A particle in the cyclotron shown in Figure 29.16a gains

Chapter 29, Problem 28

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A particle in the cyclotron shown in Figure 29.16a gains energy q DV from the alternating power supply each time it passes from one dee to the other. The time interval for each full orbit is T 5 2p v 5 2pm qB so the particles average rate of increase in energy is 2q DV T 5 q 2 B DV pm Notice that this power input is constant in time. On the other hand, the rate of increase in the radius r of its path is not constant. (a) Show that the rate of increase in the radius r of the particles path is given by dr dt 5 1 r DV pB (b) Describe how the path of the particles in Figure 29.16a is consistent with the result of part (a). (c) At what rate is the radial position of the protons in a cyclotron increasing immediately before the protons leave the cyclotron? Assume the cyclotron has an outer radius of 0.350 m, an accelerating voltage of DV 5 600 V, and a magnetic field of magnitude 0.800 T. (d) By how much does the radius of the protons path increase during their last full revolution?