Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/248

This expression for the force between two electric charges was taken by Weber as the basis of his theory. Weber's is the first of the electron-theories—a name given to any theory which attributes the phenomena of electrodynamics to the agency of moving electric charges, the forces on which depend not only on the position of the charges (as in electrostatics), but also on their velocity.

The latter feature of Weber's theory led its earliest critics to deny that his law of force could be reconciled with the principle of conservation of energy. They were, however, mistaken on this point, as may be seen from the following considerations. The above expression for the force between two charges may be written in the form

$-{\frac {dll}{dr}}+{\frac {d}{dt}}\left({\frac {\partial U}{\partial {\dot {r}}}}\right)$,

where U denotes the expression

${\frac {ee^{\prime }c^{2}}{r}}\left(1+{\frac {{\dot {r}}^{2}}{2c^{2}}}\right)^{2}$.

Consider now two material particles at distance r apart, whose mechanical kinetic energy is T, and whose mechanical potential energy is V, and which carry charges e and e′. The equations of motion of these particles will be exactly the same as the equations of motion of a dynamical system for which the kinetic energy is

$T-{\frac {ee^{\prime }{\dot {r}}^{2}}{2r}}$,

and the potential energy is

$V+{\frac {ee^{\prime }c^{2}}{r}}$.

To such a system the principle of conservation of energy may be applied: the equation of energy is, in fact,

$T+V-{\frac {1}{2r}}ee^{\prime }r^{2}+{\frac {ee^{\prime }c^{2}}{r}}=\ \mathrm {constant}$^{[errata 1]}.

Errata

1. Correction: $T+V-{\frac {1}{2r}}ee^{\prime }r^{2}+{\frac {ee^{\prime }c^{2}}{r}}=\ \mathrm {constant}$ should be amended to $T+V-{\frac {1}{2r}}ee^{\prime }{\dot {r}}^{2}+{\frac {ee^{\prime }c^{2}}{r}}=\ \mathrm {constant}$