Motion of a Charged Particle in a Perpendicular Magnetic Field: The Cyclotron

Circular motion in a perpendicular field. From Section 4.8, a charged particle moving with

speed $v$ exactly perpendicular to a uniform magnetic field $B$ feels a magnetic force of constant

magnitude $F=qvB$, always directed perpendicular to its (instantaneous) velocity. A force of constant

magnitude, always perpendicular to the velocity, is precisely the condition for uniform circular

motion, with the magnetic force itself supplying the required centripetal force:

The particle therefore moves on a circle of radius $r$ given by this expression: a larger momentum

$mv$ gives a larger circle, while a stronger field $B$ or a larger charge $q$ pulls the circle

tighter.

Cyclotron frequency (period independent of speed). The TIME PERIOD for one complete revolution

is $T = 2\pi r/v$. Substituting $r=mv/(qB)$ from above,

Strikingly, the SPEED $v$ has cancelled out completely: the period (and hence the frequency,

$f = 1/T = qB/2\pi m$, called the cyclotron frequency) depends only on the particle's own charge

$q$, mass $m$, and the field strength $B$ -- NOT on its speed or on the radius of its particular

orbit. A slower particle, moving on a smaller circle, and a faster particle, moving on a larger

circle, both complete one full revolution in exactly the SAME time. This single fact is what makes

the cyclotron, described next, work at all.

The cyclotron. A cyclotron is a device that exploits this speed-independent period to accelerate

charged particles (protons, deuterons, alpha particles) to high energies using a comparatively

modest, repeatedly reapplied electric field. It consists of two hollow, D-shaped electrodes

("dees"), separated by a narrow gap, placed inside a strong uniform magnetic field directed

perpendicular to the plane of the dees. An oscillating high-frequency voltage is applied across the

narrow gap between the two dees; INSIDE each dee itself, there is no electric field (a hollow

conductor shields its interior), so a charged particle released near the centre moves in a simple

circular arc of the kind described above while it is inside either dee.

How the acceleration accumulates. Every time the particle crosses the gap between the two dees,

it is accelerated by the electric field present there at that instant, gaining a small, fixed

increment of kinetic energy on each crossing. Having gained energy (and hence speed), it then follows

a LARGER-radius semicircular arc inside the next dee than it did before -- but, crucially, because

the period $T=2\pi m/(qB)$ does not depend on speed, this larger, faster arc still takes exactly the

same time to complete as the smaller, slower one did. If the oscillating voltage's own frequency is

tuned to match the particle's cyclotron frequency $f = qB/2\pi m$ EXACTLY, the field in the gap

reverses direction at exactly the right moment, every single time, to keep accelerating (never

decelerating) the particle on every crossing -- this matching condition is called resonance.

Repeating this process many times, the particle spirals steadily outward (as the accompanying figure

shows) until it reaches the outer edge of the dee assembly, where it is extracted as a high-energy

beam. …