London Meissner Slab Screening

The Meissner effect describes the expulsion of a magnetic field from a superconductor. When a magnetic field is applied tangentially to a superconducting slab, screening currents arise to cancel the magnetic field inside the material, causing the field to decay exponentially from the surface. This is the equilibrium Meissner response, not a frequency-dependent skin effect.

Worked Example

Let us calculate the relative magnetic field and normalized screening current for an infinite superconducting slab of half-thickness d = 2 and penetration depth λ = 1, evaluated at a position fraction u = 0.5.

The position x inside the slab is u × d = 0.5 × 2 = 1.0.
Calculate the boundary parameter cosh(d/λ) = cosh(2/1) = cosh(2) ≈ 3.76219569.
Calculate the position parameter cosh(x/λ) = cosh(1/1) = cosh(1) ≈ 1.54308063.
The relative field is B(x)/B_surface = cosh(x/λ) / cosh(d/λ) ≈ 1.54308063 / 3.76219569 ≈ 0.41015427200459836.
The normalized screening current is related to sinh(x/λ) = sinh(1) ≈ 1.17520119.
The normalized screening current is -sinh(x/λ) / cosh(d/λ) ≈ -1.17520119 / 3.76219569 ≈ -0.3123710965989933.

Experiment

We can observe the Meissner screening response using the London Meissner Slab Screening module.

Set the Penetration length to 1.
Set the Half thickness to 2.
Set the Position fraction to 0.5.
Observe the Relative field value matches the calculated 0.41015427200459836.
Observe the Normalized screening current matches -0.3123710965989933.

Limitations

This model assumes the static London equation holds within a perfectly linear superconducting regime. It assumes an infinite slab geometry with exactly equal tangential fields applied to both surfaces, resulting in a symmetric field penetration profile.

If the half thickness is set to exactly 0, it represents a formal zero-thickness limit where the relative field is 1.0 and the susceptibility is 0.0 everywhere, rather than a physical bulk superconducting volume. It does not model nonlinear phenomena such as critical field limits or temperature-dependent behavior.

References

The derivation of the linear static London equation in an infinite slab and the resulting hyperbolic field and current profiles are discussed in academic problems on superconductivity. See the University of Maryland homework set: P732HmwkSet1.pdf, which outlines the calculation of the field profile and effective susceptibility.

London Meissner Slab Screening