- Abraham-Lorentz force
In the

physics ofelectromagnetism , the**Abraham-Lorentz force**is therecoil force on an acceleratingcharged particle caused by the particle emittingelectromagnetic radiation . It is also called the**radiation reaction force**.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 theAbraham-Lorentz-Dirac force . In addition, the formula is in the domain ofclassical physics , notquantum physics , and therefore is not valid at distances of roughly theCompton wavelength ( [*http://physics.nist.gov/cgi-bin/cuu/Value?ecomwl λ*] ≈ 2.43 pm) or below. [_{C}*http://wwwphy.princeton.edu/~kirkmcd/examples/EM/rohrlich_ajp_65_1051_97.pdf F. Rohrlich: "The dynamics of a charged sphere and the electron" Am J Phys*] . "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."] (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 [**65**(11) p. 1051 (1997)*cite journal|author=PR Johnson, BL Hu|journal=Physical Review D|year=2002|volume=65|title=Stochastic theory of relativistic particles moving in a quantum field: Scalar Abraham-Lorentz-Dirac-Langevin equation, radiation reaction, and vacuum fluctuations|url=http://link.aps.org/abstract/PRD/v65/e065015|doi=10.1103/PhysRevD.65.065015|pages=065015|format=abstract*] [*PR Johnson, BL Hu: [*] and Galley and Hu. [*http://arxiv.org/abs/quant-ph/0101001v2 " Stochastic Theory of Relativistic Particles Moving in a Quantum Field: II. Scalar Abraham-Lorentz-Dirac-Langevin Equation, Radiation Reaction and Vacuum Fluctuations "*]*CR Galley and BL Hu (2005); see particularly Section VI "Discussions": [*] )*http://arxiv.org/abs/gr-qc/0505085v2 "Self-Force with a Stochastic Component from Radiation Reaction of a Scalar Charge Moving in Curved Spacetime"*]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 acyclotron , 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 Yaghjiancite book

author=Arthur D. Yaghjian

title=Relativistic dynamics of a charged sphere: Updating the Lorentz-Abraham model

publisher= Springer

location=Berlin

year=1992

page=Chapter 8

isbn=3540978879

url=http://books.google.com/books?id=keeyf1cJLjwC&printsec=frontcover&lr=#PPA10,M1] , and a fuller discussion of its resolution is made by Rohrlich [*[*] , and Medina. [*http://wwwphy.princeton.edu/~kirkmcd/examples/EM/rohrlich_ajp_65_1051_97.pdf F. Rohrlich: "The dynamics of a charged sphere and the electron" Am J Phys*]**65**(11) p. 1051 (1997)*[*]*http://arxiv.org/abs/physics/0508031v3 Rodrigo Medina "Radiation reaction of a classical quasi-rigid extended particle" J. Phys. A: Math. Gen.*]**A39**(2006) 3801-3816**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)or

::$mathbf\{F\}\_mathrm\{rad\}\; =\; \{\; 2\; over\; 3\}\; frac\{\; q^2\}\{\; c^3\}\; mathbf\{dot\{a$ (

cgs units)where::

**F**is the force,:$mathbf\{dot\{a$ is thejerk (the derivative ofacceleration , or the third derivative ofdisplacement ),: [*http://physics.nist.gov/cgi-bin/cuu/Value?mu0 μ*] and [_{0}*http://physics.nist.gov/cgi-bin/cuu/Value?ep0 ε*] are the permeability and permittivity of_{0}free space ,: [*http://physics.nist.gov/cgi-bin/cuu/Value?c "c"*] is the speed of light infree space [*The symbol "c"*] :"q" is the_{0}is used byCIPM andNIST .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 carriesmomentum 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 asrenormalization , 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 ofrenormalization . This has led to a theory that is able to make the most accurate predictions that humans have made to date. Seeprecision tests of QED . The renormalization process fails, however, when applied to thegravitational force . The infinities in that case are infinite in number, which causes the failure of renormalization. Thereforegeneral 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 $au\_1$ to $au\_2$:

:$int\_\{\; au\_1\}^\{\; au\_2\}\; mathbf\{F\}\_mathrm\{rad\}\; cdot\; mathbf\{v\}\; dt\; =\; int\_\{\; au\_1\}^\{\; au\_2\}\; -P\; dt\; =\; -\; int\_\{\; au\_1\}^\{\; au\_2\}\; frac\{mu\_0\; q^2\; a^2\}\{6\; pi\; c\}\; dt\; =\; -\; int\_\{\; au\_1\}^\{\; au\_2\}\; frac\{mu\_0\; q^2\}\{6\; pi\; c\}\; frac\{d\; mathbf\{v\{dt\}\; cdot\; frac\{d\; mathbf\{v\{dt\}\; dt$.

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:

:$int\_\{\; au\_1\}^\{\; au\_2\}\; mathbf\{F\}\_mathrm\{rad\}\; cdot\; mathbf\{v\}\; dt\; =\; -\; frac\{mu\_0\; q^2\}\{6\; pi\; c\}\; frac\{d\; mathbf\{v\{dt\}\; cdot\; mathbf\{v\}\; igg|\_\{\; au\_1\}^\{\; au\_2\}\; +\; int\_\{\; au\_1\}^\{\; au\_2\}\; frac\{mu\_0\; q^2\}\{6\; pi\; c\}\; frac\{d^2\; mathbf\{v\{dt^2\}\; cdot\; mathbf\{v\}\; dt\; =\; -0\; +\; int\_\{\; au\_1\}^\{\; au\_2\}\; frac\{mu\_0\; q^2\}\{6\; pi\; c\}\; mathbf\{dot\{a\; cdot\; mathbf\{v\}\; dt$.

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 absurd results. See the quote from Rohrlich 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

:$m\; dot\; \{mathbf\{v\}\; \}\; =\; mathbf\{F\}\_mathrm\{rad\}\; +\; mathbf\{F\}\_mathrm\{ext\}\; =\; m\; t\_0\; ddot\; \{\; mathbf$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

:$m\; dot\; \{mathbf\{v\}\; \}\; =\; \{1\; over\; t\_0\}\; int\_t^\{infty\}\; exp\; left(\; -\; \{t\text{'}-t\; over\; t\_0\; \}\; ight\; )\; ,\; mathbf\{F\}\_mathrm\{ext\}(t\text{'})\; ,\; dt\text{'}\; .$

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

:$exp\; left(\; -\{t\text{'}-t\; over\; t\_0\; \}\; ight\; )$

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.

**ee also***

Max Abraham

*Cyclotron radiation

*Radiation resistance

*Radiation damping

*Synchrotron radiation

*Wheeler–Feynman absorber theory **References****Further reading***cite book | author=Griffiths, David J.|title=Introduction to Electrodynamics|edition=3rd ed. | publisher=Prentice Hall |year=1998 |id=ISBN 0-13-805326-X See sections 11.2.2 and 11.2.3

*cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 0-471-30932-X

*Donald H. Menzel, "Fundamental Formulas of Physics", 1960, Dover Publications Inc., ISBN 0-486-60595-7, vol. 1, page 345.**External links*** [

*http://www.mathpages.com/home/kmath528/kmath528.htm MathPages - Does A Uniformly Accelerating Charge Radiate?*]

* [*http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html Feynman: The Development of the Space-Time View of Quantum Electrodynamics*]

*Wikimedia Foundation.
2010.*

### См. также в других словарях:

**Abraham-Lorentz-Dirac force**— In electrodynamics, the Abraham Lorentz Dirac force is the force experienced by a relativistic charged particle due to an electromagnetic field. It is a modification of the Abraham Lorentz force, which describes the same effect, but does not… … Wikipedia**Lorentz force**— This article is about the equation governing the electromagnetic force. For a qualitative overview of the electromagnetic force, see Electromagnetism. For magnetic force of one magnet on another, see force between magnets. Electromagnetism … Wikipedia**Force d'Abraham-Lorentz**— En électromagnétisme, la force d Abraham Lorentz est la force à laquelle est soumise un objet chargé électriquement en raison du champ électromagnétique qu il crée par son déplacement. Pour cette raison, on parle aussi de force de réaction au… … Wikipédia en Français**Force d'Abraham-Lorenz**— Force d Abraham Lorentz En électromagnétisme, la force d Abraham Lorentz est la force à laquelle est soumise un objet chargé électriquement en raison du champ électromagnétique qu il crée par son déplacement. Pour cette raison, on parle aussi de… … Wikipédia en Français**Lorentz ether theory**— What is now called Lorentz Ether theory ( LET ) has its roots in Hendrik Lorentz s Theory of electrons , which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century … Wikipedia**Lorentz transformation**— A visualisation of the Lorentz transformation (full animation). Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest with respect to that frame; the … Wikipedia**Abraham–Minkowski controversy**— The Abraham–Minkowski controversy is a physics debate concerning electromagnetic momentum within dielectric media. The preponderance of evidence in the debate suggests that the Abraham equation is correct,[1] but some investigators disagree. Two… … Wikipedia**Magnetic radiation reaction force**— In the physics of electromagnetism, one can derive an electric radiation reaction force for an accelerating charged particle caused by the particle emitting electromagnetic radiation. Likewise, a magnetic radiation reaction force can be derived… … Wikipedia**Hendrik Antoon Lorentz**— Hendrik Antoon Lorentz, gemalt von Menso Kamerlingh Onnes Hendrik Antoon Lorentz (* 18. Juli 1853 in Arnheim; † 4. Februar 1928 in Haarlem) war ein niederländischer … Deutsch Wikipedia**Electromagnetism**— Electricity · … Wikipedia