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  • $\begingroup$ Thanks. Could you provide reference for the fact that Maxwell's equations are not only valid for accelerated test but also source charges? And in the paper I linked to, Maxwell's equations are derived from Coulomb's law and the formalism of special relativity and therefore neither is limited to electrostatics nor is it derived from the action principle. Did you read it? $\endgroup$ Commented Nov 28, 2017 at 6:59
  • $\begingroup$ I do not think reference is appropriate, the validity of Maxwell's equations for general motion of charges is an accepted fact, based on 150 years of their use in ever more complex situations - AC power generation and transmission, radio antennas, accelerators and others are all about accelerated charges and are commonly analyzed with help of Maxwell's equations. No evidence of a problem with them was ever found, as far as I know. $\endgroup$ Commented Nov 29, 2017 at 2:50
  • $\begingroup$ As to derivations, the common ones assume there is an inertial frame where the field is electrostatic everywhere, then look at things from different reference frames, so they can define magnetic field and derive that the fields obey Maxwell's equations. This is discovering form of Maxwell's equations as a mathematical object, it is not deriving validity of Maxwell's equations for general situations, where there is no inertial frame in which the field is electrostatic. $\endgroup$ Commented Nov 29, 2017 at 2:55
  • $\begingroup$ Thanks, I'll think about it, do some additional search and will get back to you in a couple of days. Btw, it was not me who downvoted your answer. $\endgroup$ Commented Nov 30, 2017 at 6:19
  • $\begingroup$ However, as a first response to the comments: Yes, Maxwell's equations are extremely successful at explaining an incredible range of phenomena, however, phenomena like transmission and movement of electrons in antennas, accelerators etc that you named describe accelerated test charges, i.e. charges that react to the force of fields from sources that need not be accelerated, and that is a difference to describing effects where you include acceleration of these sources as well - for example, for many applications one must add a radiative reaction, or self-force to the Lorentz-force. $\endgroup$ Commented Nov 30, 2017 at 6:27