Where is the energy transfer from a metal ball falling from a magnet?

If the ball is made of iron

You put “magnetic” potential energy into the system when you brought the metal ball up to the magnet. The sign of that potential energy is negative indicating an attractive force.

When you drop the ball, the ball loses gravitational potential energy, but adds magnetic potential energy and kinetic energy. Therefore less energy was available for kinetic energy and the corresponding impact speed was less.


We can write this as an energy relationship: $$ \begin{align} \Delta E &= \Delta K + \Delta U \\ 0 &= (K_f-K_i) + \Delta U_G + \Delta U_{M} \\ K_f &= -\Delta U_G - \Delta U_M \end{align} $$ Where $K_i = \Delta E = 0$. In the situation where the ball falls the gravitational potential energy decreases ($\Delta U_G < 0$) but the magnetic potential energy increases ($\Delta U_M > 0$), since the metal ball has moved further from the magnet. The type of forces two magnets experience is a conservative force (since it's path independent) and so it makes sense to talk about a magnetic potential energy.

The ball would need to be iron (or one of its alloys like steel) because in terms of everday materials, iron is the only one that can be temporarily (induced) magnetized. For most other materials, $\Delta U_M \approx 0$.

My favorite version of this demonstration is dropping a magnet through a conducting tube. In that case the gravitational potential energy in the system is converted, by the electromagnetic field, into thermal energy in the tube. The mechanism is Joule heating by the induced currents as the magnetic field at different parts of the tube changes. The terminal velocity decreases if the conductivity of the tube is improved; a superconducting tube would prevent the magnet from falling altogether, such is another well-known demonstration.

In your case, with a fixed magnet and a falling metal ball, the mechanism is the same: eddy currents in the metal cause resistive heating, for which the energy comes from the electromagnetic field. But the geometry there makes computations trickier.

The magnetic force acting on the ball as a whole (ponderomotive force due to magnet) slows down the fall, thus does negative work on the ball. This means energy is taken away from the mechanical energy of the system (ball+Earth). Since the force is due to electromagnetic field, the only obvious destination where it goes is the electromagnetic energy of the system (ball+magnet). The electromagnetic field is everywhere, and due to descent of the ball, it changes everywhere. In some places, the EM energy density increases, in some other places, it decreases. If there are no losses of energy to heat, net EM energy has to increase by the same amount that mechanical energy decreased.