# Charged particle moving relativistically through E field

## Main Question or Discussion Point

Right now I'm taking an introductory E&M class that uses Purcell's book "Electricity and Magnetism." The chapter we're covering focuses on deriving the Lorentz force law for moving charges using SR arguments. This is confusing because we've never covered SR before, so I have a lot of difficulty going through the appropriate transformations and understanding what's going on. One problem that is troubling me a lot goes like this:

Consider the trajectory of a charged particle moving with a speed 0.8c in the x direction when it enters a large region in which there is a uniform electric field in the y direction. Show that the x velocity of the particle must actually decrease. What about the x component of momentum?

This is totally counterintuitive to me, and if someone could explain it for me I would really appreciate it.

Related Special and General Relativity News on Phys.org
pervect
Staff Emeritus
I think I just figured it out. The quantity $$c^2p^2 - E^2$$ is not changed by the Lorentz transformation, so it seems like a good equation to start from. Since the electric field does work on the particle the energy term gets bigger, which means the momentum term must get smaller. Because the particle gains y-velocity, the x-velocity must decrease if the momentum is going to decrease. Does this sound reasonable?