3º Finally the force acting on an element of current is not normal to this element.
In other words, the unity of the magnetic force has disappeared.
Let us see in what this unity consists. Two systems which exercise the same action on a magnetic pole will exert also the same action on an indefinitely small magnetic needle, or on an element of current placed at the same point of space as this pole.
Well, this is true if these two systems contain only closed currents; this would no longer be true if these two systems contained open currents.
It suffices to remark, for instance, that, if a magnetic pole is placed at A and an element at B, the direction of the element being along the prolongation of the sect AB, this element which will exercise no action on this pole will, on the other hand, exercise an action either on a magnetic needle placed at the point A, or on an element of current placed at the point A.
5. Induction.—We know that the discovery of electrodynamic induction soon followed the immortal work of Ampère.
As long as it is only a question of closed currents there is no difficulty, and Helmholtz has even remarked that the principle of the conservation of energy is sufficient for deducing the laws of induction from the electrodynamic laws of Ampère. But always on one condition, as Bertrand has well shown; that we make besides a certain number of hypotheses.
The same principle again permits this deduction in the case of open currents, although of course we can not submit the result to the test of experiment, since we can not produce such currents.
If we try to apply this mode of analysis to Ampère's theory of open currents, we reach results calculated to surprise us.
In the first place, induction can not be deduced from the variation of the magnetic field by the formula well known to savants and practicians, and, in fact, as we have said, properly speaking there is no longer a magnetic field.