Abraham–Lorentz force

In classical electrodynamics, energy carries momentum. Thus, when a particle emits energy, there is a force exerted on that particle according to Newton's third law. This phenomenon is known as the radiation reaction. For small velocities, the average force acting on a charged particle due to the radiation reaction is known as the Abraham-Lorentz force:

$\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} .$

The relativistic generalization of the above is the Abraham-Lorentz-Dirac force.

Derivation

We begin with the Larmor formula for radiation of a point charge:

$P={\frac {\mu _{0}q^{2}a^{2}}{6\pi c}}$.

If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham-Lorentz force is the negative of the Larmor power integrated over one period from $\tau _{1}$ to $\tau _{2}$:

$\int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=\int _{\tau _{1}}^{\tau _{2}}-Pdt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}a^{2}}{6\pi c}}dt=-\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot {\frac {d\mathbf {v} }{dt}}dt$.

Notice that we can integrate the above expression by parts. If we assume that there is periodic motion, the boundary term in the integral by parts disappears:

$\int _{\tau _{1}}^{\tau _{2}}\mathbf {F} _{\mathrm {rad} }\cdot \mathbf {v} dt=-{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} {\bigg |}_{\tau _{1}}^{\tau _{2}}+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}{\frac {d^{2}\mathbf {v} }{dt^{2}}}\cdot \mathbf {v} dt=-0+\int _{\tau _{1}}^{\tau _{2}}{\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} \cdot \mathbf {v} dt$.

Clearly, we can identify

$\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}}$.

Problems with the Abraham-Lorentz Force

If we have no external forces acting on a particle, we have

$\mathbf {F} _{\mathrm {net} }=m\mathbf {a} =\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} .$

This equation has the solution

$\mathbf {a} =\mathbf {a} _{0}e^{t/t_{0}}$

where

$t_{0}={\frac {\mu _{0}q^{2}}{6\pi mc}}.$

If we do not set $\mathbf {a} _{0}=\mathbf {0}$, then we get acceleration exponentially increasing, known as a runaway solution. However, it can be shown that if we do set $\mathbf {a} _{0}=\mathbf {0}$ in the presence of an external force, then we end up with acceleration occurring before the external force is applied, or "pre-acceleration."

References

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