Where does the energy come from?It comes from the kinetic and potential energy of bound current in atoms of a magnet.
What do the atoms provide us?The atoms produce electric and magnetic fields. If we consider the unpaired electrons contained within a magnet, the alignment of their quantum spins is responsible for the production of the magnet's collective magnetic field. When the magnet (an object possessing a magnetic dipole) moves relative to other objects, it induces upon them electric field lines (or electric fluxes) that connect back to (or terminate on) the magnet.
(That is to say the relatively moving magnet carries with it an apparent electric dipole. See the relevant Equation 3.85 below.)
Under what conditions do the electric fields generated by a moving magnet terminate on that magnet?The termination of a magnet's own electric fields back on to itself is true as long as the magnet's relative motion is not parallel to the alignment of its magnetic domains. However, if the magnet moves in parallel with the alignment of its domains, the induced electric fields will not terminate on the magnet.
(See the relevant Equation 3.85 above.)What is the role of the observer or medium in regards to the induced electric fields of a moving magnet?It is important to point out that such motion of a moving magnet is relative to the observer or medium which is subjected to the influence of the magnet. Thus not every observer or medium will detect (or be subject to) an electric field terminating on the magnet.
What do we provide for these atoms?We can use changing magnetic fields to cause the alignment of the quantum spins of the unpaired electrons in the magnet to vary with time. Thus the magnetic strength of our magnet can be made to vary with time.
After that, what do the atoms provide us?
When combined with the relative motion of the magnet, this causes the amount of electric field lines (or electric fluxes) terminating on the magnet to vary with time. These electric field lines terminating on the magnet may interact with the electric field of external electrostatic charges, building up energy which is stored in their mutual electric capacitance. The reason is that while electric fields add linearly, the energy density in the fields adds quadratically. The "interaction energy" is the energy in the mutual capacitance, and it is mathematically similar to the cosine term in the law of cosines. It is stored in the electrical capacitance through electrical/capacitive coupling.
Does all of the energy get stored in the electrical capacitance?
No.
Where else can the energy get stored?Energy can be stored in the magnetic inductance. The magnetic field strength of magnets and the magnetic field strength of electric currents add together linearly, while the density of the stored energy increases quadratically (with the square of the magnetic field strength). As with electric fields from multiple sources, for magnetic fields from multiple sources there is an "interaction energy" density analogous to the cosine term in the law of cosines. Similarly, the energy is stored by magnetic or inductive coupling (analogous to electric or capacitive coupling).
Where did the energy stored in the magnetic inductance come from?
The flow of energy through magnetic inductive coupling is subject to Lenz' law. So we must provide this energy. We must input energy to the system to get the magnetic domains to align or unalign on a repeated basis, and we must input energy to get the magnet to rotate on a repeated basis.
Where did the energy stored in the electric capacitance come from?The flow of energy through electric capacitive coupling through the "curl-free component" of the electric field is not subject to Lenz' law, so it may come from somewhere else other than our input, namely the bound currents in our magnet.
How is it possible that the electric field of the magnet allows us to obtain energy from its electrons?
The moving magnet produces displacement currents providing a wireless electrical "connection" from its bound currents to any unscreened external electric charges. This relies on the fact that the magnet in relative motion will produce electric field lines (or electric fluxes) which terminate on the magnet, and the electric potential produced is like that of a battery. If adopting the Coulomb gauge where the divergence of the magnetic vector potential is zero, the negative of the gradient of the electric potential is fully responsible for the "curl-free contribution" to the electric field. It takes an unconventional generator, such as one where static electricity is generated in windings, to benefit from the magnet's "curl-free contribution" to the electric field through capacitive coupling.
What other kind of electric field is there?The electric field of a moving magnet is actually the product of Special Relativity. Edward Purcell refers to the "pancaking" effect that relative motion has on the electric field of a moving charge. The total electric field of a magnet, of a charge, or of anything is the sum of a "curl-free contribution"
and "divergence-free contribution". The "curl-free contribution" cannot do work on closed currents. Since standard electric motors and generators operate on closed currents, they respond only to the "divergence-free contribution" of the electric field. The "divergence-free contribution" is responsible for changes in the magnetic field and therefore is responsible for enforcing Lenz' law which is applicable only to energy exchange through magnetic/inductive coupling.
How does this "electric field pancaking" differ from a battery?The difference is that we can control the potential difference across the moving magnet by changing the applied magnetic field, and thus we can oscillate the voltage across the moving magnet by simply oscillating the applied magnetic field. Also, no electric charge is made to flow into or out of the moving magnet, and instead the electric field of the magnet must be made to capacitively couple with external electrostatic charges, similarly to how a power line may capacitively couple to a fluorescent light without a conductive path in between.
The oscillation of the magnetic field can occur at the self-resonant frequency of a nearby stator electromagnet. Thus combining high resonant energy, high frequency, high magnetic permeance coefficient, high electric polarizability of the stator electromagnet, and high Q-factor should assist making our moving magnet, in essence, a high-energy density "quasi" battery - one which we "attach" to our load by bridging displacement currents through the air gap between the moving magnet and the stator.
