Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/463

from the action of other forces, and ${\frac {1}{2}}mv^{2}$ , the kinetic energy of the particle, we find as the equation of energy

${\frac {1}{2}}\left(m-{\frac {4}{3}}{\frac {\pi a\sigma e}{c^{2}}}\right)v^{2}+4\pi a\sigma e+V={\text{const.}}$

(22)

Since the second term of the coefficient of $v^{2}$ may be increased in definitely by increasing $a$, the radius of the sphere, while the surface-density $\sigma$ remains constant, the coefficient of $v^{2}$ may be made negative. Acceleration of the motion of the particle would then correspond to diminution of its vis viva, and a body moving in a closed path and acted on by a force like friction, always opposite in direction to its motion, would continually increase in velocity, and that without limit. This impossible result is a necessary consequence of assuming any formula for the potential which introduces negative terms into the coefficient of $v^{2}$.

855.] But we have now to consider the application of Weber's theory to phenomena which can be realized. We have seen how it gives Ampère's expression for the force of attraction between two elements of electric currents. The potential of one of these elements on the other is found by taking the sum of the values of the potential $\psi$ for the four combinations of the positive and negative currents in the two elements. The result is, by equation (20), taking the sum of the four values of $\left({\frac {dr}{dt}}\right)^{2}$,

$-ii'\,ds\,ds'{\frac {1}{r}}{\frac {dr}{ds}}{\frac {dr}{ds'}},$

(23)

and the potential of one closed current on another is

$-ii'\int \int {\frac {1}{r}}{\frac {dr}{ds}}{\frac {dr}{ds'}}ds\,ds'=ii'M,$

(24)

where

$M=\int \int {\frac {\cos \epsilon }{r}}ds\,ds',{\text{as in Arts. 423, 524.}}$

In the case of closed currents, this expression agrees with that which we have already (Art. 524) obtained^[1].

Weber's Theory of the Induction of Electric Currents.

856.] After deducing from Ampère's formula for the action between the elements of currents, his own formula for the action between moving electric particles, Weber proceeded to apply his formula to the explanation of the production of electric currents by

1. In the whole of this investigation Weber adopts the electrodynamic system of units. In this treatise we always use the electromagnetic system. The electro-magnetic unit of current is to the electrodynamic unit in the ratio of ${\sqrt {2}}$ to 1. Art. 526.