Magnetic screening requires more than perfect conductivity

For a homogeneous superfluid component, let q=−2e be the pair charge, m* the effective pair mass, and nₚ its effective number density. An electric field accelerates this component and produces the first London equation. Its coefficient contains q squared, so the electrical current accelerates along the field even though the carriers have negative charge. A normal component can also contribute a dissipative current at finite temperature; the London current is only the nondissipative part of that total response.

An ideal conductor with vanishing internal electric field has a time-independent magnetic field by Faraday’s law. That condition freezes an initial field rather than requiring it to vanish. The second London equation imposes an additional relation between current curl and magnetic induction. In a simply connected, vortex-free equilibrium region it selects screening rather than arbitrary frozen flux. Trapped flux in holes, pinning, metastability, and the mixed state of a type-II material require additional information, so flux expulsion should not be asserted for every superconducting configuration.

Neglect displacement current in a static sample and combine Ampère’s law with the second London equation. Taking the curl of Ampère’s law produces the Laplacian of the magnetic field because its divergence is zero. The result is a positive inverse penetration-length squared multiplying the field. This sign gives exponential spatial screening, rather than a sinusoidal bulk field. For a half-space x>0, impose a tangential induction B₀ at x=0 and require a finite field that tends to zero deep inside.

Choose the field along positive z. Its surviving profile is B₀ times exp(−x/λL); the growing exponential is excluded by the bulk boundary condition. Ampère’s law gives a positive y-directed current because the field decreases with x. Integrating that current over depth yields a sheet current B₀/μ₀. Increasing penetration depth spreads the same required sheet current over a thicker layer. Confusing the surface current density with its depth integral would incorrectly predict a different total screening requirement at fixed boundary field.

The London model holds the order-parameter amplitude fixed and treats current response as local. It is useful for long-wavelength, low-frequency electrodynamics away from vortex cores and strong pair-breaking currents. Very clean samples can display nonlocal response when the current at a point depends on fields over a coherence scale. Near a surface or near a critical field, amplitude suppression can matter. These limitations are reasons to move to a richer model, rather than to identify a measured penetration length automatically with a free-electron carrier count.

The function lab illustrates the quadratic magnetic-energy cost, not the screening equation. Reset and set both controls to a=0.5 and x=1, predicting y=0.5. Reset again and set a=0.5 and x=2, predicting y=2. Interpret x as an assigned normalized field and y as its normalized field-energy density. Doubling field costs four times the energy. The curve contains no depth coordinate, current, or superconducting transition, so an exponential London profile must be calculated from the boundary problem rather than read from this parabola.