Superconductor Levitation - 2D Meissner cross-section

Meissner effect and the image dipole

Below its critical temperature T_c a superconductor expels magnetic field (B = 0 inside) by driving surface currents of density K = B_t/μ₀. Outside, the field is exactly that of the real magnet plus an image dipole of opposite orientation placed at the mirror point below the surface, so that the normal component of B vanishes on the surface.

Levitation force

With the magnet moment m = B_rV/μ₀ a height h above the surface, the magnet and its image are d = 2h apart and repel as coaxial dipoles: F = 3μ₀m² / (2π d&sup4;) = 3μ₀m² / (32π h&sup4;). Setting F = Mg gives the floating height h_eq = [3μ₀m² / (32π M g)]^1/4. The slope of F(h) gives a vertical spring constant k = 4Mg/h, so small oscillations have ω = √(4g/h_eq).

The critical-field limit

The strongest tangential field at the surface lies a distance h/2 from the axis and equals B_max = 1.717 (μ₀/4π) m / h³ (the factor 2 is the image). The superconductor only shields while B_max < B_c(T) = B_c(0)[1 - (T/T_c)²]. Warm it or push the magnet down and shielding fails: the magnet drops. The London penetration depth λ(T) = λ₀ / √(1 - (T/T_c)⁴) shifts the mirror plane by a few hundred nanometres.

Honest simplifications: the magnet is a point dipole (good once h exceeds about its size, so the floating height of a big magnet is overestimated); the superconductor is an ideal infinite slab with a single critical field; real type-II ceramics also pin flux, which adds lateral stability and hysteresis that this model does not include; critical-field values are typical order-of-magnitude figures.

• Pick YBCO at 77 K, then slowly raise the temperature and watch the floating height stay almost constant until about 86 K, where the third graph shows the peak surface field crossing B_c(T), and the magnet drops.
• Press Drop 10 g payload: the height falls only by a few millimetres because F rises as 1/h⁴.
• Drag the magnet down below the red critical height: it falls through the surface field limit and rests on the plate.
• Set damping to zero and compare the measured oscillation period with the formula.