II. Theory of Helmholtz.—I have dwelt upon the consequences of Ampère's theory, and of his method of explaining open currents.
It is difficult to overlook the paradoxical and artificial character of the propositions to which we are thus led. One can not help thinking 'that can not be so.'
We understand therefore why Helmholtz was led to seek something else.
Helmholtz rejects Ampère's fundamental hypothesis, to wit, that the mutual action of two elements of current reduces to a force along their join. He assumes that an element of current is not subjected to a single force, but to a force and a couple. It is just this which gave rise to the celebrated polemic between Bertrand and Helmholtz.
Helmholtz replaces Ampère's hypothesis by the following: two elements always admit of an electrodynamic potential depending solely on their position and orientation; and the work of the forces that they exercise, one on the other, is equal to the variation of this potential. Thus Helmholtz can no more do without hypothesis than Ampère; but at least he does not make one without explicitly announcing it.
In the case of closed currents, which are alone accessible to experiment, the two theories agree.
In all other cases they differ.
In the first place, contrary to what Ampère supposed, the force which seems to act on the movable portion of a closed current is not the same as would act upon this movable portion if it were isolated and constituted an open current.
Let us return to the circuit C´, of which we spoke above, and which was formed of a movable wire αβ sliding on a fixed wire. In the only experiment that can be made, the movable portion αβ is not isolated, but is part of a closed circuit. When it passes from AB to A´B´, the total electrodynamic potential varies for two reasons:
1º It undergoes a first increase because the potential of A´B´ with respect to the circuit C is not the same as that of AB;
2º It takes a second increment because it must be increased by the potentials of the elements AA´, BB´ with respect to C.
It is this double increment which represents the work of the force to which the portion AB seems subjected.
If, on the contrary, αβ were isolated, the potential would undergo only the first increase, and this first increment alone would measure the work of the force which acts on AB.
In the second place, there could be no continuous rotation without sliding contact, and, in fact, that, as we have seen à propos of closed currents, is an immediate consequence of the existence of an electrodynamic potential.
In Faraday's experiment, if the magnet is fixed and if the part of the current exterior to the magnet runs along a movable wire, that movable part may undergo a continuous rotation. But this does not mean to say that if the contacts of the wire with the magnet were suppressed, and an open current were to run along the wire, the wire would still take a movement of continuous rotation.