Illustrated Physics
physics, drawn
Concept

Topology — where it appears

The properties of an arrangement that survive any continuous deformation — how many holes it has, what is connected to what, how many times a loop winds. Such quantities take whole-number values, which is why a superfluid's circulation and a superconductor's flux come in units.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

The whirlpool that comes in one size

Spin a bucket of ordinary liquid and it ends up turning with the bucket. Spin a bucket of superfluid helium slowly and it does not turn at all. Spin it faster and it does not turn either — until a threshold, at which a single line of circulation appears, carrying not some amount but exactly h/m. There is nothing in between, because the velocity is the gradient of a phase and a phase has to come back to itself.

fluids · Superfluidity The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter.

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

astrophysics · Flux freezing The rotation that mixes electricity into magnetism. The two Lorentz invariants of a field — E² − c²B² across and 2E·cB up — as the field is rotated by the duality transformation that takes E into cB and cB into −E. Every configuration moves on a circle, so the combination of the two invariants is preserved while neither is. A light wave sits at the origin and stays there, which is why a wave cannot be turned into anything else by this rotation; a static charge starts on the positive axis and is carried round to a pure magnetic field a quarter turn later. The source-free equations are unchanged by the whole family, so a universe with no charges in it has no way to say which field is which.

The symmetry one missing charge would complete

Maxwell's equations with no sources are unchanged by rotating the electric field into the magnetic one. With sources they are not, and the only thing missing is magnetic charge — which, if one existed anywhere, would force every electric charge in the universe to be a multiple of a fixed unit.

electromagnetism · Maxwell equations What comes back after one turn, and what needs two. A spin-½ pointing along z and rotated about the x axis through 720°, with the rotation integrated step by step rather than evaluated from a formula. The direction of the spin — the quantity a Stern–Gerlach magnet, a compass or any other instrument reports — is back where it started after 360°, exactly as the orientation of any other object would be. The state is not: its overlap with the state it began in has reached −1 there, and returns to +1 only after 720°. At 360° the overlap is -1.000 and ⟨σz⟩ is 1.000; At 720° the overlap is 1.000 and ⟨σz⟩ is 1.000. Both curves come off one integration of dψ/dθ = −(i/2)σx ψ whose norm is checked before anything is drawn, so the factor of two between their rates is a property of the propagation rather than of two separate formulae that were chosen to differ.

The turn that has to be made twice

Turn a spin-½ through a full circle and it does not come back. The direction it points in does, and every measurement on it does, but the state itself has changed sign — and a second full turn is needed before anything is where it started. The sign is invisible on one spin and measurable the moment a superposition has one branch turned and the other not.

quantum · Spin A potential that is lower every time round. The magnetic scalar potential along a path circling a wire carrying 10 amps, against the angle turned through, for 2 complete circuits. Away from the wire the magnetic field has no circulation round any small loop, so it is the gradient of something — and it is, except that the something does not come back to its own value. Each circuit lowers it by exactly the current, 10 amps, and a second circuit lowers it by 10 again. The potential is perfectly good locally and has no single value globally, and the amount by which it fails to close is the current threaded. So nothing has been lost in going from a circulation to a potential: Ampère's law has been rewritten as a statement about the shape of the region the potential lives in.

A potential that does not come back to itself

Where no current flows, the magnetic field has no circulation round any small loop, so it is the gradient of something and a magnetic problem becomes an electrostatic one. The catch is not that the potential fails to exist. It is that walking once round a wire lowers it by the current, and walking round again lowers it by the current again — so Ampère's law survives the translation as a statement about what the path encircles rather than about where it went.

electromagnetism · Ampere law Two straight worldlines that meet twice. A spacetime diagram of a universe whose space is a circle 10 light-years round, drawn unrolled: the shaded strip is the whole of space, and the strips beside it are the same space again, so the dashed lines are the same stay-at-home twin A seen in the copies. Twin B leaves A at 0.6c and never changes speed or direction. After 16.67 years by A's clock B has gone once round and meets A again — the line reaches A's copy at 10 light-years. Neither twin has accelerated at any moment; both worldlines are straight. A has aged 16.67 years and B 13.33, less by the factor 1/γ = 0.800. The usual resolution of the twin paradox, that the travelling twin is the one who turned round, is not available: here nobody turned.

The twin who never turns round

The twin paradox is usually settled by pointing at the turn: the travelling twin is the one who fired the engines and came back, and so the one whose worldline is not straight. Put the twins in a universe whose space closes on itself, like the surface of a cylinder, and the traveller can come home by going straight on. Neither twin ever accelerates, both worldlines are straight, and the traveller still comes back younger. What decides it is not any motion but the shape of space, which picks out one state of motion as special — and hands every observer a way to measure it.

relativity · Time dilation Four places where three charges' fields cancel. Field lines in the plane of three equal positive point charges at the corners of an equilateral triangle, with every point where the field is zero marked. There are four: one at the centre, and three more on the lines from the centre to the middle of each side, 0.164 of a side from the centre. Four is (n − 1)² for three charges, the largest number Maxwell conjectured any three charges could make. Each zero is a saddle: along some directions the field pushes a test charge back and along others away, so none of them can hold anything still.

The zeros a handful of charges can make

Between two equal charges there is a point where their fields cancel. With three there can be two such points, or one, or four. Every such point is a saddle, never a place where a charge could rest. In two dimensions the count is fixed by algebra: n equal line charges make exactly n − 1 zeros, the roots of a polynomial's derivative. In three dimensions nobody knows the answer. In 1873 Maxwell conjectured that n point charges can make at most (n − 1)² zeros. Three charges on a triangle make exactly four, and the conjecture has not been proved even for three charges.

electromagnetism · The field concept Six cells of a two-dimensional foam, by number of sides. Cells of a two-dimensional soap froth with three to eight sides, each drawn with its films meeting three at a time at 120°, as surface tension requires. A hexagon can do that with straight sides. A cell with fewer than six sides cannot: to make its corners 120° its sides must bulge outward, so its gas is at higher pressure than its neighbours' and leaks out. A cell with more than six must have sides curving inward, lower pressure, and gains gas. The rate of change of area, in units of πκγ/3 where κ is the films' permeability, is n − 6: −3 for a triangle, +2 for an octagon.

The bubble that grows by counting its sides

A froth of soap bubbles squeezed between two sheets of glass coarsens over a few days: small cells vanish and large ones swell. The obvious guess is that size decides which cells win, as it does for two bubbles joined by a tube, where the small one always empties into the large one. In a froth it is wrong. John von Neumann showed in 1952 that a cell's area changes at a rate fixed entirely by how many sides it has — shrinking if fewer than six, growing if more — and not at all by how big it is. The proof is a single sum of angles.

fluids · Surface tension

Named alongside it

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

CirculationLaplace equationQuantisationRotationSymmetryAmpere lawAngular momentumBoundary conditionsCoarseningCondensateConductivityCurrent sheet

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