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 >> λL, d << λL, and draw a picture of the distributions in these two cases (λL– 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 λL 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 Js can be found from Maxwell’s equation
Since H = Hẑ, we have
and
b) In the limiting case of a thin film, λL >> d, we have
Figure 1.2b
since cosh(x/λL) ≈ 1 + (1/2)(x/λL)2 + …. and sinh(x/λL) ≈ x/λL +…. Both cases d >> λL and d << λL are shown in Figure 1.2b.
