Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations

Static dipole-dipole magnetic interaction is a classic topic discussed in electricity and magnetism text books. Its dynamic version, however, has not been reported in scientific literature. In this article, the author presents a comprehensive analysis of the latter. We consider two identical permanent cylindrical magnets. In a practical setting, we place one of the magnets at the bottom of a vertical glass tube and then drop the second magnet in the tube. For a pair of suitable permanent magnets characterized with their mass and magnetic moment we seek oscillations of the mobile magnet resulting from the unbalanced forces of the anti-parallel magnetic dipole orientation of the pair. To quantify the observed oscillations we form an equation describing the motion of the bouncing magnet. The strength of the magnet-magnet interaction is in proportion to the inverse fourth order separation distance of the magnets. Consequently, the corresponding equation of motion is a highly nonlinear differential equation. We deploy Mathematica and solve the equation numerically resulting in a family of kinematic information. We show our theoretical model with great success matches the measured data.