Week 6: Moving Charges and Magnetic Force 201
In between the upper and lower halves of the cyclotron are two copper chambers shaped like the
letter “D”, with a narrow slit in the plane perpendicular to the field cutalong the straight segment
in the middle. An alternating electric potential is appliedbetweenthese two “Dees” that has the
same angular frequency as the cyclotron frequency of the particle being accelerated in the magnetic
field in questionso that when the particle arrives at the gap between the upper andlower Dee in
the figure above, it happens to point down (and hence speeds the particle up). When the particle
gets to the gap between the lower and the upper Dee on the right, though, the field hasswitched
directionand stillspeeds the particle upstill more. Every time the particle arrives at the gap, it
finds the field is there, aligned with its motion to give it yet another push.
This works because it takesallof the particles the same amount of time to make it around a
half-circle regardless of how fast they are going. So one can have astream of particles all falling
across the gap at once at different radii from the source (with short gaps between these ”pulses” that
are in phase and being accelerated together). As the particle is moving faster and faster, the radius
of the circle of its motion increases until it reaches an electrostaticdeflector plate at the outside
edge of the magnetic field that angles it into a beam pipe where it travels through a vacuum to hit
an eventual beam target.
Early cyclotrons played an important role in the development of nuclear physics, permitting
the creation and discovery of the first transuranic elements pastplutonium (one of which is named
Lawrencium, after the inventor of the cyclotron, another of which is named Berkelium after the
University where Lawrence worked).
Cyclotrons, alas, no longer work when the particles are accelerated enough to be moving at
relativistic velocities. At some point the time dilation of the cyclotron period in the frame of
the moving particle is enough to keep the particle from being accelerated by a Dee voltage at the
cyclotron frequency that worked for a slowly moving particle. One can “fix” this problem by sweeping
the frequency to match and acclerating only pulses of charge (in asynchrotron) but as one reaches
higher and higher energies other problems emerge.
The principal limiting factor is ultimately the fact thataccelerated charges radiate, and particles
moving in circlesare accelerated all of the timeby the centripetal magnetic force. This causes a kind
of “resistance” wherein the work done speeding the particle up in a cycle is balanced by radiative
losses in the cycle. Only the use of very large circles can minimize the latter, which is why the
extreme relativistic accelerators of modern times, such as the Large Hadron Collider (LHC) are
enormous circles, the latter being27 kilometersin circumference.
Example 6.2.3: Cloud Chamber
In a nuclear collision, a lot of “stuff” is produced – nucleons knocked out of nuclei, electrons,
positrons, gamma rays, alpha particles, and more exotic particles that help us understand the nuclear
field itself. To be able to categorize and classify all of this “stuff”, it helps to be able to “see” the
trajectory of a particle produced in the collision, and determine things like the ratio of its charge
to its mass. A cloud chamber (and more exotic bubble chambers thatwork on a similar principle)
is a device that makes a charged particle’s trajectory visible so thatit can be photographed. It
works by creating a “supersaturated” gas of e.g. alcohol, water vapor, or other substances. The
charged particle in question zips through the vapor and causes it tobounce together in its wake,
precipitating the vapor out as a condensation trial, much like the jetcontrails one can sometimes see
overhead on a clear day. In a cloud chamber the trajectories typically only last a few seconds before
re-evaporating, but that is long enough to be easily seen and/or photographed for later analysis.
By putting the chamber in amagnetic fieldand right next to a nuclear target, thepositive
particles curve one way and thenegativeparticles curve the other. The radius of curvature is related