If a charge $q$ is placed at rest at a point $P$ near a wire carrying current $I$, with some distance $x$, it experiences almost no force. However, if the charge $q$ is projected from the point $P$ in the direction of the current, it deflects towards the wire.
Now my question begins. Let us consider this event in front of two observers. One is moving with the velocity $v$ which is also the velocity of the charged body. For the observer who is at rest for him it is a pure magnetic field, and for the moving observer it is an a electric field. However both observers agree with horizontal distance traveled by a charged body as they are unaccelerated with respect to each other. I wonder why can’t the force acting on the charged body effect its vertical velocity.
Please try to understand me this entire process with the help of special relativity. How can’t the vertical velocity of the charged body have no effect.
I am pretty new to this topic. Please correct me, if am wrong completely wrong about this question.