An Analytic Solution for the Force Between Two Magnetic Dipoles

An analytic equation describing the force between two magnetic dipoles is derived in this paper. We assumed that the dipole sizes are small compared to their separation. A Taylor expansion for the first non-zero term was performed in order to derive the magnetic force analytic expression. Vector differential and path integral derivation approaches are used, and the same result is achieved with both methods. This force decreases with the fourth power of the distance between the dipoles. Accuracy estimates due to the Taylor approximation are given.