Questions: Superconductivity: Phenomenology (Meissner, London Equations)

Question 1

A perfect conductor (σ = ∞) and a superconductor both carry current with zero resistance. What experiment distinguishes them?

• A. Measuring the critical temperature
• B. Apply a magnetic field above T_c, then cool below T_c. A perfect conductor would trap the field inside (dB/dt = 0 prevents change), while a superconductor actively expels the field (Meissner effect: B = 0 regardless of history). The field expulsion on cooling is the unique signature of superconductivity
• C. Measuring the current-carrying capacity
• D. A perfect conductor has zero resistance only at T = 0

Answer: (B)

This is the crucial conceptual point. A perfect conductor (hypothetical material with σ → ∞) obeys Faraday's law: dB/dt = 0 inside, so whatever field was present when resistance vanished would be frozen in. A superconductor actively expels field (B → 0) regardless of whether the field was applied before or after cooling through T_c. This means superconductivity is an equilibrium thermodynamic phase with B = 0 as a state variable, not merely a kinetic property (zero resistance). The Meissner effect requires the London equation curl(J) ∝ -B, not just E = ρJ with ρ = 0.

Question 2

The London penetration depth λ_L sets the length scale over which magnetic fields decay inside a superconductor. What determines its magnitude?

• A. The crystal lattice spacing
• B. λ_L = √(mc²/4πn_se²) depends on the superfluid density n_s — higher n_s means more screening current available and shorter penetration depth. Typical values are 20-200 nm, much larger than atomic spacings but much smaller than macroscopic samples
• C. The mean free path of electrons
• D. The Debye temperature of the material

Answer: (B)

The London penetration depth is set by the inertia of the superconducting electrons (mass m) versus their ability to screen (charge e and density n_s). Near T_c, n_s → 0 and λ_L → ∞ (the superconductor can no longer screen fields effectively). At T = 0, λ_L has its minimum value, typically 20-50 nm for elemental superconductors. The temperature dependence λ(T)/λ(0) ≈ [1 - (T/T_c)⁴]^{-1/2} (approximately) provides experimental access to the superfluid density.

Question 3

The Meissner effect (B = 0 inside a superconductor) proves that superconductivity is a thermodynamic equilibrium state, not merely a kinetic phenomenon (zero resistance).

• T. True
• F. False

Answer: True

If superconductivity were only zero resistance (perfect conduction), the internal magnetic field would depend on the history — field applied before or after the transition. The Meissner effect shows that B = 0 inside regardless of history: the superconducting state is uniquely defined by (T, H), just like an equilibrium thermodynamic phase. This allows treatment by thermodynamics: the free energy difference between normal and superconducting states determines the critical field H_c via the condensation energy H_c²/8π. Without the Meissner effect, there would be no thermodynamic framework for superconductivity.

Question 4

Derive the London penetration depth from the London equation and explain what happens physically at the surface of a superconductor in an applied field.

Model answer: Starting from the London equation curl(J_s) = -(n_se²/mc)B and Ampere's law curl(B) = (4π/c)J_s, taking the curl of Ampere's law and substituting gives ∇²B = B/λ_L², with λ_L = √(mc²/4πn_se²). For a flat surface with field B₀ applied parallel, the solution is B(x) = B₀ exp(-x/λ_L). Physically, the applied field induces persistent screening currents in a surface layer of thickness ~λ_L. These currents produce a field that exactly cancels the applied field in the interior. The screening currents flow without resistance (supercurrent) and create the perfect diamagnetic response. The superconductor pays a kinetic energy cost (½mv²n_s per unit volume in the screening layer) to maintain these currents.

The exponential decay of the field — not abrupt cancellation — is key. The penetration depth is measurable by muon spin rotation, microwave surface impedance, or the magnetic field dependence of the London moment in rotating superconductors.