It is necessary that _U_ can be regarded as the potential energy of a system and _T_ as the _vis viva_ of the same system.
There is no difficulty as to _U_, but can _T_ be regarded as the _vis viva_ of a material system?
It is easy to show that this is always possible, and even in an infinity of ways. I will confine myself to referring for more details to the preface of my work, "electricite et optique."
Thus if the principle of least action can not be satisfied, no mechanical explanation is possible; if it can be satisfied, there is not only one, but an infinity, whence it follows that as soon as there is one there is an infinity of others.
One more observation.
Among the quant.i.ties that experiment gives us directly, we shall regard some as functions of the coordinates of our hypothetical molecules; these are our parameters _q_. We shall look upon the others as dependent not only on the coordinates, but on the velocities, or, what comes to the same thing, on the derivatives of the parameters _q_, or as combinations of these parameters and their derivatives.
And then a question presents itself: among all these quant.i.ties measured experimentally, which shall we choose to represent the parameters _q_?
Which shall we prefer to regard as the derivatives of these parameters?
This choice remains arbitrary to a very large extent; but, for a mechanical explanation to be possible, it suffices if we can make the choice in such a way as to accord with the principle of least action.
And then Maxwell asked himself whether he could make this choice and that of the two energies _T_ and _U_, in such a way that the electrical phenomena would satisfy this principle. Experiment shows us that the energy of an electromagnetic field is decomposed into two parts, the electrostatic energy and the electrodynamic energy. Maxwell observed that if we regard the first as representing the potential energy _U_, the second as representing the kinetic energy _T_; if, moreover, the electrostatic charges of the conductors are considered as parameters _q_ and the intensities of the currents as the derivatives of other parameters _q_; under these conditions, I say, Maxwell observed that the electric phenomena satisfy the principle of least action. Thenceforth he was certain of the possibility of a mechanical explanation.
If he had explained this idea at the beginning of his book instead of relegating it to an obscure part of the second volume, it would not have escaped the majority of readers.
If, then, a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others, that will render an account equally well of all the particulars revealed by experiment.
And this is confirmed by the history of every branch of physics; in optics, for instance, Fresnel believed vibration to be perpendicular to the plane of polarization; Neumann regarded it as parallel to this plane. An "experimentum crucis" has long been sought which would enable us to decide between these two theories, but it has not been found.
In the same way, without leaving the domain of electricity, we may ascertain that the theory of two fluids and that of the single fluid both account in a fashion equally satisfactory for all the observed laws of electrostatics.
All these facts are easily explicable, thanks to the properties of the equations of Lagrange which I have just recalled.
It is easy now to comprehend what is Maxwell"s fundamental idea.
To demonstrate the possibility of a mechanical explanation of electricity, we need not preoccupy ourselves with finding this explanation itself; it suffices us to know the expression of the two functions _T_ and _U_, which are the two parts of energy, to form with these two functions the equations of Lagrange and then to compare these equations with the experimental laws.
Among all these possible explanations, how make a choice for which the aid of experiment fails us? A day will come perhaps when physicists will not interest themselves in these questions, inaccessible to positive methods, and will abandon them to the metaphysicians. This day has not yet arrived; man does not resign himself so easily to be forever ignorant of the foundation of things.
Our choice can therefore be further guided only by considerations where the part of personal appreciation is very great; there are, however, solutions that all the world will reject because of their whimsicality, and others that all the world will prefer because of their simplicity.
In what concerns electricity and magnetism, Maxwell abstains from making any choice. It is not that he systematically disdains all that is unattainable by positive methods; the time he has devoted to the kinetic theory of gases sufficiently proves that. I will add that if, in his great work, he develops no complete explanation, he had previously attempted to give one in an article in the _Philosophical Magazine_. The strangeness and the complexity of the hypotheses he had been obliged to make had led him afterwards to give this up.
The same spirit is found throughout the whole work. What is essential, that is to say what must remain common to all theories, is made prominent; all that would only be suitable to a particular theory is nearly always pa.s.sed over in silence. Thus the reader finds himself in the presence of a form almost devoid of matter, which he is at first tempted to take for a fugitive shadow not to be grasped. But the efforts to which he is thus condemned force him to think and he ends by comprehending what was often rather artificial in the theoretic constructs he had previously only wondered at.
CHAPTER XIII
ELECTRODYNAMICS
The history of electrodynamics is particularly instructive from our point of view.
Ampere ent.i.tled his immortal work, "Theorie des phenomenes electrodynamiques, _uniquement_ fondee sur l"experience." He therefore imagined that he had made _no_ hypothesis, but he had made them, as we shall soon see; only he made them without being conscious of it.
His successors, on the other hand, perceived them, since their attention was attracted by the weak points in Ampere"s solution. They made new hypotheses, of which this time they were fully conscious; but how many times it was necessary to change them before arriving at the cla.s.sic system of to-day which is perhaps not yet final; this we shall see.
I. AMPERE"S THEORY.--When Ampere studied experimentally the mutual actions of currents, he operated and he only could operate with closed currents.
It was not that he denied the possibility of open currents. If two conductors are charged with positive and negative electricity and brought into communication by a wire, a current is established going from one to the other, which continues until the two potentials are equal. According to the ideas of Ampere"s time this was an open current; the current was known to go from the first conductor to the second, it was not seen to return from the second to the first.
So Ampere considered as open currents of this nature, for example, the currents of discharge of condensers; but he could not make them the objects of his experiments because their duration is too short.
Another sort of open current may also be imagined. I suppose two conductors, _A_ and _B_, connected by a wire _AMB_. Small conducting ma.s.ses in motion first come in contact with the conductor _B_, take from it an electric charge, leave contact with _B_ and move along the path _BNA_, and, transporting with them their charge, come into contact with _A_ and give to it their charge, which returns then to _B_ along the wire _AMB_.
Now there we have in a sense a closed circuit, since the electricity describes the closed circuit _BNAMB_; but the two parts of this current are very 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; it is said to be displaced by convection.
If then the current of convection is considered as altogether a.n.a.logous to the current of conduction, the circuit _BNAMB_ is closed; if, on the contrary, the convection current is not "a true current" and, for example, does not act on the magnet, there remains only the conduction current _AMB_, which is open.
For example, if we connect by a wire the two 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 sort are very difficult to produce with appreciable intensity. With the means at Ampere"s disposal, we may say that this was impossible.
To sum up, Ampere could conceive of the existence of two kinds of open currents, but he could operate 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 a current, because a current can be made to describe a closed circuit composed of a moving part and a fixed part. It is possible then to study the displacements of the moving part under the action of another closed current.
On the other hand, Ampere had no means of studying the action of an open current, either on a closed current or another open current.
1. _The Case of Closed Currents._--In the case of the mutual action of two closed currents, experiment revealed to Ampere remarkably simple laws.
I recall rapidly here those which 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 deformations and displacements whatsoever, return finally to their initial positions, the total work of the electrodynamic actions will be null.
In other words, there is an _electrodynamic potential_ of the two circuits, proportional to the product of the intensities, and depending on the form and relative position of the circuits; the work of the electrodynamic actions is equal to the variation of this potential.
2 The action of a closed solenoid is null.
3 The action of a circuit _C_ on another voltaic circuit _C"_ depends only on the "magnetic field" developed by this circuit. At each point in s.p.a.ce 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 ma.s.s of that 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 magnetic force, the magnetic moment of the needle and the sine of the dip of the needle;
(_c_) If the circuit _C_ is displaced, the work of the electrodynamic action exercised by _C_ on _C"_ will be equal to the increment of the "flow of magnetic force" which pa.s.ses through the circuit.
2. _Action of a Closed Current on a Portion of Current._--Ampere not having been able to produce an open current, properly so called, 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, the one fixed, the other movable. The movable part was, for instance, a movable wire [alpha][beta] whose extremities [alpha] and [beta] could slide along a fixed wire. In one of the positions of the movable wire, the end [alpha] rested on the _A_ of the fixed wire and the extremity [beta] on the point _B_ of the fixed wire. The current circulated from [alpha] to [beta], that is to say, from _A_ to _B_ along the movable wire, and then it returned from _B_ to _A_ along the fixed wire. _This current was therefore closed._
In a second position, the movable wire having slipped, the extremity [alpha] rested on another point _A"_ of the fixed wire, and the extremity [beta] on another point _B"_ of the fixed wire. The current circulated then from [alpha] to [beta], that is to say from _A"_ to _B"_ along the movable wire, and it afterwards returned from _B"_ to _B_, then from _B_ to _A_, then finally from _A_ to _A"_, always following the fixed wire. The current was therefore also closed.