Superconducting levitation
A Socratic walk-through of superconducting levitation — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why can a magnet hover motionless above a chilled superconductor when no arrangement of ordinary magnets can hold anything stably in mid-air?
Everyone has tried to balance one magnet above another. It hovers for an instant, then slides off sideways and slaps down. That failure is not clumsiness — it is a theorem. Yet drop a small magnet over a ceramic puck sitting in liquid nitrogen and it hangs there, indifferent, and you can nudge it and it comes back.
So the question is not "what pushes up?" — pushing up is easy. It is what forbids the sideways escape, and what a cold ceramic disc has that steel magnets do not.
Reasoning it through
REASONING #Let us find the theorem first, because knowing exactly what is forbidden tells us exactly what the superconductor must be doing differently.
Take a small permanent magnet of fixed moment m sitting in a field B made by other magnets. Its energy is U = -m·B. In a region with no currents and no charges, B obeys ∇×B = 0 and ∇·B = 0, and those two together force each Cartesian component of B to satisfy Laplace's equation, ∇²B = 0. Since m is fixed, U is a sum of those components with constant coefficients, so ∇²U = 0 as well.
What does such a function look like? Its value at the centre of any small sphere equals its average over that sphere, so it has no interior minima — a minimum would need the centre to be below the average. No minimum of energy means no stable equilibrium: every candidate resting place is a saddle. That is Earnshaw's theorem, and it is why the magnet slides off.
Notice where the proof leans: it required m to be fixed. What if the moment is induced by the field itself? A diamagnetic body acquires a moment opposing the field, roughly m = -(χV/μ₀)B, so its energy is U = +(χV/2μ₀)|B|². Take the Laplacian of that: ∇²|B|² = 2|∇B|², which is non-negative and generally strictly positive. |B|² is not harmonic, so it can have a local minimum in empty space. Diamagnets escape Earnshaw's theorem legitimately — which is why a strong enough magnet can float a frog.
A superconductor below its transition temperature is the extreme case: it expels flux entirely, a perfect diamagnet with χ = 1. That is the Meissner effect, and it is a genuine thermodynamic expulsion rather than induced screening from Lenz's law — cool the material in a field already present and the field is still pushed out. So Meissner repulsion gives lift, and in a well-shaped field even lateral stability.
But here is the crux: it only ever pushes. Lift the magnet away and nothing pulls it back; invert the assembly and it falls off. The demonstration everybody films does neither. The magnet holds a fixed gap, resists being pulled up as stubbornly as being pushed down, and can be inverted so it dangles beneath the puck.
So something must hold the field configuration itself. In a type-II superconductor — and these cuprate pucks are strongly type-II — flux above a first critical field does not stay out. It penetrates as discrete filaments, each carrying one quantum, Φ₀ = h/2e: Planck's constant 6.626 × 10⁻³⁴ divided by twice the elementary charge 1.602 × 10⁻¹⁹, about 2.07 × 10⁻¹⁵ webers. Around each filament the material is normal; between them it still superconducts.
Now the decisive step. Those filaments sit preferentially at defects — grain boundaries, oxygen vacancies — places where the material was going to superconduct poorly anyway, so parking a normal core there costs least. To move the magnet in any direction you must drag the threads out of those wells, so every displacement raises the energy. That is not a repulsion but a genuine three-dimensional trap, which is why the magnet holds a gap, holds an angle, and holds upside down.
The analogy
THE ANALOGY #Think of a coat hung on a rail of hooks. A smooth rail lets the coat slide anywhere and off the end — that is Meissner repulsion, a surface that resists penetration but offers no purchase. The hooks change everything: the coat is not merely held up, it is held there, and it takes work to move it sideways or lift it off.
a coat on hooks is passive and permanent, whereas the flux threads are pinned only while the material stays cold, and they creep slowly out of their wells even then — the gap drifts a little over hours, and collapses the instant the puck warms past its transition.
Clarifying the model
THE MODEL #Earnshaw's theorem is not violated here, it is sidestepped — twice. Once by the induced-moment loophole that diamagnets exploit, and again because a pinned superconductor is not a fixed dipole in an external field at all but a body whose internal field configuration is frozen.
So "the Meissner effect makes things levitate" is the usual shorthand, and it is half right. Meissner expulsion supplies the lift; pinning supplies the stability, and everything memorable about the demonstration — the fixed gap, the inversion, the tilted parking — is pinning. That is also why the state is prepared by field cooling: set the magnet at the gap you want and cool the puck through its transition with the field already threading it, and that configuration is the one frozen in.
The sharp test follows. Field-cool a puck with a magnet held at a gap, then pull the magnet away: if pinning is real there is a restoring force resisting upward displacement as well as downward, and the assembly can be inverted intact. If the magnet came away freely, or fell when turned over, pinning would be wrong and plain diamagnetic repulsion would be the whole story. It does not — that inversion is the observation the pure-Meissner account cannot survive.
Two numbers above are recalled rather than derived: YBa₂Cu₃O₇ transitions near 92 K and liquid nitrogen boils near 77 K, which is why these demonstrations are cheap. The flux quantum I computed in the line.
A picture of it
THE PICTURE #How to readRead right along the horizontal axis for lateral stability, and up the vertical axis for whether the arrangement pulls back when you lift the magnet away. Two ordinary magnets sit bottom-left because Earnshaw forbids both. The diamagnetic and Meissner-only cases climb rightward — laterally stable in a well-shaped field — but stay low, because repulsion never pulls. Only the field-cooled type-II puck reaches the top-right corner, and that corner is the whole difference: it is the one that can be turned upside down. The warmed puck drops to the origin, the falsification test performed by simply waiting.
What became clearer
WHAT CLEARED #The hovering magnet is two separate physical facts wearing one costume. Expulsion of field is what lets anything float at all and what evades Earnshaw's theorem — because an induced moment makes the energy go as |B|², which unlike B itself is free to have a minimum. But expulsion alone gives a spring, not a grip. What makes the levitation look impossible is flux threading the material in quantized filaments and catching on its imperfections, so the pattern is frozen and any motion costs energy. The defects, which spoil the superconductor, are what make the trick work.
Where to go next
ONWARD #- Why flux creep sets the long-term stability limit of levitated superconducting bearings.
- How artificial pinning centres are engineered into commercial tapes to raise the current they carry.
Key terms
TERMS #| Term | What it means |
|---|---|
| Earnshaw's theorem | no static arrangement of fixed charges or dipoles gives stable equilibrium, because the energy is harmonic and harmonic functions have no interior minima. |
| Meissner effect | the active expulsion of magnetic flux from a superconductor below its transition temperature. |
| Type-II superconductor | one that admits flux above a first critical field as quantized filaments while staying superconducting between them. |
| Flux pinning | the trapping of those filaments at material defects, which freezes the field configuration into a true positional trap. |
Every term the collection defines is gathered in the glossary.