Abraham–Lorentz–Dirac force

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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.

[edit] Definition

The expression for the Abraham-Lorentz-Dirac force was derived by Paul Dirac in 1938^[1] and is given by

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

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

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

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

[edit] 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 Yaghjian^[2], and a fuller discussion of its resolution is made by Rohrlich^[3] , and Medina.^[4]

[edit] Non-Relativistic form

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

[edit] Relativistic form

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

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

[edit] References

^ Paul A.M. Dirac, (1938) Classical theory of radiating electrons. Proc. Roy. Soc. of London. A929:0148-0169. JSTOR
^ 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/books?id=keeyf1cJLjwC&printsec=frontcover&lr=#PPA10,M1.
^ Fritz Rohrlich: The dynamics of a charged sphere and the electron Am J Phys 65 (11) p. 1051 (1997)
^ Rodrigo Medina, Radiation reaction of a classical quasi-rigid extended particle J. Phys. A: Math. Gen. A39 (2006) 3801-3816

[edit] See also

Abraham-Lorentz force

[0] Paul A.M. Dirac, (1938) Classical theory of radiating electrons. Proc. Roy. Soc. of London. A929:0148-0169. JSTOR

[Yaghjian-1] 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/books?id=keeyf1cJLjwC&printsec=frontcover&lr=#PPA10,M1.

[Rohrlich-2] Fritz Rohrlich: The dynamics of a charged sphere and the electron Am J Phys 65 (11) p. 1051 (1997)

[3] Rodrigo Medina, Radiation reaction of a classical quasi-rigid extended particle J. Phys. A: Math. Gen. A39 (2006) 3801-3816

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Abraham–Lorentz–Dirac force

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Paradoxes

[edit] Non-Relativistic form

[edit] Relativistic form

[edit] References

[edit] See also

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