In a perfect conductor, the conductivity is infinite, so E=0 (Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics).
(a) Show that the magnetic field is constant (∂B/∂t=0), inside a perfect conductor.
(b) Show that the magnetic flux through a perfectly conducting loop is constant.
A superconductor is a perfect conductor with the additional property that the (constant) B inside is in fact zero. (This "flux exclusion" is known as the Meissner effect.)
(c) Show that the current in a superconductor is confined to the surface.
(d) Superconductivity is lost above a certain critical temperature (Tc), which varies from one material to another. Suppose you had a sphere (radius a) above its critical temperature, and you held it in a uniform magnetic field B0z^ while cooling it below Tc. Find the induced surface current density K, as a function of the polar angle θ.