Illustrated Physics
physics, drawn
Series

Induction — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.

    The field that makes the other, and only while it is changing

    A magnet sitting next to a coil does nothing at all. Move it and a current flows. The law is not about the field but about its rate of change, and everything electrical since 1831 rests on that distinction.

    part 1 · electromagnetism
  2. 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.

    part 2 · electromagnetism
  3. A disc that generates 0.785 V with no changing flux anywhere. The force per unit charge on a carrier in a conducting disc of radius 100 mm spinning at 3000 revolutions a minute in an axial field of 0.5 tesla, against how far out the carrier is. Each carrier is moving through the field at ωr, so v × B pushes it radially with a force per unit charge of ωrB — zero at the axle and largest at the rim. The area under the line is the potential difference between the axle and the rim, which is ½BωR² = 0.7854 volts. Nothing in this calculation mentions a loop, a flux or a rate of change, and the flux through the circuit formed by the disc, the contacts and the external wire does not change at all while the machine runs.

    The rule that is two laws wearing one coat

    The flux rule folds two entirely different pieces of physics into one number, and they agree exactly, which is why nobody notices there are two. A disc spinning in a steady field generates a voltage with no changing flux anywhere, and the folding comes apart.

    part 3 · electromagnetism
  4. Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields.

    The coupling that is the same both ways

    A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

    part 4 · electromagnetism
  5. How much of μ₀N²A/ℓ a real coil actually has. The inductance of a 140-turn coil of radius 10 mm, divided by the long-solenoid formula μ₀N²A/ℓ, against how long the coil is compared with its diameter. The inductance is computed as a sum of Maxwell's mutual inductances between every pair of turns plus each turn's own, so the long-coil formula appears nowhere in it. At a length equal to the diameter the real coil has 68% of what the formula promises, and at a quarter of the diameter about a third. The reason is that the formula assumes every turn is threaded by the full interior field, and near the ends of a short coil the field has already begun to spread. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computed points agree with the table.

    The circuit that fights its own change

    Every circuit is threaded by the field its own current makes, so every circuit resists having that current altered. It is the reason a coil takes time to start and the reason breaking one makes a voltage a thousand times the supply's.

    part 5 · electromagnetism
  6. 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.

    part 6 · electromagnetism
  7. One number decides the efficiency. The best efficiency a link of two resonant circuits can reach, with its load chosen to maximise it, against the figure of merit U = k√(Q₁Q₂) on a logarithmic scale: the coupling coefficient times the geometric mean of the two coils' quality factors. The curve is η = U²/(1 + √(1 + U²))², and nothing else about the coils, the frequency or the distance enters: 0.25 per cent at U = 0.1, 17 per cent at U = 1, 52 per cent at U = 3, 82 per cent at U = 10, 94 per cent at U = 30. A weak coupling is rescued by a high Q: k = 0.003 with Q = 1,000 on each side gives U = 3 and 54 per cent, where the same coils used without tuning capacitors, as a plain transformer, deliver under one per cent, with the source driving mostly reactive current that does nothing.

    The coupling a resonance rescues

    Two coils a hand's breadth apart share a few thousandths of their magnetic flux, and as a transformer they deliver almost nothing. Tune each with a capacitor to the same frequency and the same pair can carry most of the power across. Resonance does not strengthen the coupling, which is fixed by the geometry and falls as the cube of the distance. It changes how many times the energy goes round before it is lost. The efficiency then depends on one number, the coupling times the quality factor, and that one number explains a toothbrush on its stand, a phone on its pad, a car parked over a plate in the road and a light bulb lit two metres away.

    part 7 · electromagnetism
  8. 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.

    part 8 · electromagnetism
  9. The resistance a pan offers the coil. The surface resistance — the square root of πfμρ — of five pan materials against frequency, on logarithmic axes, with the band where induction hobs work, 20 to 100 kHz, shaded. All five rise as the square root of the frequency. At 25 kHz carbon steel offers 9.9·10⁻⁴ Ω and ferritic stainless 0.0024 Ω, while austenitic stainless, which is not magnetic, offers 2.7·10⁻⁴ Ω, aluminium 5.1·10⁻⁵ Ω and copper 4.1·10⁻⁵ Ω. For the same coil current the heating goes as this resistance, so steel heats 19 times as hard as aluminium. Matching steel with aluminium by frequency alone would need 377 times the frequency — 9 MHz, where the hob's own electronics and coil would be the problem.

    The pan a better conductor will not heat

    An induction hob heats a pan with currents the pan carries itself, driven by a field oscillating twenty-five thousand times a second, and it will heat a cast-iron skillet furiously while an aluminium saucepan, which conducts electricity six times better, barely warms. The better conductor is the worse load. What a pan presents to the coil is a surface resistance that goes as the square root of permeability times resistivity, and on that measure iron beats aluminium twenty times over. The same arithmetic explains why kitchen foil on the same hob can catch fire, and why a steel tip heated this way switches itself off at a fixed temperature.

    part 9 · electromagnetism

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