Do charged particles in Uniform Rectilinear Motion also give rise to magnetic fields in respect to stationary references? From Wiki, I already know that EM waves are generated whenever charged particles are accelerated. Also, accelerated charges (as in electric current) give rise to magnetic fields near them. My question is, if this charged particle IS MOVING in respect to a stationary person, but not accelerated under any force, thus basically in uniform rectilinear motion (resultant force equal to zero) do they still give rise to a magnetic field in respect to said stationary observer?
I think its a simple enough question. English is not my first language so sorry for any weirdly constructed sentences.
 A: Potentials for arbitrary motion of a test particle have been studied, and they answer to the name of Liénard-Wiechert potentials. Basically, what they did was to take the Maxwell equations and solve them for a distribution that was corresponding to a point charge.
You can find more on Wikipedia, https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential, or on any textbook on electromagnetism (they usually are calculated after introducing relativity). I refer to the Wikipedia page for the formulas under "Definition of Liénard–Wiechert potentials". They are what you are looking for: there the potentials are calculated for a particle in arbitrary position $\vec r_s$ and with arbitrary, constant velocity $\vec \beta_s$. Look out that you have to evaluate at retarded time, that takes relativistic effects into account! It is defined as (it is not in that section of Wikpiedia)
$$
t_r=t-\frac{|\vec r-\vec r_s|}{c},
$$
where $\vec r$ is the position in which you are evaluating.
You see that the vector potential has a leading order that is proportional to velocity. That means that, when velocity goes to zero, the vector potential vanishes, and you recover the electrostatic potential.
