Helmholtz’s Theory

Ampère’s theory states his method of explaining the action of open currents.

• Ampère’s fundamental hypothesis is that the action of 2 elements of current reduces to a force along their join.

I find it difficult to disregard the paradoxical and artificial character of his propositions.

We feel bound to think “it cannot be so.”

Helmholtz has been led to look for something else.

He rejects the Ampere’s hypothesis.

He admits that an clement of current is not acted on by a single force but by a force and a couple. This is what gave rise to the celebrated polemic between Bertrand and Helmholtz.

Helmholtz replaces Ampère’s hypothesis:

Two elements of current always admit of an electro-dynamic potential, depending solely on their position and orientation. 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 do so without explicitly announcing it.

The 2 theories agree in closed currents which are accessible only to experiment.

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 that acting on the movable portion if it were isolated and if it constituted an open current.

Let us return to the circuit C 0 , of which we spoke above, and which was formed of a movable wire 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 0 B 0 , the total electro-dynamic potential varies for 2 reasons.

1. It has a slight increment because the potential of A 0 B 0 with respect to the circuit C is not the same as that of AB;
2. It has a second increment because it must be increased by the potentials of the elements AA 0 and B 0 B with respect to C.

It is this double increment which represents the work of the force acting upon the portion AB.

If, on the contrary, αβ be isolated, the potential would only have the first increment, and this first increment alone would measure the work of the force acting on AB.

In the second place, there could be no continuous rotation without sliding contact. In fact, in closed currents, it is an immediate consequence of the existence of an electro-dynamic potential.

In Faraday’s experiment, if the magnet is fixed, and if the part of the current external to the magnet runs along a movable wire, that movable wire may undergo continuous rotation.

But it does not mean 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 have a movement of continuous rotation.

An isolated element is not acted on in the same way as a movable element making part of a closed circuit. But there is another difference. The action of a solenoid on a closed current is zero according to experiment and according to the 2 theories.

Its action on an open current would be zero according to Ampère, and it would not be zero according to Helmholtz.

From this follows an important consequence.

We have given above 3 definitions of the magnetic force.

The third has no meaning here, since an element of current is no longer acted upon by a single force. Nor 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, the action exercised by an open current on an indefinite solenoid would only depend on the position of the extremity of that solenoid—i.e., that the action of a closed solenoid is zero.

But this is not the case.

On the other hand, there is nothing to prevent us from adopting the second definition which is founded on the measurement of the director couple which tends to orientate the magnetic needle.

But, if it is adopted, neither the effects of induction nor electro-dynamic effects will depend solely on the distribution of the lines of force in this magnetic field.

3. Difficulties raised by these Theories

Helmholtz’s theory is an advance on Ampère’s theory.

But every difficulty should be removed.

In both, the name “magnetic field” has no meaning, or, if we give it one by a more or less artificial convention, the ordinary laws so familiar to electricians no longer apply

Thus, the electro-motive force induced in a wire is no longer measured by the number of lines of force met by that wire.

And our objections do not proceed only from the fact that it is difficult to give up deeply-rooted habits of language and thought.

There is something more.

If we do not believe in actions at a distance, electro-dynamic phenomena must be explained by a modification of the medium.

This medium is the “magnetic field”.

The electro-magnetic effects should only depend on that field.

All these difficulties come from the hypothesis of open currents.

4. Maxwell’s Theory

Maxwell, with a stroke of the pen, caused these difficulties to vanish.

To him, all currents are closed currents.

Maxwell admits that if in a dielectric, the electric field happens to vary, this dielectric becomes the seat of a particular phenomenon acting on the galvanometer like a current and called the current of displacement.

If, then, two conductors bearing positive and negative charges are placed in connection by means of a wire, during the discharge there is an open current of conduction in that wire; but there are produced at the same time in the surrounding dielectric currents of displacement which close this current of conduction.

We know that Maxwell’s theory leads to the explanation of optical phenomena which would be due to extremely rapid electrical oscillations.

At that period such a conception was only a daring hypothesis which could be supported by no experiment.

But after 20 years, Maxwell’s ideas received the confirmation of experiment.

Hertz succeeded in producing systems of electric oscillations which reproduce all the properties of light, and only differ by the length of their wave—as violet differs from red.

In some measure he made a synthesis of light.

Hertz has not directly proved Maxwell’s fundamental idea of the action of the current of displacement on the galvanometer.

What he has shown directly is that electro-magnetic induction is not instantaneously propagated, as was supposed, but its speed is the speed of light. Yet, to suppose there is no current of displacement, and that induction is with the speed of light; or, rather, to suppose that the currents of displacement produce inductive effects, and that the induction takes place instantaneously—comes to the same thing.

This cannot be seen at the first glance, but it is proved by an analysis of which I must not even think of giving even a summary here.