|
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.
|
