OPTICS AND ELECTRICITY
FRESNEL'S THEORY.--The best example[5] that can be chosen of physics in
the making is the theory of light and its relations to the theory of
electricity. Thanks to Fresnel, optics is the best developed part of
physics; the so-called wave-theory forms a whole truly satisfying to the
mind. We must not, however, ask of it what it can not give us.
[5] This chapter is a partial reproduction of the prefaces of two
of my works: _Théorie mathématique de la lumière_ (Paris, Naud,
1889), and _Électricité et optique_ (Paris, Naud, 1901).
The object of mathematical theories is not to reveal to us the true
nature of things; this would be an unreasonable pretension. Their sole
aim is to coordinate the physical laws which experiment reveals to us,
but which, without the help of mathematics, we should not be able even
to state.
It matters little whether the ether really exists; that is the affair of
metaphysicians. The essential thing for us is that everything happens as
if it existed, and that this hypothesis is convenient for the
explanation of phenomena. After all, have we any other reason to believe
in the existence of material objects? That, too, is only a convenient
hypothesis; only this will never cease to be so, whereas, no doubt, some
day the ether will be thrown aside as useless. But even at that day, the
laws of optics and the equations which translate them analytically will
remain true, at least as a first approximation. It will always be
useful, then, to study a doctrine that unites all these equations.
The undulatory theory rests on a molecular hypothesis. For those who
think they have thus discovered the cause under the law, this is an
advantage. For the others it is a reason for distrust. But this distrust
seems to me as little justified as the illusion of the former.
These hypotheses play only a secondary part. They might be sacrificed.
They usually are not, because then the explanation would lose in
clearness; but that is the only reason.
In fact, if we looked closer we should see that only two things are
borrowed from the molecular hypotheses: the principle of the
conservation of energy and the linear form of the equations, which is
the general law of small movements, as of all small variations.
This explains why most of Fresnel's conclusions remain unchanged when we
adopt the electromagnetic theory of light.
MAXWELL'S THEORY.--Maxwell, we know, connected by a close bond two parts
of physics until then entirely foreign to one another, optics and
electricity. By blending thus in a vaster whole, in a higher harmony,
the optics of Fresnel has not ceased to be alive. Its various parts
subsist, and their mutual relations are still the same. Only the
language we used to express them has changed; and, on the other hand,
Maxwell has revealed to us other relations, before unsuspected, between
the different parts of optics and the domain of electricity.
When a French reader first opens Maxwell's book, a feeling of uneasiness
and often even of mistrust mingles at first with his admiration. Only
after a prolonged acquaintance and at the cost of many efforts does this
feeling disappear. There are even some eminent minds that never lose it.
Why are the English scientist's ideas with such difficulty acclimatized
among us? It is, no doubt, because the education received by the
majority of enlightened Frenchmen predisposes them to appreciate
precision and logic above every other quality.
The old theories of mathematical physics gave us in this respect
complete satisfaction. All our masters, from Laplace to Cauchy, have
proceeded in the same way. Starting from clearly stated hypotheses, they
deduced all their consequences with mathematical rigor, and then
compared them with experiment. It seemed their aim to give every branch
of physics the same precision as celestial mechanics.
A mind accustomed to admire such models is hard to suit with a theory.
Not only will it not tolerate the least appearance of contradiction, but
it will demand that the various parts be logically connected with one
another, and that the number of distinct hypotheses be reduced to
minimum.
This is not all; it will have still other demands, which seem to me
less reasonable. Behind the matter which our senses can reach, and which
experiment tells us of, it will desire to see another, and in its eyes
the only real, matter, which will have only purely geometric properties,
and whose atoms will be nothing but mathematical points, subject to the
laws of dynamics alone. And yet these atoms, invisible and without
color, it will seek by an unconscious contradiction to represent to
itself and consequently to identify as closely as possible with common
matter.
Then only will it be fully satisfied and imagine that it has penetrated
the secret of the universe. If this satisfaction is deceitful, it is
none the less difficult to renounce.
Thus, on opening Maxwell, a Frenchman expects to find a theoretical
whole as logical and precise as the physical optics based on the
hypothesis of the ether; he thus prepares for himself a disappointment
which I should like to spare the reader by informing him immediately of
what he must look for in Maxwell, and what he can not find there.
Maxwell does not give a mechanical explanation of electricity and
magnetism; he confines himself to demonstrating that such an explanation
is possible.
He shows also that optical phenomena are only a special case of
electromagnetic phenomena. From every theory of electricity, one can
therefore deduce immediately a theory of light.
The converse unfortunately is not true; from a complete explanation of
light, it is not always easy to derive a complete explanation of
electric phenomena. This is not easy, in particular, if we wish to start
from Fresnel's theory. Doubtless it would not be impossible; but
nevertheless we must ask whether we are not going to be forced to
renounce admirable results that we thought definitely acquired. That
seems a step backward; and many good minds are not willing to submit to
it.
When the reader shall have consented to limit his hopes, he will still
encounter other difficulties. The English scientist does not try to
construct a single edifice, final and well ordered; he seems rather to
erect a great number of provisional and independent constructions,
between which communication is difficult and sometimes impossible.
Take as example the chapter in which he explains electrostatic
attractions by pressures and tensions in the dielectric medium. This
chapter might be omitted without making thereby the rest of the book
less clear or complete; and, on the other hand, it contains a theory
complete in itself which one could understand without having read a
single line that precedes or follows. But it is not only independent of
the rest of the work; it is difficult to reconcile with the fundamental
ideas of the book. Maxwell does not even attempt this reconciliation; he
merely says: "I have not been able to make the next step, namely, to
account by mechanical considerations for these stresses in the
dielectric."
This example will suffice to make my thought understood; I could cite
many others. Thus who would suspect, in reading the pages devoted to
magnetic rotary polarization, that there is an identity between optical
and magnetic phenomena?
One must not then flatter himself that he can avoid all contradiction;
to that it is necessary to be resigned. In fact, two contradictory
theories, provided one does not mingle them, and if one does not seek in
them the basis of things, may both be useful instruments of research;
and perhaps the reading of Maxwell would be less suggestive if he had
not opened up to us so many new and divergent paths.
The fundamental idea, however, is thus a little obscured. So far is this
the case that in the majority of popularized versions it is the only
point completely left aside.
I feel, then, that the better to make its importance stand out, I ought
to explain in what this fundamental idea consists. But for that a short
digression is necessary.
THE MECHANICAL EXPLANATION OF PHYSICAL PHENOMENA.--There is in every
physical phenomenon a certain number of parameters which experiment
reaches directly and allows us to measure. I shall call these the
parameters _q_.
Observation then teaches us the laws of the variations of these
parameters; and these laws can generally be put in the form of
differential equations, which connect the parameters _q_ with the time.
What is it necessary to do to give a mechanical interpretation of such a
phenomenon?
One will try to explain it either by the motions of ordinary matter, or
by those of one or more hypothetical fluids.
These fluids will be considered as formed of a very great number of
isolated molecules _m_.
When shall we say, then, that we have a complete mechanical explanation
of the phenomenon? It will be, on the one hand, when we know the
differential equations satisfied by the coordinates of these
hypothetical molecules _m_, equations which, moreover, must conform to
the principles of dynamics; and, on the other hand, when we know the
relations that define the coordinates of the molecules _m_ as functions
of the parameters _q_ accessible to experiment.
These equations, as I have said, must conform to the principles of
dynamics, and, in particular, to the principle of the conservation of
energy and the principle of least action.
The first of these two principles teaches us that the total energy is
constant and that this energy is divided into two parts:
1º The kinetic energy, or _vis viva_, which depends on the masses of the
hypothetical molecules _m_, and their velocities, and which I shall call
_T_.
2º The potential energy, which depends only on the coordinates of these
molecules and which I shall call _U_. It is the _sum_ of the two
energies _T_ and _U_ which is constant.
What now does the principle of least action tell us? It tells us that to
pass from the initial position occupied at the instant t_{0} to the
final position occupied at the instant t_{1}, the system must take such
a path that, in the interval of time that elapses between the two
instants t_{0} and t_{1}, the average value of 'the action' (that is to
say, of the _difference_ between the two energies _T_ and _U_) shall be
as small as possible.
If the two functions _T_ and _U_ are known, this principle suffices to
determine the equations of motion.
Among all the possible ways of passing from one position to another,
there is evidently one for which the average value of the action is less
than for any other. There is, moreover, only one; and it results from
this that the principle of least action suffices to determine the path
followed and consequently the equations of motion.
Thus we obtain what are called the equations of Lagrange.
In these equations, the independent variables are the coordinates of the
hypothetical molecules _m_; but I now suppose that one takes as
variables the parameters _q_ directly accessible to experiment.
The two parts of the energy must then be expressed as functions of the
parameters _q_ and of their derivatives. They will evidently appear
under this form to the experimenter. The latter will naturally try to
define the potential and the kinetic energy by the aid of quantities
that he can directly observe.[6]
[6] We add that _U_ will depend only on the parameters _q_, that _T_
will depend on the parameters _q_ and their derivatives with
respect to the time and will be a homogeneous polynomial of the
second degree with respect to these derivatives.
That granted, the system will always go from one position to another by
a path such that the average action shall be a minimum.
It matters little that _T_ and _U_ are now expressed by the aid of the
parameters _q_ and their derivatives; it matters little that it is also
by means of these parameters that we define the initial and final
positions; the principle of least action remains always true.
Now here again, of all the paths that lead from one position to another,
there is one for which the average action is a minimum, and there is
only one. The principle of least action suffices, then, to determine the
differential equations which define the variations of the parameters
_q_.
The equations thus obtained are another form of the equations of
Lagrange.
To form these equations we need to know neither the relations that
connect the parameters _q_ with the coordinates of the hypothetical
molecules, nor the masses of these molecules, nor the expression of _U_
as a function of the coordinates of these molecules.
All we need to know is the expression of _U_ as a function of the
parameters, and that of _T_ as a function of the parameters _q_ and
their derivatives, that is, the expressions of the kinetic and of the
potential energy as functions of the experimental data.
Then we shall have one of two things: either for a suitable choice of
the functions _T_ and _U_, the equations of Lagrange, constructed as we
have just said, will be identical with the differential equations
deduced from experiments; or else there will exist no functions _T_ and
_U_, for which this agreement takes place. In the latter case it is
clear that no mechanical explanation is possible.
The _necessary_ condition for a mechanical explanation to be possible is
therefore that we can choose the functions _T_ and _U_ in such a way as
to satisfy the principle of least action, which involves that of the
conservation of energy.
This condition, moreover, is _sufficient_. Suppose, in fact, that we
have found a function _U_ of the parameters _q_, which represents one of
the parts of the energy; that another part of the energy, which we shall
represent by _T_, is a function of the parameters _q_ and their
derivatives, and that it is a homogeneous polynomial of the second
degree with respect to these derivatives; and finally that the equations
of Lagrange, formed by means of these two functions, _T_ and _U_,
conform to the data of the experiment.
What is necessary in order to deduce from this a mechanical explanation?
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, 'Électricité 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 quantities 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 quantities 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 passed 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.