on a flat disc the density goes without limit at the rim, where no curvature is defined. No local rule can be right, because where charge sits is decided by the whole shape at once.

electromagnetism · Conductors

Named alongside it

The objects these essays reach for when they reach for this one.

ConductorEquipotentialBoundary conditionsElectric fieldElectric potentialCapacitanceLaplace equationConductorsElectrostatic shieldingThe Faraday cageField energyHarmonic measure

All concepts