Coulomb's Law and relative motion

Question

This might be a repeated question but I couldn't find an answer already. I am told by my teacher that Coulomb's Law is valid for stationary charges. Here are the two validity criteria I am given;

1. either both the charges are at rest or

2. one of the charge is moving and other is at rest.

What if both the charges are moving with zero relative motion. I realise that there will be Lorentz force (magnetic and electrical forces) (in ground frame of reference, at least) but then they are still at rest in some other frame of reference. Is Coulomb's Law still valid (in relative FOR)? If yes, how will a person sitting "on" the charge account for Lorentz forces that an observe on ground calculates?

Answer 1

No, there are some misconceptions in your understanding.

Two charges with individual non-zero relative motion, cannot both be at rest, with respect to a definite reference frame (and this is true for any classical particle). And in that case, other EM Interactions are to be taken into account.

Answer 2

Let's say that relative observer at rest in the relative FOR thinks that there is a Coulomb force and a magnetic force, and the charges move side-by-side.

Then Coulomb law is correct.

Person sitting "on" the charge accounts for Lorentz forces that an observer on ground calculates, by saying that all the springs that the observer uses are effected by internal Lorentz forces. So all the observer's ideas about forces are wrong.