Electro-dynamics

Ampère’s immortal work is “Théorie des phénomènes electro-dynamiques, uniquement fondée sur expérience”.

On the other hand, his successors see them clearly enough, because their attention is attracted by the weak points in Ampère’s solution. They made fresh hypotheses, but this time deliberately.

How many times they had to change them before they reached the classic system, which is perhaps even now not quite definitive, we shall see.

Ampère’s Theory

In Ampère’s experimental study of the mutual action of currents, he has operated, and he could operate only, with closed currents.

This was not because he denied the existence or possibility of open currents. If two conductors are positively and negatively charged and brought into communication by a wire, a current is set up which passes from one to the other until the two potentials are equal.

According to the ideas of Ampère’s time, this was considered to be an open current; the current was known to pass from the first conductor to the second, but they did not know it returned from the second to the first.

All currents of this kind were therefore considered by Ampère to be open currents—for instance, the currents of discharge of a condenser; he was unable to experiment on them, their duration being too short.

Another kind of open current may be imagined. Suppose we have two conductors A and B connected by a wire AMB.

Small conducting masses in motion are first of all placed in contact with the conductor B, receive an electric charge, and leaving B are set in motion along a path BNA, carrying their charge with them. On coming into contact with A they lose their charge, which then returns to B along the wire AMB.

Now here we have, in a sense, a closed circuit, since the electricity describes the closed circuit BNAMB;

But the two parts of the current are quite different.

In the wire AMB the electricity is displaced through a fixed conductor like a voltaic current, overcoming an ohmic resistance and developing heat; we say that it is displaced by conduction. In the part BNA the electricity is carried by a moving conductor, and is said to be displaced by convection. If therefore the convection currentelectro-dynamics.

is considered to be perfectly analogous to the conduction current, the circuit BNAMB is closed; if on the contrary the convection current is not a “true current,” and, for instance, does not act on the magnet, there is only the conduction current AMB, which is open.

For example, if we connect by a wire the poles of a Holtz machine, the charged rotating disc transfers the electricity by convection from one pole to the other, and it returns to the first pole by conduction through the wire.

But currents of this kind are very difficult to produce with appreciable intensity; in fact, with the means at Ampère’s disposal we may almost say it was impossible.

To sum up, Ampère could conceive of the existence of two kinds of open currents, but he could experiment on neither, because they were not strong enough, or because their duration was too short.

Experiment therefore could only show him the action of a closed current on a closed current—or more accurately, the action of a closed current on a portion of current, because a current can be made to describe a closed circuit, of which part may be in motion and the other part fixed.

The displacements of the moving part may be studied under the action of another closed current. On the other hand, Ampère had no means of studying the action of an open current either on a closed or on another open current.

1. The Case of Closed Currents

In the case of the mutual action of two closed currents, experiment revealed to Ampère remarkably simple laws. The following will be useful to us in the sequel:

1. If the intensity of the currents is kept constant, and if the two circuits, after having undergone any displacements and deformations whatever, return finally to their initial positions, the total work done by the electrodynamical actions is zero. In other words, there is an electro-dynamical potential of the two circuits proportional to the product of their intensities, and depending on the form and relative positions of the circuits; the work done by the electro-dynamical actions is equal to the change of this potential.

2. The action of a closed solenoid is zero.

3. The action of a circuit C on another voltaic circuit C 0 depends only on the “magnetic field” developed by the circuit C.

At each point in space we can, in fact, define in magnitude and direction a certain force called “magnetic force,” which enjoys the following properties:

(a) The force exercised by C on a magnetic pole is applied to that pole, and is equal to the magnetic force multiplied by the magnetic mass of the pole.

(b) A very short magnetic needle tends to take the direction of the magnetic force, and the couple to which it tends to reduce is proportional to the product of the magnetic force, the magnetic moment of the needle, and the sine of the dip of the needle.

(c) If the circuit C 0 is displaced, the amount of the work done by the electro-dynamic action of C on C 0 will be equal to the increment of “flow of magnetic force” which passes through the circuit.

2. Action of a Closed Current on a Portion of Current

Ampère was unable to produce the open current. He had only one way of studying the action of a closed current on a portion of current. This was by operating on a circuit C composed of two parts, one movable and the other fixed. The movable part was, for instance, a movable wire αβ, the ends α and β of which could slide along a fixed wire. In one of the positions of the movable wire the end α rested on the point A, and the end β on the point B of the fixed wire.

The current ran from α to β—i.e., from A to B along the movable wire, and then from B to A along the fixed wire. This current was therefore closed.

In the second position, the movable wire having slipped, the points α and β were respectively at A 0 and B 0 on the fixed wire. The current ran from α to β—i.e., from A 0 to B 0 on the movable wire, and returned from B 0 to B, and then from B to A, and then from A to A 0 —all on the fixed wire. This current was also closed.

If a similar circuit be exposed to the action of a closed current C, the movable part will be displaced just as if it were acted on by a force.

Ampère admits that the force, apparently acting on the movable part AB, representing the action of C on the portion αβ of the current, remains the same whether an open current runs through αβ, stopping at α and β, or whether a closed current runs first to β and then returns to α through the fixed portion of the circuit.

This hypothesis seemed natural enough, and Ampère innocently assumed it.

Nevertheless the hypothesis is not a necessity, for we shall presently see that Helmholtz rejected it. However that may be, it enabled Ampère, although he had never produced an open current, to lay down the laws of the action of a closed current on an open current, or even on an element of current. They are simple:

(1) The force acting on an element of current is applied to that element; it is normal to the element and to the magnetic force, and proportional to that component of the magnetic force which is normal to the element.electro-dynamics.

(2) The action of a closed solenoid on an element of current is zero. But the electro-dynamic potential has disappeared—i.e., when a closed and an open current of constant intensities return to their initial positions, the total work done is not zero.

3. Continuous Rotations

The most remarkable electro-dynamical experiments are those in which continuous rotations are produced, and which are called unipolar induction experiments.

A magnet may turn around its axis. A current passes first through a fixed wire and then enters the magnet by the pole N, for instance, passes through half the magnet, and emerges by a sliding contact and re-enters the fixed wire. The magnet then begins to rotate continuously.

This is Faraday’s experiment.

How is it possible?

If it were a question of two circuits of invariable form, C fixed and C 0 movable about an axis, the latter would never take up a position of continuous rotation; in fact, there is an electro-dynamical potential; there must therefore be a position of equilibrium when the potential is a maximum.

Continuous rotations are therefore possible only when the circuit C 0 is composed of two parts—one fixed, and the other movable about an axis, as in the case of Faraday’s experiment.

Here again it is convenient to draw a distinction.

The passage from the fixed to the movable part, or vice versâ, 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 sliding contact, the same point of the movable part coming successively into contact with the different 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 up a position of equilibrium; but, when at the point of reaching that position, the sliding contact puts the moving part in contact with a fresh point in the fixed part.

It changes the connexions and therefore the conditions of equilibrium, so that as the position of equilibrium is ever eluding, so to speak, the system which is trying to reach it, rotation may take place indefinitely.

Ampère admits that the action of the circuit on the movable part of C 0 is the same as if the fixed part of C 0 did not exist, and therefore as if the current passing through the movable part were an open current.

He concluded that the action of a closed on an open current, or vice versâ, that of an open current on a fixed current, may give rise to continuous rotation. But this conclusion depends on the hypothesis which I have enunciated, and to which, as I said above, Helmholtz declined to subscribe.

4. Mutual Action of Two Open Currents

All experiment breaks down in the mutual action of two open currents, and in particular that of two elements of current.

Ampère falls back on hypothesis and assumes:

1. The mutual action of two elements reduces to a force acting along their join
2. The action of two closed currents is the resultant of the mutual actions of their different elements, which are the same as if these elements were isolated.

Here again Ampère makes 2 hypotheses without being aware of it. These 2, together with the experiments on closed currents, determine completely the law of 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 electro-dynamical 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 we have above defined this force in three different ways:

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.

In the case with which we are immediately concerned, not only are these three definitions not in harmony, but each has lost its meaning:—

1. A magnetic pole is no longer acted on by a unique force applied to that pole. We have seen, in fact, 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 zero. It decomposes into a director couple, properly so called, and a supplementary couple which tends to produce the continuous rotation of which we have spoken above.

3. Finally, the force acting on an element of a current is not normal to that 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 also exercise the same action on an indefinitely small magnetic needle, or on an element of current placed at the point in space at which the pole is.

Well, this is true if the two systems only contain closed currents, and according to Ampère it would not be true if the systems contained open currents.

It is sufficient to remark, for instance, that if a magnetic pole is placed at A and an element at B, the direction of the element being in AB produced, this element, which will exercise no action on the pole, will exercise an action either on a magnetic needle placed at A, or on an element of current at A.

5. Induction

The discovery of electro-dynamical induction followed not long after 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 us to deduce the laws of induction from the electro-dynamical laws of Ampère.

But on the condition, as Bertrand has shown,—that we make a certain number of hypotheses.

The same principle again enables this deduction to be made in the case of open currents, although the result cannot be tested by experiment, since such currents cannot be produced.

If we wish to compare this method of analysis with Ampère’s theorem on open currents, we get results which are calculated to surprise us. In the first place, induction cannot be deduced from the variation of the magnetic field by the well-known formula of scientists and practical men.

In fact, there is no magnetic field.

But further, if a circuit C is subjected to the induction of a variable voltaic system S, and 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, then so long as after these variations the system eventually returns to its initial position, it seems natural to suppose that the mean electro-motive force induced in the circuit C is zero.

This is true if the circuit C is closed, and if the system S only contains closed currents. It is no longer true if we accept the theory of Ampère, since there would be open currents. So that not only will induction no longer be the variation of the flow of magnetic force in any of the usual senses of the word, but it cannot be represented by the variation of that force whatever it may be.