ps07sol

# ps07sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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PS07-1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2010 Problem Set 7 Solutions Problem 1: Short Questions (a) Can a constant magnetic field set into motion an electron which is initially at rest? Explain your answer. Solution: No. Changing the velocity of a particle requires an accelerating force. The magnetic force is proportional to the speed of the particle. If the particle is not moving, there can be no magnetic force on it. (b) Is it possible for a constant magnetic field to alter the speed of a charged particle? What is the role of a magnetic field in a cyclotron? Solution: No, it is not possible. Because ( ) B q = × F v B r r r , the acceleration produced by a magnetic field on a particle of mass m is ( ) B q m = × a v B r r r . For the acceleration to change the speed, a component of the acceleration must be in the direction of the velocity. The cross product tells us that the acceleration must be perpendicular to the velocity, and thus can only change the direction of the velocity. The magnetic field in a cyclotron essentially keeps the charged particle in the electric field for a longer period of time, and thus experiencing a larger change in speed from the electric field, by forcing it in a spiral path. Without the magnetic field, the particle would have to move in a straight line through an electric field over a distance that is very large compared to the size of the cyclotron. (c) How can a current loop be used to determine the presence of a magnetic field in a given region of space? Solution: If the current loop feels a torque, it must be caused by a magnetic field. If the current loop feels no torque, try a different orientation—the torque is zero if the field is along the axis of the loop. (d) If a charged particle is moving in a straight line through some region of space, can you conclude that the magnetic field in that region is zero? Why or why not?

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PS07-2 Solution: Not necessarily. If the magnetic field is parallel or anti-parallel to the velocity of the charged particle, then the particle will experience no magnetic force. There may also be an electric force acting on the particle such that ( ) 0 q q q = × = F E + v B r r r r . (e) List some similarities and differences between electric and magnetic forces. Solution: Similarities : 1. Both can accelerate a charged particle moving through the field. 2. Both exert forces directly proportional to the charge of the particle feeling the force. Differences : 1. The direction of the electric force is parallel or anti-parallel to the direction of the electric field, but the direction of the magnetic force is perpendicular to the magnetic field and to the velocity of the charged particle. 2. Electric forces can accelerate a charged particle from rest or stop a moving particle, but magnetic forces cannot.
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