Thus, dotting this quantity with the velocity and time-averaging will give the (*) Larmor formula.
For a dumbbell, a naïve calculation of this quantity in cgs units will result in a constant factor of one over three c-cubed, half the actual value; that can be amended by recursively applying the same procedure to each pole of the dumbbell, or by using a continuous strip. This vector quantity can be derived more directly by time integrating by parts the acceleration squared over a full cycle, canceling boundary terms, and then realizing that the result minus the integral over the same interval of this quantity dotted with the velocity must vanish. Thus, dotting this quantity with the velocity and time-averaging will give the (*) Larmor formula. This quantity is given by the Abraham-Lorentz formula and results from the interaction of a charge with its own retarded field, thus changing its momentum. For 10 points, name this relativistic quantity that can be understood as the recoil of a particle emitting Bremsstrahlung photons.
For a dumbbell, a naïve calculation of this quantity in cgs units will result in a constant factor of one over three c-cubed, half the actual value; that can be amended by recursively applying the same procedure to each pole of the dumbbell, or by using a continuous strip.
the self-force [or radiation reaction force or radiation recoil force; accept Abraham-Lorentz force before Abraham-Lorentz is mentioned, prompt afterwards; prompt on partial answers]