The drag that falls as the magnet speeds up

The magnet that falls slowly drops a strong magnet down a copper pipe and watches it take seconds to fall a metre. Its moving field induces currents in the pipe, the currents make a field of their own that opposes the change, and the opposition is a force on the magnet in proportion to its speed — a viscous drag with nothing viscous in it. The magnet reaches a terminal speed where that drag equals its weight. That essay also noted, as a limit of its own model, that the proportion holds only while the magnet is slow: move faster and the induced currents cannot follow.

This essay follows the magnet past that limit. The metal is a thin conducting sheet, the magnet moves parallel to it rather than through it, and the speed runs from a crawl to that of a fast train. The drag does not keep growing. It peaks and falls. And a second force, which the slow magnet in the pipe could not have shown because it points across the motion, rises to take its place: the sheet pushes the magnet away. That is the whole physics of one of the two ways of levitating a train, and of why an eddy-current brake cannot stop one.

Drag that turns into lift #

A thin sheet of conductivity $\sigma$ and thickness $d$ has one natural speed built into it,

$$w = \frac{2}{\mu_0\sigma d},$$

and the drawing uses it as the unit. For ten millimetres of aluminium $w$ is about 4.5 metres a second; for a millimetre of copper, 27; for five millimetres of stainless steel, over two hundred. A magnet slow compared with $w$ is in the regime of the falling magnet: its drag grows in proportion to its speed, $v/2w$ of the repulsion a perfect conductor would give, and there is almost no lift. A magnet fast compared with $w$ is in the opposite regime. The drag falls as $w/v$ and the lift approaches the full repulsion of a perfect mirror image — the force a superconductor would exert.

The two curves cross at exactly $v = w$. Below that crossing the metal behaves mostly like a brake, above it mostly like a mirror, and the drag’s peak, at 1.27 $w$, is where the changeover costs most. The closed forms for a magnet of the simplest shape, derived by John Reitz in 1970, are

$$F_L = F_0\left[1 - \frac{w}{\sqrt{v^2 + w^2}}\right], \qquad F_D = \frac{w}{v}\,F_L,$$

with $F_0$ the perfect-image repulsion. The second relation is the important one, and it holds for any shape of magnet over a thin sheet: lift over drag is the speed over $w$.

Where the speed w comes from #

The speed $w$ is a decay rate in disguise. A pattern of current in a thin sheet, left to itself, dies away through the sheet’s resistance, and how fast it dies depends on how large the pattern is. A broad pattern links a lot of its own magnetic flux and so has a large inductance for its resistance; a fine pattern links little and dies quickly. Worked out for the sheet, a current pattern of size $\ell$ decays in a time of about $\ell/w$, which is exactly the statement that its field recedes from the sheet at speed $w$.

A magnet at height $h$ above the sheet makes a pattern of field on the metal about $h$ across, and it passes over any one point in a time of about $h/v$. So the ratio $v/w$ is the ratio of two times: how long the currents the magnet induces survive, over how long the magnet takes to go by. For a magnet ten centimetres above ten millimetres of aluminium the currents survive about twenty milliseconds, and a magnet moving at five metres a second takes about the same time to pass. A magnet crawling over the same sheet takes seconds, the currents it induced are long gone before it has moved its own size, and only the small part proportional to its speed is present at any moment. A magnet moving at a hundred metres a second passes in a millisecond, and the currents it induces are still almost undiminished when it has gone.

The images the sheet leaves behind #

Why the forces trade places is easiest to see through a construction James Clerk Maxwell found in 1872 for exactly this problem. A perfect conductor answers a magnet with an image on the far side of its surface, a mirror copy that makes the field inside the metal vanish. A thin sheet of finite conductivity cannot hold the currents that make such an image, because they decay by resistance. Maxwell showed that for a thin sheet the decay has a strikingly simple form: whenever the field above the sheet changes, the sheet at once produces an image of the change, and that image then moves away from the sheet at the constant speed $w$, weakening as it goes.

A magnet moving along the sheet changes the field continuously, so the sheet produces a continuous stream of images, each born at the mirror point beneath the magnet’s position at that moment and then sinking. By the time the magnet has moved on, the images it left behind have sunk, and together they form a trail sloping down behind the magnet at an angle whose tangent is $w/v$.

The two limits follow at once. A slow magnet leaves a steep trail: each image is gone before the magnet has moved far, the image that matters is always faint, and what remains is mostly behind the magnet, pulling it backwards — drag with little lift. A fast magnet leaves a nearly level trail lying just under the mirror point, and the whole trail acts like one mirror image directly beneath it, pushing it straight up — lift with little drag. Nothing about the sheet has changed between the two panels. What changed is whether the sheet’s memory, which fades at speed $w$, outlasts the magnet’s passage.

Lift over drag is a speed ratio #

The relation $F_L/F_D = v/w$ makes the engineering transparent. A vehicle that wants to be held up, not held back, needs $v$ large compared with $w$, and so needs a sheet with a small $w$: a good conductor, and a thick one. Ten millimetres of aluminium at five hundred kilometres an hour gives a lift thirty-one times the drag, comparable with a good glider’s wings. A millimetre of copper gives five. Stainless steel, a poor conductor, gives about one, and would be a brake at any speed a train reaches.

This is electrodynamic suspension, and it is the principle of the Japanese superconducting maglev, whose superconducting magnets on the train pass over coils in the guideway and induce the currents that repel them. It levitates only when moving — below about a hundred and fifty kilometres an hour the train runs on rubber wheels — and it needs no feedback, because the repulsion grows as the gap shrinks. That stability matters because a static field cannot hold anything still: a magnet over a fixed arrangement of other magnets and iron always has a direction in which it falls. Motion evades the theorem because the field the sheet makes is not static; it is manufactured continuously by the magnet’s passage.

The other kind of maglev, the German-built design running in Shanghai, works the opposite way: electromagnets on the train are attracted up towards iron rails, and the gap is held at a centimetre by feedback control a thousand times a second. It levitates at rest and needs the control loop to do so. The two designs are the two ways round the static-field theorem — feedback, or motion.

The heat that levels off #

Drag times speed is the power the eddy currents turn into heat in the sheet, and because the drag falls as $1/v$ at high speed, the power levels off. Its ceiling is $F_0 w$, and since the lift at high speed is nearly $F_0$, the power the sheet absorbs is simply the lift times $w$. The heating is a fixed toll for each newton of weight supported, set by the conductor and not by how fast the vehicle goes.

For a levitated train this is the decisive fact. Air resistance grows as the square of the speed, so at high speed the magnetic drag, levelling off, becomes a small and shrinking share of the total, and the faster the train goes the more nearly it is limited by its aerodynamics alone. At low speed the reverse holds: below the drag peak the magnetic drag dominates and the lift is too small to carry the train, which is why it needs wheels to take off and land.

The toll also explains why the sheet should be thick and highly conducting, in the sense that matters. The heat per unit weight goes as $w$, so halving $w$ halves the power at every high speed. There is no way to make it zero, because a sheet that could carry currents without loss would need infinite conductivity — which is the superconductor, where flux is frozen into the metal and the image is perfect at any speed, including zero.

A brake that is weak at both ends #

Used the other way round, the same physics is a brake with no contact and no wear, and its shape against speed is both its virtue and its limitation. A brake whose drag peaks at a speed is weak above that speed and weak below it. The drawing slows a vehicle from three hundred kilometres an hour with a brake that gives its best retardation of 1.5 metres per second squared at seventy. A friction brake of the same strength stops in under a minute. The eddy brake spends that long just getting down to its best speed, because at high speed its drag falls as one over the speed.

Below the peak it weakens again, now in proportion to the speed, and a force proportional to speed gives an exponential decay: the speed falls tenfold every fourteen seconds in the drawing and never quite reaches zero. At rest there is no force at all, because nothing is changing. An eddy brake cannot hold a stopped vehicle on a slope.

Real installations are designed around this. Roller coasters use permanent magnets on the cars and copper or aluminium fins along the track to slow trains smoothly and silently from speed, with a friction brake or a slow drive for the last metre. High-speed trains with linear eddy brakes use them for service braking at speed, where they save the wear of friction brakes, and hand over at low speed. In both cases the conductor’s thickness and conductivity are chosen to put the drag peak where the braking is wanted, which is the design freedom the formula for $w$ hands over.

The same curve, turned into a motor #

Reverse the roles and the drag becomes a thrust. Hold the conductor still and move the field over it, and the eddy currents drag the conductor after the field; let the conductor move too, and what matters is only the speed of the field relative to it. That is an induction motor, whose rotor is a conductor dragged round by a field that turns with nothing turning. The motor’s torque against the slip — the difference between the field’s speed and the rotor’s — has the shape of the drag curve here: rising in proportion to the slip, peaking at a slip set by the rotor’s resistance against its inductance, and falling beyond. Designers who want a motor that starts hard give its rotor a higher resistance, which moves the torque peak to larger slip, the same move as choosing a sheet with a larger $w$ to put a brake’s peak at higher speed.

The linear version is used where a vehicle must be pushed without touching the track. A linear induction motor under a train carriage sweeps its field along an aluminium plate between the rails and is pulled along by the plate’s eddy currents; the plate’s thickness and conductivity set the slip at which the thrust peaks. The same physics that floats one maglev pushes others along, and the same number $w$ decides the operating point of each.

Where the thin sheet stops being thin #

Everything above assumes a sheet thin enough that the induced currents fill its whole thickness. A thick conductor does not behave that way at high speed. A magnet of length $L$ passing at speed $v$ presents the metal with a field changing at a frequency of roughly $v/L$, and a changing field gets only a skin depth into metal, a depth that shrinks as the square root of the frequency. At high speed the currents crowd into a surface layer, the effective thickness falls, the effective $w$ rises, and the lift-to-drag ratio grows more slowly than in proportion to the speed — as the square root of the speed for a very thick plate. The drawing’s straight lines are the thin-sheet limit, which holds for aluminium a centimetre thick up to speeds where the skin depth becomes comparable with a centimetre.

The sheet is also assumed infinite and flat. Real guideways are made of discrete coils or strips, whose ends and gaps add losses, and a magnet near the edge of a sheet feels a sideways pull. And the magnet’s own shape enters $F_0$ and the detailed curves but not the ratio $v/w$, which is the one result that survives any magnet and any thin sheet.

Ferromagnetic conductors, such as steel rails, add a complication the model leaves out entirely: they are attracted to the magnet at the same time as their eddy currents repel it, and the attraction dominates at low speed. A rail eddy brake therefore pulls the train down onto the rail as well as braking it, which the train’s suspension has to be designed to take.

One law, three regimes #

The three essays on moving magnets and conductors are one law read at three speeds. The falling magnet in the pipe is the slow regime, where the induced field is only there while something changes and the change is gentle, so the currents decay almost as fast as they form and the force is a drag in proportion to the speed. The levitated magnet is the fast regime, where the currents cannot decay before the magnet has gone and the metal behaves as a mirror. And the flux frozen into a superconductor, or into a plasma, is the limit where $w$ is zero and the mirror is perfect at every speed, including rest.

What decides the regime is a single ratio, the magnet’s speed over the metal’s own speed $w$ — equivalently, the time the magnet takes to pass compared with the time the currents take to decay. In the language of plasma physics it is a magnetic Reynolds number, and the circuit that fights its own change is the same ratio for a single loop: an inductance over a resistance is a decay time, and whether a circuit’s current can follow a change depends on how fast the change is compared with it.

What the curves do not show #

The curves are forces on a magnet held at a fixed height over the sheet. A magnet free to move vertically would settle at a height where lift balances its weight and would oscillate about it, and the oscillation is lightly damped: the same eddy currents that give the lift give almost no damping of vertical motion at high speed, because a mirror image stores energy rather than dissipating it. Electrodynamic maglev trains need added damping in their suspensions, either passive or active, to stop the ride from bouncing, which is a problem the formulas above do not reveal.

Nor do they show the temperature of the sheet. The power levelling off at $F_0 w$ is a steady rate of heating per unit length of guideway travelled, and for a vehicle passing a given point once in a while the sheet has time to cool between passages. A conductor under a magnet held still over a spinning disc, as in some eddy-current dynamometers, heats continuously and must be cooled.

Still open: how thin a levitating conductor can be #

The attractive idea of electrodynamic levitation for transport has always run into the cost of the guideway: kilometres of thick aluminium or of coils. Designs that use a magnet array arranged to concentrate its field on one side — the Halbach array, whose strong side faces the track — can produce levitation over thin, cheap conductors or over simple grids of wire loops, and experimental systems have been built. Whether such passive systems can be made cheap enough, stable enough without active damping, and efficient enough at low speed to compete with wheels on a rail has not been established by any line in service, and the answer depends as much on the price of aluminium as on the physics.

The habit worth carrying away is to ask how long a system remembers. A conductor answers a change with currents that fade on their own time scale; when the change is slow compared with that time the answer is a drag, and when it is fast the answer is a mirror. The same sheet of metal brakes a slow magnet and floats a fast one, and the only number that says which is the speed over $w$.