The ring that jumps because its current is late

At a meeting of the American Institute of Electrical Engineers in 1887, the inventor Elihu Thomson showed a demonstration that has been repeated in physics lectures ever since. A long iron core stood upright inside a coil connected to the alternating-current mains. Over the top of the core he slipped a ring of aluminium. When the current was switched on, the ring shot up the core and off its top, sometimes several metres into the air. With the current left on, a ring could float a few centimetres above the coil, humming. A ring with a narrow cut across it did nothing at all.

The explanation usually given is short. The coil’s changing field induces a current in the ring, Lenz’s law says that current flows so as to oppose the change, so the ring is repelled. That explanation is incomplete in a way that matters, and the missing piece is the reason for the cooled ring, the cut ring, the floating ring and the choice of aluminium rather than copper.

An induced current, and a radial field to push on #

The field that makes the other found that a changing magnetic flux through a loop drives a voltage round it, in proportion to the flux’s rate of change. The coil’s alternating current makes a flux through the iron core that rises and falls sinusoidally, fifty times a second, and the ring around the core encloses that flux. So a voltage is induced round the ring, alternating at the same frequency, and since the ring is a closed loop of metal a large current flows.

A current in a magnetic field feels a force at right angles to both. Inside the core the field points along the axis, and a current going round the ring, crossed with an axial field, gives a force pointing inward or outward — radial — which a rigid ring simply resists. What lifts the ring is the part of the field that is not axial. Near the top of the coil the field lines spread out of the end of the core, so at the ring they lean outward, with a radial component. A current round the ring, crossed with a radial field, gives a force along the axis: up or down, depending on the directions of the current and the field at that instant.

A transformer with one shorted turn #

Seen from the coil’s side, the ring is the secondary winding of a transformer: a single turn of metal, shorted on itself, sharing the coil’s iron core. The coupling that is the same both ways found that the mutual inductance between two circuits is symmetric — the flux the coil drives through the ring per amp equals the flux the ring drives back through the coil — and that symmetry is why the ring’s current shows up in the coil’s: a shorted secondary draws a large current from the supply, which a demonstrator feels as the coil buzzing harder and the mains lead warming the moment the ring goes on. Lift the ring off and the coil’s current drops back.

The transformer picture also says where the ring’s energy comes from. Each cycle the supply puts energy into the coil’s field, some of it is transferred to the ring through the shared flux, and the ring spends it in two ways: as heat in its resistance, and as work against gravity when it moves. The proportion that goes into work rather than heat is set by the same ratio $x$ that sets the push. A ring with high resistance turns nearly all the energy it receives into heat; a ring with low resistance passes most of it back to the coil each half-cycle, as an inductive load does, and keeps enough to lift itself. It is the shorted-turn problem that the coupling a resonance rescues found limiting wireless power transfer, met here from the other side: there the aim was to deliver power to a load without it being lost as heat in the coils; here the ring is the load, and its job is to be pushed rather than warmed.

Pushed and pulled, every half-cycle #

Now follow the force through a cycle. The radial field at the ring is in step with the coil’s flux. The voltage induced in the ring is in step with the flux’s rate of change — a quarter-cycle ahead of the flux itself. If the ring’s current were in step with that voltage, it would be a quarter-cycle ahead of the radial field, and the product of the two, the force, would alternate equally between up and down, averaging to nothing. Lenz’s law would be satisfied at every instant, and the ring would only vibrate.

But the ring’s current is not in step with the voltage. A ring has inductance as well as resistance: its own current makes a magnetic field round itself, and the circuit that fights its own change found that such a self-inductance resists any change in current, so the current lags behind the voltage driving it. The lag is an angle $\phi$ with $\tan\phi = \omega L/R$, the ratio of the ring’s reactance to its resistance. With a lag, the current is less than a quarter-cycle ahead of the radial field — it is more nearly opposite to the flux it encircles — and the force spends more of each cycle pushing than pulling. The figure draws it for a ring whose reactance equals its resistance: the current lags by 45°, the force swings above and below zero, but the upward lobes are larger, and the average is a quarter of the peak product of flux and current.

So the ring is repelled on average, but only because its current is late. Lenz’s law sets the sign of the current; the lag sets whether it does any lifting. A ring with no inductance would not jump at all.

How much the lag buys #

The average force can be worked out in two lines. The current is the voltage, proportional to $\omega\Phi_0$, divided by the ring’s impedance $\sqrt{R^2 + \omega^2L^2}$; the average of the product of a sine and a lagging cosine is half the sine of the lag; and putting these together the average push is proportional to

$$\bar F \propto \frac{\omega^2 L}{R^2 + \omega^2 L^2} = \frac{1}{L}\,\frac{x^2}{1 + x^2}, \qquad x = \frac{\omega L}{R}.$$

The factor $x^2/(1+x^2)$ runs from nothing, for a ring with large resistance and a current nearly in step with the voltage, to one, for a ring with no resistance, whose current lags by a full quarter-cycle and sits exactly opposite the flux it encircles. That limit is the perfect conductor: a ring that keeps the flux through it constant by carrying whatever current is needed to cancel any change, which is the behaviour of the field that is pushed out of a superconductor, and which pushes against the coil’s field at every instant. A real ring sits somewhere along the curve, and where it sits depends on its resistance, its inductance and the frequency.

The same lag decides the lift on a magnet moving over a conducting sheet, which the drag that falls as the magnet speeds up followed: at low speed the induced currents are nearly in step with the changing field, and the magnet is mainly dragged; at high speed they lag, and the magnet is mainly lifted. The jumping ring and the levitating magnet are the same physics, one driven by an alternating current, the other by motion.

A cold ring jumps higher #

The most striking variation on the demonstration is to cool the ring in liquid nitrogen first. The cold ring leaps several times higher than a warm one.

The ring’s size and shape, and so its inductance, are unchanged by cooling. Its resistance is not. The resistance of a pure metal comes mostly from its electrons scattering off the vibrations of the crystal lattice, and cooling quietens the vibrations: between room temperature and the 77 kelvin of liquid nitrogen, aluminium’s resistivity falls to about a tenth. The ring’s ratio of reactance to resistance therefore rises tenfold, and it moves from the lower part of the curve, where the push is a fifth of the limit, to near the top. For the ring assumed in the figure, the average push rises nearly fivefold. Warm the ring back up, as it does within a minute or two in the air, and the effect fades.

The resistivity curve in the figure is the Bloch–Grüneisen form, which describes how lattice vibrations scatter electrons, scaled to aluminium’s room-temperature value; below about 40 kelvin it flattens towards a residual resistance set by impurities, and a ring of very pure aluminium cooled in liquid helium would reach the perfect-conductor limit almost exactly.

Nothing from a steady current #

The formula also explains why the ring needs alternating current.

At low frequency the ring’s current is small, because the flux changes slowly, and nearly in step with the voltage, so its average push grows as the square of the frequency. A steady current, frequency zero, gives none at all once it has settled. Switching a direct current on does give a single kick: while the coil’s current is rising, the flux through the ring is changing and the ring carries a current that opposes it, and a ring can be thrown that way — the principle of the electromagnetic launchers and coil guns that fire a ring or a projectile with a single pulse. But a steady field, however strong, only holds a ring still, and does not hold it up. Above a frequency of about $R/L$, on the other hand, the push stops growing, because the current is already as late, and as opposed to the flux, as it can be. Higher frequencies buy little, and they cost more, because the iron core and the coil waste more power.

Which ring jumps highest #

What matters for a jump is not the push but the push per unit weight.

In the resistive regime, where most rings sit at room temperature, the push is inversely proportional to the metal’s resistivity, and the weight is proportional to its density. Copper conducts better than aluminium by a factor of about 1.6, but it is more than three times as dense, so a copper ring of the same size is pushed harder but jumps lower: per unit weight, about half as well. Brass, a poorer conductor and dense, manages an eighth. This is why the demonstration always uses aluminium, and why induction motors’ rotor bars and the conducting plates of eddy-current brakes are often aluminium too. Cooling multiplies the figure for both metals, though at the cold values the rings are near the inductive limit and the real gain is capped well short of the figure’s simple estimate.

The cut ring is the control experiment. A saw cut across the ring breaks the path round it, so although the changing flux still induces a voltage round the ring — it appears across the cut — no current can flow, and with no current there is no force. The cut ring sits on the coil as if nothing were happening. It is a direct demonstration that the force comes from a current circulating in the ring, and not from any attraction or repulsion between the coil’s field and the metal itself.

Lenz’s law is a statement about energy #

It is worth being careful about what Lenz’s law actually says, because the jumping ring is so often offered as its illustration. The rule that is two laws wearing one coat found the flux rule to be two different physical statements — a changing field inducing an electric field, and a moving conductor feeling a magnetic force — that happen to give the same formula. Lenz’s law adds the sign: the induced current always flows so that its own field opposes the change that drove it. That sign is required by conservation of energy. If the induced current aided the change, it would strengthen the flux, which would drive a larger current, and the process would run away with no energy supplied.

The magnet that falls slowly through a copper pipe is Lenz’s law in its purest form: the induced currents oppose the magnet’s motion, the magnet’s energy goes into heating the pipe, and the magnet falls at a slow, steady speed. The jumping ring is a subtler case. Lenz’s law fixes the direction in which the ring’s current tends to flow, but the ring is pushed upward only on average and only because of the lag, and an explanation that stops at Lenz’s law predicts a ring that should be repelled even by a perfectly resistive current in step with the voltage, which it is not.

A floating ring and a molten one #

With the current left on, a ring that has jumped falls back and settles at a height where the average push balances its weight, a few centimetres above the coil, where the radial field is weaker. There it floats, humming at twice the mains frequency as the force swings through each cycle, and slowly warming, since the induced current heats it; as it warms its resistance rises, its lag shrinks, and it sinks a little. Held down by hand, it becomes hot enough to burn within a minute. That heating is the same effect on purpose in an induction hob, where a pan’s bottom is the ring.

Industrial versions use the push and the heat together. A lump of metal placed in a cone-shaped coil fed with high-frequency current is both lifted and melted, floating as a liquid droplet out of contact with any container — levitation melting, used to prepare very pure alloys and to measure the properties of molten metals without contamination. And a magnetically levitated train of the electrodynamic kind floats on the same lag: magnets on the moving train induce currents in coils or plates on the track, and above a certain speed those currents lag enough to push the train up.

A ring with one resistance and one inductance #

The figures treat the ring as a single circuit with one resistance and one self-inductance, and the radial field at the ring as simply proportional to the coil’s flux. Both are idealisations. A real ring is a thick piece of metal in which the current crowds towards the surface facing the coil, so its effective resistance rises with frequency; its inductance depends on the iron core it surrounds, which raises it considerably; and the radial field varies with the ring’s height and with how the iron saturates. The room-temperature value of $\omega L/R$ assumed in the cooling figure, 0.5, is a choice, not a measurement — real demonstration rings range over a factor of several either side of it — and the conclusion that cooling helps by a large factor holds for any ring that starts in the resistive regime.

The materials figure uses the resistive-regime scaling, which overstates the gain from cooling once a ring passes into the inductive regime, as its caption says. And nothing in the figures includes the iron core’s own losses, the ring’s heating during a jump, or the dynamics of the jump itself, which depends on how fast the push falls as the ring rises out of the region where the field is strongest.

Still open: the best shape for a launched ring #

The jumping ring is a simple linear motor, and the same physics in a more serious form drives electromagnetic launchers: coil guns that accelerate a conducting projectile through a sequence of coils fired in turn, and proposals to launch payloads into orbit along kilometres-long tracks. Their rival, the railgun of the push at the end of a pair of rails, drives its projectile with a direct current instead and pays for it in rail erosion. The coil gun’s efficiency is limited by exactly the trade-off in the figures — the projectile must have low resistance to be pushed hard, but its current also heats it, and the push falls once it leaves the region of strong radial field — and the best shape, material and timing for a projectile is an optimisation problem without a general solution, settled for each design by computation and test.

The ring on the coil carries the principle in its simplest form. An alternating field induces a current that opposes the flux, but a current that merely opposes the change, in step with the induced voltage, is pushed and pulled equally; what lifts the ring is its inductance making the current late, so that it spends more of each cycle opposing the field than aiding it. The ring jumps because its current is late — and so the colder, lighter and less resistive the ring, the higher it goes, and the ring with a cut in it does not go at all.