Nothing can be held still by a static field

A potential is one number for every point, and everything about the force follows from it. This essay is about a property of that number which rules out a whole category of devices.

The equation, and what it forbids #

In a region containing no charge, the potential satisfies Laplace’s equation:

$$\nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0$$

A stable equilibrium for a positive charge requires a minimum of $V$, which requires all three second derivatives to be positive. They sum to zero. So they cannot all be positive, and they cannot all be negative either — which rules out a maximum, and therefore rules out trapping a negative charge as well.

That is the whole proof, and its brevity is the point. It is not a statement about which arrangements have been tried; it is a statement about the equation every arrangement obeys.

The integral version is the more intuitive one. A harmonic function’s value at a point is the average of its values on any sphere around it, so it can never be lower than everything nearby — there is always some direction in which it decreases.

Earnshaw published this in 1842, while arguing about a theory of the luminiferous ether in which particles were held in place by forces. The theorem killed the model, and it has been killing proposals ever since.

The four conditions, and the four escapes #

The theorem’s premises are worth listing, because every working trap is an attack on one of them.

The field must be static. A field that varies in time is not required to satisfy Laplace’s equation instantaneously in the way the argument needs, and more usefully, a rapidly oscillating field produces an effective potential that is not harmonic at all.

That is the Paul trap. Rotate or switch the saddle fast enough and the particle is confined in every direction, because a force that averages to nothing can still hold something up when the position and the force are correlated. Every quadrupole mass spectrometer, every trapped-ion clock and every trapped-ion quantum computer is built on this evasion, and it shared the 1989 Nobel Prize.

The charge must be fixed. A particle that responds to the field — an induced dipole rather than a fixed charge — is a different problem, because its energy goes as $-\alpha|E|^2/2$ rather than as $qV$. And $|E|^2$ is not harmonic: it can have a minimum in empty space, though not a maximum.

That asymmetry is why diamagnets can be levitated and dielectrics cannot. A diamagnet is repelled from strong field, seeks a minimum of $|B|^2$, and such minima exist — which is how a live frog was levitated in a 16-tesla magnet at Nijmegen in 1997, and how a piece of pyrolytic graphite floats above permanent magnets on a desk. A dielectric seeks a maximum of $|E|^2$, and those do not exist, so the attraction that needs no charge can pull a speck toward a charge and never hold it.

The first escape is to abandon the potential. What a diamagnet seeks is a minimum of the field’s magnitude, and unlike a minimum of the potential such a thing is permitted — which is the single loophole that makes passive magnetic levitation possible at all, and why a frog can be floated in a strong enough magnet while a charge cannot be floated in any electric field whatever.

There must be no other force. Adding dissipation, or a velocity-dependent force, changes the problem entirely. Optical tweezers hold a bead in the gradient of a laser’s intensity — a $|E|^2$ minimum or maximum depending on the bead’s index — with viscous damping doing the work of stabilising it. A magnetic bottle confines charged particles using their motion, since the magnetic force depends on velocity and does no work at all.

And the region must be free of charge. Inside a distribution of charge, $\nabla^2 V = -\rho/\varepsilon_0$ is not zero, and minima can exist. That is the least useful escape, since it requires putting matter where the trapped particle is meant to be.

The other trap, and why it is static #

The Paul trap escapes by making the field vary in time, and there is a second trap for the same job that does not — which is worth setting out, because at first sight it looks like a violation.

A Penning trap has a static electric quadrupole and a uniform static magnetic field along its axis. Nothing about it changes with time. The electric field confines the ion along the axis and, being a quadrupole obeying Laplace’s equation, necessarily pushes it outward in the two directions across the axis — that is the theorem doing its work, and the trap does not pretend otherwise.

What holds the ion radially is the magnetic field, and the escape is the one already named: the magnetic force depends on velocity, so it is not derivable from a potential and the theorem says nothing about it. An ion pushed outward by the electric field is deflected by the magnetic field into a circle, and the combination is a slow drift around the axis rather than an escape.

The resulting motion has three independent frequencies — a fast circular motion set mainly by the magnetic field, a slow drift around the axis, and an oscillation along it — and all three are measurable, which is what makes the device the instrument it is. Measuring the fast one gives the ion’s charge-to-mass ratio directly, since it depends on nothing else.

That is why the most precise measurements in physics are made in Penning traps. The electron’s magnetic moment is known to about a part in $10^{13}$ from one, and the comparison of a proton’s mass with an antiproton’s to a part in $10^{11}$, both by measuring frequencies of a single particle held for months. A trap that admits it is unstable in two directions, and defeats the instability with a velocity-dependent force, turns out to be the steadiest place anybody has built.

The top that hovers #

There is a fifth escape that belongs neither with the four listed nor with feedback, and it is sold as a toy.

Set a small ring magnet spinning and release it above a larger base magnet arranged to repel it. It hovers, with nothing touching it, no power supply, no sensor, no oscillating field and no diamagnetism — which is exactly the arrangement Braunbeck’s extension of the theorem forbids.

The escape is the spin. A magnet held stationary above another has its moment either aligned or anti-aligned with the local field, and both are unstable: it flips over and falls. A magnet that is spinning about its own axis behaves gyroscopically, and if the spin is fast enough its axis follows the local field direction adiabatically as it drifts. Its energy is then $+\mu|B|$ rather than $-\boldsymbol{\mu}\cdot\mathbf{B}$ with a fixed direction — and a minimum of $|B|$ is permitted where a minimum of the potential is not.

So the spinning top is effectively a diamagnet: its moment stays pointed the wrong way for energy, it seeks a field minimum, and the field minimum exists. The theorem is satisfied throughout.

The condition is a window rather than a threshold, which is what makes the toy fussy. Too slow and the axis cannot follow the field, so the top flips and falls; too fast and the precession is slow enough that the top drifts away before it can respond. In practice it works over a range of a few hundred revolutions a minute, and the mass has to be trimmed to a fraction of a gram — which is why the toy comes with a set of small washers.

How strong a magnet a frog needs #

The diamagnetic escape is the most spectacular and it is worth a number, because the number explains why it is not more common.

The force per unit volume on a diamagnet is proportional to its susceptibility times the gradient of the field’s square. Setting that against the weight gives the condition for levitation as a requirement on the product of the field and its gradient:

$$B\frac{\mathrm{d}B}{\mathrm{d}z} = \frac{\mu_0\rho g}{|\chi|},$$

and for water, whose susceptibility is about nine parts in a million, that comes to roughly 1400 tesla squared per metre.

Which is a very demanding number. A field of a tesla falling off over a centimetre gives a hundred; a superconducting solenoid of sixteen tesla with a bore ten centimetres across gives about the right value in the region where the field is falling fastest. That is a substantial piece of laboratory equipment, and it is why the famous demonstration — a live frog, which is mostly water, hovering in a vertical bore — was done at a high-field laboratory rather than on a bench.

The one common exception is pyrolytic graphite, whose susceptibility is about forty times water’s because of its layered structure, and which therefore levitates over an array of ordinary permanent magnets on a desk. It is the only substance that does at room temperature, and the reason is entirely that one number.

Superconductors are the other case, and they are not really diamagnets of this kind: a superconductor expels the field completely, which is a susceptibility of exactly $-1$ rather than of a few parts per million, and it levitates over almost any magnet at all. The gap between $10^{-5}$ and $1$ is the whole difference between needing a national facility and needing a magnet from a scrapyard.

The feedback escape, which is the commonest #

The fifth way out is not physics but engineering, and it is how most levitation anybody sees actually works.

Measure the position, and change the field in response. An electromagnet above a steel ball is unstable — the closer the ball, the stronger the pull — but a sensor and a controller adjusting the current a thousand times a second turns the instability into stability. Maglev trains of the electromagnetic type, magnetic bearings, and the suspension of the test masses in gravitational-wave detectors all work this way.

That escape is worth separating from the others because it does not evade the theorem at all: the system is unstable at every instant, and what makes it work is that the field is not a function of position alone. The distinction between “stable” and “stabilised” is exactly the distinction between a system that has a minimum and one that is being held near a saddle by something watching it.

The feedback escape is the commonest, and it works by leaving the theorem’s scope entirely. A conductor moving through a field induces currents that oppose its motion, which is a velocity-dependent force and therefore not derivable from a potential at all. A spinning magnet above a conducting plate levitates for precisely that reason — and stops the moment it stops spinning, which is the honest signature of an escape that costs something to maintain.

How the theorem is actually used #

A theorem that forbids things is most useful as a diagnostic, and this one is used that way constantly.

The routine application is to a proposal. Somebody describes a levitation scheme, and the first question is which of the four conditions it breaks — because if it breaks none of them it does not work, whatever the drawing shows. That is a genuinely efficient way to evaluate a claim: not by finding the error in the design, but by observing that the design cannot exist.

The second application is in the other direction, as a design constraint. Wanting to trap something, one starts by choosing which condition to break, and each choice comes with a characteristic cost. Time-varying fields bring micromotion and heating; induced-dipole traps are weak, because the polarisability is small; dissipative traps need a viscous medium and cannot work in vacuum; feedback traps need bandwidth and fail when the sensor does.

The third use is pedagogical and slightly mischievous: the theorem is a standing test of whether a piece of reasoning about fields has been done or merely sketched. Any argument that concludes with a stable static electrostatic equilibrium has an error in it, and locating the error is a good exercise.

The same theorem in other fields #

In gravity. The gravitational potential in empty space obeys the same equation, so there is no point in the vacuum between two masses where a third can rest stably. The Lagrange points of a two-body system are apparent counterexamples and are not: they exist in a rotating frame, where the centrifugal term is not harmonic, and the stability of L4 and L5 comes from the Coriolis force, which is velocity-dependent.

In the magnetic scalar potential. In a current-free region the magnetic field also derives from a harmonic potential, so a permanent magnet cannot be levitated stably above another permanent magnet in any arrangement — a result proved by Braunbeck in 1939 and rediscovered periodically since by inventors of perpetual levitation devices.

And in heat and in diffusion at steady state, where the same equation governs the temperature or the concentration, and the same conclusion follows: the hottest and coldest points of a body in steady state are always on its boundary, never inside.

The same theorem governs other fields for the same reason. A concentration at steady state obeys the identical Laplace equation, so it has no interior maximum and no interior minimum and its extremes always lie on the boundary — which is why a steady-state temperature field can be computed from its boundary alone, and why the hottest point in a uniformly conducting slab is never in the middle of it.

The atom that should not have existed #

The theorem’s most consequential application was to a system nobody could redesign, and it is the reason quantum mechanics had to be invented.

Rutherford’s atom of 1911 has electrons at rest or in orbit around a nucleus, held by electrostatic attraction. Earnshaw’s theorem says that no static arrangement of them is stable: whatever the configuration, there is a direction in which an electron falls. The escape available to a planetary system — orbiting rather than sitting still — is closed by a second argument, because an accelerating charge radiates and a radiating electron spirals into the nucleus in about ten picoseconds.

So classical physics has two independent proofs that matter cannot exist, and they were both understood before 1913. The resolution is not a cleverer arrangement and not a new force; it is that an electron is not a particle with a position, and where the electron probably is is a distribution whose spread costs kinetic energy. That cost rises as the distribution is squeezed, and the balance between it and the Coulomb attraction is what fixes the size of an atom.

The atom is the case that had to be rescued, and what rescued it has no classical counterpart. The energy of a confined quantum state rises as the confinement tightens, so there is a cost for being localised that no arrangement of charges can offer. That rise is the only thing standing between this theorem and the non-existence of matter: classically, an atom is a static arrangement of charges and therefore cannot be stable, and the argument is airtight.

That is worth stating plainly because the theorem is usually presented as a curiosity about levitation. Its largest consequence is that stable matter is not classically possible, and the fact was known and unexplained for a decade and a half.

What the theorem does not say #

It does not forbid equilibrium, only stable equilibrium. Stationary points exist in abundance; they are saddles, and a particle placed exactly at one stays there until it is disturbed.

It does not forbid confinement in fewer than three dimensions. A line charge can hold a particle stably in two dimensions while it is free along the third, which is a real and useful arrangement — and the theorem’s content is precisely that the third direction cannot also be closed.

And it does not forbid metastability with a large barrier. The relevant question in practice is often not whether an arrangement is stable but how long a particle stays, and a saddle with a shallow descent can hold something for a long time.

What the theorem does not say is that nothing attracts. A charge above a conductor is pulled towards its own image, and the attraction grows without limit as it approaches — a one-directional instability of exactly the kind the theorem guarantees. There is no arrangement of conductors that removes it, because the surface always offers a downhill direction.

What the pictures cannot show #

The numerical check is over spheres in one configuration. The mean value property is a theorem and needs no checking; what the figure verifies is that the arithmetic here implements the potential correctly, which is what makes the saddle figure trustworthy.

The saddle drawn is at one point. Every point in a charge-free region has the same property, and the centre of a symmetric arrangement is chosen because it is where the stationary point is. There is nothing special about it otherwise.

The Paul trap’s stability boundary is for one plane. A real three-dimensional trap has a stability diagram in two parameters and an operating region rather than a threshold, and choosing an operating point involves several considerations the one-dimensional calculation does not have.

And the theorem is classical. It says nothing about whether a quantum particle can be bound, and quantum mechanics does not evade it — a bound state still needs a potential minimum — but the question of what happens for a particle whose position is uncertain over a region comparable with the saddle’s scale is a different one, and it is how electrons are bound in atoms despite the classical argument that they should spiral in.

The shape of the argument, which is worth stealing #

The proof has a structure that recurs across physics, and recognising it is worth more than the theorem itself.

It works by finding a quantity that is constrained by a differential identity rather than by a calculation. Nothing about any particular arrangement of charges was used; what was used is that every arrangement produces a potential satisfying one equation, and that equation has consequences for the shape of its solutions. The conclusion is therefore about a whole class of systems at once and cannot be evaded by any member of the class.

The same move appears wherever a conservation law or a field equation constrains the possible. The impossibility of a magnetic monopole in Maxwell’s equations, the impossibility of a perpetual-motion machine given the first law, the impossibility of superluminal signalling given the light cone, and the impossibility of cloning a quantum state given linearity are all of this kind: a structural fact about the equations, converted into a statement about what no device can do.

The practical habit that follows is to look, when a design seems to require ingenuity, for the identity that decides whether ingenuity can help. Often there is one, and finding it saves the effort of being ingenious.

The ladder from here #

Later rungs on this anchor: the uniqueness theorem, which says the boundary values determine the interior and is the same fact stated as a construction; the method of images as its practical consequence; the general theory of harmonic functions, including the maximum principle in its proper form; and the stability analysis of a Paul trap in two parameters, where the escape from the theorem becomes a design problem.

The neighbouring ladders are one number for every point, which is the potential this theorem constrains, the force that averages to nothing, which is the mechanism the commonest escape uses, and the attraction that needs no charge, where the asymmetry between electric and magnetic responses decides what can be levitated.