Surface electric current distributions on spheres and spheroids as sources of pure quadrupole magnetic fields

neutral atom magnetic traps and nuclear magnetic resonance imaging require internal regions with constant gradient magnetic induction fields, which are identified as pure quadrupole fields. this contribution starts from such fields in the interior of spheres and spheroids in cartesian coordinates, identifying immediately their respective scalar magnetic potentials. next, the corresponding potentials inside and outside are constructed using spherical and spheroidal harmonic functions, respectively, except for a proportionality constant to be determined by the boundary conditions at the surface of spheres r = α, prolate ξ = ξ0 and oblate ζ = ζ0 spheroids, where the electric current sources are distributed. the negative gradients of the scalar potentials yield the respective magnetic induction fields inside (r < α, ξ < ξ0, ζ < ζ0) and outside (r > α, ξ > ξ0, ζ > ζ0). gauss's law in its boundary condition form determines the normalization constant of the external potentials, while ampere's law determines the electric current source distributions on the surface of the spheres and spheroids.