PhilDSP said:
And I have to think about the apparent disconnect between classical EM and the relativistic variant that lies in the effective redefinition of very many values such as E and B ;-) Does the Minkowski EM tensor preclude you from having separate fields where one (the former B field) deals only with the rotational aspects of EM and the other only the linear aspects of EM? Does the stepping from the transformational matrices of the Lorentz and Poincare groups into the tensor formulation force a reinterpretation of the distinction between rotation, boost and translation operations? (But this is certainly off-topic for this thread)
I think your reply is relevant to the OP's question. Many posts here make a distinction between electric fields and magnetic fields.
The controversy seems to concern the "fact" that a magnetic field can't do work on an electric charge, while an electric field can do work on an electric charge. However, relativity shows that the "percentage" of electric field energy and magnetic field energy varies with the observer. So your question is really whether the "fact" that a magnetic field can't do work on an electric charge is consistent with relativity. I would answer "yes", but I have to explain why.
The "fact" that a magnetic field can't do work on a charge is literally true only for one observer in an inertial frame (special relativity) or in a geodesic frame (general relativity). The OP wasn't asking about an accelerating frame or a frame that wasn't in fall. Relativistic transforms describe how different observers interpret the same series of events.
In anyone inertial frame, there is a big distinction between electric fields and magnetic fields. The diagrams shown by the OP and others were for an inertial frame by default. There was no description of a gravitational field, or accelerating magnets, or anything that implies more than one inertial frame. Because there is only one inertial frame presented in the problem, one must respect the sharp distinction between electric fields and magnetic fields.
The Lorentz force law is precisely satisfied only in an inertial frame. The fraction of force that is electric or magnetic may vary with different inertial frames. However, the Lorentz force law is precise and accurate in an inertial frame. The Lorentz force law and the work-energy theorem imply that the magnetic field can't be doing work on a free electric charge.
The questions which this thread started with really amount to asking whether the Lorentz force law is ever inaccurate in an inertial frame. I interpreted the OP's question about "classical electrodynamics" as being the self consistency of the Lorentz force law in an inertial frame. If there is any other interpretation, then maybe someone should explain it to me.
Analysis has shown that the Lorentz force law is consistent with relativity given relativistic modifications in mass. Therefore, a magnetic field can not do work on an electric charge in an inertial frame. If a magnetic field could do work on an electric field, the Lorentz force law would have to be modified.
Relativity does not say what the direction of the electric current is. The direction of the electric current is actually determined by constitutive relationships between the fields. Relativity does put constraints on what the constitutive relationships can be. However, changing the constitutive relationships can not change the basic fact. A magnetic field can not do work on an electric charge.
The key to this conundrum may be the direction of the electric current. I do believe the OP had a valid point many posts ago. A magnetic field can change the direction of an electric current without doing work. An electric field pointing in the direction of the new electric current can do work on the electric charges in this current. So a magnetic field can "enable" an electric field to do work. That is very different from the magnetic field doing work.