Classical Electrodynamics · Problem 16.10

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-damping-classical-models-charged-particles

Problem

The Abraham--Lorentz equation of motion (16.9) can be replaced by an integro-differential equation if the external force is considered a function of time. (a) Show that a first integral of (16.9) that eliminates the possibility of ``runaway'' solutions is mv˙(t)=∫0∞e−sF(t+τs) dsm\dot{\mathbf{v}}(t) = \int_0^\infty e^{-s}\mathbf{F}(t + \tau s)\,ds (b) Show that a Taylor series expansion of the force for small τ\tau leads to mv˙(t)=∑n=0∞τndnF(t)dtnm\dot{\mathbf{v}}(t) = \sum_{n=0}^\infty \tau^n \frac{d^n\mathbf{F}(t)}{dt^n} The approximate equation (16.10) contains the first two terms of the infinite series. (c) For a step-function force in one dimension, F(t)=F0Θ(t)F(t) = F_0\Theta(t), solve the integro-differential equation of part a for the acceleration and velocity for t<0t < 0 and t>0t > 0 for a particle at rest at t=−∞t = -\infty. Plot ma/F0ma/F_0 and mv/F0τmv/F_0\tau in units of t/τt/\tau. Compare with the solution from (16.10). Comment.

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