  Recent proposals suggest using magnetically trapped superconducting spheres
in the Meissner state to create low-loss mechanical oscillators with long
coherence times. In these proposals the derivation of the force on the
superconducting sphere and the coupling to the sphere typically relies on a
vanishing penetration depth $\lambda$ as well as a specific symmetry (i.e.
restricting the position of the sphere to one axis) or heuristic methods (e.g.
assigning an equivalent point magnetic dipole moment to the sphere). In this
paper we analytically solve the Maxwell-London equations with appropriate
boundary conditions for a superconducting sphere in a quadrupole field. The
analytic solutions provide the full field distribution for arbitrary $\lambda$
and for an arbitrary sphere position as well as the distribution of shielding
currents within the sphere. We furthermore calculate the force acting on the
sphere and the maximum field over the volume of the sphere. We show that for a
certain range of $\lambda$ the maximum field experienced by the superconducting
sphere is actually lower than it is for a non-magnetic sphere.
