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The first step here is to draw a diagram - draw the velocity vector for
Here's a very elementary example that I find helpful.
Suppose you have a hollow tube oriented along the y-axis. Inside the
tube is a small marble that can slide without friction. Neglect
gravity. The marble carries a net positive charge and there is a
uniform B field in the negative z direction.
Imagine you apply an external force to the tube so that the tube slides
with a constant speed in the positive x direction with the tube always
remaining oriented parallel to the y-axis. Initially, let the marble be
at the "lower" end of the tube with zero y-component of velocity. The
marble will slide up the tube with a nonzero y-component of acceleration
as the tube slides to the right. When the marble reaches the top of the
tube it will have gained KE.
Consider the work done by the various forces in the reference frame of
the fixed x-y-z coordinate system.
First take the marble alone as the "system" of interest so that the tube
is part of the "environment". There are only two forces that act on the
marble: the magnetic force and the normal force from the wall of the
tube. The magnetic force does zero work on the marble and the normal
force does an amount of work equal to the gain in KE of the marble.
This last sentence does not parse - the marble has gained KE in the y
direction, but the normal force acts only in the x direction. See below.
From the velocity triangle, sub in vy/v for sin(theta) and vtube/v forcos(theta):