Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/249

The first objection made to Weber's theory is thus disposed of; but another and more serious one now presents itself. The occurrence of the negative sign with the term $-ee^{\prime }{\dot {r}}^{2}/2r$ implies that a charge behaves somewhat as if its mass were negative, so that in certain circumstances its velocity might increase indefinitely under the action of a force opposed to the motion. This is one of the vulnerable points of Weber's theory, and has been the object of much criticism. In fact,^[1] suppose that one charged particle of mass μ is free to move, and that the other charges are spread uniformly over the surface of a hollow spherical insulator in which the particle is enclosed. The equation of conservation of energy is

${\tfrac {1}{2}}(\mu -ep)v^{2}+V=$ constant,

where e denotes the charge of the particle, v its velocity, V its potential energy with respect to the mechanical forces which act on it, and p denotes the quantity

$\iint {\frac {\sigma }{r}}\cos ^{2}({\hat {v.r}})dS$,

where the integration is taken over the sphere, and where σ denotes the surface-density; p is independent of the position of the particle μ within the sphere. If now the electric charge on the sphere is so great that ep is greater than μ, then v² and V must increase and diminish together, which is evidently absurd.

Leaving this objection unanswered, we proceed to show how Weber's law of force between electrons leads to the formulae for the induction of currents.

The mutual energy of two moving charges is

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

or ${\frac {ee^{\prime }c^{2}}{r}}\left[1-{\frac {\{(\mathbf {r.v^{\prime }} )-(\mathbf {r.v} )\}^{2}}{2c^{2}r^{2}}}\right]$, where v and v′ denote the velocities of the charges; so that the

1. This example was given by Helmholtz, Journal für Math. lxxv (1873), p. 35; Phil. Mag. xliv (1872), p. 530.