Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/208

We shall return again to this method of Faraday. In the mean time we must enumerate the theories of induction which are founded on other considerations.

Lenz's Law.

542.] In 1834, Lenz^[1] enunciated the following remarkable relation between the phenomena of the mechanical action of electric currents, as defined by Ampère's formula, and the induction of electric currents by the relative motion of conductors. An earlier attempt at a statement of such a relation was given by Ritchie in the Philosophical Magazine for January of the same year, but the direction of the induced current was in every case stated wrongly. Lenz's law is as follows.—

If a constant current flows in the primary circuit $A$; and if, by the motion of $A$, or of the secondary circuit $B$, a current is induced in $B$, the direction of this induced current will be such that, by its electromagnetic action on $A$, it tends to oppose the relative motion of the circuits.

On this law J. Neumann^[2] founded his mathematical theory of induction, in which he established the mathematical laws of the induced currents due to the motion of the primary or secondary conductor. He shewed that the quantity $M$, which we have called the potential of the one circuit on the other, is the same as the electromagnetic potential of the one circuit on the other, which we have already investigated in connexion with Ampère's formula. We may regard J. Neumann, therefore, as having completed for the induction of currents the mathematical treatment which Ampère had applied to their mechanical action.

543.] A step of still greater scientific importance was soon after made by Helmholtz in his Essay on the Conservation of Force^[3], and by Sir W. Thomson^[4], working somewhat later, but independently of Helmholtz. They shewed that the induction of electric currents discovered by Faraday could be mathematically deduced from the electromagnetic actions discovered by Órsted and Ampère by the application of the principle of the Conservation of Energy.

Helmholtz takes the case of a conducting circuit of resistance $R$, in which an electromotive force $A$, arising from a voltaic or thermo-

1. Pogg., Ann. xxxi. 483 (1834).
2. Berlin Acad., 1845 and 1847.
3. Read before the Physical Society of Berlin, July 23, 1847. Translated in Taylor's 'Scientific Memoirs,' part ii. p. 114.
4. Trans. Brit. Ass., 1848, and Phil. Mag., Dec. 1851. See also his paper on Transient Electric Currents, Phil. Mag., 1853.