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I have a particle with charge q and mass m with a relativistic velocity v·u_x in a region of electric field E = E_0 cos (wt)·u_z. I want to calculate the time evolution of the velocity.

I would use the 2nd law of Newton, so that dp / dt = F

where p = gamma·m·v·u_x and F = q (E + v·u_x ^ B).

The thing is that as the electric field is variable, it is assumed that there will be an induced magnetic field acting on the particle (or not?). The first thing is that I don't know if I have to consider a field B acting on the particle and the second is that, in that case, when I try to calculate B, I use Maxwell's equations but I get results that do not fit.

So is there only electric force or no electric and magnetic force?

Thank you.