Abraham–Lorentz force

The Abraham-Lorentz force is the average force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is applicable when the particle is travelling at small velocities. The extension to relativistic velocities is known as the Abraham-Lorentz-Dirac force.

Definition and Description

Mathematically, the Abraham-Lorentz force is given by:

$\mathbf {F} _{\mathrm {rad} }={\frac {\mu _{0}q^{2}}{6\pi c}}\mathbf {\dot {a}} ={\frac {q^{2}}{6\pi \epsilon _{0}c^{3}}}\mathbf {\dot {a}} \,{\mbox{ (SI units)}}$

$\mathbf {F} _{\mathrm {rad} }={2 \over 3}{\frac {q^{2}}{c^{3}}}\mathbf {\dot {a}} \,{\mbox{ (cgs units)}}$

for small velocities. According to the Larmor formula, an accelerating charge emits radiation, which carries momentum away from the charge. Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. The Abraham-Lorentz force is the average force on an accelerating charge due to the emission of radiation.

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

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed. ed.). Prentice Hall. ISBN 013805326X. {{cite book}}: |edition= has extra text (help)

• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X.