Abraham-Lorentz-Dirac force


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 account for the effects of special relativity.

Definition

The expression for the Abraham-Lorentz-Dirac force was derived by Paul Dirac in 1938 [Dirac, P.A.M. (1938) Classical theory of radiating electrons. Proc. Roy. Soc. of London. A929:0148-0169. [http://www.jstor.org/view/00804630/ap000639/00a00000/0 JSTOR] ] and is given by :F^{mbox{rad_mu = frac{mu_o q^2}{6 pi m c}left(frac{d^2 p_mu}{d au^2}+frac{p_mu}{m^2 c^2} ,left(frac{d p_ u}{d au}frac{d p^ u}{d au} ight) ight)

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 extbf{F} cdot extbf{v} dt

Larmor's Formula describes the power of a system in a non-relativistic interpretation.

Paradoxes

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 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 [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)] ] , and Medina. [ [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] ]

Non-Relativistic form

:P = frac{mu_o q^2 a^2}{6 pi c}

Relativistic form

Liénard generalized Larmor's formula into a relativistic formulation in the co-moving frame.

:P = frac{mu_o q^2 a^2 gamma^6}{6 pi c}

References

ee also

*Abraham-Lorentz force


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