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I found the following formula for the electric field produced by an arbitrarily moving charge in Feynman lectures (Vol. 1 eqn (28.3) and Vol 2 eqn (21.1))

\begin{equation} \vec{{\bf E}}=\frac{q}{4 \pi \varepsilon_0}[\frac{\bf e_{r^{'}}}{r^{'2}} + \frac{r^{'}}{c}\frac{d}{dt}(\frac{{\bf e}_{r^{'}}}{r^{'2}})+\frac{1}{c^2}\frac{d^2}{dt^2}{\bf e}_{r^{'}}] \end{equation}

The first part is the well-known coulomb's law in retarded form.

Is there a way to derive the second and third term from Maxwell's equations?

I know the third term is related to the acceleration of the charges and gives rise to radiation. What is the role of the second term of the equation? Are there names for second and third parts of the equation? Is there another reference giving these terms?

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    $\begingroup$ Well if you want the whole derivation you can find it in chapter 6 of Jackson's book, Classical Electrodynamics, 3ed. That equation you have there is 6.60 in the chapter and Maxwell's equations are equation 6.1 so you better be prepared for the ride. I'm assuming there is some odd convention explaining the minus sign in front of your equation. $\endgroup$
    – secavara
    Commented Feb 21, 2018 at 15:38
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    $\begingroup$ You might also check up on the Jefimenko equations, for example see en.wikipedia.org/wiki/Jefimenko%27s_equations $\endgroup$
    – jim
    Commented Feb 21, 2018 at 18:05
  • $\begingroup$ This is known as the Heaviside-Feynman formula, if you search under that name you should find some literature on the subject. $\endgroup$
    – Ruvi Lecamwasam
    Commented Mar 10, 2019 at 23:26

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