Superconducting Sphere in Magnetic Field
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 77
11-8-2016
1796
Superconducting Sphere in Magnetic Field
A superconducting (Type I) spherical shell of radius R is placed in a uniform magnetic field B0 (B0 << Hc, Hc, the critical field). Find
a) the magnetic field everywhere outside the shell
b) the surface current density
Hint: Inside, B = 0.
SOLUTION
a) Prom symmetry considerations, it is clear that the current would flow on the surface of the shell perpendicular to the applied magnetic field. As for any ellipsoid in a uniform electric or magnetic field, we can assume that the field outside the shell produced by these currents is equivalent to a magnetic dipole moment m placed in the center of the shell. For
The total field outside is then B = Bm + B0. The boundary condition on the surface at an arbitrary point gives
The normal component of B is continuous and inside B = 0. From the boundary conditions on the surface at an arbitrary angle θ between the direction of B0 and the normal n (see Figure 1.1) we have
(1)
Figure 1.1
Hence
(2)
At m = -(R3/2) B0, where R is the radius of the spherical shell, the boundary conditions are satisfied on the surface of the shell. Therefore,
b) The surface current density Js can be found by using tangential H component continuity:
and therefore Js(θ) = 3cB0 sin θ/8π. This solution is only true while B < 2/3 Hc, and the whole sphere is superconducting. When B > 2/3 Hc the field at the equator exceeds Hc, and the sphere goes into an intermediate state.
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