I've already got the electric fields and magnetic fields derived from the Lienard-Wiechert potentials:

$${\bf E}=\frac{q}{4\pi\epsilon_0}\frac{R}{(\bf R\cdot u)^3}[(c^2-v^2){\bf u}+\bf R\times(u\times a)]$$

$${\bf B}=\frac{\bf R}{cR}\times\bf E$$

where ${\bf R=r-r'}$ and ${\bf u}=\frac{c\bf R}{R}-\bf v$.

I wonder if they satisfy Maxwell's equations, I've tried to derive Gauss's law, but in vain. So do they? Or is there something wrong in my derivation?

  • 1
    $\begingroup$ The first term of $\mathbf{E}$ is not a vector. $\endgroup$ – SuperCiocia Jun 28 '20 at 3:31
  • $\begingroup$ Thank you! I've rectified it. $\endgroup$ – Soluty Jun 28 '20 at 12:15
  • $\begingroup$ Isn't $u$ a vector? So isn't the first term of $E$ a vector? $\endgroup$ – J Thomas Jun 29 '20 at 17:32

They are derived from Maxwell's equations, so they satisfy Maxwell's equations, but taking vector derivatives is very complicated with retardation.


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