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When there is a charge moving at a constant speed $\vec{u}$, the induced electric field $\vec{E}$ and magnetic field $\vec{B}$ would be:

$$\beta=\frac{u}{c}\tag{01}$$

$$\vec{E}=\frac{q}{4\pi\epsilon r^2} \frac{1−\beta^2}{\left(1−\beta^2\sin^2\theta\right)^\frac32}\hat{r}\tag{02}$$

$$\vec{B}=\frac{\vec{u}\times\vec{E}}{c^2}\tag{03}$$

How to derive them?

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    $\begingroup$ Like this? $\endgroup$
    – secavara
    Commented Jun 26, 2022 at 8:49
  • $\begingroup$ @secavara: Looks like that, but it goes beyond what I can understand. $\endgroup$
    – user1058602
    Commented Jun 26, 2022 at 11:48
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    $\begingroup$ Alternatively, you can start with a frame in which the charge is static (where you know how to compute the fields) and then move with a constant relative velocity with respect to the charge. The fields in the moving frame can be computed using the transformation rules for boosts (see here). This approach is taken in a few references, the one I remember is Electrodynamics, by Harald JW Muller-kirsten, in section 17.6, "Transformation from Rest Frame to Inertial Frame". $\endgroup$
    – secavara
    Commented Jun 26, 2022 at 17:02
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    $\begingroup$ Take a look in my answer here Charges and relative motion to "...compute using the transformation rules for boosts..." as in the comment by @secavara. $\endgroup$
    – Voulkos
    Commented Jun 27, 2022 at 2:41
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    $\begingroup$ @Frobenius: Thanks for your answers!! $\endgroup$
    – user1058602
    Commented Jun 28, 2022 at 16:53

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