200 Week 6: Moving Charges and Magnetic Force
The forceF~acting on this charge is:F~=q(~v 0 ×B~ 0 ) (409)which has magnitude
F=qv 0 B 0 (410)
and which acts so that it isalways perpendicularto the velocity of the particle! If you think back
to your studies ofcircular motion, you should be able to easily see that this sort of force:
- Does no work. This in turn means that thespeedof the particle is unchanged by the magnetic
field. - Acts to bend the particle’s trajectory into a constant speedcircle, with the magnetic field
providing the necessary centripetal force.
That is:
Fr=qv 0 B 0 =mv^20
r(411)
We can, of course, solve this equation for any single unknown given the rest of the variables, but
its mostcommonuse is to derive the so-calledcyclotron frequencyfor the circulating particle:
ωcyclotron=v 0
r=
qB 0
m(412)
Note well that this frequencydoes not depend on the speed of the particle!It isfixedby the charge
of the particle, its mass, and the strength of the magnetic fieldonly, which means that identical
particles take thesame amount of timeto complete a circuit of their motion independent of their
energy or velocity. This is the basis of the design of thecyclotron, one of the original particle
accelerators (still) used to probe the structure of the atomic nucleus.
Example 6.2.2: The Cyclotron
DEEDEESourceAlternating E−fieldBeam PipeDeflector PlateProtonFigure 64: The schematic layout of a cyclotron. The electric field/potential difference between the
“Dees” of the cyclotron oscillates with the same period as the periodof the cyclotron frequency of
the particles moving in the field, so that it always pushes in the direction that speeds it up.
In figure 64 you can see the general design of a cyclotron. A suitable charged ion, e.g. a hydrogen
nucleus (proton) is produced by e.g. an electrical arc in a source in the very center of the cyclotron
with a low velocity. A powerful magnetic field bends the initial trajectory into a circular arc in the
plane perpendicular to the field.