The Foundations of Science: Science and Hypothesis, The Value of Science Science and Method Part 28

If a like current is subjected to the action of a closed current _C_, the movable part will be displaced just as if it were acted upon by a force. Ampere _a.s.sumes_ that the apparent force to which this movable part _AB_ seems thus subjected, representing the action of the _C_ on the portion [alpha][beta] of the current, is the same as if [alpha][beta] were traversed by an open current, stopping at [alpha] and [beta], in place of being traversed by a closed current which after arriving at [beta] returns to [alpha] through the fixed part of the circuit.

This hypothesis seems natural enough, and Ampere made it unconsciously; nevertheless _it is not necessary_, since we shall see further on that Helmholtz rejected it. However that may be, it permitted Ampere, though he had never been able to produce an open current, to enunciate the laws of the action of a closed current on an open current, or even on an element of current.

The laws are simple:

1 The force which acts on an element of current is applied to this element; it is normal to the element and to the magnetic force, and proportional to the component of this magnetic force which is normal to the element.

2 The action of a closed solenoid on an element of current is null.

But the electrodynamic potential has disappeared, that is to say that, when a closed current and an open current, whose intensities have been maintained constant, return to their initial positions, the total work is not null.

3. _Continuous Rotations._--Among electrodynamic experiments, the most remarkable are those in which continuous rotations are produced and which are sometimes called _unipolar induction_ experiments. A magnet may turn about its axis; a current pa.s.ses first through a fixed wire, enters the magnet by the pole _N_, for example, pa.s.ses through half the magnet, emerges by a sliding contact and reenters the fixed wire.

The magnet then begins to rotate continuously without being able ever to attain equilibrium; this is Faraday's experiment.

How is it possible? If it were a question of two circuits of invariable form, the one _C_ fixed, the other _C'_ movable about an axis, this latter could never take on continuous rotation; in fact there is an electrodynamic potential; there must therefore be necessarily a position of equilibrium when this potential is a maximum.

Continuous rotations are therefore possible only when the circuit _C'_ is composed of two parts: one fixed, the other movable about an axis, as is the case in Faraday's experiment. Here again it is convenient to draw a distinction. The pa.s.sage from the fixed to the movable part, or inversely, may take place either by simple contact (the same point of the movable part remaining constantly in contact with the same point of the fixed part), or by a sliding contact (the same point of the movable part coming successively in contact with diverse points of the fixed part).

It is only in the second case that there can be continuous rotation.

This is what then happens: The system tends to take a position of equilibrium; but, when at the point of reaching that position, the sliding contact puts the movable part in communication with a new point of the fixed part; it changes the connections, it changes therefore the conditions of equilibrium, so that the position of equilibrium fleeing, so to say, before the system which seeks to attain it, rotation may take place indefinitely.

Ampere a.s.sumes that the action of the circuit on the movable part of _C'_ is the same as if the fixed part of _C'_ did not exist, and therefore as if the current pa.s.sing through the movable part were open.

He concludes therefore that the action of a closed on an open current, or inversely that of an open current on a closed current, may give rise to a continuous rotation.

But this conclusion depends on the hypothesis I have enunciated and which, as I said above, is not admitted by Helmholtz.

4. _Mutual Action of Two Open Currents._--In what concerns the mutual actions of two open currents, and in particular that of two elements of current, all experiment breaks down. Ampere has recourse to hypothesis.

He supposes:

1 That the mutual action of two elements reduces to a force acting along their join;

2 That the action of two closed currents is the resultant of the mutual actions of their diverse elements, which are besides the same as if these elements were isolated.

What is remarkable is that here again Ampere makes these hypotheses unconsciously.

However that may be, these two hypotheses, together with the experiments on closed currents, suffice to determine completely the law of the mutual action of two elements. But then most of the simple laws we have met in the case of closed currents are no longer true.

In the first place, there is no electrodynamic potential; nor was there any, as we have seen, in the case of a closed current acting on an open current.

Next there is, properly speaking, no magnetic force.

And, in fact, we have given above three different definitions of this force:

1 By the action on a magnetic pole;

2 By the director couple which orientates the magnetic needle;

3 By the action on an element of current.

But in the case which now occupies us, not only these three definitions are no longer in harmony, but each has lost its meaning, and in fact:

1 A magnetic pole is no longer acted upon simply by a single force applied to this pole. We have seen in fact that the force due to the action of an element of current on a pole is not applied to the pole, but to the element; it may moreover be replaced by a force applied to the pole and by a couple;

2 The couple which acts on the magnetic needle is no longer a simple director couple, for its moment with respect to the axis of the needle is not null. It breaks up into a director couple, properly so called, and a supplementary couple which tends to produce the continuous rotation of which we have above spoken;

3 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 s.p.a.ce 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 Ampere.

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 Ampere. 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 a.n.a.lysis to Ampere'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.

But, further, if a circuit _C_ is subjected to the induction of a variable voltaic system _S_, if this system _S_ be displaced and deformed in any way whatever, so that the intensity of the currents of this system varies according to any law whatever, but that after these variations the system finally returns to its initial situation, it seems natural to suppose that the _mean_ electromotive force induced in the circuit _C_ is null.

This is true if the circuit _C_ is closed and if the system _S_ contains only closed currents. This would no longer be true, if one accepts the theory of Ampere, if there were open currents. So that not only induction will no longer be the variation of the flow of magnetic force, in any of the usual senses of the word, but it can not be represented by the variation of anything whatever.

II. THEORY OF HELMHOLTZ.--I have dwelt upon the consequences of Ampere'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 Ampere's fundamental hypothesis, to wit, that the mutual action of two elements of current reduces to a force along their join. He a.s.sumes 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 Ampere'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 Ampere; 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 Ampere 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 const.i.tuted an open current.

Let us return to the circuit _C'_, of which we spoke above, and which was formed of a movable wire [alpha][beta] sliding on a fixed wire. In the only experiment that can be made, the movable portion [alpha][beta]

is not isolated, but is part of a closed circuit. When it pa.s.ses from _AB_ to _A'B'_, the total electrodynamic potential varies for two reasons: