NONE OF THE ANIMATIONS SHOWN HEREIN ARE CARTOONS--all of the field configurations and dynamics are calculated quantitatively.
In the case where an experimenter controls the motion of the magnet as he or she pushes it into the coil or pulls it out of the coil, we have solved a single differential equation for the current in the coil given the motion of the magnet. In the case where the magnet is falling through the conducting ring, the current induced in the ring by the motion of the magnet in turn affects the motion of the magnet, so that we must solve a coupled set of equations. The equations can be reduced to a set of three first order ordinary coupled differential equations for the current in the ring, the position of the magnet, and the speed of the magnet.
Given the current in the ring and the dipole moment of the magnet, we can then construct the magnetic fields. For details of the mathematics, including a development of the differential equations to be solved, see the pdf document referenced above.
If the resistance of the ring is zero, for some values of the parameters the magnet will be levitated above the ring by the eddy currents. For other values the magnet will fall through the ring. We show both cases in the animations below.
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