The cyclotron is a device that provides a beam of high velocity charged particles. Consider the electron-accelerating cyclotron depicted in Figure P20.99. Two semicircular magnets provide a uniform magnetic field of 5.0 mT between the poles; in addition, an electric potential difference of 220 V is placed on each D-shaped section so that an accelerating electric field is produced in the gap between the semicircles. An electron will accelerate as it crosses the gap and enter the magnetic field, producing a circular orbit so that the electron is directed back into the gap. By the time the electron returns to the gap, circuitry will have reversed the direction of the electric field so that the electron is again accelerated. Assume Newton’s second law and the expression for kinetic energy apply in this case.
(a) How much does the energy of an electron increase each time it crosses the gap? If the radius of each D-shaped magnet is 5.0 cm and the electrons start in the center of the magnets at one edge of the gap, what are the
(b) maximum speed and maximum kinetic energy that an electron can attain?
(c) When the electrons are traveling at their maximum speed, how often must the accelerating electric potential be reversed to make the device work?

Figure 20.99

Top view Side view Electric accelerating field between the magnetic field regions. This electric field direction changes back and forth. Injection of electrons Output beam of high-velocity electrons