Solution Found!
(a) Repeat Prob. 11.19, but this time let the external
Chapter 11, Problem 31P(choose chapter or problem)
Problem 31P
(a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: F(t) = kδ(t) (for some constant k).25 [Note that the acceleration is now discontinuous at t = 0 (though the velocity must still be continuous); use the method of Prob. 11.19 a) to show that ; Δa = −k/mτ . In this problem there are only two intervals to consider: (i) t < 0, and (ii) t > 0.]
(b) As in Prob. 11.30, check that energy is conserved in this process.
Reference prob 11.19
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 prob 11.30
Assuming you exclude the runaway solution in Prob. 11.19, calculate
(a) the work done by the external force,
(b) the final kinetic energy (assume the initial kinetic energy was zero),
(c) the total energy radiated.
Check that energy is conserved in this process.
Questions & Answers
QUESTION:
Problem 31P
(a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: F(t) = kδ(t) (for some constant k).25 [Note that the acceleration is now discontinuous at t = 0 (though the velocity must still be continuous); use the method of Prob. 11.19 a) to show that ; Δa = −k/mτ . In this problem there are only two intervals to consider: (i) t < 0, and (ii) t > 0.]
(b) As in Prob. 11.30, check that energy is conserved in this process.
Reference prob 11.19
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 prob 11.30
Assuming you exclude the runaway solution in Prob. 11.19, calculate
(a) the work done by the external force,
(b) the final kinetic energy (assume the initial kinetic energy was zero),
(c) the total energy radiated.
Check that energy is conserved in this process.
ANSWER:Problem 31P
(a) Repeat Prob. 11.19, but this time let the external force be a Dirac delta function: 
b) As in Prob. 11.30, check that energy is conserved in this process.
Reference prob 11.19
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 
(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 
(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: 
(e) Plot a(t) and v(t), both for an uncharged particle and for a (non runaway) charged particle, subject to this force.
Reference prob 11.30
Assuming you exclude the runaway solution in Prob. 11.19, calculate
(a) the work done by the external force,
(b) the final kinetic energy (assume the initial kinetic energy was zero),
(c) the total energy radiated.
Check that energy is conserved in this process.
Step by Step Solution
Step 1 of 4
Part (a)
The expression for the external force can be written as:
The expression for the acceleration can be written as:
The above equation can be written as:
If the velocity is continuous, so 
When 
When 
Therefore,
The expression for the general solution can be written as:
