Electrodynamics and Relativity — David J. Griffiths
Physics · radiation-reaction-mechanism
Problem
Deduce Eq. 11.100 from Eq. 11.99. Here are three methods:
(a) Use the Abraham-Lorentz formula to determine the radiation reaction on each end of the dumbbell; add this to the interaction term (Eq. 11.99).
(b) Method (a) has the defect that it uses the Abraham-Lorentz formula—the very thing that we were trying to derive. To avoid this, let F(q) be the total d-independent part of the self-force on a charge q. Then
F(q)=Fint(q)+2F(q/2),
where Fint is the interaction part (Eq. 11.99), and F(q/2) is the self-force on each end. Now, F(q) must be proportional to q2, since the field is proportional to q and the force is qE. So F(q/2)=(1/4)F(q). Take it from there.
(c) Smear out the charge along a strip of length L oriented perpendicular to the motion (the charge density, then, is λ=q/L); find the cumulative interaction force for all pairs of segments, using Eq. 11.99 (with the correspondence q/2→λdy1, at one end and q/2→λdy2 at the other). Make sure you don't count the same pair twice.