Eddy current effects for magnetic dipoles in arbitrary motion

Eddy currents are induced whenever magnets move near a conductive material, however computing the exact currents and their interaction with the magnets is notoriously difficult. Even for simple dipole magnets, and when ignoring the recursive self-induction from the eddy currents’ own magnetic field, few analytical solutions exist. Here we derive three closed-form solutions for the forces and torques on uniformly magnetized spheres and other dipole magnets that result from eddy currents when the velocities are low enough to neglect self-induction : (1) the eddy current coupling that occurs between two uniformly magnetized, conductive spheres, (2) the damping when a dipole magnet is moving near the center of a spherical cavity inside a large conductor and (3) the damping from motion near a large, stationary, metal plate. Additionally, for cases 1 and 3 we compare to finite element numerical solutions to test our models range of validity. These calculations generalize results from the literature to arbitrary orientations of the magnetization, linear- and angular velocity vectors, such that one may directly implement the results in dynamical simulations and compute the resulting magnet trajectories.