8
Maxwell's Theory of Action by Contact
We
have already stated that soon after Coulomb's law had been
established, electrostatics and magnetostatics were brought
into the form of pseudocontiguous action. Maxwell now
undertook to fuse this theory with Faraday's ideas, and to
elaborate it so that it also included the newly discovered
phenomena of dielectric and magnetic polarization, of
electromagnetism and magnetic induction.
Alizzed substituted * and letter for the
symbols used by Brother Born.
Must we said the scribe needs must said Zed Aliz.
Maxwell took as the starting point of his theory the idea
already mentioned above that an electric field E is
always accompanied by an electric displacement D =
*E not only in matter, for which * is different for
one, but also in the ether, where * = 1. We
explained how the displacement can be visualized as the
separation and flowing of electric fluids in the molecules.
And we have found a differential law, which connects the
charge density p in every space point with the
divergence of D =*E:
Div *E=
4 pi
p (58)
Exactly the same considerations apply to magnetism, but with
one important difference: According to Ampere no
real magnets exist, no magnetic quantities, but only
electromagnets. The magnetic field is always to
be produced by electric currents, whether they be conduction
currents in wires or molecular currents in the molecules.
From this it follows that the magnetic lines of force never
end, that is they are either closed or stretch to infinity.
This is so in the case of an electromagnet, a coil through
which a current is flowing (Fig 97a,b);
the magnetic lines of force are straight lines in the
interior of the coil, but outside they are partly closed
and partly going off into infinity. If we consider the coil
between two planes A and B,
/
Page 182 /
Formal
agreement of this kind is by no means a matter of
indifference. It exhibits the underlying
simplicity of phenomena in nature, which remains
hidden from direct perception because of the limitations of
our senses and reveal itself only to our analytical
faculty.
In general a conduction and a
displacement current will be present simultaneously. For the
former, Ohm's law, jc =
o
E,
Holds (52) p162; for the latter, Maxwell's law, jd =
*
E If both are present
simultaneously
4 pi t
we thus have J =
*
E E + o
E.
4
pi t
There is no conduction current for magnetism, so we always
have
I = u H
4 pi t
If we insert this in our symbolic equations (58) to (61) we
get:
(a)
div *E = 4 pi
p,
(b)
div u
H = O,
(c)
c curl H = * E
+ 4 pi o
E,(62)
t
(d)
c curl E = u H
.
t
These are Maxwell's laws which have remained the foundation
of all electromagnetic and optical theories up to our own
time. To the mathematician they are precise differential
equations. To us they are precise differential equations. To
us they are mnemonics which state:
(a) Wherever an electric charge occurs an
electric field arises of such a kind that in every volume
the charge is exactly compensated by the displacement.
(b) Through every closed surface Just as much
magnetic displacement passes outwards as comes inwards
there are no free magnetic charges).
(c) Every electric current, be it a conduction
or a displacement current is surrounded by a magnetic
field.
(d) A magnetic displacement current is
surrounded by an electric field in the reverse sense. "
/
The Alizzed reminded those who needed to know, of the
needs must, symbol changes that had had to be made.
These yonder scribe had in the main emphasised,.advising
consulting the original oracle Brother Born, for missing
hieroglyphics and total accuracy of interruption.
Then in a sort of apology the scribe writ. That there wasn't
much call for that sort of thing in our part of the
world.
Then out of the blue the scribe writ the words Karmic
magnet.
You are a caution scribe said Alizzed.
And you Zed Aliz said the scribe, adding, and you.
Einstein's
Theory Of Relativity
1924  1962
Max Born
Page
183 /
"Maxwell's
"field equations," as they are called, constitute a true
theory of contiguous action or action by contact, for, as we
shall presently see, they give a finite velocity of
propagation for electromagnetic forces
At the time they were
set up, however, faith in direct action at a distance,
according to the model of Newtonian attraction, was still so
deeply rooted that a considerable interval elapsed before
they were accepted, for the theory of action at a distance
had also succeeded in mastering the phenomena of induction
by means of formulae. This was done by assuming that moving
charges exert, in addition to the Coulomb attraction,
certain actions at a distance that depend on the amount and
direction of the velocity. The first hypotheses of this
kind were due to Neumann (1845). Another famous law is that
set up by
Wilhelm Weber (1846); similar formulae were given by Riemann
(1858) and Clausius (1877). These theories have in common
the idea that all electrical and magnetic actions are to be
explained by means of forces between elementary electrical
charges, or as we say nowadays, "electrons." They were thus
precursors of the presentday theory of electrons, with
however an essential factor omitted: the finite velocity of
propagation of the forces. These theories of
electrodynamics, based on action at a distance, gave a
complete explanation of the electromotive forces and
induction currents that occur in the case
Of closed conduction currents. But in the case of "open"
circuits that is, condenser charges and discharges, they
were doomed to failure, for here the displacement currents
come into play, of which the theories of action at a
distance know nothing. It is to Helmholtz that we are
indebted for appropriate experimental devices allowing us
to decide between the theories of action at a distance and
action by contact. He succeeded in carrying out the
experiment with a certain measure of success, and he himself
became one of the most zealous pioneers of Maxwell's theory.
But it was his pupil Hertz who secured the victory for
Maxwell's theory by discovering electromagnetic waves.
9
The Electromagnetic Theory of Light
"
We have already mentioned (V, 4 p. 163 ) the impression
which the coincidence,
established by Weber and Kohlrausch, of the
electro
/
Page 184 /
magnetic
constant c with the velocity of light made upon the
physicists of the day. And there were still further
indications that there is an intimate relation between light
and electromagnetic phenomena. This was shown most
strikingly by Faraday's
discovery ( 1834
)
that a polarized ray of light which passes through a
magnetized transparent substance is influenced by it. When
the beam is parallel to the magnetic lines of force, its
plane of polarization becomes turned. Faraday concluded from
this that the luminiferous ether and the carrier of
electromagnetic lines of force must be identical. Although
his mathematical powers were not sufficient to allow him to
convert his ideas into quantitative laws and formulae, his
ideas were of a most abstract type and far surpassed the
trivial view which accepted as known what was familiar.
Faraday's ether was no elastic medium. He derived its
properties, not by analogy from the apparently known
material world, but from exact experiments and systematic
deductions from them. Maxwell's talents were akin to those
of Faraday, but they were supplemented by a complete mastery
of the mathematical means available at the time.
We shall now show how
the propagation of electromagnetic forces with finite
velocity arises out of Maxwell's field laws (62). In doing
so we shall confine ourselves to events that occur in
vacuo or in the ether. The latter has no conductivity,
that is, o
= O, and no true charges, that is p = O
and its dielectric constant and permeability are equal to 1,
that is, * = 1, u
= 1. The first two field equations (62) then assert that
div
E = O, div H
=
O
(63)
or that all lines of force are either closed or run off to
infinity. To obtain a rough picture of the processes we
shall imagine individual, closed lines of force.
The other two field
equations are
then
(a) E
= c curl
H, (b) H
= c curl
E. (64)
t t
We now assume that, somewhere in a limited space, there is
an electric field E which alters by the amount
E in the small interval of time
T; then E is its rate of change.
According to the first equation,
a /
t
Page 185 /
magnetic
field immediately coils itself around this electric field,
and its strength is proportional
to E
t
The magnetic field, too, will alter in time, say by
H during each successive small
interval t
Again, in accordance with the second equation, its rate of
change H
t
immediately induces an interwoven electric field. In the
following interval of time the latter again induces an
encircling magnetic field, according to the first equation,
and so this chainlike process continues with finite
velocity (Fig. 98)
Fig.
98 Electric
and magnetic fields linked by induction.
This is, of course, only
a rough description of the process, which actually
propagates itself in all directions continuously. Later we
shall sketch a better picture.
What particularly
interests us here is the following: We know from mechanics
that the finite velocity of propagation of elastic waves is
due to the delays that occur as a result of the inertia
which comes into play when the forces are transmitted in the
body from point to point. We have formulated this in
equation (36) pb =
pf and with (37) c2
= p
p
b = c2
f.
(36a)
Here c2 means the square of the velocity of the
elastic waves, b the acceleration of the mass
particles in the elastic body
(i.e., the secondorder differential coefficient with
respect to time), and f is the secondorder
differential coefficient with respect to space.
Now, in the
electromagnetic field, the case is nearly the same. The only
difference is that instead of the dependency of the
displacement on space and time in the elastic case we have
two quantities
/
Page 186 /
E
and H depending on space and time. The rate of
change of the electric field E
t
first determines the magnetic field H, and then the
rate of change H of
the latter determines the electric field E at a
t
neighbouring point. The equations (64) contain only
differential quantities of the first order, for instance
E ,
t
a firstorder differential coefficient with respect to time,
and curl H, a first order differential coefficient
with respect to space. One gets an equation similar to (36)
by the following procedure: To begin with, take the
firstorder differential coefficient of equation (64a) with
respect to time. Then we have on the lefthand side the
secondorder differential coefficient of E with
respect to time which is analogous to b in
(36a) and which we will
call bE. On the righthand
side we have a mixed second order differential coefficient
(forming first the difference in space and then in time or
vice versa). One gets the same mixed coefficient
from (64b) by forming the firstorder differential
coefficient with respect to space. Then one sees that the
mixed coefficient is equal to the product of c into
the secondorder differential coefficient in space of E,
which is analogous to f in
(36a) and may therefore be called fe.
Now one can eliminate the mixed coefficient in the equations
and
one gets bE = c2 f
E.
(65)
This equation is in complete analogy with (36a) and shows
the existence of electric waves with
velocity c. By the same method one may
derive a corresponding equation for the magnetic
field H(bH=c2fH). If
one of the two partial effects would happen without loss of
time, no propagation of the electric force in the forms of
waves would occur. This helps us to realize the importance
of Maxwell's
displacement current, for it provides just this rate of
change E of the electric
field."
t
And so, guided by the good brother Born, and the
intermittent company of Elders. The Alizzed, Scribe, and
shaded glow of those socalled, followed the good
brother to the best of their abilities, through the trick
and treat of the hear, there, and everywhere, of that long
subterranean nights journey into light. Holding within that
minds eye the golden thread of eventual escape.
The page continued to read.
"We shall now give a
description of the propagation of an electromagnetic
wave which is somewhat nearer to the actual
process. Let two metal spheres have large, opposite, equal
charges + e and  so that a b electric field exists
between them Next let a
/
Page 187 /
spark
occur between the spheres. The charges then neutralize each
other; the field collapses at a great rate of change
E . The figure shows how the magnetic and
electric lines of force then encircle each other alternately
(Fig.
99).
t
In our diagram the magnetic lines of
Fig
99 The
electromagnetic field surrounding a discharge spark between
two spheres. This field expands with velocity c of
light in all directions.
force are drawn only in the median plane between the
spheres, the electric lines of force only in the plane of
the paper, perpendicular to the median plane. The whole
figure is, of course, radially symmetrical about the line
connecting the centre of the spheres. Each successive loop
of the lines of force is weaker than its immediate
predecessor because it lies further outwards and has a
larger circumference. Accordingly, the inner part of a loop
of electric force does not quite counterbalance the outer
part of its predecessor, especially since it enters into
action a little late.
If we pursue the process
along a straight line which is perpendicular to the line
connecting the centers of the spheres, say along the xaxis,
then we see that the electric and magnetic forces are always
perpendicular to this axis; moreover, they are
perpendicular to each other.
/
Page 188 /
FUNDAMENTAL
LAWS OF ELECTRODYNAMICS
"This
is true of any direction of propagation. Thus, the
electro
magnetic wave is precisely transversal. Furthermore, it is
polarized,
but we still have the choice of regarding either the
electric or the
magnetic intensity of field as the determining factor of the
vibration.
Thus we have shown that the
velocity of the waves is equal to the
constant c and the waves are
transversal. Further since according
to Weber and Kohlrausch the value of c is equal to
that of the
velocity of light c, Maxwell was able to
conclude that light waves are
nothing other than electromagnetic waves."
The
Alizzed had yonder scribe count in descending order of a
kind , the number of the lines on which dances the following
words "light
waves are nothing other than electromagnetic
waves."
The scribe noted in passing, that if the title of the
Chapter v, counted as line 1 then the phrase
occupied lines 9
and 10
or
alternatively if the heading
be
discounted az iz usually the case Edgar,
then
lines
8 and
9 carry
the crown.
Either way the reference contains
9
lines
And we all know what that means, writ the scribe,
wonderingly wandering.
Page 188 continues
"
One of the inferences which Maxwell drew was soon confirmed
experimentally to a certain extent, for he calculated the
velocity of light c1 for the case of an insulator
(o
= O) and no free charge (p = O).
Maxwell's equations (62c and d) show that
we get nearly the same equations as (64) but with other
cvalues. In (64a) c is to be released by
c
*
and in (64b) by c . The same reasons which
led us to equation (65) show now that the square of the
velocity c 1 2 of the
electromagnetic waves must be equal to the product of
c and c : c 1 2 =
c2 .
u *
u
*u
Many materials are not noticeably magnetizable, so we can
set u = 1, which means that the velocity of light in an
insulator with dielectric constant * is given by c1
= c. This leads to
the value n =
c = / *
/
* c1
for the refractive index.
Thus it should be possible to
determine the refrangibility of light from the dielectric
constant as given by purely electrical measurements. For
some gasesfor example, hydrogen, carbon dioxide, airthis
is actually the case, as was shown by L Boltzmann. For other
substances Maxwell's relation n = / * is not
correct, but in all these cases the refractive index is not
constant but depends on the colour (frequency) of the light.
This shows that dispersion of the light introduces a
disturbing effect. We shall return to this fact
later and deal with it from the point of view of the theory
of electrons. At any rate, it is clear that the slower the
vibrations or the longer the waves of light that is used,
the more closely the dielectric constant as determined
statically, agrees with the square of the refractive index.
Waves of an infinite time of vibration are,
of" /
The nightmare transition of mind realization that out is
in within the womb of the mother.
The scribe recorded the 7 lettered name
Page 189
1 x 8 x 9 = 72 7
+ 2 = 9 1+
8 + 9 = 18
1 + 8 = 9
"course
identical with a stationary state. Researches into the
region of long waves (lengths of the order of centimeters)
have completely confirmed Maxwell's formula.
Concerning the more
geometrical laws of optics, reflection, refraction, double
refraction, and polarization in crystals and so forth, the
electromagnetic theory of light resolves all the
difficulties that were quite insuperable for the theories of
the elastic ether. In the latter, the greatest obstacle was
the existence of longitudinal waves which appeared when
light crossed theboundary between two media and which could
be removed only by making quite probable hypotheses about
the constitution of the ether. The electromagnetic waves are
always strictly transversal. Thus this difficulty
vanishes.
Maxwell's theory is almost identical formally with the ether
theory of MacCullagh, as we mentioned above (IV,6. P 117);
without repeating the calculations we can take over most of
his deductions.
We cannot here enter
into the later development of electrodynamics. The bond
between light and electromagnetism became ever closer. New
phenomena were continually being discovered which showed
that electric and magnetic fields exerted an influence on
light. Everything proved to be in accordance with Maxwell's
laws, the certainty of which continued to grow.
But the striking proof
of the oneness of optics and electrodynamics was given by
Henrich Hertz (1888) when he showed that the velocity of
propogation of electromagnetic waves was finite and when he
actually produced electromagnetic waves. He made sparks jump
across the gaps between two charged spheres and by this
means generated waves such as are represented by our diagram
(Fig 99
). When they encountered a circular wire with a small gap in
it, they produced in it currents which manifested themselves
by small sparks at the gap. Hertz succeeded in reflecting
these waves and in making them interfere. This enabled him
to measure their wave length. He knew the frequency of the
oscillations and thus could calculate the velocity of the
waves which came out equal to c, that of light.
This directly confirmed Maxwell's hypothesis. Nowadays the
Hertzian waves of wireless stations travel over the earth
without cessation and bear their tribute to the two great
scientists Maxwell and Hertz, one of whom predicted the
existence of electromagnetic waves while the other actually
produced them.
Page
190
1
x 9
x
0
= 9
1
+ 9
= 10
10 The
Electromagnetic Ether
From
this time on there was only one ether, which was the carrier
of all electric, magnetic, and optical phenomena."
Either or either, either or neither, writ yon
scribe.
Page 190
"We
know its laws, Maxwell's field equations, but we know little
of its constitution. Of what do the electromagnetic fields
actually consist, and what is it that executes vibrations in
the waves of light ?
We recall that Maxwell
took the concept of displacement as the foundation of his
argument, and we interpreted this visually as meaning that
in the smallest parts or molecules of the ether, just as in
the molecules of matter, an actual displacement and
separation of the electric (or magnetic) fluid occur. So far
as this idea concerns the process of electric polarization
of matter, it is well founded: it is also adopted in the
modern modifications of Maxwell's theory, the theory of
electrons, for numerous experiments have rendered certain
that matter has a molecular structure and that every
molecule carries displaceable charges. But this is by no
means the case for the free ether: here Maxwell's idea of
displacement is purely hypothetical, and its only value is
that it provides a visualizable image for the abstract laws
of the field.
These
laws state that with every change of displacement in time
there is associated an electromagnetic field of force. Can
we form a mechanical picture of this relationship ?
Maxwell himself designed mechanical
models for the constitution of the ether and applied them
with some success. "...Kelvin was particularly
inventive in this direction and strove unceasingly to
comprehend electromagnetic phenomena as actions of concealed
mechanisms and forces."
Good Lord! exclaimed the scribe at all this
activity.
Page 190
"The
rotational character of the relationship between electric
currents and magnetic fields, and its reciprocal character,
suggests that we regard the electric state of the ether as a
linear displacement, the magnetic state as a rotation about
an axis, or conversely. In this way we arrive at ideas that
are related to MacCullagh's ether theory. According to this
the ether was not to generate elastic resistences against
distortions in the ordinary sense, but resistances against
the absolute rotation of its elements of volume. It would
take us too far to count the numerous and sometimes very
fantastic hypotheses that have been put forward about the
constitution of the ether. If
/
Page 191 /
we
were to accept them literally, the ether would be a
monstrous mechanism of invisible cogwheels, gyroscopes, and
gears intergripping in the most complicated fashion, and of
all this confused mass nothing would be observable but a few
relatively simple features which would present themselves as
an electronic field.
There are also less
cumbersome, and in some cases, ingenious theories in which
the ether is a fluid whose rate of flow represents, say, the
electric field, and whose vortices represent the magnetic
field. Bjerknes has sketched a theory in which the electric
charges are imagined as pulsating spheres in the ether
fluid, and he has shown that such spheres exert forces on
one another which exhibit considerable similarity with the
electromagnetic forces.
If we inquire into the meaning and
value of such theories, we must grant them the credit of
having suggested (though rather seldom) new experiments and
of having led to the discovery of new phenomena. More often,
however, elaborate and laborious experimental researches
have been carried out to decide between two ether theories
equally improbable and fantastic. In this way much effort
has been wasted. Even nowadays there are people who regard a
mechanical explanation of the electromagnetic ether as
something demanded by reason. Such theories continue to crop
up, and naturally they became more and more abstruce as the
abundance of facts to be explained grows; hence the
difficulty of the task increases without cessation. Heinrich
Hertz deliberately turned away from all mechanistic
speculations. We give the substance of his own words:
"The interior of all bodies, including the free ether, can,
from an initial state of rest, experience some disturbances
which we call electrical and others which we call magnetic.
We do not know the nature of these changes of state, but
only the phenomena which their presence calls up." This
definite renunciation of a mechanical explanation is of
great importance from the methodical point of view. It opens
up the avenue for the great advances which have been made by
Einstein's researches. The mechanical properties of solid
and fluid bodies are known to us from experience, but this
experience concerns only their behaviour in a crude sense.
Modern molecular researches have shown that these visible,
crude properties are a sort of appearance, an illusion, due
to our clumsy methods of observation,
/
Page 192
1 x 9 x 2 + 18 1
+ 8 = 9
/
whereas
the actual behaviour of the smallest elements of structure,
the atoms, molecules, and electrons follows quite different
laws. It is therefore, naïve to assume that every
continuous medium, like the ether, must behave like the
apparently continuous fluids and solids of the crude world
accessible to us through our coarse senses. Rather, the
properties of the ether must be ascertained by studying the
events that occur in it independent of all other
experiences. The result of these researches may be expressed
as follows: The state of the ether may be described by two
directed magnitudes, which bear the names electric
and magnetic strength of field, E and
H, and whose changes in space and time are
connected by Maxwell's equations. Under certain
circumstances such an ether phenemon produces mechanical,
thermal, and chemical actions in matter that are capable of
being observed.
Everything that goes beyond these
assertions is superfluous hypothesis and fancy. It may be
objected that such an abstract view undermines the inventive
power of the investigator, which is stimulated by visual
pictures and analogies, but Hertz's own example contradicts
this opinion, for rarely has a physicist been possessed of
such wonderful ingenuity in experiment, although as a
theorist he recognized only pure abstraction as valid."
11.
Hertz's Theory of Moving Bodies
A
more important question than the pseudo problem of the
mechanical interpretation of the ether is that concerning
the influence of the motions of bodies (among which must be
counted, besides matter, the ether) on electromagnetic
phenomena.
This brings us back, but from a more general standpoint, to
the investigations which we made earlier
(IV,7)into
the optics of moving bodies. Optics is now a part of
electrodynamics, and the luminiferous ether is identical
with the electromagnetic ether. All the inferences that we
made earlier from the optical observations with regard to
the behaviour of the luminiferous ether must retain their
validity since they are obviously quite independent of the
mechanism of light vibrations; for our investigation
concerned only the geometrical characteristics of a light
wave, namely, frequency (Doppler effect), velocity
(convection), and direction of propagation
(abberation).
/
Page 193
/
We
have seen that up to the time when the electromagnetic
theory of light was developed only quantities of the first
order in B = v
c
were open to measurement. The result of these observations
could be expressed briefly as the "optical principle of
relativity":
Optical events depend on the relative motions of the
involved material bodies that emit, transmit or receive the
light. In a system of reference moving with constant
velocity relative to the ether all inner optical events
occur just as if it were at rest.
Two theories were
proposed to account for this fact. That of Stokes assumed
that the ether inside matter was completely carried along by
the latter; the second, that of Fresnel, assumed only a
partial convection, the amount of which could be derived
from experiments. We have seen that Stoke' theory, when
carried to its logical conclusion, became involved in
difficulties, but that Fresnel's represented all the
phenomena satisfactorily.
In the electromagnetic
theory the same two positions are possible, either complete
convection, as advocated by Stokes, or the partial
convection of Fresnel. The question is whether purely
electromagnetic observations will allow us to come to a
decision about these two hypotheses.
Hertz was the first to
apply the hypothesis of complete convection to Maxwell's
field equations. In doing so, he was fully conscious that
such a procedure could be only provisional, because the
application to optical events would lead to the same
difficulties as those which brought Stoke's theory to grief.
But the simplicity of a theory which required no distinction
between the motion of ether and that of matter led him to
develop and to discuss it in detail. This brought to light
the fact that the induction phenomena in moving
conductors, which are by far the most important for
experimental physics and technical science, are correctly
represented by Hertz's theory. Disagreements with
experimental results occur only in finer experiments in
which the displacements in nonconductors play a part. We
shall investigate all possibilities in
succession:
1.
Moving conductors (a) in the electrical field.
(b) in the magnetic field.
2.
Moving insulators (a) in the
electrical field
(b)
in the magnetic field.
/
Page 194
1a.
A conductor acquires surface charge in an electric field. If
it is moved, it carries them along with itself. But moving
charges must be equivalent to a current, and hence must
produce a surrounding magnetic field according to the law of
Biort and Savert. To picture this to ourselves we imagine a
plate condenser whose plates are parallel to the xzplane
(Fig.100).
Let them be be oppositely charged
Fig.100 A
charged plate of a condenser moving with velocity v
perpendicular to the electric
field.
With density of charge o
on the surface. This means e =
of
is the amount of electricity on an area f of the plate. Now
let one plate be moved with respect to the ether in the
direction of the xaxis with the velocity v. Then a
convection current arises.
The moving plate is displaced with velocity v, that
is, by a length vr
in
the time t
.
If its width in zdirection is a, then an
amount of electricity e =
oavt passes
in time
t through
a plane that is parallel to the
yzplane,
Hence a current J = e =
oav flows. This must exert exactly the
same
t
magnetic action as a conduction current of magnitude J
flowing through the plate when it is at rest.
This was confirmed
experimentally in Helmholtz's laboratory by H. A. Rowland
(1875), and later, more accurately, by
A. Eichenwald. Instead of a plate moving rectilinearly, a
rotating metal disk was used.
1b. When conductors are moved about in a magnetic
field, electric fields arise in them, and hence currents are
produced. This is the
/
Page 195 /
phenomenon
of induction by motion, already discovered by Faraday and
investigated quantitatively by him. The simplest case is
this: Let the magnetic field H produced, say by a
horseshoe magnet be parallel to the zaxis (Fig. 101). Let
there be a straight piece of wire of length l
parallel to the yaxis, and let
this be moved with the velocity v in
the direction of the xaxis. If the wire is now made part
of
Fig.101 Motion
of a wire of length 1, which is part of a closed
circuit, in a magnetic field between the ends of a large
horseshoe magnet.
A closed circuit by sliding it on the two opposite arms of a
Ushaped piece of wire in such a way that the Utakes no part
in the motion (seefigure). Then an induction current
J flows in the wire. This is given most simply by
stating Faraday's law of induction thus: The current induced
in a wire which forms part of a closed circuit is
proportional to the change per second of the number of lines
of force enclosed by the wire loop. This number is measured
by the magnetic displacement per unit area
uH
multipied by the area f of the loop
fuH.
In the section on magnetic induction (p.176) the change of
this quantity was considered to be due to a change of H
by H
in
the short time interval t
Here
it is due to a change of the area f produced by the
movement of the wire. If its length is l and its
velocity perpendicular to its extension is v, then
it sweeps out the
/
Page 196 /
area
lv each second, and this is the change of f.
The change of the number of lines of force per second
is therefore vluH.
According to Faraday's induction law an electric
current J is induced in the wire.
Instead of speaking of the current J it is better
to express the effect in terms of the potential difference
V produced between the ends of the wire. The
experiment gives V proportional to the quantity just
discussed, vluH.
Concerning the factor of proportionality, a remarkable law
of symetry has been revealed. If one measures all quantities
in units here used, this factor turns out to be 1 , so
that
c
one has the equation V=1 v l u H. Seen from
the wire this corresponds to an electric field E=
V = v
c l
c u H. If the same piece of wire were to move without
being part of a closed circuit, there would appear charges
at the end of the wire corresponding to this field as long
as the movement went on.
This law is the basis of
all machines and apparatus of physics and electrotechnical
science in which energy of motion is transformed
by induction into electromagnetic energy; these
include, for example, the telephone, and dynamo machines of
every kind. Hence the law may be regarded as having been
confirmed by countless experiments.
Fig.
102 A
charged condenser is filled with a diskshaped insulator. In
the insulator the displacement has induced charges on the
surface of the disk. One part of the displacement (dipoles
+ ) is caused by the ether, the other part (dipoles
+
)
by the insulator. If the insulator is moved, only the
insulator dipoles are moved with it..
/
Page 197 /
2a.
We
suppose the motion of a nonconductor in an electric field to
be realized thus: A moveable disk composed of the substance
of the nonconductor is placed between the two plates of the
condenser of Fig. 100 (see Fig. 102). The disk shall fill
the space between the condenser plates so that the distance
a marked in Fig. 100 measures also the
corresponding width of the disk. If the condenser is now
charged, an electric field E arises in the disk,
and a displacement*E is induced which is
perpendicular to the plane of the plates, that is, parallel
to the ydirection. This causes the two boundary
faces of the insulating disk to be charged equally and
oppositely to the metal plates facing them respectively. The
surface charge has a density o
which
is proportional to the displacement D in the
insulator: 4 pio
=
D E. D consists of two parts,
De = E. the displacement of the ether, and
D =D  e,
the displacement of matter alone.
If the insulating layer
is now moved in the direction of the xaxis with
the velocity v, then, according to Hertz, the ether
in the layer will be carried along completely. Hence the
field E and the charges of density
o
= *E produced by it on the bounding planes will
also be carried
along. 4pi
Therefore the moving charge of a
bounding surface again represents a current
*E av and must generate, according to Biot
and Savert,s law, a magnetic
field.
4pi
W.C. Rontgen proved experimentally
(1885) that this was the case, but the deflection of the
magnet needle that he observed was much smaller than it
should have been from Hertz's theory. Rontgen's experiments
show that only the excess of the charge density over the
displacement of the ether alone (i.e, D

De
= E (*)1
= Dm,
the displacement of the insulator alone) participates in the
motion of the matter. We shall interpret this result later
in a simple way. Here we merely establish that, as was to be
expected according to the well known facts of optics,
Hertz's theory of complete convection also fails to explain
purely electromagnetic phenomena.
Eichenwald (in 1903) confirmed
Rontgen's result very strikingly by allowing the charged
metal plates to take part in the motion. These
give a convection current of the amount
oav
= *E av; according /
4pi
Page 198
1 x 9 x 8 = 72 2
+ 7 = 9 8
+ 9 +1= 18 8
+ 1 = 9
/
to
Hertz this insulating layer ought, on account of the
opposite and equal charges, exactly to compensate this
current. But Eichenwald found that this was not the case.
Rather, he obtained a current which was entirely independent
of the material of the insulator. This is exactly what is to
be expected according to Rontgen's results described
above."
The Alizzed removes information for one reason or
another.
Page 198 "he obtained a current which was entirely
independent of the material of the insulator."
" For the current due to the insulator is
(*E
 E
) av, of which the first
term is compensated by the convection of the
plates, 4pi
4pi
and so we are left with the current
E av, which is
independent of the dielectric constant *.
4 pi
Fig.
103 A
piece of an insulator is moved in a magnetic field to
measure the induced displacement charges at the surface of
the disk.
2b. We assume a magnetic field parallel to the
zaxis, produced, say, by a horseshoe magnet, and a
disk of nonconducting material moving through the field in
the direction of the xaxis ( Fig. 103). Let the
insulator be not magnetizable (u = 1).
Let the two bounding faces of the disk which are
perpendicular to the yaxis be covered with metal,
and let these surface layers be connected to an
electrometer by means of sliding contacts so that the
charges that arise on them can be measured.
Page
199
This
experiment corresponds exactly with the induction experiment
discussed under (1b), except that a moving dielectric now
takes the place of the moving conductor. The law of
induction is applicable in the same way. It demands the
existence of an electric field E = H
v , acting in the magnetic
direction of the yaxis on the moving insulator. Hence,
according to
Hertz's
c
theory, the two superficial layers must exhibit opposite
charges of surface density *E
= * H v
4pi
4pi c
which cause a deflection of the electrometer. The experiment
was carried out by H.F. Wilson, in
1905,
with a rotating dielectric, and it did, indeed, confirm the
existence of the charge produced, but again to a lesser
extent, namely, corresponding to a surface density
(* 1) H v .
Thismeans that there is only an effect of the moving matter
and none of the ether. Here too, then Hertz's theory
fails.
4
pi
c
In all these four typical phenomena
what counts is clearly only the relative motion of the
fieldproducing bodies with respect to the conductor or
insulator investigated. Instead of moving this in the x
direction, as we have done, we could have kept it at rest
and moved the remaining parts of the apparatus in the
negative direction of the xaxis. The result would
have been the same. For Hertz's theory recognisezes only
relative motions of bodies, the ether being also reckoned as
a body. In a system moving with constant velocity everything
happens, according to Hertz, as if it were at rest; that is
the classical principle of relativity holds.
But Hertz's theory is
incompatible with the facts, and it soon had to make way for
another which took exactly the opposite point of view with
regard to relativity."
Hearing a sighting, on the nearside of an outside
sounding wind the Zed Aliz Zed thanked, the me, mi'self, and
I, of the far yonder scribe for the, az iz transcription, of
Brother Born's most important exposition. Telling the wah
scribe that reight or wrong it would be right az ninepence
on the night, and not to hurry a worry.
Page 199
1
x 9 x 9
= 81
8
+ 1
= 9
1
+ 9 + 9 = 19
1
x 9 = 9
1
+ 9 = 1
1
x 9
= 9
9
+ 1
= One
x
the
NINE
of the that.
