Thin Superconducting Plate in Magnetic Field

A very long, thin plate of thickness d is placed in a uniform magnetic field H₀ parallel to the surface of the plate (see Figure 1.1).

 Figure 1.1

a) Find the distribution of the magnetic field and the current inside the plate.

b) Consider two limiting cases d >> λ_L, d << λ_L, and draw a picture of the distributions in these two cases (λ_L– London penetration depth).

SOLUTION

a) Choose x = 0 at the center of the plate (see Figure 1.2a). We know that the external field penetrates to a depth λ_L into the superconductor; this can be described in our case by the

Figure 1.2a

equation:

Because of the symmetry of the problem, H = H(x) ẑ where the H field inside the the superconductor will be in the z direction and depend only on x. So we have

(2)

The general solution of (1) is

Using the boundary conditions

we obtain

So

The supercurrent density J_s can be found from Maxwell’s equation

Since H = Hẑ, we have

and

b) In the limiting case of a thin film, λ_L >> d, we have

Figure 1.2b

since cosh(x/λ_L) ≈ 1 + (1/2)(x/λ_L)² + …. and sinh(x/λ_L) ≈ x/λ_L +…. Both cases d >> λ_L and d << λ_L are shown in Figure 1.2b.