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With the inclusion of the radiation reaction force (Eq.

Introduction to Electrodynamics | 4th Edition | ISBN: 9780321856562 | Authors: David J. Griffiths ISBN: 9780321856562 45

Solution for problem 19P Chapter 11

Introduction to Electrodynamics | 4th Edition

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

Introduction to Electrodynamics | 4th Edition

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Problem 19P

With the inclusion of the radiation reaction force (Eq. 11.80), Newton’s second law for a charged particle becomes

where F is the external force acting on the particle.

(a) In contrast to the case of an uncharged particle (a = F/m), acceleration (like position and velocity) must now be a continuous function of time, even if the force changes abruptly. (Physically, the radiation reaction damps out any rapid change in a.) Prove that a is continuous at any time t, by integrating the equation of motion above from  and taking the limit

(b) A particle is subjected to a constant force F, beginning at time t = 0 and lasting until time T . Find the most general solution a(t) to the equation of motion in each of the three periods: (i) t < 0; (ii) 0 < t < T ; (iii) t > T .

(c) Impose the continuity condition (a) at t = 0 and t = T . Show that you can either eliminate the runaway in region (iii) or avoid pre acceleration in region (i), but not both.

(d) If you choose to eliminate the runaway, what is the acceleration as a function of time, in each interval? How about the velocity? (The latter must, of course, be continuous at t = 0 and t = T .) Assume the particle was originally at rest: v(−∞) = 0.

(e) Plot a(t) and v(t), both for an uncharged particle and for a (nonrunaway) charged particle, subject to this force.

Reference equation 11.80

Step-by-Step Solution:

a.)

Step 1 of 10</p>

We have to prove that the acceleration of a charged particle given by Newton’s second law of motion is a continuous function of time.

where  is the external force acting on the particle.

We can prove that is a continuous function of time by integrating the above equation of motion from to and taking the limit .

 

Integrating from () to with limit

where, is the average force during the time interval.

Now, is continuous as long as is not a delta function . So, we are left (in the limit ) with  

Thus, is also a continuous function of time.

Therefore, we have proved that is continuous function at anytime

b.)

Step 2 of 10</p>

We have to find the general solution to the equation of motion for a particle subjected to a constant force , beginning at time  and lasting until time  in the time periods , and

The equation of motion from Newton’s second law is given by,

For the time period :

 

 

 

       

     

or

where, is a constant.

Therefore, the general solution in the region is

Step 3 of 4

Chapter 11, Problem 19P is Solved
Step 4 of 4

Textbook: Introduction to Electrodynamics
Edition: 4
Author: David J. Griffiths
ISBN: 9780321856562

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With the inclusion of the radiation reaction force (Eq.

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