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The paper Direct derivation of Lienard–Wiechert potentials, Maxwell’s equations and Lorentz force from Coulomb’s law by Hrvoje Dodig, purports to derive electrodynamics from Coulomb's law, without regard to special relativity, including the Lorentz force. This he does by, first, generalizing the Helmholtz decomposition:

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This seems to fly in the face of explanations of the Lorentz Force as deriving from Lorentz contraction of charged particles in motion resulting in apparent changes of density of electric charge. Does this derivation, indeed, cover the observed phenomena?

Urb
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James Bowery
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2 Answers2

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You do not need special relativity to derive electrodynamics or the lorentz force law. special relativity is derived FROM electrodynamics( ish)

jensen paull
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    Disagree. It is possible to reconstruct the Lorenz force law from special relativity given Coulomb’s law. – QuantumDot Dec 24 '21 at 02:30
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    Then its possible to derive special relativity purely from the lorentz force and coulombs law? Even without knowledge from the 3 other maxwell equations? – jensen paull Dec 24 '21 at 02:36
  • I do not think it is appropriate to say Special Relativity is derived from Electrodynamics. SR assumes that all of Physics is the same in every inertial reference frame, and Electrodynamics does not include this assumption. Notice that aether was ruled out experimentally, for example. ED certainly can be said to suggest SR, but to say it logically implies SR seems to be pushing a little too far – Níckolas Alves Dec 24 '21 at 03:04
  • Also, if one assumes the Coulomb Law and SR, E&M pretty much follows as a consequence in the sense explained in this question. I believe this is what OP was referring to originally – Níckolas Alves Dec 24 '21 at 03:12
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The whole point of that paper is not to use special relativity to derive electrodynamics from Coulomb's law. It is understood that in that paper Coulomb's law is considered an experiment (the result of the measurement), and it is remarkable that using only that information one can derive all of the electrodynamics.

Regarding the use of SR to derive electrodynamics, one should be aware that SR is valid only for objects moving with constant velocity along a straight line. That being said, the results in that paper are more general than the ones obtained by combining SR and Coulomb's law.

In conclusion, SR has nothing to do with electrodynamics.

If one is still not convinced that electrodynamics has nothing to do with SR, try to derive electromagnetic fields of the charge moving along a circular path using SR. The result would be very different than one obtained by using Lienard-Wiechert potentials.

gallieo1985
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  • -1. The phrase "SR has nothing to do with electrodynamics" is blatantly wrong. It doesn't make sense to speak of time or spatial derivatives (which E&M definitely uses) without specifying a spacetime. If you try Galilean spacetime, you'll get to contradictions. You need SR (or GR) to do E&M. Even if the method of derivation of charges along circular paths won't work with the SR trick, it is incorrect to try to disassociate E&M and SR. Furthermore, SR is not valid only for objects in uniform motion: it holds for any relativistic motion, but the math gets more complicated. The specific + – Níckolas Alves Aug 02 '22 at 13:55
  • derivation of Maxwell's equations from Coulomb's law will fail because for non-uniform motion the reference frame is no longer inertial, which means simply that Coulomb's law doesn't apply in the charge's reference frame (in fact, Maxwell's equations in their usual vector Calculus expressions do not apply: they assume an inertial frame implicitly). – Níckolas Alves Aug 02 '22 at 13:57
  • Dear Nickolas, the paper contradicts you. In that paper there was no mention of space-time, Lorentz transformation, length contraction, etc...LW potentials for arbitrary motion were derived from Coulomb law directly without using any of SR concepts. You cannot do this using SR and Coulomb law except for uniform motion along straight line. However, you can derive Lorentz transformation from LW potentials for a charge moving uniformly along a straight line (look at Feynmann proof). In conclusion it seems that SR is only a special case for uniform motion along a straight line – gallieo1985 Aug 04 '22 at 10:24
  • I apologize and retract my comments on the specific derivation, for I had not checked the paper. There is a common derivation of Maxwell's equations from Coulomb's Law (in the form of Gauss's Law) + SR, which is the one I assumed the author did, but it seems he attempted at a different one. Still, my comments on the fact that E&M and SR go together as a matter of consistency remains (regardless of that paper) – Níckolas Alves Aug 04 '22 at 16:29