The cyclotron (Fig. 20-72) 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 semicircular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass through the gap between the metal "dees," where there is an electric field E. (There is no electric field inside the hollow metal dees where the electrons move in circular paths.) The electric field changes direction each half-cycle, owing to an ac voltage V = V0 sin 2Ï€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/2Ï€m, 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 2.0 m and the magnetic field strength is 0.50 T, what will be the maximum kinetic energy of accelerated protons in MeV?

Gap "Dees" to