Interaction potentials for mutually induced dipoles in uniform fields

Dipolar interactions govern the structure and dynamics of many soft-matter systems, from molecular to colloids assemblies. When dipole moments are induced by an external field, mutual interactions lead to a many-body magnetization response that cannot be described by fixed-dipole models. Here, we derive the interaction potential for a system of mutually interacting induced dipoles in a uniform external field using a force-based approach. By accounting for the displacement-induced variation of the dipole moments, we obtain an interaction potential consisting of the classical dipole-dipole term supplemented by two- and three-body corrections arising from mutual induction. Comparisons with simplified models that neglect mutual magnetization reveal significant errors in the interaction potential, particularly in anisotropic particle assemblies. We also discuss an efficient $\mathcal{O}(N^2)$ iterative scheme for computing the mutual magnetization, enabling accurate simulations of large dipolar systems.