A superconductor does not merely stop resisting — it actively pushes the magnetic field out. Cool it below its transition and the flux is expelled; the interior goes to B = 0, a perfect diamagnet with susceptibility χ = −1. That expulsion, not zero resistance, is the true fingerprint of the state, and it is runnable: field goes in, the engine screens it over a London depth and expels the rest, and out comes the phase boundary. Blue builds it; red tries to break it.
source Meissner, W. & Ochsenfeld, R., Ein neuer Effekt bei Eintritt der Supraleitfähigkeit, Naturwissenschaften 21 (44), 787–788 (1933) — doi.org/10.1007/BF01504252. Rendered, not quoted.
In the superconducting state the field obeys three laws, each computed live below:
L1 perfect diamagnetism — χ = −1, so the deep interior is B = 0 for any applied field. L2 the field is not banished at the surface: it penetrates a thin London depth λ, decaying as e−x/λ. L3 the state survives only below the critical field Hc(T) = Hc0(1 − (T/Tc)²) and below Tc — a parabola.
Live, for the current (T, Bapp) on the panel:
| quantity | value |
|---|
Meissner & Ochsenfeld (1933) measured that the field is pushed out at the transition — B = 0 inside, χ = −1, screened over a London depth. This is a distinct thermodynamic phase, not a memory of past currents.
London theory says how the field is screened but not why the response is rigid. That answer is the-bcs-theory (1957): paired electrons with a gap 2Δ ≈ 3.53 kBTc, a wavefunction stiff enough to hold χ = −1. Each sphere is the next one's premise.
The blue team's live check: re-derive χ, the interior field, the 1/e screening and the Hc(T) parabola from the pure functions. If red freezes the field instead of expelling it, this badge turns red.
Three inputs feed the engine: the applied field Bapp (Tesla), the temperature T (kelvin), and the material's two constants — its transition temperature Tc and thermodynamic critical field Bc0. Defaults are niobium:
| constant | symbol | value |
|---|---|---|
| transition temp. | Tc | 9.30 K |
| critical field (0 K) | Bc0 | 0.200 T |
| superfluid density | ns | 4×10²⁸ m⁻³ |
From ns the engine computes the London depth λ = √(me/(μ₀ ns e²)); from T it computes Bc(T). That is what you feed the panel below.
Left: field lines meeting the slab — expelled (curved around) or admitted. Right: the Hc(T) parabola with the live point. Every number is computed from the pure functions on the spot, never looked up.
What the machine proves: below Tc and Hc(T) the interior is B = 0 (χ = −1) for any applied field — expulsion, not a frozen memory; the surface admits a 1/e skin one London depth deep; and the state dies on the parabolic critical-field boundary. The current point's state is above; the boundary is the output.
The blue witness (left) re-derives these live; the red team (right) tries to make the field freeze instead of expel.
And the ideal is a type-I slab. Real type-II superconductors admit quantized flux vortices in a mixed state between Hc1 and Hc2 — partial penetration, not clean expulsion. Demagnetizing geometry (a sphere's equator) drives an intermediate state even in type-I. The parabola and χ = −1 are the clean limit, marked AMBER where a real sample departs.
"A superconductor is just a perfect conductor." Cut. A perfect conductor freezes flux (dB/dt = 0) — a field present when it cools stays. The Meissner state expels: field-cooled, it still reaches B = 0. Distinct phases; the engine's tamper button shows the difference.
"The field never enters at all." Cut. It penetrates a London depth λ (tens of nm), decaying e−x/λ. B = 0 holds only in the deep interior.
"Expulsion holds up to Hc for every superconductor." Kept, corrected. Type-I, yes. Type-II admit vortices above Hc1 — a mixed state, modelled here only as the type-I ideal.
The red team's move: make the sample a mere perfect conductor (χ = 0) — freeze the field instead of expelling it, so B ≠ 0 inside. The blue witness (window 7) is watching.
Set χ = 0 and the interior field is no longer zero — the field is frozen, not expelled. The witness recomputes, finds B ≠ 0 in the Meissner state, and turns red. Nothing is faked; the attack is real and it is caught.