A particle, a neutral wire and two relative inertial reference frame

Case 1 ($O_1$):

We have wire which has current $I$, in this condition the wire is neutral, and we have a charged particle $+q$

I am put them as following.

There will be no attraction since wire is neutral and particle is not moving relative to wire. (and for simplicity we can say wire and $+q$ is stationary to the ground reference system (let say $O_1$))

Case 2($O_2$): Now we are taking this system inside a train, and the train then moves with constant velocity $v$. $O_2$(observer 2) sees this motion outside the train stationary relatively to the ground.

$O_2$ observes that there must be magnetic force between wire and particle $+q$ since particle is moving and there is current in the wire.

Since there is nothing different for $O_1$, there must be no net force between wire and particle, and since $O_2$ outside train sees magnetic force there must be a force equalize the magnetic force for $O_2$.

Question: I do not understand what this equalizer force should be and why this occurs? I thought it must be electric force, but how electric force occurs although it was a neutral wire. In addition, the wire and the whole train will contracted due to Lorentz Transformations. However, how this contraction causes a breaking of a symmetry in current density (or something else related to it because I do not know ).

• Write down the effect of a Lorentz transformation on a current and you will see. May 6, 2018 at 14:07

There will not be any force on the q with respect to O2 also, because O2 will observe both magnetic and electric forces on the charge which cancel each other.

This is what you might be looking for : What would happen to a charge in uniform magetic field when observed from a moving reference frame?

According to this source