The magnet that falls slowly

A neodymium magnet dropped down a length of copper pipe does not fall. It descends, at something between a walk and a crawl, and arrives at the bottom several seconds after an identical unmagnetised slug of steel dropped at the same moment. Nothing in the apparatus is unusual. The copper is not magnetic — a compass laid on the pipe does not move — the magnet never touches the wall, and there is no circuit anywhere: no wire, no battery, no loop of anything.

The explanation is Faraday’s, and the interesting part is not that it works but what shape the answer has: the retarding force turns out to be strictly proportional to the magnet’s speed, and to depend on the tube’s radius to the fourth power. Neither is obvious, and both can be computed exactly.

Where the current is, when there is no circuit #

A changing magnetic flux drives an electromotive force around any closed path. Usually that path is a wire, because a wire is where the free charges are made to go.

The textbook arrangement is a loop of wire leaving a field region, with an emf around the loop equal to the rate at which the flux through it falls. The loop makes the path explicit, and that is exactly what a solid tube does not: in a continuous conductor the current picks its own path, and the whole difficulty of the problem is that nothing says in advance where it goes.

A copper tube is a conductor with no ends. Any circle drawn round its axis is a closed path, and every such path has the magnet’s flux through it. As the magnet moves, that flux changes, an EMF appears around the circle, and — because the copper conducts — a current flows. These are the eddy currents, and they need no circuit because the metal is the circuit, everywhere at once — the same currents that decide how far an alternating field gets into a metal at all.

The whole calculation follows from writing down the flux through one such ring. For a magnet far enough from the ring to be treated as a point dipole of moment $m$ on the axis, at axial offset $z$ from a ring of radius $a$,

$$\Phi(z) = \frac{\mu_0 m a^2}{2\left(a^2 + z^2\right)^{3/2}}.$$

That second curve settles something the picture of a magnet “pushing against” the metal gets wrong. The ring doing the braking is never the ring the magnet is inside. At the moment the magnet is level with a ring, the flux through it is at its greatest and changing not at all, so no EMF appears there. The current is in the rings above and below — one being entered and one being left — and it runs in opposite senses in the two, so the arrangement grips the magnet from both ends rather than pushing it from behind.

Which way the current goes, and why it always opposes #

The direction is fixed by Lenz’s law, which is not an extra postulate but a consequence of energy conservation: the induced current must oppose the change producing it, because a current that reinforced the change would grow without limit and deliver energy from nothing.

Each ring of eddy current in the tube wall makes a field of its own, and the directions are what Lenz’s law fixes. Ahead of the falling magnet the induced field opposes the increase and behind it opposes the decrease — so both act to slow the magnet, and neither can be arranged to help it. That is the sign that makes the effect a brake rather than an accelerator, and it follows from energy conservation rather than from any detail of the geometry.

The force on the magnet is the reaction to the force the magnet’s field exerts on the induced currents, and that one is the plain magnetic force on a moving charge — perpendicular to both the current and the field, and adding up around each ring to a net axial push.

The elementary force it all reduces to is a charge moving across a magnetic field being pushed sideways. In the tube wall the charges are conduction electrons and the field is the magnet’s own, so the current is not driven by a battery or by a changing flux through any particular circuit — it is the direct Lorentz force on carriers that happen to be moving relative to the magnet.

Adding up the whole wall #

Each ring of height $\mathrm{d}z$ in a wall of thickness $w$ and conductivity $\sigma$ has resistance $2\pi a/(\sigma w\,\mathrm{d}z)$. A magnet moving at speed $v$ makes an EMF of $-v\,\mathrm{d}\Phi/\mathrm{d}z$ in it, so the power dissipated there is $(\mathrm{d}\Phi/\mathrm{d}z)^2 v^2 \sigma w\,\mathrm{d}z/2\pi a$. Summing over the whole tube and setting the total equal to $Fv$ gives the retarding force:

$$F = \frac{\sigma w}{2\pi a}\, v \int_{-\infty}^{\infty}\left(\frac{\mathrm{d}\Phi}{\mathrm{d}z}\right)^{2}\mathrm{d}z.$$

Two features of that expression matter more than its value. The integral does not contain $v$, so the force is exactly proportional to speed — this is a dashpot and not a friction, and the difference is measurable rather than a matter of description. And the integral contains the tube’s radius in a way that turns out to be severe.

Evaluating the integral gives $45\mu_0^2m^2\sigma w/1024\,a^4$, and it is worth saying plainly what the figure does with that: it does not use it. The integral is evaluated numerically from the flux profile every time, so the coefficient $45/1024$ is something the site’s own gate can recover from the drawn force and compare against, rather than a number the figure was told.

The fourth power, which decides whether the demonstration works #

A fourth power is brutal. A tube sixty per cent wider than another brakes six times less hard, which is why the demonstration needs a magnet that only just goes down the pipe and why the same magnet dropped down a drainpipe falls like a stone. It is also why the effect is so often described as requiring “a strong magnet”: what it really requires is a close-fitting one, and a modest magnet in a snug tube outperforms a powerful one in a loose tube by a wide margin.

Where the four powers come from is worth separating, because two quite different things contribute two each. The flux through a ring falls as $a^{-1}$ at fixed dipole moment once the ring is larger than the magnet, and the derivative of that flux with respect to position falls as $a^{-2}$, since the length over which the flux changes appreciably is itself the radius. Squaring the derivative in the dissipation gives four powers down. The ring’s resistance then supplies one power up, because a bigger ring is longer, and the prefactor $1/a$ in the sum over rings supplies the last one down. Four powers, assembled from three separate pieces of geometry, none of which is the obvious one.

The magnet’s own strength enters as $m^2$, because the moment appears once in the EMF and once in the force. Doubling the magnet quadruples the braking, and since the weight only doubles, a bigger magnet of the same shape falls more slowly — the opposite of the usual intuition about heavy things falling fast.

A real magnet is not a point dipole, and the flux expression used throughout is the dipole one. The approximation is worth testing rather than assuming: it holds where the tube radius is large compared with the magnet’s length, and it fails for the short fat magnets that make the best demonstrations — which is why the measured drag exceeds the calculated one in most classroom versions.

Where the energy goes, and how quickly it gets there #

At terminal speed the magnet’s kinetic energy is constant, so all the work gravity does is being dissipated as it is released. For the metre drop in the first figure that is 118 millijoules — small, but deposited in a region of wall only a few centimetres long that travels down the pipe with the magnet.

Energy does not travel along a wire with the current; it flows in through the sides, from the field outside. That is worth stating because it answers where the magnet’s lost potential energy goes: it crosses into the tube wall through the field surrounding it, and appears as heat distributed through the metal rather than delivered along any path a current follows.

This is the direction of time entering an otherwise reversible piece of electromagnetism. Maxwell’s equations run backwards perfectly well; a film of a magnet rising out of a copper tube while the tube cooled would satisfy every one of them. What forbids it is that the energy has become heat, spread over the enormously larger number of ways a lattice has of holding it, and the current in a resistive metal is where that conversion happens. A superconducting tube would brake the magnet too, but reversibly: the currents would not decay, and the magnet would bounce.

The tube as an instrument #

The force expression contains exactly one property of the wall — its conductivity — and everything else in it is a geometry. Run that backwards and the falling magnet is a conductivity meter.

That is not a curiosity; it is an industry. A coil driven with alternating current held near a metal surface induces eddy currents in it, and those currents react back on the coil and change its impedance. Measuring the change gives the conductivity of whatever is underneath, without contact, without preparation, and through a coat of paint.

The two things it is used for follow from the two ways the reading can go wrong. A change in conductivity means a change in material, so the technique sorts alloys and grades heat treatments — two visually identical pieces of aluminium can be told apart in a second. And a discontinuity in the metal forces the eddy currents to detour round it, which changes the impedance too, so the same instrument finds cracks. Every aircraft fastener hole and every steam-generator tube is inspected this way.

The frequency is the control. Eddy currents at frequency $f$ penetrate to a depth that falls as $1/\sqrt f$, so a low frequency reads deep and coarsely and a high one reads a thin surface layer with fine resolution. Sweeping the frequency therefore profiles the material with depth, which is a remarkable thing to get from a coil held against a surface.

None of that needs a magnet falling down a tube, and the tube is the cleanest demonstration that the underlying quantity really is the conductivity alone. Three tubes of the same dimensions in copper, aluminium and brass give three terminal speeds in the ratio of three conductivities, and nothing else about the metals — their density, their hardness, their magnetic susceptibility — appears anywhere.

The same effect, four orders of magnitude larger #

The most surprising place this argument reappears is not in a laboratory.

A loop of highly conducting fluid contracting, with field lines threading it, obeys the same law: a change in flux drives a current that opposes the change. Four orders of magnitude up in scale that is flux freezing, and the magnet in the tube is the same physics in a form that fits in a hand — the only difference being how nearly perfect the conductor is.

A copper tube and a collapsing star differ in this only by a number. Both are conductors in which a change of flux drives currents that resist the change; the copper resists weakly, so the magnet still falls, and the stellar plasma resists almost perfectly, so the field is carried with the collapse and multiplied by ten billion. The magnet in the pipe is a laboratory demonstration of flux freezing, run at a conductivity low enough to see it fail.

One consequence of that comparison is worth stating as a number. The quantity deciding which regime a conductor is in is the product of its conductivity, its size and the speed of whatever is moving through it — the magnetic Reynolds number, in the notation of people who work on stellar plasmas. For the copper tube it is of order a hundredth: the currents decay far faster than the magnet moves, so the flux slips through freely and the braking is a small correction to a fall. For the collapsing core it is of order $10^{20}$, and the flux cannot slip at all. The two situations differ in a dimensionless number by twenty-two orders of magnitude and in the physics not at all.

The same physics is also, at the engineering end, how a train stops. An electromagnetic brake is a magnet moved past a conducting rail, with a force proportional to speed — smooth, silent, without contact and without wear, and useless at the last few metres per hour, because a force proportional to speed goes to zero exactly when the vehicle does.

Where the model stops #

The dipole approximation is the first casualty. Close to a real magnet the field has structure the point-dipole flux misses, and the tube’s wall sits a few millimetres from a magnet a centimetre long, so the numbers here are good to some tens of per cent rather than to per cent.

The thin-wall assumption is the second. The derivation gives every ring of the wall the same flux, which requires the wall to be thin compared with the tube’s radius. A thick-walled tube shields its own outer layers — the induced currents in the inner wall reduce the field reaching the outer — and the drag then grows more slowly than in proportion to the thickness. There is a skin depth here even though nothing oscillates, set by the magnet’s speed and the tube’s conductivity, and doubling the wall of an already-thick tube buys little.

Third, the magnet’s own field is treated as unaffected by the currents it induces. That is right when the induced field is small compared with the magnet’s, and it fails for very high conductivities — at which point the problem becomes the flux-freezing one above rather than this one, and the magnet does not fall at all.

How thin is thin #

The thin-wall assumption deserves a number rather than a caution, and the number comes from asking how far a field gets into a conductor before the currents it induces shut it out.

For a field changing at rate $\omega$ the penetration depth is $\sqrt{2/\mu_0\sigma\omega}$, and here the changing is done by the magnet’s own passage: a ring sees the flux rise and fall over the time the magnet takes to travel about one tube radius, so the effective rate is the speed over the radius. For a magnet at half a metre a second in a copper tube of seven and a half millimetres’ radius, that gives a penetration depth of about two centimetres.

So a wall a millimetre or two thick is thin by a wide margin, every ring of it sees essentially the same flux, and the calculation in this essay applies. A wall of two centimetres would not: its outer layers would be shielded by the currents in its inner ones, and adding more copper would buy less and less.

The estimate also explains something about the demonstration that would otherwise look like luck. Because the depth goes as one over the square root of the speed, a faster magnet penetrates less — so a tube that is thin for a slow magnet becomes effectively thick for a fast one, and the drag grows more slowly than in proportion to speed at the top end. The linearity the force figure shows is a low-speed statement, and it is comfortably satisfied at the tens of centimetres per second the demonstration runs at.

That is worth generalising. The linear dashpot behaviour and the shielded behaviour are the two ends of one parameter, and which end a given arrangement sits at is decided by comparing a diffusion depth with a thickness. It is the same comparison that decides whether a plasma freezes its flux, differing only in which of the two lengths is larger.

The ring that jumps #

There is a classic demonstration of the same law that behaves oppositely, and putting the two side by side isolates what each depends on.

Slip an aluminium ring over the core of a coil and switch on an alternating current. The ring flies off, sometimes to the ceiling. The mechanism is the same one — a changing flux through the ring drives a current, and the current’s field opposes the change, so the ring and the coil repel — and the difference is that the change is being supplied by the source rather than by the ring’s own motion.

Two consequences follow that the falling magnet does not show. The force does not vanish at zero speed: a stationary ring is repelled just as hard, because the flux is changing whether the ring moves or not. And the force does not need the ring to be near anything in particular — it is set by the drive, and the ring accelerates rather than reaching a terminal speed.

The demonstration also has a version that fails informatively. Cut the ring, and nothing happens at all: the current has nowhere to go, and a ring with a gap in it is not a circuit. That is the one experiment that distinguishes an induced current from any static magnetic property of the aluminium, since cutting a ring changes nothing about the metal and everything about the topology. The falling magnet has no equivalent test available, because a tube cannot be cut along its length without ceasing to be a tube — which is, incidentally, exactly what a slotted tube demonstrates when the magnet falls straight through it.

What the pictures cannot show #

None of these figures shows the currents. They are in the wall, they close on themselves, and they have no beginning or end to draw an arrow at; every figure here draws the consequences of the currents — a flux, a force, a speed — because the currents themselves are a vector field in a metal shell and a legible picture of one has not been found.

Nor does the descent figure show heat. The tube warms by a fraction of a degree, spread over the region the magnet passed, and there is no axis on which that could be drawn beside a position in metres.

And the force curve is a steady-state statement. It assumes the currents reach their pattern instantly, which is true to the extent that the tube’s own inductive time constant is short compared with the time the magnet takes to pass — true here by a wide margin, and not true for a very thick, very conducting tube, where the currents lag and the drag is smaller than the calculation gives.

Where the ladder goes next #

Induction began as a law about a loop of wire and has now been applied to a solid with no wire in it. Two further rungs are in view. One is the self-inductive case, where the current a circuit induces in itself opposes its own change — the same argument turned inward, and the reason a switched inductor sparks. The other is what happens when the flux is made to change periodically rather than once: the currents then confine themselves to a skin whose depth falls as the square root of frequency, which is why a microwave heats a metal only on its surface and why an induction hob works at all.

The habit worth carrying away is the one the second figure gave. Asked where the braking happens, the intuitive answer is at the magnet, and the derivative of the flux says plainly that the answer is half a radius on either side of it, where nothing is happening to the flux at all but where it is changing fastest. In a subject built on rates of change, the place where a quantity is largest is rarely the place where anything is going on.