A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
Superconductors exhibit two defining properties: zero DC resistance below a critical temperature T_c, and the Meissner effect — complete expulsion of magnetic flux from the interior (B = 0, not just dB/dt = 0). The London equations, curl(J_s) = -(n_s e2/mc)B and partial J_s/partial t = (n_s e2/m)E, describe these phenomena phenomenologically. They predict that magnetic fields penetrate only a distance lambda_L = sqrt(mc2/(4pi n_s e2)) ~ 10-100 nm into the superconductor, decaying exponentially. The Meissner effect proves that superconductivity is a thermodynamic state (not merely perfect conduction), characterized by a macroscopic quantum wavefunction.
Superconductivity, discovered in 1911 by Kamerlingh Onnes in mercury, is defined by two phenomena. The first, zero resistance, means that a current once established in a superconducting loop persists indefinitely — experiments have verified persistent currents lasting years with no measurable decay. The second, the Meissner effect (discovered 1933), is the complete expulsion of magnetic flux from the interior of a superconductor: B = 0 inside. The Meissner effect is not a consequence of zero resistance — a perfect conductor would freeze any pre-existing flux, not expel it. Flux expulsion proves that B = 0 is an equilibrium property of the superconducting state.
The London brothers (1935) captured both phenomena in two equations. The first London equation, partial J_s/partial t = (n_s e2/m) E, says the supercurrent accelerates freely in an electric field (zero resistance). The second London equation, curl J_s = -(n_s e2/mc) B, relates the supercurrent directly to the magnetic field (not its time derivative), which forces B = 0 in the bulk. Combined with Maxwell's equations, the London equations predict that magnetic fields penetrate only a characteristic distance lambda_L into the superconductor, decaying exponentially: B(x) = B_0 exp(-x/lambda_L). The London penetration depth lambda_L = sqrt(mc2 / 4 pi n_s e2) is typically 20-200 nm.
The Meissner effect has a direct thermodynamic consequence: expelling the field costs magnetic energy (H2/8pi per unit volume of expelled field), so there is a critical field H_c above which it is energetically favorable for the material to return to the normal state. The condensation energy — the free energy gained by entering the superconducting state — equals H_c2/8pi. This thermodynamic framework, developed by Gorter and Casimir, allows the superconducting transition to be analyzed like any other phase transition, with specific heat jumps, latent heat (at finite field), and critical exponents.
The London equations are phenomenological — they describe what happens but not why. The deeper question of why electrons form a superconducting condensate was answered by the BCS theory (1957) and the Ginzburg-Landau theory (1950). But the London equations remain the starting point for understanding superconducting electrodynamics and are exact in the appropriate limits. They also introduce the concept of a macroscopic quantum wavefunction: the supercurrent is proportional to the gradient of the phase of a single quantum state that extends across the entire superconductor, a concept that leads directly to flux quantization and the Josephson effect.