Magnet and energy conservation

Question

If we consider a steel ball falling under gravity in a cup (potential well) and being stopped at the bottom by an obstacle then energy conservation implies that the gravitational potential energy has first been transformed into kinetic energy and then in heat. After the experiment the potentiel well's height is decreased because there is a steel ball standing at the bottom. So I can repeat the experiment and each time I will extract less and less energy until I extract nothing when the well is filled up with balls. Now consider a magnet and a steel ball. The magnet accelerates the steel ball until they stick together and the kinetic energy is dissipated by heat emission. Energy conservation implies that the energy of the magnet+field+ball has changed (decreased by the amount of heat produced). My question is how is it decreased ? Is it the magnetic moment of the magnet which has been decreased, or should we take into account the induced magnetic moment in the steel ball to recompute the energy of the field ?

The problem can be simplified (for computation) but remains open if instead of a magnet and a steel ball one uses two coils, each one connected to a generator. If someone can write down the details of the calculus, it would be much appreciated. (Problem solved for the coils in Griffiths p211. The generators produce the energy dissipated by heat.) So it remains to solve the problem for a magnet and the steel ball. Is there a domain reconfiguration ?

Answer 1

This is only a partial answer but, when considering magnets, you need to consider the creation and annihilation of magnetic fields. For instance, recal that in electromagnetism we treat the energy as being stored in the magnetic field itself.

Thus, if we separate two magnets we creat field in a region of space where there was no field before. So this process requires us to do work (which is intuitive) and two magnets kept apart configure a system with positive energy. Conversely, by joining two magnets we destroy magnetic field and hence reduce the systems's energy.

So, in your example this is another ingredient to be considered. Perhaps other features are also relevant, such as the fact that the heat dissipated upon the collision will also partially demagnetized the magnet (alter the domain configuration); note that magnetizing a permanent magnet requires a lot of energy.

I hope this helps.

Answer 2

This is ultimately a clarification (i.e. extended comment) of Gabriel's post:

Static magnetic fields can do no work, and if the ball is at rest then no Lorentz force is produced. What occurs is that you bring the magnet close enough to the ball, and the ball feels a changing B-field ... this field does work and alters the ball's internal energy (i.e. the current loops / magnetic moments in the metal tend to re-align like those in the magnet). We're ignoring other quantum effects, like ferromagnetism.

Up to here, conservation of energy is fine.

Now the ball strikes the magnet, and so the compression of the ball and friction/heat provides the energy loss. Loss from what? The original energy was the energy stored in the B-field, plus the negligible interaction energy from the magnetic moments of the ball with the B-field (potential term $\mu\cdot B$).

Thus it must be that the magnet loses some strength due to domain reconfiguration, or the negligible interaction energy of the ball-with-magnet got even more negligible (due to perturbed realignment of any $\mu$ with $B$).