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DEFINITION AND DESCRIPTION

Mathematically, the Abraham-Lorentz force is given by:

\mathbf{F}_\mathrm{rad} = rac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}}

for small velocities. According to the Larmor Formula , an accelerating charge emits radiation, 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. The Abraham-Lorentz force is the average force on an accelerating charge due to the emission of radiation.


DERIVATION

We begin with the Larmor formula for radiation of a point charge:

:P = rac{\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} rac{\mu_0 q^2 a^2}{6 \pi c} dt = - \int_{ au_1}^{ au_2} rac{\mu_0 q^2}{6 \pi c} rac{d \mathbf{v}}{dt} \cdot rac{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:





PROBLEMS WITH THE ABRAHAM-LORENTZ FORCE


If we have no external forces acting on a particle, we have

:\mathbf{F}_\mathrm{net} = m \mathbf{a} = \mathbf{F}_\mathrm{rad} = rac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}}.

This equation has the solution

:\mathbf{a} = \mathbf{a}_0 e^{t / t_0}

where

:t_0 = rac{\mu_0 q^2}{6 \pi m c}.

If we do not set \mathbf{a}_0 = \mathbf{0}, then we get acceleration exponentially increasing, known as a Runaway Solution . However, it can be shown that if we do set \mathbf{a}_0 = \mathbf{0} in the presence of an external force, then we end up with acceleration occurring before the external force is applied, or "pre-acceleration."


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