If you recall your studies of uniform circular motion and centripetal forces, this may all
sound familiar. A constant force that points toward the center causes uniform circular
motion. In other words, the force exerted by the magnetic field is a centripetal force.
An interesting aspect of this kind of motion is that the field does no work on the particle.
One way to conclude this is by noting that the particle’s energy does not change. Its
speed is constant (which means its KE is constant) and its PE does not change in this
uniform field. Since there is no change in energy, no work occurs. This is quite distinct
from the situation of a charged particle in an electric field: Electric fields can do work on
charged particles.
Now we summarize the equations on the right. In Equation 1, we take the centripetal
force in uniform circular motion to be the magnetic force, and use Newton’s second law
to state that this must equal the mass of the particle times its centripetal acceleration
v^2 /r. We then solve for the radius r.
We can state other equations that further describe the motion of the particle. First, we
calculate the period T of the particle’s motion: The result is shown in Equation 2. We
derive this equation by starting with the equation for the period of an object in circular
motion (the circumference divided by the speed). Then we substitute the formula for the
speed of the particle in the magnetic field (obtained by solving the radius equation in
Equation 1 for v).
Notice the interesting fact that the period is not a function of the speed, but only of the
mass and charge of the particle, as well as the strength of the magnetic field. A faster
moving particle moves in a circle of greater circumference, but the period does not
change.We have also included in Equation 3 the equations for the frequency f and the angular
frequency Ȧ of the particle’s motion. These can be derived using the relationship of
period and frequency, and of frequency and angular frequency.
Without discussing it further in this section, we do note that if the velocity of the charged
particle is not perpendicular to the field, but has a component parallel to the field, then it
will move in helical motion. If you would like to observe helical motion, go to the
simulation in the introduction to this chapter, move the particle’s velocity vector to a
direction that is not perpendicular to the field, press GO, and use the viewing angle
slider to watch the resulting motion from various vantage points.·Circular motion results
Radius
|q|vB = ma = m(v^2 /r), so
r = radius, m = mass
v = speed, q = charge
B = magnetic field strength
Period
T = period
Frequency, angular frequency
(^516) Copyright 2000-2007 Kinetic Books Co. Chapter 28