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

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