Eddy currents of a magnet falling through a copper pipe with slits

We present a theoretical model for the eddy currents induced by a cylindrical magnet falling through a copper pipe with vertical slits and the resulting inductive drag force. This system maintains the mathematically convenient cylindrical geometry of the classic intact pipe problem while introducing boundary conditions that result in fascinating eddy current patterns. Previous experiments have demonstrated that the inductive drag force is smaller in a slit pipe than in an intact pipe of the same radius, suggesting that the presence of slits reduces but does not eliminate the horizontal induced eddy currents required to produce a vertical drag force. Our theoretical model suggests that the horizontal eddy currents are reduced in slit pipes partly because the formation of a slit requires the removal of conducting material where eddy currents would otherwise have formed, and partly because the induced currents must flow purely vertically along the slit walls. We show that our theoretical model is consistent with our experimental data for the terminal velocities of magnets falling through copper pipes with different size slits and different numbers of slits. This experiment might serve as an advanced undergraduate laboratory activity.