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Superconductor and magnet
A magnet floats where the lift from its image in the superconductor equals its weight. Cooled in place (field-cooled), trapped flux pins it laterally; cooled away from the magnet (zero-field-cooled) it floats freely but is not held.
Cooling history
Model checks
The science
What is modelled
A point magnet of moment m above a superconducting half-space. The Meissner response is represented by an image dipole, giving a lift that falls steeply with height. A shielding fraction rises from 0 at the critical temperature (92 K) to 1 about 15 K below it. In field-cooled mode a saturating spring pulls the magnet back to the position where it was cooled, with slow creep of the anchor if the displacement exceeds a critical value.
Equations
Flift = s(T) · 3 μ0 m² / (32 π (h + δ)4), with softening δ = 2 mm.- Equilibrium:
Flift(h) = m g. - Vertical stiffness:
k = 4 m g / (h + δ), soω = √(4 g / (h + δ)). - Pinning:
F = -Fcap tanh(kxy r / Fcap),kxy = 2.4 s p.
Checks run in the page
- The equilibrium height against the closed-form root.
- The shielding fraction at 77 K, 84.5 K and the critical temperature.
- The small-displacement pinning stiffness.
- The measured bounce period against 2 pi / sqrt(4 g / (h + delta)).
Honest simplifications
A reduced-order educational model: an image-dipole lift with an empirical softening, a made-up shielding curve and a tanh pinning law. It is not a critical-state or finite-element superconductor calculation and does not give quantitative forces for a real YBCO sample.
What to try
- Raise the temperature to 88 K and watch the magnet sink as shielding falls.
- Press Strong pinning, push: the magnet is held near its anchor against a steady sideways force.
- Untick Field-cooled and drag the magnet: it floats freely and does not return.