6 The Electrical
Displacement Current
"Suppose the poles of a
galvanic cell to be connected with the plates of a condenser
by means of two wires, one of them containing a switch (Fig.
93). If the switch is pressed down , a current
flows..."
Fig.
93 When
the condenser is charged by a convection current
Jc, the electric field inside the condensers
changes and gives rise to a displacement current of the same
amount as Jc
Page
175
"...which charges the two
plates of the condenser; an electric field E is
thereby introduced between them. Before Maxwell's time, this
phenomenon was regarde as an "open circuit." Maxwell
however realized that during the growth of the field
E a displacement current flows between the
condenser plates, and thus the circuit becomes closed. As
soon as the condenser plates are completely charged, both
currents, the conduction and displacement current
cease..."
Fig
94 Both
the convection current Jc and the displacement
current Jd produce a surrounding magnetic
field.
"...Now
the essential point is Maxwell's affirmation that the
displacement current just like the conduction current,
produces a magnetic field according to Biot and Savart's
law. That this is actually so has not only been proved by
the success of Maxwell's theory in predicting numerous
phenomena but was also later confirmed directly by
experiment
The magnitude of the displacement
current can easily be computed...."
"... Therefore, following Maxwell,
the whole current density is the sum j = jc + jd
where jc is the current density of the free
moveable charges and jd is the displacement
current. Both kinds of current are surrounded by a magnetic
field in the usual way
(Fig.94)
Page 176
7 Magnetic
Induction
"After Oersted had
discovered that a conduction current produces a magnetic
field and Biot and Savart had formulated this fact as an
action at a distance, Ampere discovered (1820) that two
voltaic currents exert forces on each other, and he
succeeded in expressing the law underlying this phenomenon
again in terms of an action at a distance. This discovery
had far reaching consequences, for it made it possible to
regard magnetism as an effect of moving electricity.
According to Ampere small closed currents are supposed to
flow in the molecules of magnetized bodies. He showed that
such currents behaved exactly like elementary magnets. This
idea has stood the test of thorough examination; from his
time on magnetic fluids became superfluous. Only electricity
was left, which, when at rest, produced the electrostatic
field, and when flowing, produced the magnetic field
besides. Ampere's discovery may also be expressed in the
following way: According to Oersted a wire in which the
current J 1 is flowing produces a magnetic field in
its neighborhood. A second wire in which the current
J 2 is flowing is then pulled by forces
due to this magnetic field. In other words, a field produced
by one current tends to deflect or accelerate
flowing electricity.
Hence the following question
suggests itself: Can the magnetic field also set electricity
that is at rest into motion? Can it produce or "induce" a
current in the second wire which is initially without a
current?
Faraday found the answer to this
question (1831) He discovered that a static
magnetic field is not able to produce an electric
current
/ Page
177 1
x 7 x 7 =
49 /
but that a field which
varies in time is able to. For example, when he quickly
brought a magnet close to a loop of wire made of conducting
material, a current flowed in the wire as long as the magnet
moved. In particular, when he produced the magnetic field by
means of a primary current, a short impulse of current
occurred in the secondary wire whenever the first current
was started or stopped.
From this it is clear that the
induced electric force depends on the velocity of alteration
of the magnetic field in time. Faraday succeeded in
formulating the quantitive law of this phenomenon with the
help of his concept of lines of force. Using Maxwell's ideas
we shall give it such a form that its analogy with Biot and
Savert's law comes out clearly
Fig.
95 A
changing magnetic field which represents a magnetic
current 1 is surrounded by an electric
field.

Fig.
96
Direction of the electric field E induced by a
magnetic current 1 (compare with Fig. 84).

We imagine a bundle of
parallel lines of magnetic force that constitute a magnetic
field H. We suppose a circular conducting wire
placed around this sheath (Fig 95). If the intensity of
field H changes in the small interval of time..."
"...by the amount H we call..." "...its
velocity of change or the change in the number of lines of
force. If in analogy to the electrical displacement we
represent the lines of force as chains of magnetic dipoles (
which, however, according to Amphere
/ Page 178 /
then with the change of
H a displacement of the magnetic quantities will
occur in every ether molecule, or a magnetic displacement
current" will flow whose current strength per unit of area
or current density is given by I =..."
If the field H
is not in the ether but ii a substance of permeability..."
"...the density of the magnetic displacement current is
i =..." "...Thus the magnetic displacement current
is I =..." Thus the magnetic displacement current
is I =..." Thus the magnetic current I = fi =
f ..." "...passes through the cross section f,
that is, through the surface of the circle formed by the
conducting wire.
Now
according to Faraday, this magnetic current produces all
round it an electric field E, which encircles the
magnetic current exactly as the magnetic field H
encircles the electric current in Oersted's experiment but
in the reverse
direction. It is this electric field E that drives
the induced current around in the conducting wire; it is
also present even if there is no conducting wire in which
the current can form.
We see that the magnetic induction
of Faraday is a perfect parallel to the electromagnetic
discovery of Oersted. The quantitative law too, is the same.
According to Biot and Savert, the magnetic field H
produced by a current element of length l and of
strength j (compare fig. 84) in the middle plane
perpendicular to the connecting line r and to the
current direction, and has the value H = J1
(formula (54)).
Cr2
Exactly the same holds when electric and magnetic quantities
are exchanged and when the sense of rotation is reversed
(Fig.96). The induced electric intensity of field in the
central plane is given by
E
= I l .
Cr2
In it the same constant
c, the ratio of the electromagnetic to the
electrostatic unit of current, occurs which was found by
Weber and Kohlrausch to be equal to the velocity of light.
It can easily be seen from considerations about the energy
involved that this must be so.
A great number of the physical and
technical applications of electricity and magnetism depend
on the law of induction. The transformer, the induction
coil, the dynamo and innumerable
other
/ Page
179 1
x 7 x 9 =
63 6
+3 =
9 /
apparatus and machines
are appliances for inducing electric currents by means of
changing magnetic fields. But however interesting these
things may be, they do not lie on the road of our
investigation, the final goal of which is to examine the
relationship of the ether with the space problem. Hence we
turn our attention at once to the theory of Maxwell, whose
object was to combine all known magnetic phenomena into one
uniform theory of contiguous action.
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)
