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12.
The Electron theory of Lorentz
"It is the theory of H.A. Lorentz (proposed
in 1892)
that signi-fied the climax and the final step of the physics
of the material ether.
It is a one-fluid theory of
electricity that had been developed atomistically, and it is
this feature which, as we shall presently see, determines
the part allocated to the ether.
The fact that electric charges
have an atomic structure, that is,
/ Page
200 /
occur
in very small indivisible quantities, was first stated by
Helm-holtz (in 1881)
in order to make intelligible Faradays laws of eloctro-lysis
(p.157) Actually, it was only necessary to assume
that every atom in an electrolytic solution enters into a
sort of chemical bond with an atom of electricity or an
electron in order to make intelligible the fact that a
definite amount of electricity always separates out
equivalent amounts of substances.
The atomic structure of
electricity proved of particular value for explaining the
phenomena which are observed in the passage of the electric
current through a rarefied gas. Here it was first discovered
that positive and negative electricity behave quite
differently. If two metal electrodes are introduced into a
glass tube and if a current is made to pass between
them
(Fig. 104). Very complicated pheno-mena are produced so
long as gas is still present at an appreciable
Fig.
104
A
tube to produce cathode rays.
K cathode, A anode.
Pressure in the tube. But if the gas is pumped out more
and more, the phenomena becomes increasingly simple. When
the vacuum is very high, the negative electrode, the cathode
K, emits rays which pass through a hole in the positive
pole, the anode A, and are observed behind
A through fluorescence produced on a screen (as
known from any television set). These rays are called
cathode rays. It was shown that they could be deflected by a
magnet in the manner of a stream of negative electricity.
The greatest share in investigating the nature of cathode
rays was taken by Sir J. J. Thomson and P.L. Lenard. The
negative charge of the rays could also be directly
demonstrated by collecting it in a hollow conductor.
Furthermore, the rays are deflected by an electric field
applied per
/
Page 201 /
pendicular
to their path, and this deflection is opposite to the
direc-tion of the field, which again proves the charge to be
negative.
The conviction that the nature
of cathode rays is corpuscular became a certainty when
physicists succeeded in deducing quantita-tive conclusions
concerning their velocity and their charge.
If we picture the cathode ray
as a stream of small particles of mass m
el,
then clearly it will be the less deflected by a definite
elec-tric or magnetic field the greater its velocity - just
as the trajectory of a rifle bullet is straighter the
greater its velocity. Now it is possible to produce cathode
rays that can be bly deflected - that is, slow cathode rays
- by using a small difference of potential between cathode
and anode. If one chooses a b difference of potential
between the two poles, the rays are bly accelerated by the
field from K to A, which may be calculated from the
fundamental equation of
mechanics
m
el
b = K = e E,
where e is the charge and E is the field strength. We
are here clearly dealing with a case analogous to that of
"falling" bodies, in which the acceleration is not equal to
that of gravity g but to
e E. If the ratio e were
known, the velocity v could be
found mel m
el
from the laws for falling bodies. But there are two
unknowns, e and v, and hence another measurement is
necessary if
they
mel
are to be determined. This is obtained by applying a
lateral magnetic force. In discussing Hertz's theory (V,11,
1b, p. 194)
we saw that a magnetic field H sets up in a body moving
perpendicular to H an electrical field E= v
H,
c
which is perpendicular both to H and to v. Hence a
deflecting force eE = e v H will act on
every cathode ray particle so that
c
there will be an acceleration b =
e v H perpendicular to
the original motion. This may be found by measuring the
lateral
Mel c
deflection of the ray.
/
Page 202 /
Hence
we have a second equation for determining the two
unknowns e and v.
mel
The determinations carried out by this or a
similar method have led to the result that, for velocities
that are not two great e
Mel
and v. has a definite constant
value: e =
5.31 x 10 17 electrostatic
units per gm.
(66) mel
On the other hand, in dealing with electrolysis (V, 2,
formula (48),p.159), we stated that hydrogen carries an
amount of electricity Co = 2.90 x 10
14 electrostatic
units per gm. If we now make the readily suggested
assumption that the charge of a particle is in each case the
same, namely an atom of electricity or an electron, we must
conclude that the mass of the cathode ray particle
mel
must
bear the following ratio to that of the hydrogen
atom mH:
mel
= e
: e = 2.90 x 10
14
= 1 .
mH mH
mel 5.31
x 10 17 1830
Thus, the cathode ray particles are nearly 2000 times
lighter than hydrogen atoms, which are the lightest of all
chemical atoms. This leads us to conclude that cathode rays
are a current of pure atoms of electricity.
This view has stood the test of innumerable
researches Negative electricity consists of freely moving
electrons, but positive electricity is bound to matter and
never occurs without it. Thus recent experi-mental
researches have confirmed and given a precise form to the
old hypothesis of the one-fluid theory. The amount of the
charge e of the individual electron has also been
successfully determined. The first experiments of this type
were carried out by Sir J. J. Thomson
(1898).
The underlying idea is : Little drops of oil or water, or
tiny spheres of metal of microscopic or submicroscopic
dimensions, which are produced by condensation of a vapour
or by spraying of a liquid in air, fall with constant
velocity, since the fric-tion of the air prevents
acceleration. By measuring the rate of fall the size of the
particle can be determined, and then their mass M
is
/
Page203 /
obtained
by multiplying their size by the density. The weight of such
a particle is then Mg, where g =
981
cm./sec.2 is the acceleration due to gravity. Now such
particles may be charged electrically by sub-jecting the air
to the action of x-rays or the rays of radioactive
sub-stances. If an electric field E, which is directed
vertically upwards, is then applied, a sphere carrying the
positive charge e is pulled upwards by it, and if the
electric force eE is equal to the weight Mg,
The sphere will remain poised in the air. The charge e
may then be calculated from the equation eE = Mg.
Millikan (1910),
Who carried out the most accurate experiments of this
sort, found that the charge of the small drops is always an
exact multiple of a definite minimum charge. Thus we shall
call this the elementary electrical quantum Its value is
e = 4.77 x 10-10
electrostatic units. (67)
The absolute value of the
elementary charge plays no essential part in Lorentz's
theory of electrons. We shall now describe the physical
world as suggested by Lorentz.
The material atoms are the
carriers of positive electricity, which is indissolubly
connected with them. In addition they also contain a number
of negative electrons, so that they appear to be
electrically neutral with respect to their
surroundings.
In nonconductors the electrons are tightly bound to the
atoms; they may only be displaced slightly out of the
positions of equilibrium so that the atom becomes a dipole.
In electrolytes and conducting gases it may occur that an
atom has one or more electrons too many or two few; it is
then called an ion or a carrier, and it wanders in the
electric field carrying elec-tricity and matter
simultaneously. In metals the electrons move about freely
and experience resistance only when they collide with the
atoms of the substance. Magnetism comes about when the
electrons in certain atoms move in closed orbits and hence
represent Amperian molecular currents.
The electrons and the positive
atomic charges swim about in the sea of ether in which an
electromagnetic field exists in accordance with Maxwell's
equations. But we must set
*
= 1, u
= 1 in them, and, in place of the density of the conduction
current, we have the convec-tion current
pv
of the electrons. The equations thus
become
/
Page 204 /
v div E = 4 pi
p; curl H -
1 E =
4
p-
, ]
(68)
c
pi
c
]
div
H =
0; curl
e + 1
H = ]
c
pi ]
and include the laws of Coulomb, Biot and Savert, and
Faraday in the usual way.
Thus all electromagnetic events
consist fundamentally of the motions of electrons and of the
fields accompanying them.
All matter is an electrical phenomenon. The various
properties of matter depend on the various possibilities of
motion of the electrons with respect to atoms, in the manner
just described. The problem of the theory of electrons is to
derive the ordinary equations of Maxwell from the
fundamental laws (68) for the individual, invisible
electrons and atoms, that is, to show that material bodies
appear to have, according to their nature, respectively, a
conductivity o,
a dielectric constant *,
and a permeability u.
Lorentz has solved this problem
and has shown that the theory of electrons not only gives
Maxwell's laws in the simplest case, but, more than this,
also explains numerous facts which were inexplicable for the
descriptive theory or could be accounted for only with the
aid of artificial hypotheses. These facts comprise, above
all, the more refined phenomena of optics, color dispersion,
the magnetic rotation of the plane of polarization (p. 184)
discovered by Faraday, and similar interactions between
light waves and electric or magnetic fields. We shall not
enter further into this extensive and mathe-matically
complicated theory, but shall restrict ourselves to the
ques-tion which is of primary interest to us : What part
does the ether play in this concept of matter
? Lorentz proclaimed the
very radical thesis which had never before been asserted
with such definiteness:
The ether is at rest in
absolute space
In principle this identifies
the ether with absolute space. Absolute space is no vacuum,
but something with definite properties whose state is
described with the help of two directed quantities, the
elec-trical field E and the magnetic field H, and, as such,
it is called the ether.
This assumption goes
still farther than the theory of Fresnel.
In
/
Page 205 /
the
latter the ether of astronomic space was at rest in a
special inertial system, which we might regard as being
absolute rest. But the ether inside material bodies is
partly carried along by them.
Lorentz dispenses with even this partial convection and
arrives at practically the same result. To see this, we
consider the phenomenon that occurs in a dielectric between
the plates of a condenser. When the latter is charged, a
field perpendicular to the plate arises (Fig.
89),
and this displaces the electrons in the atoms of the
dielectric sub-stance and transforms them into dipoles, as
we explained earlier (pp. 197 and
198).
The dielectric displacement in Maxwell's sense
*E,
but only a part of it is due to the actual displacement of
the
Fig.
105 A
light ray with its electric vibration E and its magnetic
vibration H travelling in an insulator with velocity c1. The
insulator moves with velocity v.
Electrons. For a vacuum has the dielectric
constant*
= 1, and hence the displacement E; consequently the true
value of the electronic displacement is *E - E = (* - 1) E.
Now we have seen that the experiments of Rontgen and Wilson
on the phenomenona in moving insulators affirm that actually
only this part of the displacement takes part in the motion.
Thus Lorentz's theory gives a correct account of
electromagnetic facts without having recourse to hypo-theses
about the ether carried along by matter.
The fact that the convection of
light comes out in exact agreement with Fresnel's formula
(44), (p.134) is made plausible by the following
argument:
/
Page 206 /
As
in Wilson's experiment, we consider a dielectric body which
moves in the x-direction with the velocity v and in which a
light ray travels in the same direction (Fig. 105). Let this
ray consist of an electrical vibration E parallel to the
y-axis and a magnetic vibration parrallel to the z-axis. Now
we know from Wilson's experiment that such a magnetic field
in the moving body produces a corres-ponding displacement of
the value (*- 1) v H in
the y-direction
c
From this we get a superposed electrical field if we
divide by *. Thus the total electrical field
is E + *
-
1 v H.
* c
If the convection were
complete, as is assumed in Hertz's theory. We should have
only * in place of * -
1,
thus the total field would have the value E
+ v -
H. We
see that in our formula v is replaced
by *
-
1
v.
c
*
Therefore, this value should correspond to the absolute
velocity of the ether within matter according to Fresnel's
theory, that is, to the convection coefficient
called in optics (compare formula (44)). This is
precisely the case, for, according to Maxwell's
electromagnetic theory (p.188), the dielectric constant is
equal to the square of the index of refraction n,
i.e., * = n2. If we insert this value, we
get
* - 1 v =
n2
- 1
v = (
1 - 1 )
v =
* n2
n2
in agreement with formula (44).
We recall that Fresnel's theory
encountered difficulties through color dispersion; for if
the refractive index n depends
On the fre-quency (color) of the light, so also will the
convection coefficient . But the ether can be
carried along in only one definite way, not differently for
each color. This difficulty vanishes for the theory of
electrons, since the ether remains at rest and it is the
electrons situ-ated in the matter that are carried along:
color dispersion is due to their being forced into vibration
by light and reacting, in turn, on the velocity of
light.
/
Page 207
We
cannot enter further into the details of this theory and its
many ramifications, but we shall summarize the results as
follows: Lorentz's theory presupposes the existence of an
ether that is absolutely at rest. It then proves that, in
spite of this, all electromagnetic and optical phenomena
depend only on the relative motions of translation of
material bodies so far as terms of the first order in come
into account. Hence it accounts for all known phenomena,
above all for the fact that the absolute motion of the earth
through the ether cannot be demonstrated by experiments on
the earth involving only quantities of the first order (this
is the optica, or rather the electro-magnetic principle of
relativity). There is,
however, one experiment of the first order which can no more
be explained by Lorentz's theory than by any of the other
theories previously discussed: this would be a failure of
the experiment to find an absolute motion of the whole solar
system with Romer's method (see pp. 91 and
129). The deciding point
for Lorentz's theory is whether it stands the test of
experiments that allow quantities of the second order in to
be measured. For they should make it possible to establish
the absolute motion of the earth through the ether. Before
we enter into this question, we have yet to discuss an
achievement of Lorentz's theory of electrons through which
its range became greatly extended, namely, the
electrodynamic interpretation of inertia."
For pages 91
and 129
see original whatsit.
13.
Electromagnetic Mass
" The reader will have remarked that from
the moment when we left the elasic ether and turned our
attention to the electrodynamic ether, we havehad little to
say about mechanics. Mechanical and electrodynamic phenomena
each form a realm for themselves. The former take place in
absolute Newtonian space, which is defined by the law of
inertia and which betrays its existence through centrifugal
forces; the latter are states of the ether which is at rest
in absolute space. Acomprehensive theory, such as Lorentz's
aims at being, cannot allow these two realms to exist side
by side unassociated. We have seen that physicists in spite
of incredible effort and ingenuity were not able to reduce
electrodynamics to terms of
/
Page 208 /
mechanics.
The converse then boldly suggests itself: can mechanics be
reduced to terms of electrodynamics ?
If this could be carried out
successfully the absolute abstract space of Newton would be
transformed into the concrete ether. The iner-tial
resistances and centrifugal forces would appear as physical
actions of the ether, say, as electromagnetic fields of
particular form, but the principle of relativity of
mechanics would lose its strict validity and would be true,
like that of electrodynamics, only approxi-mately, for
quantities of the first order in =
v
c
Science has not hesitated to take this step, which
entirely reverses the order of rank of concepts. And,
although the doctrine of an
Fig.
106 A
circuit with condenser K, coil S, and spark gap F used to
demon-strate the oscillations of electricity.
Ether absolutely at rest later had to be dropped, this
revolution, which ejected mechanics from its throne and
raised electrodynamics to sovereign power in physics was not
in vain Its results have retained their validity in a
somewhat altered form.
We saw already (p. 185) that the
propagation of electromagnetic waves comes about through the
mutual action of electrical and magnetic fields producing an
effect analogous to that of mechanical inertia.
An electromagnetic field has a power of persistence quite
similar to that of matter. To generate the field, work must
be per-formed, and when it is destroyed, this work again
appears. This is
/
Page 209 2
x 9 = 18 1
+8 = 9 /
observed
in all phenomena that are connected with electromagnetic
vibrations - for example, in the various forms of wireless
trans-mitters. An old-fashioned wireless
transmitter of the Marconi type contains an electric
oscillator, consisting essentially (Fig. 106) of a spark gap
F, a coil S, and a condenser K (two metal plates that
are
Seperated from each other) connected by wires to form an
"open" circuit. The condenser is charged until a spark jumps
across the gap at F. This causes the condenser to become
discharged and the quantities of electricity that have been
stored flow away. They do not simply neutralize each other,
but shoot beyond the state of equilibrium and again become
collected on the condenser plates, but with reversed signs,
just as a pendulum swings past the position of equilibrium
to the opposite side. When the condenser has thus
Fig.107
The electric field around a charge
at rest. Fig.
108 The
electric field around a charge is supplemented by
a
magnetic field when it is moved.
Been charged up afresh, the electricity again flows back
causing another spark to jump the gap, and thus the system
oscillates to and fro until its energy has been used up in
warming the conducting wires or in being passed on to other
parts of the apparatus, for example, the emitting antenna.
Thus the oscillation of the electricity proves the inertial
property of the field, which exactly corresponds to the
inertial property of the field, which exactly corresponds to
the inertia of mass of the pendulum bob. Maxwell's theory
represents this fact correctly in all details. The
electromagnetic vibrations that
/
Page 210 /
occur
in a definite apparatus can be predicted by calculation from
the equations of the field.
This led J.J.Thompson to infer
that the inertia of a body must be increased by an electric
charge which is imparted to it. Let us con-sider a charged
sphere first at rest and then moving with the velocity v.
The stationary sphere has an electrostatic field with lines
of force directed radially outwards (Fig.
107);
the moving sphere has, in addition, a magnetic field with
circular lines and forces that encircles the path of the
sphere (Fig. 108); the
moving sphere has, in addition, a magnetic field with
circular and forces that encircle the path of the
sphere (Fig. 108). For
a moving charge is a convec-tion current (combined with a
displacement current) and produces a magnetic field in
accordance with Biot and Savert's law. Both states have the
inertial property above described. The one can be
transformed into the other only by the application of work.
The force that is necessary to set the stationary sphere
into motion is thus greater for the charged than for the
uncharged sphere.
To accelerate the moving charged sphere still further,
the magnetic field H must be clearly be strengthened. Thus,
again, an increase of force is necessary.
We remember that a force K that
acts for a short time t
represents
an impulse
J
= Kt which
produces a change of velocity
w in a mass m in accordance with the formula
(7)
(11.
9,
p. 34):
mw = J
If the mass carries a charge, a definite impulse J will
produce a smaller change of velocity and the remainder J'
will be used to change the magnetic field. Thus we
have
mw
= J - J'.
Now, calculation gives the rather obvious result that
the impulse J' necessary to increase the magnetic field is
greater, the greater the change of velocity w, and, indeed,
it is approximately proportional to this change of velocity.
Thus we may set
J' = m'w, where m' is a factor of proportionality which,
moreover, may depend on the state, that is, the velocity v,
before the change of velocity occurs. We then
have mw
= J - m' w
or
(m + m' ) w = J.
/
Page 211 /
Thus,
it is as if the mass m were augmented by an amount m' ,
which is to be calculated from the electromagnetic field
equations and which may be depende on the velocityv. The
exact value of m' for any velocity v may be calculated only
if assumptions are made about the distribution of the
electric charge over the moving body. But the limiting value
for velocities that are small compared with that of light c,
that is, for small values of , is obtained
independently of such assumptions
as
m' = 4 S
, (69) 3
c2
where S is the electrostatic energy of the charges on
the body.
We have seen that the massof
the electron is about 2000 times smaller than that of the
hydrogen atom. Hence the idea occurs that the electron has,
perhaps, no "ordinary" mass at all, but is nothing other
than an "atom of electricity," and that its mass is entirely
electromagnetic in origin. Is such an assumption
reconcil-able with the knowledge that we have of the size,
charge, and mass of the electron?
Since the electrons are to be
the structural elements of atoms, they must at any rate be
small compared with the size of atoms. Now we know from
atomic physics that the radius of atoms is of the
order 10-8
cm.
Thus the radius of the electron must be smaller than
10-8
cm. If we imagine the electron as a sphere of radius a with
charge e distributed over its surface, then, as may be
derived from Coulomb' law, the electrostatic energy
is S =
1 e2.
Hence, by (69),
2 a
the electromagnetic mass
becomes
el
=
4 S = 2 e2
.
3 c2 3 ac
2
From this we can calculate the radius a:
a =
2 e e
3 c2 mel
On the right-hand side we know all the
quantities, e from the deflection of
the cathode rays (formula (66), p. 202), e from
Milikan's
/Page
212 /
measurements
(formula (67), p. 203, and c the velocity of light. If we
insert the values given, we get
a
= 1.88 x 10-13 cm.
A length which is about 100,000 times smaller than the
radius of an atom.
Thus the hypothesis that the
mass of the electron is electromagnetic in origin does not
conflict with the known facts. But this does not prove the
hypothesis.
At this stage the theory found
b support in refined observa-tions of cathode rays and of
the -rays of radioactive substances, which are also ejected
electrons. We explained above how electric and magnetic
action on these rays allows us to determine the ratio of the
charge to mass, e , and also
their velocity v, and that at first a definite
valuefor e was obtained,
mel mel
which was independent of v. But,
on proceeding to higher velocities, a decrease
of e was found. This effect
was particularly clear and could be measured quantitatively
in the case of the -rays of radium, which are only slightly
slower than light. The assumption that an
electric charge should depend on the velocity is
incompatible with the ideas of the electron theory. But that
the mass should depend on the velocity was certainly to be
expected if the mass was to be electromagnetic in origin. To
arrive at a quantitative theory, it is true, definite
assumptions had to be made about the form of the electron
and the distribution of the charge on it. M.
Abraham
(1903)
regarded the electron as a rigid sphere, with a charge
distributed on the one hand, uniformly over the interior,
or, on the other, over the surface,and he showed that both
assumptions lead to the same dependence of the
electromagnetic mass on the velocity, namely, to an increase
of mass with increasing velocity. The faster the electron
travels, the more the electromagnetic field resists a
further increase of velocity. The increase
of mel explains
the observed decrease of e ,
and Abraham's
theory agrees quantitatively very well with the
mel
results
of measurement
/
Page 213 /
of
Kaufmann (1901)
if it is assumed that there is no "ordinary" mass
present.
Thus, the object of tracing the
inertia of electrons back to electro-magnetic fields in the
ether was attained. At the same time a further perspective
presented itself. Since atoms are the carriers of positive
electricity, and also contain numerous electrons, perhaps
their mass is also electromagnetic in origin? In that case,
mass as the measure of the inertial persistence would no
longer be a primary phenomenon, as it is in elementary
mechanics, but a secondary consequence of the ether.
Therefore, Newton's absolute space, which is defined only by
the mechanical law of inertia, becomes super-fluous; its
part is taken over by the ether whose electromagnetic
properties are well known.
We shall see (V, 15, P. 221)
that new facts contradict this view. But the relationship
between mass and electromagnetic energy, which was first
discovered in this way, constitutes a fundamental discovery
the deep significance of which was brought into prominence
only when Einstein proposed his theory of
relativity".
The Zed Aliz Zed made a nine from the
anagram EINSTEIN
"We have yet to add that, besides Abraham's theory of the
rigid electron, other hypothesi were set up and worked out
mathematically. The most important is that of Lorentz (1904)
which is closely connected with the theory of relativity.
Lorentz assumed that every moving electron contracts in the
direction of motion, so that from a sphere it becomes a
flattened
spheroid of revolu-tion, the amount of flattening
depending in a definite way on this velocity. This
hypothesis seems at first sight strange. It certainly gives
a simpler formula for the way electromagnetic mass depends
on velocity than does Abraham's theory, but this in itself
does not justify it. The actual confirmation came from the
course Lorentz's theory of electrons took when it had to
consider quantities of the second order in the discussion of
experimental researches, to which we shall presently direct
our attention. Lorentz's formula then turned out to have a
universal significance in the theory of relativity. The
experimental decision between it and Abraham's theory will
be discussed later (VI, 7, p 278).
At the beginning of the new century, after the theory of
electrons had reached the stage above described, the
possibility of forming a uniform physical picture of the
world seemed at hand - a picture
/
Page 214 /
which
would reduce all forms of energy, including mechanical
inertia, to the same root, to the electromagnetic field in
the ether. Only one form of energy - gravitation - seemed
still to remain outside the system; yet it could be hoped
that that, too would allow itself to be interpreted as an
action of the ether.
14.
Michelson and Morley's Experiment
Twenty
years before this period, however, the base of the whole
structure had already cracked and, while the building was
going on above, the foundations needed repairing and
strengthening.
We have several times
emphasized that any decisive experiment in regard to the
theory of the stationary ether had to be precise enough to
determine quantities of the second order in . Only then
could it be ascertained whether or not a
fast- moving body is swept by an ether wind which blows
away the light waves as is demanded by theory.
Michelson and Morley
(1881)
were the first sucecessfully to carry out the most important
experiment of this type They used Michel-son's
interferometer (IV, 4, p. 102) which they had refined to a
precision instrument of unheard-of efficiency.
In investigating the influence
of the earth's motion on the velocity of light (IV,
9,
P. 130), it has been found that the time taken by a ray of
light to pass back and forth along a distance l parallel to
the earth's motion differs only by a quantity of the second
order from the value it has when the earth is at rest. We
found earlier that this time
was
t1= l (
1 +
1 )
= 21c ,
c
+ v c -
v
c2
-
v2
for which we may also write
t1
=
21 1 .
c 1-
2
If this time could be so accurately
measured that the fraction
1 could be distinguished from 1
in spite of the extremely
1-
2
small value of the quantity 2, we should have means of
proving the exis-tence of an ether
wind.
/
Page 215 /
It
is by no means possible, however, to measure the short time
taken by a light ray to traverse a certain
distance. Interferometric methods give us rather
only differences of the times taken by light to traverse
different routes between two given points. But they give
these with amazing accuracy.
For this reason Michelson and
Morly caused a second ray of light to traverse a path AB of
the same length l backwards and forwards, but perpendicular
to the earth's orbit (Fig, 109). While
Fig.
109 The
path of the light in Michelson's experiment.
the light passes from Ato B, the earth moves a short
distance forward so that the point B arrives at the point B'
of the ether. Thus the true path of the light in the ether
is AB', and if it takes a time t to cover this distance,
then AB, = ct. During the same time A has moved on to the
point A' with the velocity v, thus AA' = vt. If we now apply
Pythagoras' theorem to the right-angled
triangle AA'B, we get
c2t2
=l 2
+ v2t2
or
t2
(c2-
v2)
=l2,
t2
= l2 = l2 1 , c2
- v2 c2
1- 2
t = l 1 .
c
/ 1-
2
/
Page 216 /
The
light requires exactly the same time to make the return
journey, for the earth shifts by the same amount, so that
the initial point A moves
from A'
to
A ".
Thus the light takes
the following time for the journey backwards and
forwards:
t = 2l 1 .
c / 1-
2
The difference between the times taken to cover the same
distance parallel and perpendicular to the earth's motion is
thus
t1-
t2 =
21 ( 1
. -
1
).
c 1-
2
/
1-
2
Now, by neglecting terms of higher order than the second
in (similar to what was done on p. 126) we may
approximate by replacing
1 by 1 +
2,
and 1
by
1
+ 2* .
/1 - 2
2
Hence we may write to a sufficient degree of
approximation t1-
t2 =
2l [
(
1 + 2)
- ( 1
+ 2 ) ] = 21
2 =
1 2
.
c 2
c 2
c
The retardation of the one
light wave compared with the other is thus a quantity of the
second order.
This retardation may be
measured with the help of Michelson's interferometer (Fig
110). In this the light coming from the source Q
is divided at the half silvered-plate P into two rays which
run in perpendicular directions to the mirrors
S1
and S2,
At which they are reflected and sent back to the
plate P. From P onwards they run parallel into
the telescope F where they interfere. If the distances
S1P
and S2P
are equal and if one arm of the apparatus is placed in the
direction of the earth's motion, one duplicates
the /
*
To
show the approximate validity of the
equation
1 = 1 +
2 write
it 1 =
(1- 2)
(
1 + 2)
-
4,
1-
2
which is correct
if 4
is
neglected. In the same way
squaring 1 =
1 + 2
,
one
obtains 1
= 1 + 2
+
4; if
the /
1- 2
2
1-
2
4
last term is neglected one has the same formula as
above.
Page 217 /
case
just discussed. Thus the two rays reach the field of vision
with a difference of time
l 2. Hence
the c
interference fringes are not situated precisely where
they would be if the earth were at rest. But if we now turn
the apparatus through 90º
until the other arm is paral-lel to the direction of the
earth's motion, the interference fringes will be displaced
by the same amount but in the opposite direction.
Fig.
110
Michelsons's interfero-meter.
Fig. 111
Two
waves of the same wave length , one shifted against the
other by
2 l2.
Hence if we observe the position of the interference
fringes while the apparatus is being rotated, a displacement
should be measured which corresponds to the double
retardation 2 1
2. c
If T is the period of vibration
of the light used, the ratio of the retardation to the
period is
2l 2
and
cT
since
by formula
(35) (p.99)
the wave length = Ct, we may write this ratio as
2 1 2.
Hence, when the apparatus is
rotated, the two interfering trains of waves experience a
relative displacement whose ratio to the wave length is
given by 2l
2 (Fig.
111). The interference fringes them-selves arise because the
rays which leave
the
source in different
/
Page 218 /
directions
have to traverse somewhat different paths. The distance
between two fringes corresponds to a path difference of one
wave length, hence the observable displacement of the
fringes is the fraction 2l
2
of the width of the
fringe.
Now Michelson, in a repetition
of his experiment with Morley (1887) carried out on a larger
scale, extended the length of the path traversed by the
light by means of several reflections forward and back to
11m. = 1.1 x
10 3
cm. The wave length of the light used was about = 5.9 x
10-5
cm.
We know that is approximately equal to 10
-
4,
and hence 2
= 10 -
8.
So we get
2l 2 =
2 x 1.1 x 103
x 10 -
8 =
0.37, 5.9
x 10 -5
that is, the interference fringes must be displaced by
more than one-third of their distance apart when the
apparatus is turned through
90º.
Michelson was certain that the one-hundredth part of this
displacement would still be observable.
When the experiment
was carried out, however, not the slightest sign of the
expected displacement manifested itself, and later
repe-titions with still more refined means led to no other
result. From this we must conclude that the ether wind does
not exist. The velocity of light is not influenced by the
motion of the earth even to the extent involving quantities
of the second order.
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