Illustrated Physics
physics, drawn
Concept

Eddy currents — where it appears

Currents induced in the body of a conductor by a changing magnetic field, circulating in closed loops rather than along a wire. They screen the interior, dissipate energy, and set the depth a field reaches: 9.2 mm into copper at fifty hertz and 2.1 microns at a gigahertz.

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

A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start.

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

electromagnetism · Induction Skin depth against frequency, over eleven decades. The skin depth of copper, stainless steel, seawater, mu-metal against frequency, both axes logarithmic. Every line has the same slope, −½, because the depth goes as the inverse square root of the frequency for all of them; what separates them is conductivity and permeability, which enter the same way. Three points are marked and each is a practical fact. Copper at mains frequency has a skin depth of 9.22 mm, so a busbar of that thickness carries current throughout and a thicker one does not. Copper at a gigahertz has 2.1 µm, so the current in a microwave component runs in a layer thinner than the plating and the surface finish becomes the conductor. Seawater at the frequency navies use to signal submerged submarines has 28.9 m, which is why that link is measured in tens of hertz and carries a few characters a minute. Mu-metal is the outlier and the reason it exists: its conductivity is forty times worse than copper's and its permeability twenty thousand times better, so at low frequency it is the one material on the chart that is thin.

How far a field gets into metal

Every static argument about conductors says the field inside one is zero, and every one assumes the electrons have had time to move. Give them less time and the field gets in — 9.2 millimetres into copper at mains frequency, 2.1 microns at a gigahertz — and the metal box that silences a radio does almost nothing about the cable running past it.

electromagnetism · Conductors 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 A ring pushed and pulled, and pushed a little more. Over two cycles of the alternating supply, for a ring whose reactance equals its resistance (ωL/R = 1): the flux through the ring (grey), which also sets the radial field that pushes the ring along the core; the voltage induced round the ring (dashed); the current, which lags that voltage by 45° because of the ring's own inductance (blue); and the force along the axis, current times radial field (red). The force pushes the ring away from the coil for part of each half-cycle and pulls it back for the rest; with the lag, the pushes outweigh the pulls, and the average (dotted) is 0.250 of the peak flux–current product. With no lag the pushes and pulls would cancel exactly. The ring jumps because its current is late.

The ring that jumps because its current is late

Slip an aluminium ring over the iron core of a coil, switch on the alternating current, and the ring leaps off the top. The usual explanation is Lenz's law: the ring's induced current opposes the change, so the ring is repelled. But an alternating field changes both ways every cycle, and an induced current that simply opposed the change would be pushed and pulled equally and stay put. The ring jumps only because its current runs late, held back by its own inductance — which is why a cooled ring jumps far higher and a ring with a saw cut through it does not move.

electromagnetism · Induction Two descriptions of gold that agree everywhere but at zero. How strongly gold answers a slowly varying field, (ε(iξ) − 1)ξ²/ωₚ² in the two forms of its permittivity at imaginary frequency ξ, against ξ on a logarithmic axis: the plasma model (dashed), which ignores the electrons' collisions and gives exactly 1, and the Drude model (solid), which includes them at the measured rate ħγ = 35 meV and falls to zero below about γ = 5.3·10¹³ per second. The Lifshitz force samples this curve only at the Matsubara frequencies 2πlkT/ħ (dots, 300 K): the first, 2.47·10¹⁴ per second, lies above γ, and there the two descriptions differ by 18 per cent, at the next by 10, and less beyond. At l = 0, ξ = 0, they part completely: 1 against 0. Everything the two models disagree about is in that one point.

The static field a metal may or may not reflect

The Casimir force between two gold plates at room temperature has been computed two ways for twenty-five years, and the two answers differ by forty per cent at three micrometres. Both use the same measured optical properties of gold. They part company over a single number: whether gold reflects a magnetic field that does not change in time. Common sense and every electrical engineer say it does not, since a static field passes straight through a copper sheet. The most precise force measurements say it does. The disagreement is not about the Casimir effect at all; it is about what a metal's electrons do with a fluctuation slower than their collisions.

electromagnetism · Retardation

Named alongside it

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

ConductivityElectromagnetic inductionLenz's lawMagnetic levitationSkin depthCasimir effectConductorsConservation of energyDampingDiffusionDissipationDrag

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