Chapter 27 Problem 66

Problem

model cyclotron

The cyclotron is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in circular orbits in the magnetic field $B$. The particles are accelerated to higher speeds each time they pass in the gap between the metal “dees,” where there is an electric field $E$. (There is no electric field within the hollow metal dees.) The electric field changes direction each half-cycle, due to an ac voltage $V=V_{0}sin2\pi ft$, so that the particles are increased in speed at each passage through the gap.

(a) Show that the frequency $f$ of the voltage must be $f=Bq/2\pi m$, where $q$ is the charge on the particles and $m$ their mass.

(b) Show that the kinetic energy of the particles increases by $2qV_{0}$ each revolution, assuming that the gap is small.

(c) If the radius of the cyclotron is 0.50 m and the magnetic field strength is 0.60 T, what will be the maximum kinetic energy of accelerated protons in MeV?