Superconductors — Question 6

Question 6

Magnetic screening has a finite penetration depth

A superconductor occupies $x>0$ in the Meissner regime. Its magnetic field satisfies the supplied equation $d^2B/dx^2=B/\lambda^2$, with $B(0)=B_0$ and $B(\infty)=0$.

1. Solve for B(x) and find the depth where it is one percent of its surface value.

2. Find the integrated field per unit transverse width, $\int_0^\infty B\,dx$, and explain why exact zero resistance alone does not specify this equilibrium screening profile.

Question 6 - Solution

Magnetic screening has a finite penetration depth

(a) Method and calculation

$$B=C_1e^{x/\lambda}+C_2e^{-x/\lambda},\quad C_1=0,\ C_2=B_0.$$$$B(x)=B_0e^{-x/\lambda},\quad x_{1\%}=\lambda\ln100=4.60517\lambda.$$

(b) Interpretation and further calculation

$$\int_0^\infty B(x)dx=B_0\lambda.$$$$R=0\ \text{alone supplies no equation for }d^2B/dx^2.$$Meissner screening is an additional property; a perfect-conductor idealization can retain a previously present field.

Physical check

The field is very small deep in the material but does not jump discontinuously to zero at the surface in this model.