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Wikipedia (https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential) says

"The calculation is nontrivial and requires a number of steps".

Nice but a link would be good to add showing that calculation. Can anyone point me to a suitable place online or a book that contains this calculation/derivation fully?

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  • $\begingroup$ I have never seen (or done) the full calculation, but from what I gathered from Feynman's lectures (see my comment below), it is just "turning the crank", so it does not add to the understanding of these things. OTOH, Feynman uses the L-W fields when discussing EM radiation and the classical derivation of the index of refraction in Vol. I, Ch. 28-31 and I've always found those chapters fascinating. $\endgroup$
    – NickD
    Commented Apr 30, 2020 at 17:56
  • $\begingroup$ Why didn't you look first here? en.wikipedia.org/wiki/Jefimenko%27s_equations $\endgroup$
    – DanielC
    Commented Apr 30, 2020 at 18:45
  • $\begingroup$ @DanielC Short but interesting read. I concur with the statement "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. ...." and the discussion of causality is interesting, though not the way I take it. I would say the induced electric field component - the change in vector potential in time - is the momentum change caused by the other fields of the observer charge. This is akin to the F=ma equation. $\endgroup$
    – Lamaan
    Commented Apr 30, 2020 at 23:30
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    $\begingroup$ I wrote an undergraduate textbook on Special Relativity, called Relativity made Relatively Easy (the name because it was precisely my aim to simplify derivations such as this where possible); pub OUP. The potentials are derived in the main text, and the derivation of the fields is set out in full in an appendix. $\endgroup$
    – Andrew Steane
    Commented Nov 10, 2021 at 14:16
  • $\begingroup$ Trying to prove the Heaviside-Feynman formula : Deriving Heaviside-Feynman formula for the electric field of an arbitrarily moving charge from Lienard-Wiechert potential. $\endgroup$
    – Voulkos
    Commented Nov 10, 2021 at 14:42

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I would recommend Introduction to Electrodynamics by D. J. Griffiths. He works through the derivation of the Liénard-Wiechert potentials using diagrams and clear geometric arguments. The book should be very accessible provided you have taken an elementary course in electrodynamics and special relativity.

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  • $\begingroup$ The OP is asking for a derivation of the fields, not the potentials. Feynman's lectures derive the potentials but leave the calculation of the fields to the interested reader: "To close the ring back to Eq. (21.1) it is only necessary to compute E and B from these potentials ... It is now only arithmetic. The arithmetic, however, is fairly involved, so we will not write out the details." Then he adds some hints in a footnote: "If you have a lot of paper and time you can try to work it through yourself. We would, then, ..." $\endgroup$
    – NickD
    Commented Apr 30, 2020 at 17:46
  • $\begingroup$ "... make two suggestions: First, don’t forget that the derivatives of r′ are complicated, since it is a function of t′. Second, don’t try to derive (21.1), but carry out all of the derivatives in it, and then compare what you get with the E obtained from the potentials (21.33) and (21.34)." $\endgroup$
    – NickD
    Commented Apr 30, 2020 at 17:47

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