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Physics / Introduction to Electrodynamics 4 / Chapter 11 / Problem 17P

Introduction to Electrodynamics | 4th Edition | ISBN: 9780321856562 | Authors: David J. Griffiths

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(a) A particle of charge q moves in a circle of radius R

Chapter 11, Problem 17P

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QUESTION:

Problem 17P

(a) A particle of charge q moves in a circle of radius R at a constant speed v. To sustain the motion, you must, of course, provide a centripetal force mv2/R; what additional force (Fe) must you exert, in order to counteract the radiation reaction? [It’s easiest to express the answer in terms of the instantaneous velocity v.] What power (Pe) does this extra force deliver? Compare Pe with the power radiated (use the Larmor formula).

(b) Repeat part (a) for a particle in simple harmonic motion with amplitude A and angular frequency Explain the discrepancy.

(c) Consider the case of a particle in free fall (constant acceleration g). What is the radiation reaction force? What is the power radiated? Comment on these results.

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QUESTION:

Problem 17P

(a) A particle of charge q moves in a circle of radius R at a constant speed v. To sustain the motion, you must, of course, provide a centripetal force mv2/R; what additional force (Fe) must you exert, in order to counteract the radiation reaction? [It’s easiest to express the answer in terms of the instantaneous velocity v.] What power (Pe) does this extra force deliver? Compare Pe with the power radiated (use the Larmor formula).

(b) Repeat part (a) for a particle in simple harmonic motion with amplitude A and angular frequency Explain the discrepancy.

(c) Consider the case of a particle in free fall (constant acceleration g). What is the radiation reaction force? What is the power radiated? Comment on these results.

ANSWER:

a.)

Step 1 of 6

We have to find the additional force that must be exerted, in order to counteract the radiation reaction force from a particle of charge $q$ which is moving in a circle of radius $R$ at a constant speed $v$ and also the power delivered by this extra force.

The additional force ${F}_{e}$ to counteract the radiation reaction force can be found using the Abraham- Lorentz formula for the radiation reaction force.

where, $\vec{a}$ is the acceleration of the charge.

Now for a circular motion,

so,

Thus,

Therefore, the additional force that must be exerted is

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