Illustrated Physics
physics, drawn
Series

Potential — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

    One number for every point, and nothing at all is lost

    The electric field is three numbers at every point of space. Replacing it with one number loses nothing — and the reason it loses nothing is the same reason a hill can be drawn as a contour map.

    part 1 · electromagnetism
  2. Every stationary point has a way out. The potential along four lines through the most symmetric point of a symmetric arrangement of 4 equal charges at the corners of a square — the one place a trap might be expected. In the plane of the square the potential rises in every direction; out of the plane it falls. The point is stationary and it is a saddle, which is what Laplace's equation forces: the three second derivatives must sum to zero — computed here as 1.4e-5 against the individual values of order 2e+0 — so if two of them are positive the third must be negative. A particle released here rolls away along the direction that falls. No amount of ingenuity in placing the charges changes this, because the constraint is on the equation rather than on the arrangement, and it is why every real trap for a charged particle either uses time-varying fields, or a magnetic field with a velocity, or a material with a negative response.

    Nothing can be held still by a static field

    However many charges are arranged, however cleverly, a charge placed among them has somewhere to fall. The reason is one line of arithmetic — the potential in empty space satisfies Laplace's equation, and a solution of that equation has no interior maximum or minimum — and the consequence is that every real trap for a charged particle works by breaking one of the assumptions rather than by being cleverer.

    part 2 · electromagnetism
  3. 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.

    part 3 · electromagnetism
  4. 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.

    part 4 · electromagnetism
  5. Three cylinders that focus electrons. An einzel lens: three coaxial tubes of radius R, the outer two at the potential the electrons arrived with and the centre one, 2R long, at 4 times it. Electrons enter from the left parallel to the axis at five heights, from 0.1R to 0.7R, and are traced through the field without approximation. In each gap the equipotentials bulge between the tubes, and an electron crossing them is refracted as at a curved glass surface: pushed towards the axis where the potential is low and away from it where the potential is high. It is slower where it is pushed inward, so it spends longer there, and each gap converges on balance; the electron leaves with the speed it came in with. The rays cross the axis at 4.00R, 3.92R, 3.78R, 3.56R, 3.25R beyond the lens centre: the outer ones are bent too much and cross first, which is spherical aberration.

    The voltage that bends an electron like glass

    An electron that has fallen through a potential difference V moves at a speed proportional to √V. Crossing a surface of constant potential, it keeps the part of its momentum that runs along the surface and changes the part across it, and the two rules together are Snell's law with √V in place of the refractive index. Every arrangement of charged electrodes is therefore a piece of glass for electrons. Three metal tubes make a lens that focuses whichever way its middle tube is charged — and, by a theorem proved in 1936, no lens built from round electrodes can be free of spherical aberration. That theorem kept electron microscopes some fifty times coarser than their wavelength for sixty years.

    part 5 · electromagnetism
  6. 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.

    part 6 · electromagnetism
  7. Water drops that charge two cans to a spark. The voltage between the two cans of a Kelvin water dropper against time, on a logarithmic axis, starting from a stray difference of 1 V, for 10, 20, 40 drops a second in each stream; drops 3 mm across, each with a capacitance of 0.17 pF to its ring, cans of 20 pF. Each drop carries a charge induced by the opposite can's potential into its own can, which raises the opposite can's induced charge on the next drop: the difference grows exponentially, multiplying by e every 6.0 s at 20 drops a second. It passes the 10 kV that will jump a few millimetres of air after 55 s — from one volt to a spark, with nothing but falling water.

    The water drops that charge themselves to a spark

    Two streams of water drip from one earthed pipe through two metal rings into two metal cans, and each ring is wired to the opposite can. Nothing is charged to begin with, nothing is rubbed, nothing is plugged in. Within a minute a spark cracks between the cans at ten thousand volts. A stray volt between them makes the drops forming in one ring pick up a slight charge, which they carry to the can that makes the other ring's drops pick up the opposite charge — a loop in which every drop strengthens the imbalance that charged it. Kelvin built it in 1867; it is a dynamo made of water and gravity.

    part 7 · electromagnetism
  8. An electron held above helium by its own reflection. The potential energy of an electron at height z above liquid helium (ε = 1.0572), attracted by its image, −Λe²/4πε₀z with Λ = 0.00695; the surface itself is a wall the electron cannot cross. The lowest three levels (horizontal lines) are −0.657, −0.164, −0.073 meV — hydrogen's 1/n² series at 0.657 meV instead of 13.6 eV — and each is drawn with its probability density above it. The ground state's electron hovers 11.4 nm above the surface on average, about thirty helium atoms' diameters; the first excited state 45.7 nm.

    The atom whose nucleus is a reflection

    An electron set down on liquid helium cannot go in, because the helium atoms' closed shells repel it, and it cannot go far, because the liquid polarises and pulls it back with the field of an image charge. The image is weak, three-hundredths of the electron's charge. So the electron sits in a one-dimensional hydrogen atom whose nucleus is its own reflection: its levels follow hydrogen's 1/n² series at 0.66 milli-electronvolts instead of 13.6 electronvolts, and it hovers eleven nanometres above the surface. Microwaves at 126 gigahertz lift it to the next level, and a field pressing down on it tunes the line.

    part 8 · electromagnetism

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