Nondipole interaction between two uniformly magnetized spheres and its relation to superconducting levitation

TL;DR

This work analytically solves the magnetostatic problem for two axis-tiltedly aligned, uniformly magnetized spheres using bispherical coordinates, deriving the external field from a Laplace potential expanded in axisymmetric harmonics. It shows that the field outside the spheres is not equivalent to the sum of two dipole fields, a non-equivalence that becomes evident when one sphere is nonmagnetized (a superconducting Meissner sphere) and a repulsive force arises due to magnetic pressure. An infinite series of image dipoles is shown to underlie the exact solution, explaining the nondipole interaction and providing convergence assurances via the Weierstrass M-test. The results have implications for understanding magnetic interactions in composite sphere systems and for superconducting levitation phenomena, where finite-size effects preclude a naive dipole-dipole description.

Abstract

Analytically solving the magnetostatic Maxwell equations in the bispherical coordinates, we calculate the magnetic field around two uniformly magnetized spheres oriented so that their magnetic moments are parallel to the axis passing through the centers of the spheres. We demonstrate that, contrary to what is often claimed in the literature, the magnetic interaction between such spheres is not equivalent to the interaction between two point magnetic dipoles placed in the centers of the spheres. The nonzero levitation force acting on a uniformly magnetized sphere or a point magnetic dipole above a superconducting sphere in the ideal Meissner state is a clear manifestation of the non-equivalence.

Figures (2)

• Figure 1: Bispherical coordinates.
• Figure 2: Repulsion of a superconducting sphere from a uniformly magnetized sphere due to the magnetic pressure difference.