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Solutions to the Equation of Motion

Part of the Lecture Notes in Physics Monographs book series (LNPMGR, volume 11)

Abstract

As a preliminary to solving the equation of motion (7.1) for the uniformly charged sphere of radius a and total charge e, write the magnitude of the four acceleration in (7.1) as
$$ \frac{{du_j }} {{ds}}\frac{{du^j }} {{ds}} = \frac{{(w \cdot w')^2 }} {{\gamma ^2 c^6 }} - \frac{{w'^2 }} {{c^4 }} $$
(8.1a)
where w is defined in terms of the velocity of the center of the shell by
$$ w = \gamma u, \gamma = (1 - u^2 /c^2 )^{ - 1/2} = (1 + w^2 /c^2 )^{1/2} $$
(8.1b)
and the primes denote derivatives with respect to the proper time
$$ d\tau = dt/\gamma . $$
(8.1c)
Insertion of (8.1) into (7.1) yields the three-vector equation for w
$$ \gamma F_{ext} = \frac{{e^2 }} {{8\pi \varepsilon _0 c^2 }}\left[ {\frac{{w'}} {a} - \frac{4} {{3a}}w'' + \frac{4} {{3c^3 }}\left( {w'^2 - \frac{{(w \cdot w')^2 }} {{c^2 \gamma ^2 }}} \right)w} \right] + O(a). $$
(8.2)

Keywords

External Force Electromagnetic Force Asymptotic Condition Radiation Reaction Corrected Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

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