Chapter 27, Problem 66GP

The cyclotron (Fig. 27–50) 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 = V₀ sin 2πft, 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 πm, where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by 2qV₀ 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?