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.

Background

I classical electrodynamics problems are typically divided into two classes:

1. problems in which the charge and current sources of fields are specified and the fields are calculated, and
2. problems in which the fields are specified and the motion of particles are calculated.

In some fields of physics, such as plasma physics, the fields generated by the sources and the motion of the sources are solved self-consistently. In this case, however, the motion of a source is calculated from fields generated by all other sources. Rarely is the motion of a particle (source) due to the fields generated by the same particle calculated. The reason for this is twofold:

1. Neglect of the " self-fields" usually leads to answers that are accurate enough for many applications, and
2. Inclusion of self-fields leads to currently unsolved problems in physics that relate to the very nature of matter and energy.

This conceptual problems created by self-fields are highlighted in a standard graduate text. [Jackson]

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limted areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the clasical ones. It is one of the triumphs of comparatively recent years (~ 1948 - 1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.

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.