World Library  
Flag as Inappropriate
Email this Article

Abraham–Lorentz force

Article Id: WHEBN0003209246
Reproduction Date:

Title: Abraham–Lorentz force  
Author: World Heritage Encyclopedia
Language: English
Subject: Wheeler–Feynman absorber theory, Jerk (physics), Lorentz force, List of Dutch inventions and discoveries, Electromagnetic radiation
Collection: Electrodynamics, Electromagnetic Radiation, Radiation
Publisher: World Heritage Encyclopedia

Abraham–Lorentz force

In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham 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 or the self force.

The formula predates the theory of special relativity, and is not valid at velocities of the order of the speed of light. Its relativistic generalization is called the "Abraham–Lorentz–Dirac force". Both of these are in the domain of classical physics, not quantum physics, and therefore, may not be valid at distances of roughly the Compton wavelength 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.

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 pre-acceleration solutions. One resolution of this problem was discussed by Yaghjian,[3] and is further discussed by Rohrlich[1] and Medina.[4]


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

Definition and description

Mathematically, the Abraham–Lorentz force is given in SI units 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}}

or in cgs units by

\mathbf{F}_\mathrm{rad} = { 2 \over 3} \frac{ q^2}{ c^3} \mathbf{\dot{a}}.

Here Frad is the force, \mathbf{\dot{a}} is the jerk (the derivative of acceleration, or the third derivative of displacement), μ0 is the magnetic constant, ε0 is the electric constant, c is the speed of light in free space, and q is the electric charge of the particle.

Note that this formula is for non-relativistic velocities; Dirac simply renormalized the mass of the particle in the equation of motion, to find the relativistic version (below).

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.


In 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. 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:

  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 problems in physics such as renormalization, some of which is 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 an unsolved self-field problem. String theory and loop quantum gravity are current attempts to resolve this problem, formally called the problem of radiation reaction or the problem of self-force.


The simplest derivation for the self-force is found for periodic motion from the Larmor formula for radiation of a point charge:

P = \frac{\mu_0 q^2}{6 \pi c} \mathbf{a}^2.

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}{6 \pi c} \mathbf{a}^2 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}}.

A more rigorous derivation, which does not require periodic motion, was found using an Effective Field Theory formulation.[5][6] An alternative derivation, finding the fully relativistic expression, was found by Dirac.

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} + \mathbf{F}_\mathrm{ext} = m t_0 \ddot { \mathbf} + \mathbf{F}_\mathrm{ext} .


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

To find the relativistic generalization, Dirac renormalized the mass in the equation of motion with the Abraham–Lorentz force in 1938. This renormalized equation of motion is called the Abraham–Lorentz–Dirac equation of motion.[7]


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

F^{\mathrm{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.


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 is further discussed by Rohrlich[1] and Medina.[4]

See also


  1. ^ a b c d (11) p. 1051 (1997)65 Am J Phys The dynamics of a charged sphere and the electronF. Rohrlich: . "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."
  2. ^ 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".  
  3. ^ a b Yaghjian, Arthur D. (2006). Relativistic Dynamics of a Charged Sphere: Updating the Lorentz–Abraham Model. Lecture Notes in Physics 686 (2nd ed.). New York: Springer. Chapter 8.  
  4. ^ a b Rodrigo Medina (2006). "Radiation reaction of a classical quasi-rigid extended particle". Journal of Physics A: Mathematical and General 39 (14): 3801–3816.  
  5. ^ "Radiation reaction at the level of the action" by Ofek Birnholtz, Shahar Hadar, and Barak Kol
  6. ^ "Theory of post-Newtonian radiation and reaction" by Ofek Birnholtz, Shahar Hadar, and Barak Kol
  7. ^ Paul A.M. Dirac, (1938) Classical theory of radiating electrons. Proc. Roy. Soc. of London. A929:0148-0169.

Further reading

  • Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. See sections 11.2.2 and 11.2.3  
  • Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley.  
  • Donald H. Menzel (1960) Fundamental Formulas of Physics, Dover Publications Inc., ISBN 0-486-60595-7, vol. 1, page 345.
  • Stephen Parrott (1987) Relativistic Electrodynamics and Differential Geometry, § 4.3 Radiation reaction and the Lorentz-Dirac equation, pages 136–45, and § 5.5 Peculiar solutions of the Lorentz-Dirac equation, pages 195–204, Springer-Verlag ISBN 0-387-96435-5 .

External links

  • MathPages – Does A Uniformly Accelerating Charge Radiate?
  • Feynman: The Development of the Space-Time View of Quantum Electrodynamics
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.