Abraham–Lorentz force

In the physics of electromagnetism, the Abraham–Lorentz force is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force.

There are three important conditions that must be met for the Abraham–Lorentz force expression to be valid. First, the formula for the Abraham–Lorentz force is applicable only when the particle is traveling at non-relativistic velocities (that is, much slower than the speed of light); the extension to relativistic velocities is known as the Abraham-Lorentz-Dirac force. Second, the motion of the particle must be periodic (an assumption that is explicitly made in the derivation of the force). Third, the formula is in the domain of classical physics, not quantum physics, and therefore, is not valid at distances of roughly the Compton wavelength (λ_C ≈ 2.43 pm) or below.^[1] There is, however, an analogue of the formula which is both fully quantum and relativistic, called the "Abraham-Lorentz-Dirac-Langevin equation". See Johnson and Hu.^[2]

The force is proportional to the square of the object's charge, times the so-called "jerk" (rate of change of acceleration) that it is experiencing. The force points in the direction of the jerk. For example, in a cyclotron, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action.

In an antenna, it is responsible for the phenomenon of radiation resistance.

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. For example, there are pathological solutions using the Abraham–Lorentz-Dirac equation in which a particle accelerates in advance of the application of a force, so-called preacceleration solutions! One resolution of this problem was discussed by Yaghjian,^[3] and a fuller discussion of its resolution is made by Rohrlich,^[4] Medina.,^[5] and Ribarič and Šušteršič.^[6]

1 Definition and description
2 Background
3 Derivation
4 Signals from the future
5 Abraham–Lorentz–Dirac Force
- 5.1 Definition
- 5.2 Paradoxes
6 See also
7 References
8 Further reading
9 External links

Definition and description

Mathematically, the Abraham–Lorentz force is given by:

$\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}} = \frac{ q^2}{6 \pi \epsilon_0 c^3} \mathbf{\dot{a}}$ (SI units)

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

where:

F is the force,

$\mathbf{\dot{a}}$ is the jerk (the derivative of acceleration, or the third derivative of displacement),

μ₀ and ε₀ are the permeability and permittivity of free space,

c is the speed of light in free space^[7]

q is the electric charge of the particle.

Note that this formula applies only for non-relativistic velocities; for relativistic velocities, see Abraham-Lorentz-Dirac force.

Physically, an accelerating charge emits radiation (according to the Larmor formula), 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. In fact the formula above for radiation force can be derived from the Larmor formula, as shown below.

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
The reverse situation, problems in which the fields are specified and the motion of particles are calculated.

In some fields of physics, such as plasma physics and the calculation of transport coefficients (conductivity, diffusivity, etc.), the fields generated by the sources and the motion of the sources are solved self-consistently. In such cases, however, the motion of a selected source is calculated in response to fields generated by all other sources. Rarely is the motion of a particle (source) due to the fields generated by that same particle 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 problems in physics such as renormalization, some of which still unsolved, 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 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 led to a theory that is able to make the most accurate predictions that humans have made to date. See precision tests of QED. 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.

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 $\tau_2$ :

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

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:

$\int_{\tau_1}^{\tau_2} \mathbf{F}_\mathrm{rad} \cdot \mathbf{v} dt = - \frac{\mu_0 q^2}{6 \pi c} \frac{d \mathbf{v}}{dt} \cdot \mathbf{v} \bigg|_{\tau_1}^{\tau_2} %2B \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \frac{d^2 \mathbf{v}}{dt^2} \cdot \mathbf{v} dt = -0 %2B \int_{\tau_1}^{\tau_2} \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}} \cdot \mathbf{v} dt$ .

Clearly, we can identify

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

Signals from the future

Below is an illustration of how a classical analysis can lead to surprising results. The classical theory can be seen to challenge standard pictures of causality, thus signaling either a breakdown or a need for extension of the theory. In this case the extension is to quantum mechanics and its relativistic counterpart quantum field theory. See the quote from Rohrlich ^[1] in the introduction concerning "the importance of obeying the validity limits of a physical theory".

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

$m \dot {\mathbf{v} } = \mathbf{F}_\mathrm{rad} %2B \mathbf{F}_\mathrm{ext} = m t_0 \ddot { \mathbf{{v}}} %2B \mathbf{F}_\mathrm{ext} .$

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 it takes for a light wave to travel across the "size" of an electron.

Abraham–Lorentz–Dirac Force

An extension of the Abraham–Lorentz force for relativistic particles that accounts for special relativity was derived by Paul Dirac in 1938 and is called the Abraham–Lorentz–Dirac Force.^[8]

Definition

The expression derived by Dirac is given in signature (−, +, +, +) by

$F^{\mbox{rad}}_\mu = \frac{\mu_o q^2}{6 \pi m c} \left(\frac{d^2 p_\mu}{d \tau^2}-\frac{p_\mu}{m^2 c^2} \, \left(\frac{d p_\nu}{d \tau}\frac{d p^\nu}{d \tau}\right) \right)$

With Liénard's relativistic generalization of Larmor's formula in the co-moving frame,

$P = \frac{\mu_o q^2 a^2 \gamma^6}{6 \pi c}$ ,

one can show this to be a valid force by manipulating the time average equation for power.

$\frac{1}{\Delta t}\int_0^t P dt = \frac{1}{\Delta t}\int_0^t \textbf{F} \cdot \textbf{v} dt$

Paradoxes

Similar to the non-relativistic case, there are pathological solutions using the Abraham–Lorentz-Dirac equation that anticipate a change in the external force and according to which the particle accelerates in advance of the application of a force, so-called preacceleration solutions. One resolution of this problem was discussed by Yaghjian,^[3] and a fuller discussion of its resolution is made by Rohrlich^[1] and Medina.^[9]

Ribarič and Šušteršič ^[10] made preacceleration totally irrelevant by interpreting the Lorentz-Abraham-Dirac equation as the lowest order, asymptotic differential relation for the velocity of the radiating pointlike mass. Such a relation can be used to determine experimentally its kinetic constants, but not to compute its trajectory.

References

^ ^a ^b ^c F. Rohrlich: The dynamics of a charged sphere and the electron Am J Phys 65 (11) p. 1051 (1997). "The dynamics of point charges is an excellent example of the importance of obeying the validity limits of a physical theory. When these limits are exceeded the predictions of the theory may be incorrect or even patently absurd. In the present case, the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength…Only when all distances involved are in the classical domain is classical dynamics acceptable for electrons."
^ PR Johnson, BL Hu (2002). "Stochastic theory of relativistic particles moving in a quantum field: Scalar Abraham-Lorentz-Dirac-Langevin equation, radiation reaction, and vacuum fluctuations". Physical Review D 65 (6): 065015. arXiv:quant-ph/0101001. Bibcode 2002PhRvD..65f5015J. doi:10.1103/PhysRevD.65.065015.
^ ^a ^b Arthur D. Yaghjian (1992). Relativistic dynamics of a charged sphere: Updating the Lorentz-Abraham model. Berlin: Springer. Chapter 8. ISBN 3540978879. http://books.google.com/?id=keeyf1cJLjwC&printsec=frontcover#PPA10,M1.
^ F. Rohrlich: The dynamics of a charged sphere and the electron Am J Phys 65 (11) p. 1051 (1997)
^ Rodrigo Medina (2006). "Radiation reaction of a classical quasi-rigid extended particle". Journal of Physics A: Mathematical and General 39 (14): 3801–3816. arXiv:physics/0508031. Bibcode 2006JPhA...39.3801M. doi:10.1088/0305-4470/39/14/021.
^ Marijan Ribaric; Luka Sustersic (2010). "Improvement on the Lorentz-Abraham-Dirac equation". arXiv:1011.1805 [physics.gen-ph].
^ The symbol c₀ is used by CIPM and NIST.
^ Paul A.M. Dirac, (1938) Classical theory of radiating electrons. Proc. Roy. Soc. of London. A929:0148-0169. JSTOR
^ Rodrigo Medina, Radiation reaction of a classical quasi-rigid extended particle J. Phys. A: Math. Gen. A39 (2006) 3801-3816
^ M. Ribarič and L. Šušteršič, Improvement on the Lorentz-Abraham-Dirac equation, arxiv:1011.1805

Abraham–Lorentz force

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