Question

2. A cyclotron is a device used to accelerate charged particles to high speeds. It consists of two hollow containers -called dees because of their shape facing each other and separated by a small gap. They are immersed in a uniform magnetic field, B, and are attached to a source of alternating voltage. A charged particle is projected from the center of the cyclotron into Dee #1, and the magnetic force causes it to turn in a cirele. When it completes its semicircular path within one dee, the polarity of the voltage is reversed, and the particle is accelerated across the gap into the adjacent dee. This process continues, and the particle spirals outward at faster and faster speeds, until it emerges from Dee 02, Notice that the voltage must be alternated twice dur- ing each revolution of the particle. The figure below shows a view- looking down from above-of a cyclotron. Dec #1 Dee #2 region containing magnetie field charged particle exit alternating voltage (a) Explain why the electric field in the gap must be used to increase the speed of the particle. Assume that the charged particle to be accelerated by the cyclotron is a proton. (b) Should the magnetic field point into the plane of the page or out of the plane of the page in order to cause the particle to rotate clockwise as shown in the figure? For each of the following, write your answer in terms of the magnetic field strength B, the proton's charge (+e) and mass (m), and fundamental constants Show that the time to complete one revolution does not depend on the speed of the proton, and determine th orbital period (c) (d) How many revolutions does the proton make per second? e) What must be the frequency (in Hz) of the alternating voltage? (0 If the maximum radius of the proton's orbit is R, what is its maximum kinetic energy upon exiting? (You answer should also include R.)

Answer 1

Here cocepts from motion of charged particle in uniform magnetic field will be used.

Magnetic force doesnt do any work on charged particle as it is always perpendicular to velocity of charged particle.So electric field is used to increase the speed.

Here cocepts from motion of charged particle in uniform magnetic field will be used. Magnetic force doesnt do any work on charged particle as it is always perpendicular to velocity of charged particle.So electric field is used to increase the speed Radius = R = mv/q B so, R times V m rightarrow mass q rightarrow charge B rightarrow magnetic field we see radius of charged particle in proportional to speed To make exit particle must be moving in circular path of greater radius so its speed must increase B should be out of the plane since F^vector = q(v^vector times B^vector) Time period = 2 pi m/q B clearly independent of speed so orbital period = 2 pi m/q B q = e \r\n in 2 pi m/q B time it performs one revolution q = e so, eB/ 2 pi m rightarrow Number of revolutions per second frequency = f = e B/e pi m R = mv/eB v = e B R/m kinetic = 1/2 mv^2 = e^2 B^2 R^2/2 m