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If one goes to the electron's instantaneous rest frame (which is not
inertial) the magnetic field does not affect the electron. Imagining
the field to be moving clearly misleads one in that case. In that
frame, however, there exists an electric field. It is that electric
field which is chiefly responsible for the deflection of the electron.
If one looks instead at the inertial frame in which the electron is
instantaneously stationary then the motion can be entirely understood
as a consequence of that electric field. At that instant of time the
magnetic field does not exert a force, and it is constant and uniform.
The electron itself is accelerated, but like a ball thrown upward it
is stationary at the instant in question, but is still accelerated.
It is possible (and feasible) to analyze the system completely by
considering the effects of sources (charges and currents) which act
on the electron, and all the fields produced will be static. I think
that, contrary to what Michael Edmiston suggests, the system is quite
understandable in this way. I would generate my uniform magnetic
field with a hypothetical infinite current sheet rather than a
hypothetical cyclotron, however.