Illustrated Physics
physics, drawn
Concept

Induction — where it appears

The generation of an electric field, and so of currents in conductors, by a changing magnetic field. The induced currents oppose the change that makes them, which is why a moving magnet is braked by nearby metal and a coil resists changes in its own current.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges.

The angular momentum that is in nothing at all

A charged ring and a solenoid, both at rest, with nothing moving anywhere. Switch the solenoid off and the ring starts to turn. Angular momentum is conserved, so it was there before the switch was thrown — and it was not in the matter, because nothing was moving. It was in the field, and it is qΦ over 2π whatever the geometry.

electromagnetism · Field energy The drag peaks and then gives way to lift. The two forces on a magnet moving parallel to a thin conducting sheet, as fractions of the repulsion a perfect mirror image would exert, against the magnet's speed divided by the sheet's characteristic speed w = 2/μ₀σd, on a logarithmic axis. Drag grows in proportion to the speed at first, peaks at 1.27 w with 0.300 of the full repulsion, and then falls as w/v. Lift starts as the square of the speed, equals drag at v = w, where both are 0.293, and approaches the full repulsion at high speed: 0.900 at 10 w. The same eddy currents cause both. What changes with speed is how much of the image they form has time to sink away before the magnet has passed.

The drag that falls as the magnet speeds up

Drop a magnet down a copper pipe and it creeps, because the currents it stirs up in the metal pull back on it in proportion to its speed. Push it faster and the proportion fails: the drag rises to a peak and then falls away, while a second force, pointing straight up, grows to replace it. A magnet moving fast enough over a sheet of aluminium is not braked but held up, with lift thirty times its drag at the speed of a fast train, and the reason is that at high speed the metal has no time to forget the magnet has passed.

electromagnetism · Induction The ring where an axisymmetric field vanishes. Field lines of the slowest-decaying dipole field of a conducting sphere, drawn in a plane through its axis (the axis is the left edge, the sphere's surface the arc). Each line is a contour of the flux function, which is zero on the axis and at infinity and positive between, so it must have a maximum. Here the maximum is on the equator at 0.873 of the radius: there the lines shrink to a point, which in three dimensions is a ring round the axis on which the meridional field is exactly zero. 7 of the 11 drawn lines leave the sphere and close outside; the other 4 close inside, round the ring. The toroidal current that makes the field is not zero on the ring: it is 89 per cent of its largest value.

The field no symmetric flow can keep

The Earth's core would forget its magnetic field in about forty thousand years, and the field is at least three and a half thousand million years old, so something must be making it again all the time. The obvious candidate is the flow of liquid iron. In 1934 Thomas Cowling proved that the simplest version cannot work: no motion of a conductor, however fast or however arranged, can keep up a field that is symmetric about an axis. The reason is a single ring inside the field where the current it needs cannot be driven.

astrophysics · Flux freezing A disc that makes its own field once it turns fast enough. A self-exciting disc dynamo started from rest, its motor's torque set so that, with no current, the disc would run at 0.8, 1.5, 2.5 times the critical speed R/M. Dashed: the disc's speed, in units of the critical speed. Solid: the current in its coil, in units of √(G/M), starting from a stray seed of a millionth, of either sign. Time is in units of the circuit's own L/R. At 0.8 the disc settles at its free-running speed, below critical, and the seed dies. Above it the disc spins up past the critical speed and nothing happens for a while — the current is growing exponentially from a millionth — until it reaches a tenth, at t ≈ 29 for 1.5 and 11 for 2.5 and becomes large enough to brake the disc. Speed and current then ring and settle at exactly the critical speed and a current of ±√(g − 1): 0.71 and 1.22, of the sign the seed happened to have.

The dynamo that reverses with nothing pushing it

A copper disc spinning in a magnetic field drives a current from its axle to its rim. Wire that current through a coil wound round the disc and the coil supplies the field the disc needs — and above one definite speed, a resistance divided by an inductance, the arrangement makes its own field out of any stray one, with nothing else magnetic anywhere. It cannot choose which way the field points; the stray field chooses. Couple two such discs, each feeding the other's coil, and with the motors' torque held perfectly steady the field flips direction again and again at intervals that follow no pattern. The simplest dynamo that works is also the simplest one that reverses.

astrophysics · Flux freezing

Named alongside it

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

DynamoMagnetic fluxAngular momentumChaosConductivityConservationDissipationDragEddy currentsField energyField momentumFlux

All concepts