  We show how to calculate the magnetic-field and sheet-current distributions
for a thin-film superconducting annular ring (inner radius a, outer radius b,
and thickness d<<a) when either the penetration depth obeys lambda < d/2 or, if
lambda > d/2, the two-dimensional screening length obeys Lambda = 2 lambda^2/d
<< a for the following cases: (a) magnetic flux trapped in the hole in the
absence of an applied magnetic field, (b) zero magnetic flux in the hole when
the ring is subjected to an applied magnetic field, and (c) focusing of
magnetic flux into the hole when a magnetic field is applied but no net current
flows around the ring. We use a similar method to calculate the magnetic-field
and sheet-current distributions and magnetization loops for a thin,
bulk-pinning-free superconducting disk (radius b) containing a dome of magnetic
flux of radius a when flux entry is impeded by a geometrical barrier.

Superconductivity