Cyclotron

A cyclotron (invented by Lawrence and Livingston, 1934) is a high-energy accelerator: a charged particle moving perpendicular to a magnetic field feels the Lorentz force, which is used here to repeatedly re-accelerate it. Construction: two hollow semicircular "Dees" sit in an evacuated chamber inside a uniform field normal to their plane, separated by a small gap, with a source $S$ at the centre and a high-frequency alternating potential connected across the Dees. Working: a positive ion ejected from $S$ is first accelerated toward whichever Dee is momentarily negative; inside a Dee it simply coasts in a semicircle (the field alone bends it, at constant speed); each time it crosses the gap the Dees' polarity has just reversed, so it is accelerated again and enters the other Dee with higher speed, hence, from $r=mv/qB$, a larger radius. The particle thus spirals outward through ever-larger semicircles until, near the edge, a deflector plate extracts it toward a target.

The scheme only works if the oscillator's frequency exactly matches the particle's own cyclotron frequency ($f_{osc} = qB/2\pi m$, from §3.10.2) -- the resonance condition -- since that frequency, remarkably, does not change as the particle speeds up. The particle's final kinetic energy, once it reaches the edge (radius $r$), is

$KE = \frac{1}{2}mv^2 = \frac{q^2B^2r^2}{2m}$ …