When electron accelerates, there occurs a propagated ripple on it's electric field. But when it moves constantly, does the field "follow it", i.e. changes instantly? How does it deals with the fact that nothing can travel faster than speed of light?
Answer
Yes, in a sense, the field "instantly" moves together with it's source (if this source moves uniformly).
There is no aberration of forces. For example, an illuminated charged sphere and you are both approaching each other in uniform inertial motion along paths that do not collide. Light coming from the sphere will appear approaching you at relativistic aberration angle $\sin\alpha = v/c$. However, the electromagnetic force of attraction to the sphere does not experience aberration. It points directly toward the actual position.
So, theoretically you can always know the actual position of the charge and follow it.
It is very simple to understand it by swapping frames. Just think about a charge "at rest" , which disseminates the field and an observer (a test particle), who moves in this field. It makes clear, that the direction of the electrical force exerted by the sphere on the test particle points directly toward the actual position of the sphere.
That does not mean that the force propagates infinitely fast. The force on a test particle at any given instant is due to the electromagnetic field in the immediate vicinity of the particle at that instant.
Aberration of Forces and Waves
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