UVa EM:Meissner effect in super conducting sphere

Problem

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.