Which equations indeed describe moving/accelerated charge field? The thing that I have not understood completely, for 100%, is how does the field of the electric charge is "updated".

*

*If to consider a frame of reference where the charge is static, does not move at all, it is simple - the value and the direction of field at point can be calculated from Coulomb's force:
$$F = k\frac{q_1  q_2 }{r^2}$$
The important thing is that the value at some point is constant in time (for static charge).


*When the charge is moving with constant speed, as far, as I understand, because of space contraction, the field is flattened in the direction of the electron move. The results of that contraction in frame of reference are called magnetic field.
What I do not understand at this point is how does the field of constantly moving charge depends on time?
If to consider an accelerating charge, the information about the field is propagates with the speed of light, but what about a constant moving charge?
Say, there is a constantly moving charge and an observer, situated on 300 millions meters from the charge. Will the field at far point be the same as it should be from Coulomb's law (corrected for Lorentz boost), or when the charge will left some point, the far observer still will be "seeing" it there? Is there a "delay" as for accelerated charge? If yes, why?


*What I do not understand the most in accelerated charge field is why the initial field is "retarded", so it is need a time for new field propagation to reach the observer, but the new "new" field of now already constantly moving charge is changes instantly? Why? Because of what? Or it is not? Maybe even for constant moving charge there is delay, and it has that perpendicular retarded component, but it is tiny?

For example this famous animation, from which equations it was build?
I also intuitively understand but at the same time do not understand why lines at propagation are perpendicular to initial field?

I will read an entire book, if it will have the answers, but at least point me which one.
 A: In the electrostatic case you have $\boldsymbol{E}\equiv\boldsymbol{E}(\boldsymbol{r})$, but for isotropy actually $\boldsymbol{E}\equiv\boldsymbol{E}(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}|)$, where $\boldsymbol{r}_{\text{charge}}$ is the position of the point-charge.
Suppose that $c\to\infty$, so if your charge is moving with velocity $\boldsymbol{v}_{\text{charge}}$ you will have
$$
\boldsymbol{E}\equiv\boldsymbol{E}\left(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}|\right)
$$
at some instant, and you will have
$$
\boldsymbol{E}\equiv\boldsymbol{E}\left(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}-t\,\boldsymbol{v}_{\text{charge}}|\right)
$$
some instant later. The field would depend on $t$, but it's still an electrostatic one and points towards the actual position of the charge.
Suppose now that $c$ is finite, you will have
$$
\boldsymbol{E}\equiv\boldsymbol{E}\left(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}|\right)
$$
at some instant, but now you have to confront $|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}-t\,\boldsymbol{v}_{\text{charge}}|$ with the distance $tc$. Turns out is not very easy, but at least you could say that, when the field is "updated" in the point $\boldsymbol{r}$, is updated with an "electrostatic field of the past" and points to the direction the charge was in this past, not the direction the charge is now.
Given this different direction you can understand why the field lines get deformed, and this exactly causes the magnetic field to be. In fact magnetic field exists just by the fact $c$ is finite. In a way magnetism exists because special relativity does and is in fact a relativistic phenomenon!
About the third question, I really don't know how to answer that without citing Liénard-Wiechert potential and the associated theory, that you can find in some books like Jackson for example. Hope this helps.
