How is this classical "paradox" resolved in electromagnetism? A magnet and a coil move relative to each other. In the frame of reference of the magnet, there is a magnetic field and consequently a force acting on the charges in the coil according to the Lorentz force $F=qv\times B$ but there is no net electric field. In the frame of reference of the coil, there is a magnetic field and also an electric field, induced by the magnet, $E'$ that moves the charges in the coil, producing a current. But, in the first case no work is done on the charges, since the force is perpendicular to the velocity. In the second case, the force $qE'$ does work on the charges. How is this "paradox" resolved in classical electromagnetism?
 A: When the magnet is moving, the electric field of the magnet is doing the work, pushing the current carriers around the wire. When the magnet is still and the wire is moving, the magnetic field produces a force in the current carriers, but this force does no work, it is the constraint force that keeps the electrons in the wire that is doing the work. The paradox is resolved by noting the the wire is moving, so the constraint is not time-independent.
The constraint force is perpendicular to the surface of the wire pushing on the charge carriers in the direction of motion (because the whole thing is moving). This force is doing the work on the charge carriers in this frame (although it is somewhat strange to think of a constraint force doing work).
The push of the current carriers against the wire's constraint force gives the breaking force on the wire, which slows it down so as to conserve energy, as the resistance gives off heat.
A: There is no paradox: the two reference frames have different answers to the question "how much work is being done", indeed. It's because "work being done" is nothing else than energy and energy isn't a relativistic invariant; it is the time component of a 4-vector. According to relativity, various quantities are observer-dependent i.e. relative – a justification of the name "relativity" – and energy is one of them.
From the magnet's viewpoint, the electrons' energy may be conserved but the momentum is not. Because the energy from the coil's perspective is a mixture of the energy and the momentum in the magnet's reference frame, and because the momentum is being changed from the magnet's viewpoint, it follows that energy of the electron is being changed from the coil's viewpoint.
With this being said, one should still emphasize that the increased/decreased energies of individual electrons get averaged out around the coil.
A: I'm not quite sure about this one, but I would explain it this way: (let $\mathfrak{M}$ the frame of reference of the magnet, $\mathfrak{C}$ 〃〃〃〃〃 the coil)


*

*No work is done initially in $\mathfrak{C}$, either: the electrons are at rest in the beginning. Only once the force has started built up some speed there is a finite power.

*In $\mathfrak{M}$, the electrons also build up velocity. Even in the parts of the wire which enter the field perpendicularly (sort of), so that the electrons are initially deflected in the direction of the wire, the force will eventually (when the electrons move in the wire) point in a direction other than the wire alignment. Once this happens, only the force component in wire direction will take action, so the effective force direction is not perpendicular to the electrons' moving direction any more. To put it another way: when the electrons are pushed to one side of the wire, $\rho$ becomes macroscopically nontrivial, resulting in an electric field due to $\nabla\mathbf{D}\propto\rho$.

A: "There is no work because the force is perpendicular to the velocity". I don't think that's right. The v in your formula qv x B is the velocity of the coil, but it is not the velocity of the electrons. The electrons move inside the coil, and they move in the direction that the force is pushing them: perpendicular to the velocity of the coil. So how is there not a force pushing the electrons?
EDIT: In response to Ron's comment about the constraint force, I've worked it out and I think the following picture will show what is happening:
The wire is treated as a pipe full of electrons, and the blue electron shown is actually right against the pipe wall: as the pipe moves upwards, the electron rides along the pipe wall as shown by the blue arrow. The two forces are the vxB force, perpendicular to the electron's true motion, and the constraint force, perpendicular to the wall of the pipe. I think the picture shows that the actual work is done by that COMPONENT of the vxB force which is in line with that COMPONENT of the electron's velocity which is along the direction of the pipe.
