Consider the trajectory of a charged particle that is 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

Explanation:

The electric field applies a force qE, but only in the y direction. This means that the x component of the

momentum must be conserved:

0 = Fx =dpx/dt

=⇒ px is conserved

pxi = γimvxi = pxf = γfmvxf (see the attactments)

the particle will acquire a velocity in the y

direction, and vxf <vxi. The x-component of the velocity, and thus the x-component of the momentum

must decrease.

The quantity is not changed by the Lorentz transformation.

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.