D_x = 0, D_y = a cos n(t - xv^(-1) + l), D_z = 0,
H_x = 0, H_y = 0 , H_z = avc^(-1) cos n(t - xv^{-1} + l) (16)
For the free aether the velocity has the value c. Now it had been
found that the ratio c between the two units of electricity agrees
within the limits of experimental errors with the numerical value of
the velocity of light in aether. (The mean result of the most exact
determinations[17] of c is 3,001·10^10 cm./sec., the largest
deviations being about 0,008·10^10; and Cornu[18] gives 3,001·10^10 ±
0,003·10^10 as the most probable value of the velocity of light.) By
this Maxwell was led to suppose that light consists of transverse
electromagnetic disturbances. On this assumption, the equations (16)
represent a beam of plane polarized light. They show that, in such a
beam, there are at the same time electric and magnetic vibrations,
both transverse, and at right angles to each other.
It must be added that the electromagnetic field is the seat of two
kinds of energy distinguished by the names of electric and magnetic
energy, and that, according to a beautiful theorem due to J. H.
Poynting,[19] the energy may be conceived to flow in a direction
perpendicular both to the electric and to the magnetic force. The
amounts per unit of volume of the electric and the magnetic energy are
given by the expressions
½(E_x D_x + E_y D_y + E_z D_z), (17)
and
½(H_x B_x + H_y B_y + H_z B_z) = ½H², (18)
whose mean values for a full period are equal in every beam of light.
The formula (15) shows that the index of refraction of a body is given
by [root][epsilon], a result that has been verified by Ludwig
Boltzmann's measurements[20] of the dielectric constants of gases.
Thus Maxwell's theory can assign the true cause of the different
optical properties of various transparent bodies. It also leads to the
reflection formulae (9) and (10), provided the electric vibrations of
polarized light be supposed to be perpendicular to the plane of
polarization, which implies that the magnetic vibrations are parallel
to that plane.
Following the same assumption Maxwell deduced the laws of double
refraction, which he ascribes to the unequality of [epsilon]1,
[epsilon]2, [epsilon]3. His results agree with those of Fresnel and
the theory has been confirmed by Boltzmann,[21] who measured the three
coefficients in the case of crystallized sulphur, and compared them
with the principal indices of refraction. Subsequently the problem of
crystalline reflection has been completely solved and it has been
shown that, in a crystal, Poynting's flow of energy has the direction
of the rays as determined by Huygens's construction.
Two further verifications must here be mentioned. In the first place,
though we shall speak almost exclusively of the propagation of light
in transparent dielectrics, a few words may be said about the optical
properties of conductors. The simplest assumption concerning the
electric current C in a metallic body is expressed by the equation C =
[sigma]E, where [sigma] is the coefficient of conductivity. Combining
this with his other formulae (we may say with (12) and (13)), Maxwell
found that there must be an absorption of light, a result that can be
readily understood since the motion of electricity in a conductor
gives rise to a development of heat. But, though Maxwell accounted in
this way for the fundamental fact that metals are opaque bodies, there
remained a wide divergence between the values of the coefficient of
absorption as directly measured and as calculated from the electrical
conductivity; but in 1903 it was shown by E. Hagen and H. Rubens[22]
that the agreement is very satisfactory in the case of the extreme
infra-red rays.
In the second place, the electromagnetic theory requires that a
surface struck by a beam of light shall experience a certain pressure.
If the beam falls normally on a plane disk, the pressure is normal
too; its total amount is given by c^{-1}(i1 + i2 - i3), if i1, i2 and
i3 are the quantities of energy that are carried forward per unit of
time by the incident, the reflected, and the transmitted light. This
result has been quantitatively verified by E. F. Nicholls and G. F.
Hull.[23]
Maxwell's predictions have been splendidly confirmed by the
experiments of Heinrich Hertz[24] and others on electromagnetic waves;
by diminishing the length of these to the utmost, some physicists have
been able to reproduce with them all phenomena of reflection,
refraction (single and double), interference, and polarization.[25] A
table of the wave-lengths observed in the aether now has to contain,
besides the numbers given in § 11, the lengths of the waves produced
by electromagnetic apparatus and extending from the long waves used in
wireless telegraphy down to about 0.6 cm.
17. _Mechanical Models of the Electromagnetic Medium._--From the results
already enumerated, a clear idea can be formed of the difficulties which
were encountered in the older form of the wave-theory. Whereas, in
Maxwell's theory, longitudinal vibrations are excluded _ab initio_ by
the solenoidal distribution of the electric current, the elastic-solid
theory had to take them into account, unless, as was often done, one
made them disappear by supposing them to have a very great velocity of
propagation, so that the aether was considered to be practically
incompressible. Even on this assumption, however, much in Fresnel's
theory remained questionable. Thus George Green,[26] who was the first
to apply the theory of elasticity in an unobjectionable manner, arrived
on Fresnel's assumption at a formula for the reflection coefficient A_n
sensibly differing from (10).
In the theory of double refraction the difficulties are no less serious.
As a general rule there are in an anisotropic elastic solid three
possible directions of vibration (§ 6), at right angles to each other,
for a given direction of the waves, but none of these lies in the
wave-front. In order to make two of them do so and to find Fresnel's
form for the wave-surface, new hypotheses are required. On Fresnel's
assumption it is even necessary, as was observed by Green, to suppose
that in the absence of all vibrations there is already a certain state
of pressure in the medium.
If we adhere to Fresnel's assumption, it is indeed scarcely possible
to construct an elastic model of the electromagnetic medium. It may be
done, however, if the velocities of the particles in the model are
taken to represent the magnetic force H, which, of course, implies
that the vibrations of the particles are parallel to the plane of
polarization, and that the magnetic energy is represented by the
kinetic energy in the model. Considering further that, in the case of
two bodies connected with each other, there is continuity of H in the
electromagnetic system, and continuity of the velocity of the
particles in the model, it becomes clear that the representation of H
by that velocity must be on the same scale in all substances, so that,
if [xi], [eta], [zeta] are the displacements of a particle and g a
universal constant, we may write
[dP][xi] [dP][eta] [dP][zeta]
H_x = g --------, H_y = g ---------, H_z = g ----------. (19)
[dP]t [dP]t [dP]t
By this the magnetic energy per unit of volume becomes
_ _
| /[dP][xi]\² /[dP][eta]\² /[dP][zeta]\² |
½g² | ( -------- ) + ( --------- ) + ( ---------- ) |,
|_ \ [dP]t / \ [dP]t / \ [dP]t / _|
and since this must be the kinetic energy of the elastic medium, the
density of the latter must be taken equal to g², so that it must be
the same in all substances.
It may further be asked what value we have to assign to the potential
energy in the model, which must correspond to the electric energy in
the electromagnetic field. Now, on account of (11) and (19), we can
satisfy the equations (12) by putting D_x = gc ([dP][zeta]/[dP]y -
[dP][eta]/[dP]z), &c., so that the electric energy (17) per unit of
volume becomes
_
| 1 /[dP][zeta] [dP][eta]\²
½g²c² | ---------- ( ---------- - --------- ) +
|_[epsilon]1 \ [dP]y [dP]z /
1 /[dP][xi] [dP][zeta]\²
---------- ( -------- - ---------- ) +
[epsilon]2 \ [dP]z [dP]x /
_
1 /[dP][eta] [dP][xi]\² |
---------- ( --------- - -------- ) |.
[epsilon]3 \ [dP]x [dP]y / _|
This, therefore, must be the potential energy in the model.
It may be shown, indeed, that, if the aether has a uniform constant
density, and is so constituted that in any system, whether homogeneous
or not, its potential energy per unit of volume can be represented by
an expression of the form
_
| /[dP][zeta] [dP][eta]\²
½ | L ( ---------- - --------- ) +
|_ \ [dP]y [dP]z /
/[dP][xi] [dP][zeta]\²
M ( -------- - ---------- ) +
\ [dP]z [dP]x /
_
/[dP][eta] [dP][xi]\² |
N ( --------- - -------- ) |, (20)
\ [dP]x [dP]y / _|
where L, M, N are coefficients depending on the physical properties of
the substance considered, the equations of motion will exactly
correspond to the equations of the electromagnetic field.
18. _Theories of Neumann, Green, and MacCullagh._--A theory of light in
which the elastic aether has a uniform density, and in which the
vibrations are supposed to be parallel to the plane of polarization, was
developed by Franz Ernst Neumann,[27] who gave the first deduction of
the formulas for crystalline reflection. Like Fresnel, he was, however,
obliged to introduce some illegitimate assumptions and simplifications.
Here again Green indicated a more rigorous treatment.
By specializing the formula for the potential energy of an anisotropic
body he arrives at an expression which, if some of his coefficients
are made to vanish and if the medium is supposed to be incompressible,
differs from (20) only by the additional terms
_
| /[dP][zeta] [dP][eta] [dP][eta] [dP][zeta]\
2 | L ( ---------- --------- - --------- ---------- ) +
|_ \ [dP]y [dP]z [dP]y [dP]z /
/[dP][xi] [dP][zeta] [dP][zeta] [dP][xi]\
M ( -------- ---------- - ---------- -------- ) +
\ [dP]z [dP]x [dP]z [dP]x /
_
/[dP][eta] [dP][xi] [dP][xi] [dP][eta]\ |
N ( --------- -------- - -------- --------- ) |. (21)
\ [dP]x [dP]y [dP]x [dP]y / _|
If [xi], [eta], [zeta] vanish at infinite distance the integral of
this expression over all space is zero, when L, M, N are constants,
and the same will be true when these coefficients change from point to
point, provided we add to (21) certain terms containing the
differential coefficients of L, M, N, the physical meaning of these
terms being that, besides the ordinary elastic forces, there is some
extraneous force (called into play by the displacement) acting on all
those elements of volume where L, M, N are not constant. We may
conclude from this that all phenomena can be explained if we admit the
existence of this latter force, which, in the case of two contingent
bodies, reduces to a surface-action on their common boundary.
James MacCullagh[28] avoided this complication by simply assuming an
expression of the form (20) for the potential energy. He thus
established a theory that is perfectly consistent in itself, and may
be said to have foreshadowed the electromagnetic theory as regards the
form of the equations for transparent bodies. Lord Kelvin afterwards
interpreted MacCullagh's assumption by supposing the only action which
is called forth by a displacement to consist in certain couples acting
on the elements of volume and proportional to the components
½{([dP][zeta]/[dP]y) - ([dP][eta]/[dP]z)}, &c., of their rotation from
the natural position. He also showed[29] that this "rotational
elasticity" can be produced by certain hidden rotations going on in
the medium.
We cannot dwell here upon other models that have been proposed, and most
of which are of rather limited applicability. A mechanism of a more
general kind ought, of course, to be adapted to what is known of the
molecular constitution of bodies, and to the highly probable assumption
of the perfect permeability for the aether of all ponderable matter, an
assumption by which it has been possible to escape from one of the
objections raised by Newton (§ 4) (see AETHER).
The possibility of a truly satisfactory model certainly cannot be
denied. But it would, in all probability, be extremely complicated. For
this reason many physicists rest content, as regards the free aether,
with some such general form of the electromagnetic theory as has been
sketched in § 16.
19. _Optical Properties of Ponderable Bodies. Theory of Electrons._--If
we want to form an adequate representation of optical phenomena in
ponderable bodies, the conceptions of the molecular and atomistic
theories naturally suggest themselves. Already, in the elastic theory,
it had been imagined that certain material particles are set vibrating
by incident waves of light. These particles had been supposed to be
acted on by an elastic force by which they are drawn back towards their
positions of equilibrium, so that they can perform free vibrations of
their own, and by a resistance that can be represented by terms
proportional to the velocity in the equations of motion, and may be
physically understood if the vibrations are supposed to be converted in
one way or another into a disorderly heat-motion. In this way it had
been found possible to explain the phenomena of dispersion and
(selective) absorption, and the connexion between them (anomalous
dispersion).[30] These ideas have been also embodied into the
electromagnetic theory. In its more recent development the extremely
small, electrically charged particles, to which the name of "electrons"
has been given, and which are supposed to exist in the interior of all
bodies, are considered as forming the connecting links between aether
and matter, and as determining by their arrangement and their motion all
optical phenomena that are not confined to the free aether.[31]
It has thus become clear why the relations that had been established
between optical and electrical properties have been found to hold only
in some simple cases (§ 16). In fact it cannot be doubted that, for
rapidly alternating electric fields, the formulae expressing the
connexion between the motion of electricity and the electric force take
a form that is less simple than the one previously admitted, and is to
be determined in each case by elaborate investigation. However, the
general boundary conditions given in § 16 seem to require no alteration.
For this reason it has been possible, for example, to establish a
satisfactory theory of metallic reflection, though the propagation of
light in the interior of a metal is only imperfectly understood.
One of the fundamental propositions of the theory of electrons is that
an electron becomes a centre of radiation whenever its velocity changes
either in direction or in magnitude. Thus the production of Röntgen
rays, regarded as consisting of very short and irregular electromagnetic
impulses, is traced to the impacts of the electrons of the cathode-rays
against the anti-cathode, and the lines of an emission spectrum indicate
the existence in the radiating body of as many kinds of regular
vibrations, the knowledge of which is the ultimate object of our
investigations about the structure of the spectra. The shifting of the
lines caused, according to Doppler's law, by a motion of the source of
light, may easily be accounted for, as only general principles are
involved in the explanation. To a certain extent we can also elucidate
the changes in the emission that are observed when the radiating source
is exposed to external magnetic forces ("Zeeman-effect"; see
MAGNETO-OPTICS).
20. _Various Kinds of Light-motion._--(a) If the disturbance is
represented by
P_x = 0, P_y = a cos (nt - kx + f), P_z = a´ cos (nt - kx + f´),
so that the end of the vector P describes an ellipse in a plane
perpendicular to the direction of propagation, the light is said to be
elliptically, or in special cases circularly, polarized. Light of this
kind can be dissolved in many different ways into plane polarized
components.
There are cases in which plane waves must be elliptically or
circularly polarized in order to show the simple propagation of phase
that is expressed by formulae like (5). Instances of this kind occur
in bodies having the property of rotating the plane of polarization,
either on account of their constitution, or under the influence of a
magnetic field. For a given direction of the wave-front there are in
general two kinds of elliptic vibrations, each having a definite form,
orientation, and direction of motion, and a determinate velocity of
propagation. All that has been said about Huygens's construction
applies to these cases.
(b) In a perfect spectroscope a sharp line would only be observed if
an endless regular succession of simple harmonic vibrations were
admitted into the instrument. In any other case the light will occupy
a certain extent in the spectrum, and in order to determine its
distribution we have to decompose into simple harmonic functions of
the time the components of the disturbance, at a point of the slit for
instance. This may be done by means of Fourier's theorem.
An extreme case is that of the unpolarized light emitted by
incandescent solid bodies, consisting of disturbances whose variations
are highly irregular, and giving a continuous spectrum. But even with
what is commonly called homogeneous light, no perfectly sharp line
will be seen. There is no source of light in which the vibrations of
the particles remain for ever undisturbed, and a particle will never
emit an endless succession of uninterrupted vibrations, but at best a
series of vibrations whose form, phase and intensity are changed at
irregular intervals. The result must be a broadening of the spectral
line.
In cases of this kind one must distinguish between the velocity of
propagation of the phase of regular vibrations and the velocity with
which the said changes travel onward (see below, iii. _Velocity of
Light_).
(c) In a train of plane waves of definite frequency the disturbance is
represented by means of goniometric functions of the time and the
coordinates. Since the fundamental equations are linear, there are
also solutions in which one or more of the coordinates occur in an
exponential function. These solutions are of interest because the
motions corresponding to them are widely different from those of which
we have thus far spoken. If, for example, the formulae contain the
factor
e^(-rx) cos (nt - sy + l),
with the positive constant r, the disturbance is no longer periodic
with respect to x, but steadily diminishes as x increases. A state of
things of this kind, in which the vibrations rapidly die away as we
leave the surface, exists in the air adjacent to the face of a glass
prism by which a beam of light is totally reflected. It furnishes us
an explanation of Newton's experiment mentioned in § 2. (H. A. L.)