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.

1 Definition
2 Paradoxes
3 Non-Relativistic form
4 Relativistic form
5 References
6 See also

[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]