Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/267

Discussing next the mutual energy due to the approach of a permanent magnet and a circuit carrying a current, he arrived at the remarkable conclusion that in this case there is no electrokinetic energy which depends on the mutual action; the energy is simply the sum of that due to the permanent magnets and that due to the currents. If a permanent magnet is caused to approach a circuit carrying a current, the electromotive force acting in the circuit is thereby temporarily increased; the amount of energy dissipated as Joulian heat, and the speed of the chemical reactions in the cells, are temporarily increased also. But the increase in the Joulian heat is exactly equal to the increase in the energy derived from consumption of chemicals, together with the mechanical work done on the magnet by the operator who moves it; so that the balance of energy is perfect, and none needs to be added to or taken from the electrokinetic form. It will now be evident why it was that Helmholtz escaped in this case the errors into which he was led in other cases by his neglect of electrokinetic energy; for in this case there was no electrokinetic energy to neglect.

Two years later, in 1853, Thomson^[1] gave a new form to the expression for the energy of a system of permanent and temporary magnets.

We have seen that the energy of such a system is represented by

${\tfrac {1}{2}}\iiint \rho _{0}\phi dx\ dy\ dz$,

where ρ_e denotes the density of Poisson's equivalent magnetization for the permanent magnets, and φ denotes the magnetic potential, and where the integration may be extended over the whole of space. Substituting for ρ₀ its value - div I₀,^[2] the expression may be written in the form

$-{\tfrac {1}{2}}\iiint \phi \mathrm {div} \ \mathbf {I} _{0}\ dx\ dy\ dz$;

1. Proc. Glasgow Phil. Sou. iii (1853), p. 281; Kelvin's Math, and Phys. Papers, i, p. 521.
2. Cf. p. 64.