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Recent proposals suggest using magnetically trapped superconducting spheresin the Meissner state to create low-loss mechanical oscillators with longcoherence times. In these proposals the derivation of the force on thesuperconducting sphere and the coupling to the sphere typically relies on avanishing 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 thispaper we analytically solve the Maxwell-London equations with appropriateboundary conditions for a superconducting sphere in a quadrupole field. Theanalytic solutions provide the full field distribution for arbitrary $\lambda$and for an arbitrary sphere position as well as the distribution of shieldingcurrents within the sphere. We furthermore calculate the force acting on thesphere and the maximum field over the volume of the sphere. We show that for acertain range of $\lambda$ the maximum field experienced by the superconductingsphere is actually lower than it is for a non-magnetic sphere.

Analytic solutions to the Maxwell-London equations and levitation force for a superconducting sphere in a quadrupole field