A point magnet near a ring magnet: Magnetostatic interaction energy.

We show that the magnetostatic interaction energy between a point magnet and a ring magnet admits exact solutions in terms of complete elliptic integrals when the point magnet is off the axis of the ring magnet and has a simple solution in terms of rational functions when the point dipole is on the axis of symmetry of the ring magnet. The exact expression for the interaction energy exhibits an equilibrium point close to the axis of symmetry with a saddle point instability. The stability analysis requires us to find the force on the dipole, which is given in terms of the derivatives of the elliptic integrals in the energy. Our solution will be useful in investigations concerning magnetic levitation and Casimir levitation.