Abraham-Lorentz force

From Wikipedia, the free encyclopedia

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.

It was thought that the solution of the Abraham-Lorentz force problem predicts that signals from the future affect the present, thus challenging intuition of cause and effect. Attempts to resolve this problem touch on many areas of modern physics, although Yaghjian has shown that the solution is actually much simpler.

1 Definition and Description
2 Background
3 Derivation
4 Signals from the future
5 See also
6 References

[edit] 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.

[edit] Background

In classical electrodynamics, problems are typically divided into two classes:

problems in which the charge and current sources of fields are specified and the fields are calculated, and
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 due to the fields generated by the same particle (source) calculated. The reason for this is twofold:

Neglect of the "self-fields" usually leads to answers that are accurate enough for many applications, and
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 limited 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 classical 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.

The Abraham-Lorentz force is the result of the most fundamental calculation of the effect of self-generated fields. It arises from the observation that accelerating charges emit radiation. The Abraham-Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The immediate difficulty that arises in the calculation is that the force calculation requires the future to have an effect on the present, thus violating our intuition on the nature of cause and effect. If one accepts that signals from the future can affect the present and one maintains a classical (non-quantum) description of the problem then one is lead to the problem of answering the question of why time flows from past to future and not vice-versa.[1] The introduction of quantum effects leads one to quantum electrodynamics. The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of renormalization. This has lead to a theory that is able to make the most accurate predictions that humans have made to date. In quantum electrodynamics, signals from the future are interpreted as antimatter. The renormalization process fails, however, when applied to the gravitational force. The infinities in that case are infinite in number, which causes the failure of renormalization. Therefore general relativity has unsolved self-field problems. String theory is a current attempt to resolve these problems for all forces.

However, Yaghjian has shown that the derivation of the Abraham-Lorentz force implicitly relies on an adiabatic assumption, which places limits on how fast forces can change. When the the forces are made sufficiently continuous, the solutions no longer have the preacceleration term that relied on future forces. [Rohlich].

[edit] 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 $τ 1$ to $τ 2$ :

$\int_{\tau_1}^{\tau_2} \mathbf{F}_\mathrm{rad} \cdot \mathbf{v} dt = \int_{\tau_1}^{\tau_2} -P dt = - \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2 a^2}{6 \pi c} dt = - \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \frac{d \mathbf{v}}{dt} \cdot \frac{d \mathbf{v}}{dt} dt$ .

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

$\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}}$ .

[edit] Signals from the future

For a particle in an external force $\mathbf{F}_\mathrm{ext}$ , we have

$m \dot {\mathbf{v} } = \mathbf{F}_\mathrm{rad} + \mathbf{F}_\mathrm{ext} = m t_0 \ddot { \mathbf{{v}}} + \mathbf{F}_\mathrm{ext} .$

where

$t_0 = \frac{\mu_0 q^2}{6 \pi m c}.$

This equation can be integrated once to obtain

$m \dot {\mathbf{v} } = {1 \over t_0} \int_t^{\infty} \exp \left( - {t'-t \over t_0 }\right ) \, \mathbf{F}_\mathrm{ext}(t') \, dt' .$

The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. The future values are weighted by the factor

$\exp \left( -{t'-t \over t_0 }\right )$

which falls off rapidly for times greater than $t 0$ in the future. Therefore, signals from an interval approximately $t 0$ into the future affect the acceleration in the present. For an electron, this time is approximately $10 - 24$ sec, which is the time is takes for a light wave to travel across the "size" of an electron.