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Title: Clerk Maxwell's electromagnetic theory

Author: Hendrik Antoon Lorentz

Release Date: April 3, 2023 [eBook #70453]

Language: English

Produced by: Laura Natal Rodrigues (Images generously made available by
             Hathi Trust Digital Library.)

*** START OF THE PROJECT GUTENBERG EBOOK CLERK MAXWELL'S
ELECTROMAGNETIC THEORY ***


                        CLERK MAXWELL'S
                    ELECTROMAGNETIC THEORY

                   The Rede Lecture for 1923




                             BY

                        H. A. LORENTZ



                          CAMBRIDGE
                   AT THE UNIVERSITY PRESS

                            1923




CLERK MAXWELL'S
ELECTROMAGNETIC THEORY


WHEN I had the honour to be invited to deliver a lecture in the Rede
Foundation, I thought I might perhaps present to you a brief review of
the electromagnetic theory of your great physicist James Clerk Maxwell.
The choice seemed the more appropriate as it is now exactly fifty years
ago that the work which raised him at once to the very first rank of
investigators of all ages, the Treatise on Electricity and Magnetism,
was published. In this work it was proved beyond all doubt that electric
and magnetic actions can be conceived as being transmitted through a
medium and the theory was crowned by the wonderful revelation that light
is an electromagnetic phenomenon.

Maxwell's theory was also a great simplification. Indeed, before his
time there was much uncertainty and confusion in this part of physics
and many contending theories were in the field. In electrodynamics, for
instance, we had the laws of Ampère and Grassmann for the actions
between elements of current, and, when we went further, we found the
speculations of Weber, Riemann and Clausius about the mutual actions of
particles of electricity. In connexion with these theories there was a
good deal of discussion on the phenomena that were to be expected in the
case of closed and in that of open circuits. It was thought in those
days that the current in a wire by means of which a metallic conductor
is charged, ends on that conductor, and even the discharge current of a
condenser was considered not to be closed; there was a gap in the
circuit, because we had no idea that something is going on in the
insulating layer between the coatings.

In optics we had no less trouble. It is true that the general principles
of the undulatory theory of light had been firmly established and
physicists were justly proud of the success that had been achieved in
the explanation of interference and diffraction, double refraction and
polarization. Yet, when we tried to penetrate somewhat deeper, we were
confronted with serious difficulties. When we wanted to account for the
different optical properties of various substances, of air and water for
instance, we had the choice between two assumptions. Fresnel had sought
the cause of the difference in an inequality of the density of the ether
in the two substances, the elasticity being the same in both. F. E.
Neumann, on the other hand, had supposed the densities to be the same,
but the elasticities to be different. On either of these suppositions,
and in no other way, it had been found possible to deduce the right
value for the ratio between the amplitude of the reflected and that of
the incident light. You know that in this problem two principal cases
must be distinguished, the vibrations being normal to the plane of
incidence in the one case and parallel to that plane in the other. The
two values for the ratio in question are


    sin (_i_ — _r_) / sin (_i_ + _r_)

and

    tg (_i_ - _r_) / tg (_i_ + _r_),


if _i_ is the angle of incidence and _r_ the angle of refraction; and it
is remarkable that, of the two rival theories, one led to the expression
with the sines when the other required that with the tangents, and
conversely. In connexion with this Fresnel supposed the vibrations of
plane polarized light to be at right angles to the plane of
polarization, whereas Neumann wanted them to be parallel to that plane.

Here was a problem that long baffled the efforts of physicists, and many
attempts were made to determine experimentally the direction of the
vibrations. One cannot say that the result has been very satisfactory
and the question remained open until Maxwell's theory settled it once
for all.

I have enumerated some only of the difficulties with which we had to
struggle. I could have mentioned similar problems that arose in the
theory of double refraction and I may add that in some cases
longitudinal vibrations intruded themselves and complicated the theory.

Maxwell relieved us of all these doubts and uncertainties. By his bold
assumption that in a non-conducting body, in a dielectric as he called
it, there can exist what is truly a motion of electricity, and that, if
this motion, the dielectric displacement, is taken into account,
electricity can always be said to move as an incompressible fluid, the
open currents and the longitudinal vibrations that were closely allied
to them were made to vanish from the scene. Further, the optical
behaviour of non-conducting substances was shown to depend on two
properties, each characterized by a physical constant, the dielectric
constant, or Faraday's specific inductive capacity, and the magnetic
permeability. It is true that the way in which these constants are
determined by the constitution of matter, by the structure of molecules
and atoms, was not considered and that, so far, they were no less
inaccessible than the ethereal density and elasticity of the old
theories, but there was this important difference that, whereas these
latter constants had no connexion with any other phenomena, the
dielectric constant and the magnetic permeability can be measured by
means of statical experiments, so that, at least in certain simple
cases, we can deduce the optical properties of a substance from wholly
different data. It was found that in the new theory the treatment of the
reflexion problem was much like that in the old one; one is led to the
two formulae which I recalled to you, if one supposes either the
dielectric constant or the magnetic permeability to be the same in the
two substances. The choice between these alternative suppositions again
entailed a decision concerning the direction of the vibrations with
respect to the plane of polarization, but the choice was not doubtful
now, as it had been ascertained experimentally that the ratios between
the magnetic permeabilities of transparent substances are little
different from unity, whereas the dielectric constants diverge to a much
greater extent. It was therefore at once established that the electric
vibrations are normal to the plane of polarization. This implies that
the magnetic vibrations are in that plane, so that, in a sense, the
contending parties both had their will.

In the case of crystals it became certain that their double refraction
is due to an inequality of the dielectric properties in different
directions.

So, many difficulties and outstanding problems melted away as snow
before the sun. Indeed, a reviewer in _Nature_ actually compared
Maxwell's work to the sun, his only criticism being that there are spots
on the sun itself, which, however, "are not visible save to those whose
eyes can bear the full glare of the glowing orb." My eyes certainly were
not as strong as that. I could not see the spots, but what I could see
was that the sun was not entirely unclouded; what sun always was? It was
not always easy to grasp Maxwell's ideas, and one feels a want of unity
in his book, due to the fact that it faithfully reproduces his gradual
transition from old to new ideas. When we read what Maxwell says of
Ampère and Faraday, of the former having removed all traces of the
scaffolding by which he had built up a perfect demonstration of his law,
whereas Faraday "shews us his unsuccessful as well as his successful
experiments, and his crude ideas as well as his developed ones," we feel
that, great though the difference may be between the _Experimental
Researches_ and Maxwell's largely mathematical Treatise, yet the two
works were written in the same spirit. In fact, Maxwell repeatedly
expresses his indebtedness to Faraday, from whom he had borrowed part of
his fundamental ideas, so that, when there is question of Maxwell's
theory, we must often think of Faraday also.

Maxwell's followers, of whom there were many, in this country and
elsewhere, have perfected the theory in its form and extended it by the
introduction of new ideas. Think, for instance, of Poynting's beautiful
and important theorem on the flow of energy, determined at every point
by the electric and the magnetic force existing in the field, a theorem
that has produced more clearness perhaps than any other and which is now
so essential that we can hardly recall the state in which physics was
when we did not know it. Yet, notwithstanding all innovations of this
kind, we always speak, and with full justice, of "Maxwell's Theory." We
continue to do so now that we have been led to introduce electric
charges supposed to exist in the interior of molecules and atoms, by
which we have come to the theory of electrons. And when we refer to
those wonderfully simple equations in which the fundamental laws of
electromagnetism are embodied with a conciseness that could never have
been dreamed of before, we call them "Maxwell's Equations." Surely,
though Maxwell did not use them in their modern form, no name could be
more appropriate, for the general relations which they express are those
that were constantly in his mind.

Time does not permit me to dwell at length on the verifications of
Maxwell's theory, but I should like to make an exception for two of
them.

Allow me, in the first place, to say some words on the optical
properties of metals.

"If the medium," so we read in Maxwell, "instead of being a perfect
insulator, is a conductor, the disturbance" (viz. that which is produced
by an incident beam of light) "will consist not only of electric
displacements but of currents of conduction, in which electric energy is
transformed into heat, so that the undulation is absorbed by the
medium." After having stated in these words one of the most important
consequences drawn from his theory, Maxwell goes on to calculate the
coefficient of absorption as a function of the conductivity, and he
proceeds: "Gold, silver and platinum are good conductors, and yet, when
formed into very thin plates, they allow light to pass through them.
From experiments which I have made on a piece of gold leaf, it appears
that its transparency is very much greater than is consistent with our
theory, unless we suppose that there is less loss of energy when the
electromotive forces are reversed for every semi-vibration of light than
when they act for sensible times, as in our ordinary experiments." Later
researches have amply confirmed what Maxwell says here; obviously,
bodies, both conductors and dielectrics, behave in general differently
towards rapidly alternating electric forces and towards stationary ones.
Yet, Hagen and Rubens have been able to show that when, instead of
working with visible light, one uses infra-red rays of sufficiently
great wave-length, the properties of metals will, in the limit, exactly
conform to the theory, if we reckon with the ordinary conductivity.

Hagen and Rubens did not measure the amount of radiation that is
transmitted through a thin plate but the coefficient of reflexion of a
thick mirror. For the case of normal incidence, this coefficient and
therefore also the loss of energy, i.e. the quantity that is absorbed by
the mirror, can easily be calculated as a function of the conductivity.
For the residual rays of sylvin, whose wavelength is 12 μ, and for
silver, copper, gold and platinum, the absorbed energy was found to be
respectively 1·15, 1·6, 2·1 and 3·5 per cent, of the incident
energy, whereas it ought to have been 1·3, 1·4, 1·6, and 3·5 per
cent, according to the theoretical formula.

The agreement became still better when the residual rays of fluorite
with a wave-length of 25·5 μ were used. Since, however, for waves of
this length the reflexion becomes nearly complete, it was not possible
to determine the loss of energy with sufficient precision. Hagen and
Rubens overcame this difficulty by measuring the emissivity of the
different metals, or rather the ratio between this emissivity and that
of a perfectly black body at the same temperature, a ratio which, by
Kirchhoff's law, is equal to that between the absorbed and the incident
energy for a beam falling on the metal from the outside, so that it can
be calculated by the same formula as this latter ratio. For the four
metals just mentioned (at a temperature of 170° C.) the ratio in
question was found to be (after multiplication by 100) 1·13, 1·17,
1·56 and 2·82. The theoretical values were 1·15, 1·29, 1·39 and
2·96.

These numbers show conclusively that, however complicated things may be
for shorter waves, we can calculate the optical properties of metals in
the extreme infra-red by means of Maxwell's equations, simply
substituting for the conductivity the value that has been deduced from
experiments with constant or slowly alternating currents. This is
certainly a most splendid confirmation, the counterpart to the
verification, which for gaseous bodies at least has been very
satisfactory, of Maxwell's relation of the dielectric constant to the
index of refraction.

The phenomenon of the pressure of radiation may serve as a second
example of verification. That a beam of light falling, say in the normal
direction, on a mirror exerts on it a pressure proportional to the
intensity of the beam was deduced by Maxwell from his formulae, and he
calculated the force that may be expected in the case of sunlight. It
lasted a quarter of a century before Lebedew succeeded in observing this
small force, which, for sunlight, amounts to no more than about a ten
millionth part of a gramme weight per cm.^2, and which it is therefore
difficult to disentangle from other forces that are caused by the
surrounding gas, even when this is highly rarefied. Some years later E.
F. Nichols and Hull repeated the experiment with the utmost care and
were able to measure the pressure and to prove that its intensity agrees
with Maxwell's calculation.

We are now quite sure of this phenomenon which has come to play a great
part in stellar physics. When we are concerned with very small particles
near or in a star, the radiation pressure may very well become greater
than the force of gravitation, and it is taken into account by many
astronomers in their speculations about the state of heavenly bodies.

The forces exerted by rays of light or heat are a special case of what
we call ponderomotive forces, i.e. of the forces with which the
electromagnetic field acts on material bodies. Maxwell showed how, in
general, these forces can be deduced from the values of the
electromagnetic energy corresponding to different positions of a system
of bodies, or from a consideration of certain stresses which exist in
the electromagnetic field and of which he taught us to determine the
direction and the intensity. Every student, even of rather elementary
physics, now knows that the mutual attraction of two conducting plates
between which there is a difference of potential, e.g. of the plates of
an absolute electrometer, may be considered as due to stresses along the
lines of force, that the same may be said of the attraction between a
magnetic pole and a piece of iron, and that in the case of an
electromagnetic motor we are concerned with the tangential stresses
acting at the surface of the revolving system. Here again there has been
a great deal of later development, but we continue to speak of
"Maxwell's stresses."

The notion of the electromagnetic momentum, which Maxwell seems not to
have had, though he was quite near it, has also proved very fruitful. A
beam of light has a definite momentum, much like a moving ball, and when
the beam is normally reflected by a mirror, so that the momentum is
inverted, we can deduce the force acting on the mirror from the change
of the momentum, exactly as we can do in the case of the ball or of a
stream of material particles.

In modern theory--I allude to the theory of relativity--one has found
good reasons for combining into one unity, which we call the
stress-energy-tensor, all these quantities of which I have spoken, viz.
the energy, the flow of energy, Maxwell's stresses and the
electromagnetic momentum. In Einstein's theory of gravitation this
tensor determines the gravitation field that is produced by an
electromagnetic system and in virtue of which such a system has an
influence on the motion of material particles, unfortunately much too
small to be observed.

I should be led too far astray if I dwelt on these questions, but what I
want to point out is this, that we could never have gone so far if we
had contented ourselves with the actions at a distance, if we had not
fixed our attention on the intervening medium, localizing the energy in
it and considering it as the seat of momenta and stresses which manifest
themselves in the observed motions of bodies. All these modern ideas
have their origin in Maxwell's work.

We are also concerned with a stress-energy-tensor, similar to the
electromagnetic one, when we consider a system of material particles,
whether unconnected like the molecules of a gas, or held together by
internal forces as in an elastic body or a fluid. The question naturally
arises: are these stress-energy-tensors, the electromagnetic and, let me
say, the material one, wholly independent or can one be reduced to the
other? One has often tried to do so and, more particularly, to imagine
electromagnetic phenomena as produced by some invisible mechanism moving
according to the laws of dynamics.

This was a favourite idea of Maxwell's and one of his most brilliant
chapters is devoted to the dynamical theory of electromagnetism. It is
the more important because it shows that such a theory can be developed
on very general lines, it not being necessary to make definite
assumptions regarding the underlying mechanism. Maxwell showed that in
the case of linear circuits carrying electric currents we can account
for the ponderomotive forces and for the phenomena of self and mutual
induction by Lagrange's or Hamilton's equations of motion, provided that
we introduce, besides the coordinates which determine the positions of
the material circuits, a certain number of new coordinates, one for each
circuit, the velocities belonging to these coordinates for the several
circuits being proportional to the current intensities. In fact, the new
or "internal" coordinate for each circuit represents the total quantity
of electricity that has traversed some section since a fixed instant
that is chosen as the origin of time. When the internal coordinates are
given this meaning, the magnetic energy becomes the kinetic energy of
the system, whereas the electric energy has to be identified with the
potential energy and is comparable to the energy of deformation of an
elastic body.

While he was applying the laws of dynamics in this very general way,
Maxwell was led to discuss certain phenomena that might perhaps be
expected to exist and some of which have been actually observed in our
days though Maxwell was not able to detect them with the means at his
disposal.

In the theory of dynamical systems there are as many velocities as there
are coordinates and the kinetic energy is a homogeneous quadratic
function of these velocities, in which in general not only the squares
but also the products of the velocities appear. When we have one or more
circuits carrying electric currents, we can distinguish in the kinetic
energy one part that depends on the material velocities only, and this
is the kinetic energy of ordinary mechanics, and a second part
containing only the velocities corresponding to the internal
coordinates; this is the magnetic energy that manifests itself in so
many ways. Now, is this all? There would certainly be a third part of
the kinetic energy if an electric current consisted in a real motion of
some substance along the conducting wire, for if the wire were moving,
say in the direction of its length, with the velocity _v_ and if _v'_
were the internal velocity proportional to the current, the total
velocity of the moving substance would be _v_ + _v'_ and in its square
we should have the term 2_vv'_. One is led to a similar conclusion on
other less simple assumptions and so, independently of an special
conception, the question arises whether any part of the kinetic energy
consists of products of ordinary velocities and strengths of electric
currents. Maxwell thinks this question to be of great importance and
deems it "desirable that experiments should be made on the subject with
great care."

He then proceeds to examine different ways in which the terms in
question might be made to reveal themselves, the first of which he
explains as follows:


If any part of the kinetic energy depends on the product of an ordinary
velocity and the strength of a current, it will probably be most easily
observed when the velocity and the current are in the same or in
opposite directions. We therefore take a circular coil of a great many
windings, and suspend it by a fine vertical wire, so that its windings
are horizontal, and the coil is capable of rotating about a vertical
axis, either in the same direction as the current in the coil, or in the
opposite direction.

We shall suppose the current to be conveyed into the coil by means of
the suspending wire, and, after passing round the windings, to complete
its circuit by passing downwards through a wire in the same line with
the suspending wire and dipping into a cup of mercury. A vertical mirror
is attached to the coil to detect any motion in azimuth.

Now let a current be made to pass through the coil in the direction
N.E.S.W. If electricity were a fluid like water, flowing along the wire,
then, at the moment of starting the current, and as long as its velocity
is increasing, a force would require to be supplied to produce the
angular momentum of the fluid in passing round the coil, and as this
must be supplied by the elasticity of the suspending wire, the coil
would at first rotate in the opposite direction or W.S.E.N., and this
would be detected by means of the mirror. On stopping the current there
would be another movement of the mirror, this time in the same direction
as that of the current.


It does not appear that Maxwell actually tried the experiment; he only
says: "no phenomenon of this kind has yet been observed."

Now, if for Maxwell's coil we substitute a rod of iron, the
magnetization and demagnetization of which are comparable to the
starting and stopping of a current in the coil, we have exactly the
Richardson-Einstein-de Haas effect that was really observed by Einstein
and de Haas and by some other physicists. You know that it amounts to
this, that a cylindrical rod of iron suspended in a vertical direction
is set rotating, with a sudden jerk, when it is rapidly magnetized or
demagnetized. When the magnetization is periodically reversed, the rod
is made to oscillate and the amplitude of this motion may be increased
by adjusting the frequency of the reversals to that of the free
oscillations of the rod.

Whereas Maxwell seems not to have tried the above experiment with the
coil, he tried to observe another effect. A coil, through which a
current could be passed and which could be provided with an iron core,
was placed in a system rapidly revolving about a vertical axis, the
arrangement being such that the coil was free to rotate in the revolving
system, so that the axis of the coil could be inclined to different
degrees with respect to the vertical axis of rotation. If in the formula
for the kinetic energy there were terms of the kind which Maxwell wanted
to detect, there would be a tendency for the coil to place itself with
its axis parallel to the axis of rotation; it would behave as a
gyroscope and might be called an electromagnetic gyroscope. No trace of
a phenomenon of this kind could, however, be observed. I may insert the
remark that, if the tendency of which I spoke existed to an appreciable
extent, a magnetic needle would, even in the absence of the terrestrial
magnetic field, still take a definite position that would be determined
by the rotation of the earth; in fact, in virtue of its internal motions
the magnetic needle would be comparable to a gyroscopic compass, such as
has lately come into use in navigation.

Now that we know the intensity of the Einstein effect we can say with
certainty that, if only his instrumental means had been more refined.
Maxwell's experiment would have had a positive result. We can evaluate
the magnitude of the effect and we can also calculate that the force
which a magnetic needle experiences on account of its internal motions
and of the rotation of the earth is thousands of millions of times
smaller than the force due to the earth's magnetic field, so that you
need not fear from this cause any error in measurements in terrestrial
magnetism.

In the experiments just discussed we were concerned with forces acting
on the material bodies. Maxwell next considers cases in which, always on
account of the terms in question, not the material system but the
electricity contained in it is set in motion. Here he reverts again to
the suspended circular coil, and he points out that when a rotation is
suddenly imparted to it there could be produced a transient electric
current. Similarly, there would be a current, but now in the opposite
direction, when the motion of the coil is stopped.

A very simple experiment may serve to give you an idea of these
phenomena. We take a cylindrical tumbler, partly filled with water, and
set it suddenly in rotation about its vertical axis. Then the friction
between the glass and the water will make the fluid rotate likewise, but
this will take some time; during a certain period the water will lag
behind. Let us next suppose that the motion of the vessel has been kept
constant for a sufficient length of time, so that the water has acquired
the full angular velocity and then let the vessel be suddenly brought to
rest. It is clear that the circulation of the water will continue for a
certain time, until it is exhausted by the friction.

Similar phenomena could be observed with a closed circular tube capable
of rotating about its axis, and there must be a corresponding
electromagnetic phenomenon if a metallic wire contains something like
movable electricity, as we are now in a position to assert, because we
have good reasons for believing that an electric current consists in a
motion of negative electrons. Though the experiment is much more
delicate with electricity than with water, Tolman and Stewart have
performed it with very satisfactory results. Using a coil with a great
number of windings, rapidly rotating about its axis, they were able to
observe with a sensitive galvanometer the transient electric current
that was produced on stopping the motion by means of a brake. The
electrons continued to move over some distance just as the water did in
the experiment with the circular tube. The direction of the current
showed that the movable particles really have negative charges and the
observed deflections agreed with what can be inferred from the ratio
between the charge and the mass of the electrons, a ratio that was found
for the first time by Zeeman and Sir J. J. Thomson somewhat more than
twenty-five years ago and has been repeatedly determined in later years.

No less remarkable than Tolman and Stewart's experiments are those made
by Mr and Mrs Barnett. They found that a cylindrical rod of iron,
rotating about its geometrical axis, becomes thereby magnetized in the
direction of its length. Like the Einstein effect, to which it forms a
counterpart, this new phenomenon can be predicated on the assumption
that magnetization consists in a motion of electrons in the molecules of
the metal. This allows us to assign the direction of the effects, but
for the sake of truth I must add that both the rotation produced by the
magnetization of rods and the magnetization caused by a rotation are,
for some reason which we do not yet understand, only about half what the
theory of electrons had led us to expect.

In Maxwell's time the electron was unknown and the mechanism of
conduction was even more mysterious than it is now. It was precisely for
this reason that he attached so much importance to the phenomena to
which I have drawn your attention. He says in this connexion:


It appears to me that while we derive great advantage from the
recognition of the many analogies between the electric current and a
current of a material fluid, we must carefully avoid making any
assumption not warranted by experimental evidence, and that there is, as
yet, no experimental evidence to shew whether the electric current is
really a current of a material substance, or a double current, or
whether its velocity is great or small as measured in feet per second.

A knowledge of these things would amount to at least the beginning of a
complete dynamical theory of electricity, in which we should regard
electrical action, not, as in this treatise, as a phenomenon due to an
unknown cause, subject only to the general laws of dynamics, but as the
result of known motions of known portions of matter, in which not only
the total effects and final results, but the whole intermediate
mechanism and details of the motion, are taken as the object of study.


Maxwell did not always express himself so cautiously: at other times he
did not shrink from imagining an elaborate mechanical model. All
physicists know its principal features. The magnetic energy is
considered as a true _vis viva_, the magnetic field being the seat of
invisible motions, rotations of small particles about the lines of
force. The system of these particles may be compared to a wheelwork, and
Maxwell has to explain how it can be that all the wheels in an element
of volume are rotating in the same direction. This shows that the motion
is not transmitted directly from one wheel to the next. So Maxwell is
led to assume that between these wheels of the magnetic field and in
contact with them, there are smaller ones which transmit the motion in
the manner of friction wheels.

What I called wheels might in reality be spheres capable of turning
about axes in any direction and, as Maxwell showed, a system of this
kind is amenable to mathematical analysis. In his image the friction
wheels represent what we call electricity; in a conductor we must
conceive them to be freely movable, whereas the centres of the larger
wheels or balls have fixed positions. It is easily seen how a motion of
translation imparted to the friction wheels can give rise to rotations
of the larger balls; this is how a current produces a magnetic field.
Other phenomena can be explained on the same lines; it is sufficient for
this that Maxwell's equations can be deduced by means of the model.

Similar models have been invented, with more or less success, by other
physicists, in such quantities, even, that nearly all that could be done
has been tried. The outcome of these various attempts is, in my opinion,
this: that we must admit the possibility of mechanical representations,
a possibility that is already shown by the fact that the formulae of the
electromagnetic field can be given the form of the general equations of
dynamics. On the other hand it cannot be denied that, when we desire
them to be applicable to a comparatively wide range of phenomena, the
theories that have been proposed become so complicated that they can
give us but little satisfaction. So, I think, the majority of physicists
will agree to attach less importance to mechanical models and even to
the general dynamical analogies, however helpful these may be to us,
than to Maxwell's equations that summarize so admirably all that is
essential.

Will it be possible to maintain these equations? I am not thinking here
of the comparatively slight modifications that have been found necessary
in the theory of relativity; and by which we explain, for instance, the
curvature of a ray of light in a gravitation field. A greater and really
serious danger is threatening from the side of the quantum theory, for
the existence of amounts of energy that remain concentrated in small
spaces during their propagation, to which several phenomena seem to
point, is in absolute contradiction to Maxwell's equations. However this
may be, even if further development should require profound alterations,
Maxwell's theory will always remain a step of the highest importance in
the progress of physics.

But I must not trespass any longer on your patience. Let me rather thank
you for the attention with which you have listened to what I had to say,
which was mainly intended as a tribute to the memory of one of the
greatest Masters of Science.

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