Below a critical magnetic field $ H_c $, type I superconductors exhibit the Meissner effect, in which the magnetic induction is excluded from the volume of the superconductor. Assume that in this case the superconductor in question has no holes – i.e., it is topologically like a sphere and not like a ring.
(a)Explain why the component of $ \vec B $ normal to the surface vanishes just outside the superconductor.
(b)Explain why the magnetic induction outside the superconductor can be derived from a scalar potential, $ \vec B = -\vec \nabla \phi $. State the boundary value problem (differential equation and boundary conditions) satisfied by $ \phi $.
(c)Consider a superconducting sphere of radius $ a $ placed in an otherwise uniform magnetic induction $ \vec B_0 $. Find the magnetic induction outside the sphere.