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.
I have just said in fact that an isolated element is not acted upon in the same way as a movable element making part of a closed circuit.
Another difference: The action of a closed solenoid on a closed current is null according to experiment and according to the two theories. Its action on an open current would be null according to Ampère; it would not be null according to Helmholtz. From this follows an important consequence. We have given above three definitions of magnetic force. The third has no meaning here since an element of current is no longer acted upon by a single force. No more has the first any meaning. What, in fact, is a magnetic pole? It is the extremity of an indefinite linear magnet. This magnet may be replaced by an indefinite solenoid. For the definition of magnetic force to have any meaning, it would be necessary that the action exercised by an open current on an indefinite solenoid should depend only on the position of the extremity of this solenoid, that is to say, that the action on a closed solenoid should be null. Now we have just seen that such is not the case.
On the other hand, nothing prevents our adopting the second definition, which is founded on the measurement of the director couple which tends to orientate the magnetic needle.