|
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.
15. The
Contraction Hypothesis
Michelson and Morley concluded from their experiment that
the ether is carried along completely by the moving earth,
as is main-tained in the elastic theory of Stokes and in the
electromagnetic theory of Hertz. But this conclusion
contradicts the numerous experiments which prove partial
convection. Michelson then investigated whether it was
possible to establish a difference in the velocity of light
at different heights above the earth's surface, but without
a positive result. He concluded from this that the motion of
the ether that is carried along by the earth must extend to
very great heights above the eart's surface. Thus, then, the
ether would be
/ Page
219 /
influenced by a moving
body at considerable distances. But this is in fact not the
case, for Oliver Lodge showed
(1892)
that the velocity of light in the neighborhood of rapidly
moving bodies is not influen-ced in the slightest, not even
when the light passes through a b electric or magnetic
field, carried along by the body. But all these efforts seem
superfluous, for even if they led to an unobjection-able
explanation of Michelson's experiment, the rest of
electrodyna-mics and optics of moving bodies which speaks in
favour of partial convection would remain
unexplained.
We see now Lorentz's
theory of electrons placed in a very difficult position by
Michelson and Morley's experiment. The doctrine of the
stationary ether seems to demand that an ether wind exist on
the earth, and hence stands in contradiction to the results
of Michelson and Morley's experiment. The fact that it did
not at once succumb to this challenge proves the inherant
strength of the theory, a strength deriving from the
consistency and completeness of its physical picture of the
world. Finally, it overcame even this difficulty to a
certain extent, although by a a very strange hypothesis,
which was proposed by Fitzgerald
(1892)
and at once taken up and elaborated by Lorentz.
Let us recall the reflections on
which Michelson and Morley's experiment were based. We found
that the time taken by a light ray to travel to and fro
along a distance l differs according to
whether the ray travels parallel or perpendicular to the
earth's motion. In the former
case
t1
= 2l
1 ,
c
1-
2
in the second,
t2
=
2l
1 .
c /
1-
2
If we now assume that the arm of the interferometer which is
directed parallel to the direction of the earth's motion is
shortened in the ratio
/ 1-
2
:
1, the time
t1
would become
reduced in the same ratio,
namely, t1
= 21
/
1-
2 =
21
1
. c(1-
2 c /
1-
2
Thus we should have
t1
=t2.
/ Page
220 /
This suggests the
following general hypothesis, the crudeness and boldness of
which is startling indeed:
Everybody which has the velocity v with respect
to the ether contracts in the direction of motion by the
fraction
.
. .
. / 1-
2
= /
1
- v2
.
/ c2
Michelson and Morley's experiment
must actually, then, give a negative result, since for both
positions of the
interferometer t1
=
t2.
Furthermore - and this is the important point
-
such a
contraction could not be ascertained by any means on earth,
for every earthly measuring rod would be contracted in just
the same way. An observer who was at rest in the ether
outside the earth would, it is true, observe the
contraction. The whole earth would be flattened in the
direction of motion and likewise all things on it.
The contraction hypothesis seems so
remarkable -
indeed, almost
absurd -
because the
contraction is not a consequence of any forces but appears
only as a companion circumstance to motion. Lorentz,
however, did not allow this objection to keep him from
absorbing this hypothesis into his theory, particularly as
new experiments con-firmed that no second-order
effect of the earth's motion through the ether could be
detected.
We cannot
describe
all these
experiments
or even outline them. They are partly optical and concern
the events involved in
re-flection
and
refraction,
double
refraction,
rotation of the plane of
polarization,
and so forth; and they are partly electromagnetic and
concern induction phenomena, the distribution of the current
in wires and the like. The improved technique of physics
allows us nowadays to establsh unambiguously the existence
or
absence of second-order effects in these phenomena. A
particularly note-worthy experiment is that of Trouton and
Noble
(1903),
which was intended to detect a torsional force which should
occur in a suspended plate condenser in consequence of the
ether wind.
These experiments produced without
exception a negative result. There could no longer be any
doubt that a motion of translation through the ether cannot
be detected by an observer sharing in the motion. Thus the
principle of relativity which holds for mechanics is also
valid for all electromagnetic phenomena. /
Page 221 /
Lorentz next proceeded to
bring this fact into harmony with his ether theory. To do
this there seemed no other way than to assume the
contraction hypothesis and to fuse it into the laws of the
electron theory so as to form a consistent whole free from
inner contradictions. He first observed that a system of
electric charges which keep in equilibrium only through the
action of their electrostatic forces con-tracts of itself as
soon as it is set into motion; or, more accurately the
electromagnetic forces that arise when the system is moving
uni-formly change the configuration of equilibrium in such a
way that every length is contracted in the direction of its
motion by the
factor
|
