The window a spinning magnet floats in

Nothing can be held still by a static field proved that a charge placed among fixed charges always has somewhere to fall, because a potential obeying Laplace’s equation has no interior minimum, and that the same is true of a magnet of fixed orientation among fixed magnets. It named the escapes: change the sign of the response, as a diamagnet does; make the field vary in time; or let the orientation move. The last was dispatched in two paragraphs. A spinning magnet’s axis follows the local field, its energy depends on the field’s strength rather than on one component of it, and the strength of a field can have a minimum where its components cannot.

That is correct, and it is about a tenth of the story. A magnetic top floating above a ring magnet — sold since the 1990s as the Levitron, after a mechanism Roy Harrigan patented in 1983 — is famous among physicists for how hard it is to get working. The base must be levelled, the top weighed out with washers to a fraction of a gram, spun up within a range, and it floats for a couple of minutes before air friction slows it out of the range and it falls. Every one of those fussy requirements is the escape from Earnshaw’s theorem being narrow, and each can be computed.

The device had an awkward reception. Harrigan, an inventor working on his own, found that physicists he approached assumed the theorem settled the matter and that a floating magnet must conceal a battery or a trick, and his patent went largely unnoticed. Bill Hones and his father Edward developed a commercial version from the idea in the early 1990s and sold it from 1994, and it was the toy on laboratory desks, rather than the patent, that set theorists to work out why it worked.

A strength, not a component #

A magnet with moment $\boldsymbol\mu$ in a field $\mathbf B$ has energy $-\boldsymbol\mu\cdot\mathbf B$. If its orientation is held fixed — vertical, say — the energy is $\mp\mu B_z$, a component of the field, and one component of a field in empty space satisfies Laplace’s equation just as an electrostatic potential does. It has no minimum. A fixed magnet always slides off somewhere.

Now spin the magnet about its axis. A spinning body’s axis resists being turned; a torque on it produces not a turn but a sideways precession. The field exerts a torque $\boldsymbol\mu \times \mathbf B$ on the top, and the axis precesses about the local field direction. If the precession is fast compared with how quickly the field direction changes as the top drifts about, the axis keeps its angle to the field as the field turns — exactly the adiabatic following that the drift that does not care what the charge is found keeps a gyrating particle’s magnetic moment constant as it moves through a slowly changing field. The top’s moment then stays anti-parallel to the field wherever it goes, and its energy is

$$U = +\mu\,|\mathbf B| + m g z.$$

The magnitude $|\mathbf B|$ is not a harmonic function. Its square can have minima, and so can it. The top behaves, in effect, like a diamagnet: a body that seeks low field.

A well fifty micrometres deep #

The base is a magnetised ring, here modelled as the field of a single current loop ten centimetres across, which is what a loop that behaves like a needle found any small magnet to be in disguise. On its axis the field falls with height as $(1 + z^2/a^2)^{-3/2}$, and a twenty-gram top with a moment of 0.6 ampere-square-metres, held against the field, feels an upward push equal to $\mu$ times the field’s rate of fall.

The top floats 2.82 centimetres up, where the push equals the weight. Along the axis that balance is a minimum of energy, but a remarkably shallow one: half a centimetre lower the energy has risen by only ten microjoules and then falls away towards the base. Ten microjoules is the energy a twenty-gram top gains falling fifty-three micrometres. A careless lift that lets the top dip half a centimetre is enough to lose it. The magnetic forces involved are not small — the push holding the top up is its whole weight — but the curvature of the balance is, because the top is floating close to the height at which the field’s fall stops steepening, and that is not an accident. It is where the trap has to be.

Two curvatures, one band #

Being a minimum along the axis is not enough. The top must also be pushed back when it drifts sideways, and the two conditions pull in opposite directions.

Close to the axis the field strength at height $z$ and distance $\rho$ off the axis is, to second order,

$$|\mathbf B| \approx B + \rho^2\left(\frac{B'^2}{8B} - \frac{B''}{4}\right),$$

where $B$ and its derivatives are taken along the axis. Vertical stability needs the field’s fall to be decelerating, $B'' > 0$. Sideways stability needs the bracket to be positive, $B'^2 > 2BB''$. For a loop the first holds only above half a radius, the inflection point of the axial field; the second holds only below $\sqrt{2/5}$ of a radius, 0.632, above which the field’s own curvature overwhelms the term that makes the strength rise off-axis.

So the trap is a band. Below it the top is laterally stable but falls; above it the top holds its height but slides off. For a ring ten centimetres across the band is 6.6 millimetres thick. Within it the restoring forces are soft — the top swings to and fro sideways at less than once a second and bobs vertically at a little more — which is why a floating Levitron visibly wanders before it settles.

The sideways stiffness is the surprising half, and it is worth checking directly rather than through the expansion. The figure below computes the field of the ring off its axis, by adding up the contributions of each element of the loop, and draws the top’s energy against sideways displacement at its floating height, for two cases.

The same magnet at the same height sits on a hill if its axis is clamped vertical and in a valley if its axis follows the field. Earnshaw’s theorem is intact in the first case and irrelevant in the second, and nothing changes between them but whether the orientation is allowed to move. The theorem never said the magnet could not float. It said the magnet could not float holding still.

How exactly it must weigh #

The band of stable heights translates directly into a band of weights, because the height at which the top floats is wherever the field’s upward push equals its weight.

The upward push $\mu|B'|$ is largest at the inflection, half a radius up, so a top heavier than the push there can be held anywhere finds no balance and drops onto the base. A top only slightly lighter floats higher and higher, until above 0.632 radii it is laterally unstable and slides away. The usable range is less than five per cent of the weight — under a gram for a twenty-gram top. The Levitron ships with a set of thin washers, and finding the right combination is most of what makes it hard.

Making the base larger does not help as much as it seems to. Every condition in the band is a ratio — of heights to the ring’s radius, of the push at one height to the push at another — so a base twice as wide gives a band twice as thick in millimetres but the same ±2.4 per cent tolerance in weight, and the same sensitivity to warming. The tolerance is a property of the shape of the field, not of its size or strength, and the only way to widen it is to shape the base’s field differently, which the commercial bases do with a region of reversed magnetisation at their centre.

The window also explains the toy’s best-known foible: it works for a while in a cool room and then stops. Ferrite magnets lose about 0.2 per cent of their magnetisation per kelvin of warming. Both the top and the base weaken, so the push falls by about 0.4 per cent per kelvin, which to the trap is the same as the top gaining that fraction of its weight. A top tuned to sit 1.3 per cent from the heavy edge of the window, as the model top is, leaves it after about three kelvin. A hand’s warmth, the base sitting in sunlight, or the room heating through an afternoon is enough, and users learn to add and remove washers as the day goes on.

A diamagnet stronger than any material #

The comparison with a diamagnet can be made quantitative, and it shows how much the spin buys. A diamagnetic body — water, graphite, a frog — acquires a moment proportional to the field it sits in, pointing against it, so its energy per unit volume is $|\chi|B^2/2\mu_0$ and the upward force per unit volume is $|\chi|BB'/\mu_0$. Water’s susceptibility is about $-9 \times 10^{-6}$, and to hold it up against gravity the product of the field and its gradient must reach about 1,400 square teslas per metre: the reason the magnetism classical physics forbids can float a frog only in the bore of a sixteen-tesla research magnet.

The spinning top’s moment does not grow with the field. It is permanent, about 0.6 ampere-square-metres from a few cubic centimetres of ferrite, and the spin merely keeps it pointed the diamagnetic way. Its upward force is $\mu B'$, and to hold its twenty grams it needs a gradient of only a third of a tesla per metre in a field of thirteen millitesla — fields a fridge magnet makes. Measured as if it were a diamagnet, the top behaves like a material whose susceptibility is about minus twenty: millions of times water’s, and larger in magnitude even than a superconductor’s minus one, which is the most negative value any real material can have. That is the whole commercial point. A field-following permanent magnet does better than a superconductor’s levitation, at room temperature, bought with a spin.

It also explains why the trap is so much shallower than the forces in it. A diamagnet’s energy is quadratic in the field, so it always seeks the weakest field and a field minimum is easy to make. The top’s energy is linear in the field strength, and the minimum it needs is a minimum of $|\mathbf B| + mgz/\mu$ — of field plus height — which exists only where the field’s fall is levelling off at just the rate the weight requires. The narrowness of the band is the price of using a linear response to do a quadratic one’s job.

A spin that is neither too slow nor too fast #

The adiabatic following on which everything depends needs the spin to be in a range, and both ends can be estimated.

Too slow, and the top simply flips. Its moment points the unfavourable way, against the field, so the field’s torque tries to turn it over, like gravity acting on a top balanced upside down. Spin holds it up for the same reason a sleeping top stays upright: the top that nods before it settles found that the axis is stable when the spin’s angular momentum is large enough against the torque, $I_3^2\omega_s^2 > 4I_1\mu B$. For the model top that needs about twelve revolutions a second.

Too fast, and the following fails. The axis precesses about the field at $\mu B/I_3\omega_s$, which falls as the spin rises. When it is slower than the rate at which the field direction changes under the top as it swings about in the trap — about once a second — the axis cannot keep up. It stays fixed in space, the top becomes a magnet of fixed orientation, and Earnshaw’s theorem returns. For the model top that happens above about ninety revolutions a second.

These are estimates of where each mechanism fails, not the boundaries of the full dynamics, which couple the spin, the precession and the swing; the published analyses of the Levitron, by Michael Berry in 1996 and by Martin Simon, Lee Heflinger and Stuart Ridgway in 1997, find the real window inside them and measure it in the laboratory. The practical consequence is in the way the toy is used: the top is spun up on a plastic plate above the base, faster than it needs, lifted to the floating height, and released. It floats while air resistance slows it through the window and falls when it reaches the bottom. Spun too fast and released at once, it drifts off sideways; spun too slow, it flips immediately.

The same trap for atoms #

The mechanism is not confined to toys. A neutral atom with a magnetic moment has an angular momentum too — its spin — and in a field it precesses about the field direction at the Larmor frequency, exactly as the top’s axis does. An atom whose moment is anti-parallel to the field has energy $+\mu|B|$, seeks low field, and can be held at a minimum of $|B|$ made by a set of coils. Every magnetic trap for cold atoms, and the magnetic bottles in which ultracold neutrons are stored for many minutes to measure the neutron’s lifetime, is a Levitron with the spin supplied by quantum mechanics instead of a finger.

The two failure modes come too. Where the field is zero the Larmor precession stops, the atom’s spin can no longer follow the field direction as the atom passes through, and it flips into the state that is expelled — the atomic version of a top spun too fast for its swing. These losses, named after Ettore Majorana, who calculated them in 1932, drained the first simple traps, whose field minimum was a zero. The cure was to add a uniform bias so the minimum of $|B|$ is not zero but small — the Ioffe–Pritchard trap — or to rotate the zero around the atoms faster than they can follow it, which is a cousin of the vibrating pivot that holds up a pendulum by a force that averages to nothing. Both escapes from Earnshaw’s theorem that a spinning top uses, orientation that moves and fields that move, reappear in the instruments that made Bose–Einstein condensates.

What the one-loop model leaves out #

The base is drawn as a single circular current. A real Levitron base is a magnetised ferrite plate with a central region of reversed or missing magnetisation, whose field near the axis has the same shape but different numbers; published fields of real bases put the stable band at a similar fraction of the base’s size, a few millimetres thick, and the top’s weight tolerance at about one per cent rather than two. The model’s top is a uniform disc with a point dipole at its centre, and its trap frequencies, precession rates and spin window inherit those simplifications.

The pictures also leave out what makes the spin estimates estimates. The top’s axis does not follow the field exactly: it lags, tilts and nutates, and the energy $\mu|B|$ is the first term of an expansion in the ratio of the precession period to the swing period. The next term shifts the trap’s stiffness and is what sets the true upper spin limit. Air drag on the spin and on the swing slowly changes everything, and the toy works only because the spin decays slowly compared with the swing — a separation of time scales that the static figures cannot show. And the base must be level to a fraction of a degree, because a tilted base adds a sideways component of gravity that the soft sideways stiffness cannot resist; the figures assume perfect symmetry.

The domain of the argument is a magnet whose spin is fast enough to hold its axis against the field’s torque and slow enough to let the axis follow the field as the magnet swings, in a static field whose strength has a minimum where gravity is balanced. Within it, Earnshaw’s theorem does not apply, because its premise, a fixed orientation, is false.

Still open: how small a spin-stabilised trap can be #

The Levitron’s balance scales: the stable band is a fixed fraction of the base, the weight tolerance a fixed percentage, the spin window set by ratios of moments and frequencies. Smaller versions have been built, and spin-stabilised levitation of micrometre-sized magnetic particles, spun by rotating fields rather than by hand, is being explored as a way to make frictionless rotors and sensitive detectors of rotation and force. How far down the window survives — where thermal agitation becomes comparable to a trap only microjoules deep at desk scale and correspondingly shallower at small scale, and where the spin itself must be maintained against damping by a driving field — is being worked out experimentally rather than predicted.

What can be predicted is the shape of the escape. A magnet whose axis follows the field has energy $\mu|B| + mgz$, and $|B|$ can have a minimum where no field component can; above a ring that minimum exists only between half a radius and $\sqrt{2/5}$ of one, so the weight must be right to about two per cent, a few kelvin of warming moves it out, and the spin must be fast enough to stop the top flipping but slow enough for its axis to keep turning with the field. The theorem forbids the stillness, and the top is never still: it is a gyroscope tracking a field direction about once a second, inside a well fifty micrometres deep.