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Haskell derives in his great work Maxwell's equations from Coulomb's law and the formalism of special relativity: http://richardhaskell.com/files/Special%20Relativity%20and%20Maxwells%20Equations.pdf

(And by doing so answers this question.)

Intuitively, this may be understood as follows: If we have a number of reference frames in which the respective source charges are at rest, then these reference frames may move with different relative velocities with respect to another frame and thus the charges appear to fly by that frame with a constant velocity creating the effects described by the dynamic laws of electrodynamics.

In particular, if there happens to be a static electric field $E_k'$ in a frame moving with relative velocity $u_i=ua_i$, where $a_i$ are the components of the unit vector of the velocity, then the magnetic field in the frame in which the charges move by with constant speed is given by $B_i:= \gamma u/c^2 \epsilon_{ijk}a_jE_k'$ where $\gamma=1/\sqrt{1-u^2/c^2}$. This is a definition of the magnetic field in terms of the static electric field and the relative velocity to its corresponding frame.

However, even though Maxwell's equations come out if one defines the magnetic field like this, I wonder whether this is the most general form that a magnetic field can have. What happens if the source charges are accelerated? If they are accelerated by gravity, then one can use Maxwell's equations in curved spacetime. But what if the acceleration happens due to electromagnetic forces? Then a Lorentz transformation which always only involves constant relative velocities can not account for describing the resulting magnetic field of this accelerated charge. Consequently, the magnetic field probably can not be defined as above. Would Maxwell's equations nevertheless be valid?

If not, the question is how Maxwell's equations would have to be changed in order to describe accelerated source charges (note that the usual Maxwell theory has no problems with describing accelerated test charges which amounts e.g. to the idealised concept of an accelerated charge in an electric or magnetic field created by non-accelerated source charges, etc).

Haskell also discusses this question at the end of the document and considers the possibility that the amendment could consist of a non-linear power-series but he does not come to a definite conclusion.

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What happens if the source charges are accelerated? If they are accelerated by gravity, then one can use [Maxwell's equations in curved spacetime][2]. But what if the acceleration happens due to electromagnetic forces? Then a Lorentz transformation which always only involves constant relative velocities can not account for describing the resulting magnetic field of this accelerated charge. Consequently, the magnetic field probably can not be defined as above. Would Maxwell's equations nevertheless be valid?

According to current knowledge, yes Maxwell's equations are valid even for accelerated charges, provided the coordinates used in them are from an inertial reference frame.

And the EM theory based on Maxwell's equations is routinely used for systems where charges accelerate. It allows for generalized theorem of conservation of energy. There would not be any conversion between matter energy and EM energy in theory if it could not describe accelerated charges.

Maxwell's equations are natural laws inferred from experiments, they cannot be derived from something simpler or more general. The common "derivations" are either limited to electrostatics or use some other assumptions, equivalent to Maxwell's equations (e.g. action principle).

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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? $\endgroup$ – exchange Nov 28 '17 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. $\endgroup$ – Ján Lalinský Nov 29 '17 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. $\endgroup$ – Ján Lalinský Nov 29 '17 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. $\endgroup$ – exchange Nov 30 '17 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. $\endgroup$ – exchange Nov 30 '17 at 6:27

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