[sigmoid10]

You can plug arbitrary values in, but you can not expect to gain any valid predictions or reasonable physical insight from Coulomb's law as soon as you are no longer dealing with static point charges. That's because B and E are not independent quantities but actually closely intertwined components of the electromagnetic field strength Tensor F. As soon as you start dealing with motion, these components will mix, preserving only certain quantities like the tensor contraction E^2-B^2. So even if you construct a case where B=0 at time t=0, that will no longer be true once you had any acceleration of your charge carriers.

In the fundamental quantum field theory picture you don't even have forces and particles in the original sense anymore. The dynamics are then described by interaction between the em field and charged fermionic fields. Stuff like Coulomb's law (or any other force potential) only emerges as a macroscopic low energy approximation for specific field configurations.

[greysphere]

In a classical view of 2 particles accelerating towards each other v x r will always be 0 so B will always be 0 even if the particles are accelerating towards each other. I believe all this holds under QFT [1].

Looking further a redefinition E is necessary when including the \beta factor [2]. So that was a mistake on my part - relativity does change the rhs of Coulomb's law.

Admittedly the problem as stated (two particles falling towards each other) constrains things in such a way that there is no off-axis contribution. Or to put it another way, 1d electromagnetism doesn't have magnetism.

[1] https://physics.stackexchange.com/questions/142159/deriving-...

[2] https://en.wikipedia.org/wiki/Coulomb%27s_law#In_relativity

[sigmoid10]

>I believe all this holds under QFT

It doesn't even hold in ordinary QM (because there are no point charges anymore, only charge densities) and it fails completely in QFT for interacting fields. What you linked is an intro example from many textbooks that shows how the tree level diagram in the nonrelativistic limit can indeed yield the Coulomb potential. At the tree level you will often see such "classical" behaviour. But if you consider higher order corrections, the picture changes rapidly. See the Uhling potential for a practical example, but for higher loops this gets analytically intractable very quickly. The world starts to work very differently once you reach these length scales.

[pdonis]

> In a classical view of 2 particles accelerating towards each other v x r will always be 0

Only if the impact parameter is exactly zero. But that is an extremely unlikely edge case.