Electrodynamic effect of closed, circular currents

II. Example: Electrodynamic effect of closed, circular currents on the potential law (*).

Jb is the current intensity of the bth circular current

pa is the coordinates of the ponderable masses, whose vis viva we shall neglect.

The function H has the form: (5) H = −

in which Qb,c are functions of pa , and each of the two successive indices b and c refer to all current circles. The induced electromotive forces, which I would like to denote by Eb , are:

If a permanent magnet whose position is given by the coordinate p0 is involved then a series of linear terms will be added to H that I would like to denote by h, and which will have the form:

in which the Rb are functions of the coordinates pa and p0 .

The calculation of the forces comes about here by the same methods. The total electrodynamic energy is:

The function h will vanish, since:

The E = − H that enters into this, like the vis viva on ponderable masses, is a necessarily positive quantity for closed currents (*).

In addition, E is a homogeneous function of degree two in the Ja , and the same considerations can be applied to it as the ones that were discussed at the end of § 1, in which the determinant of the quantities

cannot be equal to zero identically.