Computation of the magnetic potential induced by a collection of spherical particles using series expansions

In Magnetic Resonance Imaging there are several situations where, for simulation purposes, one wants to compute the magnetic field induced by a cluster of small metallic particles. Given the difficulty of the problem from a numerical point of view, the simplifying assumption that the field due to each particle interacts only with the main magnetic field but does not interact with the fields due to the other particles is usually made. In this paper we investigate from a mathematical point of view the relevancy of this assumption and provide error estimates for the scalar magnetic potential in terms of the key parameter that is the minimal distance between the particles. A special attention is paid to obtain explicit and relevant constants in the estimates. When the "non-interacting assumption" is deficient, we propose to compute a better approximation of the magnetic potential by taking into account pairwise magnetic field interactions between particles that enters in a general framework for computing the scalar magnetic potential as a series expansion.