Abraham–Lorentz force

Remarks

The force is fixed by energy balance with the Larmor formula. If the motion is periodic, or the acceleration vanishes at the start and at the end, an integration by parts gives $\int F_{\mathrm{rad}} \cdot v \, dt = -\frac{q^2}{6 \pi \varepsilon_0 c^3} \int \abs{a}^2 \, dt$. So the work done by $F_{\mathrm{rad}}$ is minus the energy radiated according to the Larmor formula.

The equation is of third order in the position and has unphysical solutions. With $F_{\mathrm{ext}} = 0$ it has the runaway solutions $a(t) = a(0) e^{t/\tau}$, whose acceleration grows without bound. Excluding them by a condition at $t \to \infty$ leads to pre-acceleration: the charge starts to accelerate a time of order $\tau$ before the external force acts. For an electron, $\tau$ is about $6 \times 10^{-24}$ seconds, so these effects lie far below the scales where classical electrodynamics applies.

Going deeper

For a student

The Abraham–Lorentz equation is a standard example of the difficulties of point particles in classical field theory. In practice one uses it in reduced form, replacing $\dot a$ by the derivative of $F_{\mathrm{ext}} / m$, which removes runaways and is accurate when the radiation force is small. Its relativistic generalization is the Abraham–Lorentz–Dirac equation.