Solutions to the Equation of Motion

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)