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Abstract

Displacement current was originally conceived by James Clerk Maxwell in 1861 in connection with linear polarization in a dielectric solid which he believed to pervade all of space. Modern textbooks however adopt a different approach. The official teaching today is that displacement current is a consequence of extending the original solenoidal Ampère’s Circuital Law to embrace the conservation of electric charge. Yet, unless either of these two methods leads to a displacement current that is related to Faraday’s Law of Induction, then it cannot serve its main purpose, which is to provide a bridge between Ampère’s Circuital Law and Faraday’s Law, hence enabling the derivation of the electromagnetic wave equations. This matter will be investigated in both the Coulomb gauge and the Lorenz gauge.
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Maxwell’s Displacement Current in the Two Gauges
Frederick David Tombe
Belfast, Northern Ireland,
United Kingdom,
sirius184@hotmail.com
15th October 2021
Abstract. Displacement current was originally conceived by James Clerk Maxwell
in 1861 in connection with linear polarization in a dielectric solid which he believed
to pervade all of space. Modern textbooks however adopt a different approach. The
official teaching today is that displacement current is a consequence of extending
the original solenoidal Ampère’s Circuital Law to embrace the conservation of
electric charge. Yet, unless either of these two methods leads to a displacement
current that is related to Faraday’s Law of Induction, then it cannot serve its main
purpose, which is to provide a bridge between Ampère’s Circuital Law and
Faraday’s Law, hence enabling the derivation of the electromagnetic wave
equations. This matter will be investigated in both the Coulomb gauge and the
Lorenz gauge.
Ampère’s Circuital Law
I. The original derivation of Ampère’s Circuital Law assumes that an
electric circuit will be closed, and that there will be no accumulation of
electric charge at any point along the circuit. Since this condition does not
hold in the case of a charging or discharging capacitor, then Ampère’s
Circuital Law, in its original solenoidal form, cannot hold in that context.
However, when we take into consideration the equation of continuity of
electric charge, we can expand Ampère’s circuital law into the extended
form,
×B = µ(J + εES/∂t) (1)
where the electrostatic term, ES, satisfies Gauss’s Law,
2
·ES = ρ/ε (2)
The extra εES/∂t term is known as Maxwell’s Displacement Current,
and it will ensure that the divergence of the right-hand-side of the
equation remains zero. It is important to note though, that equation (1) is
not derived in the textbooks from first principles, but rather by force-
fitting with the benefit of hindsight, and it will be realized later, that just
as the Lorenz gauge is also a force-fit, the displacement current term,
εES/∂t, is in fact itself a result of extending Ampère’s circuital law to
incorporate the Lorenz gauge, where it had previously only incorporated
the Coulomb gauge. It is therefore ironic that Maxwell, the architect of
the displacement current, was opposed to the Lorenz gauge, which was
named after Danish physicist Ludvig Lorenz, who proposed it in 1867, [1].
It will be seen later though, that the displacement current that is involved
in wireless electromagnetic radiation, will actually emerge from the
original solenoidal version of Ampère’s circuital law.
Meanwhile, if we are looking for a derivation of equation (1) from
first principles, rather than simply a retrospective justification, we can
look to a paper written by Dr. Zhong-Cheng Liang of the Nanjing
University of Posts and Telecommunications entitled “Dark matter and
real-particle field theory”, [2]. The subject matter of this paper is actually
more fundamental than electromagnetism. In Dr. Liangs paper, he fills all
of space with what he refers to as elastic electrons, and from two
fundamental field equations, he derives the parent equation (seen as
equations (36) and (B17) in his paper) that underlies Ampère’s circuital
law in its full form. It’s important to note however, that in Dr. Liang’s
paper, the speed of light has not yet entered the proceedings. The speed of
light is something that will follow later from the elasticity and the density
of the particle field.
The Speed of Light
II. Although the primary justification for the displacement current term in
equation (1) lies in the conservation of charge, this was not the basis upon
which James Clerk Maxwell first conceived of the idea. Maxwell first
conceived of displacement current in Part III of his 1861 paper “On
Physical Lines of Force”, [3], in conjunction with an all-pervading elastic
dielectric solid. In 1855, Wilhelm Eduard Weber and Rudolf Kohlrausch,
by discharging a Leyden Jar (a capacitor), demonstrated that the ratio of
the electrostatic and electrodynamic units of charge is equal to c√2, where
c is the directly measured speed of light, [4]. On converting from
electrodynamic units into electromagnetic units, Maxwell exposed the
3
speed of light directly, and by comparing the Weber-Kohlrausch ratio
with the ratio of the transverse elasticity to the density of his dielectric
solid, Maxwell concluded that his dielectric solid is the all-pervading
luminiferous medium that is responsible for electric, magnetic, and optical
phenomena. Meanwhile, back in Part I of the same paper, Maxwell had
already derived Ampère’s circuital law hydrodynamically in its original
solenoidal form. Then, by employing the concept of linear polarization in
his dielectric solid, Maxwell presented displacement current in the form
εES/∂t, where the electric displacement, D, is equal to εES, and where
the displacement current, JD, is equal to ∂D/∂t. The electric permittivity,
ε, is inversely related to the dielectric constant, which is in turn a measure
of the transverse elasticity. Maxwell had therefore assembled equation (1)
above from two separate parts. And since linear polarization and charge
separation in a capacitor are closely related topics, Maxwell was probably
dealing with a phenomenon that involves the conservation of charge at a
deeper level.
Wireless Electromagnetic Radiation
III. In his 1865 paper, “A Dynamical Theory of the Electromagnetic
Field”, [5], Maxwell’s displacement current, which had originally been
tied up with linear polarization and the electrostatic force, ES, instead
became associated with the time-varying electromagnetic induction force,
EK. It’s a major omission on Maxwell’s part that he made no attempt to
physically justify this transfer of association. Nevertheless, the
mathematical justification alone is sufficient indication that Maxwell was
on the right tracks, further indicating that displacement current comes in
two distinct varieties, and that for the purposes of deriving the
electromagnetic wave equations, we are not interested in an electrostatic-
based displacement current, but rather in one that is based on time-
varying electromagnetic induction. This requires that the dielectric nature
of the luminiferous medium is no longer sufficient on its own to explain
the elasticity that is associated with a magnetization-based displacement
current. We need to refer back to the all-pervading sea of tiny molecular
vortices, [3], [6], [7], that Maxwell used in Part II of his 1861 paper in order
to explain electromagnetic induction.
We will identify the vector field, AC, with the circumferential
momentum circulating around the edge of these fine-grained vortices. As
such, the divergence of AC will be zero, and this is the essence of the
Coulomb gauge. If we define A in general as,
4
A = μ/4πV (JdV)/r (3)
then the Coulomb gauge is the transverse component of A within the
context of a single vortex. Since the electric field in the displacement
current needs to be interchangeable with the electric field in Faraday’s
Time Varying Law of Induction, if it is to be used to derive the
electromagnetic wave equations, this means that it should take the
mathematical form, εEK/∂t, such that,
EK = −∂AC/∂t (4)
where B is the vorticity of this circulating current, AC, as in,
×AC = B (5)
Then further taking the curl of B, this expands to,
××AC = (·AC) − 2AC (6)
In Dr. Liang’s paper, [2], if we equate αs with magnetic permeability,
μ, while equating c with the speed of light, then Dr. Liang’s equation
(B17) becomes equivalent to equation (1). Hence, equation (6) becomes
the special case of Dr. Liang’s equation (B16), in the Coulomb gauge, and
since,
∇(·A) = εES/∂t = 0 (7)
being in the Coulomb gauge, equation (1) then reduces to,
×B = μJ (8)
which is the original solenoidal form. In the solenoidal context of the
perimeter momentum of one of Maxwell’s tiny molecular vortices, this
results in Ampère’s circuital law adopting the mathematical form,
×B = μεEK/∂t (9)
with the Coulomb gauge guaranteeing that both sides of the
equation will have zero divergence.
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If we consider Maxwell’s vortices to be dipolar, each comprising of
an aether sink (electron) and an aether source (positron), then the
induction-based displacement current (in the Coulomb gauge) will be an
oscillatory phenomenon tangential to these tiny rotating electron-positron
dipoles that fill all of space, and such that pure electric fluid (aether)
swirls across from the positron of one dipole into the electron of its
neighbour, with this repeating again indefinitely with respect to the next
neighbour along the line until the wave is absorbed by a target, [8], [9]. We
know from equation (4) that displacement current in this context is equal
to −ε2AC/∂t2, and from the oscillatory nature we know that,
AC = −ε2AC/∂t2 (10)
which means that displacement current is one and the same thing as
the circumferential momentum, [10]. Maxwell referred to the
circumferential momentum as the electromagnetic momentum and he
identified it with Faraday’s electrotonic state, yet he never identified it
with his displacement current, as he should have done. In modern
textbooks, AC is referred to as the magnetic vector potential.
In the preamble to Part III of Maxwell’s 1861 paper, where his sea of
molecular vortices gradually gives way to a dielectric solid, he says, “I
conceived the rotating matter to be the substance of certain cells,
divided from each other by cell-walls composed of particles which are
very small compared with the cells, and that it is by the motions of these
particles, and their tangential action on the substance in the cells, that
the rotation is communicated from one cell to another.”
Conclusion
IV. The Coulomb gauge and the Lorenz gauge are mutually
perpendicular aspects of a single phenomenon. This can be explained
within the context of one of the tiny molecular vortices that James Clerk
Maxwell presumed to fill all of space. The Coulomb gauge pertains to the
transverse aether flow, whereas the Lorenz gauge pertains to the radial
flow. It’s therefore ironic that the Coulomb gauge does not relate to the
radial electrostatic Coulomb force, ES, but rather to the transverse
electromagnetic force, EK, that is involved when these tiny vortices are
angularly accelerating (or precessing). The transverse force is the force
that is associated with time-varying electromagnetic induction and with
wireless electromagnetic radiation. The radial electrostatic Coulomb force
on the other hand is associated with the Lorenz gauge. In the dynamic
state when radiation is passing through, these vortices are undergoing an
6
oscillatory angular acceleration, and the electric fluid (aether) of which
the dipolar vortices are comprised, is being swirled from vortex to vortex,
[9], [11].
When Maxwell first conceived of the concept of displacement
current in his1861 paper, [3], he did so in the context of dielectric
polarization and the electrostatic Coulomb force, hence he was working
inadvertently in the Lorenz gauge. Yet, when he came to deriving the
electromagnetic wave equation in the magnetic disturbance, H, in his
1865 paper, [5], he switched to the Coulomb gauge by eliminating the
electrostatic Coulomb force in the derivation. Hence displacement current
as it is used in the derivation of the electromagnetic wave equations is an
induction effect, not directly measurable by experiment. It is an action in
its own right, capable of self-propagation in a wave mechanism, and it is
not the displacement current originally derived by Maxwell, and neither
is it the displacement current that is derived in the textbooks in
connection with capacitors. The textbooks therefore teach the wrong
displacement current for the purposes of deriving the electromagnetic
wave equations. The Lorenz gauge-based displacement current which is
taught in the textbooks is not an action in its own right, but rather the
reaction to an externally applied electric field, and so it could not be
involved in the mechanism of a self-propagating wave. Maxwell believed
that Lorenz had missed the point entirely and that we should be using the
Coulomb gauge.
Both gauges are of course valid, depending on the context. The
Coulomb gauge is the relevant gauge when it comes to the wireless
electromagnetic wave propagation mechanism, whereas, in DC
transmission line pulses, we would be operating in the Lorenz gauge.
References
[1] Lorenz, L., “On the Identity of the Vibrations of Light with Electrical Currents”
Annalen der Physik, Volume 131, page 243. English translation in Philosophical Magazine
Volume 34, pages 287-301 (1867)
https://www.researchgate.net/publication/323867067_Ludvig_Lorenz_1867_on_Light_and_E
lectricity
[2] Liang, Z.-C., “Dark matter and real-particle field theory” Derivation of field equations,
2. Divergence field, “Appendix B”, (2021)
https://www.researchgate.net/publication/350620139_Dark_matter_and_real-
particle_field_theory
[3] Clerk-Maxwell, J., “On Physical Lines of Force”, Philosophical Magazine, Volume XXI,
Fourth Series, London, (1861)
http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf
7
[4] Tombe, F.D., “The 1856 Weber-Kohlrausch Experiment”, (2015)
http://gsjournal.net/Science-Journals/Research%20Papers-
Mathematical%20Physics/Download/6314
[5] Maxwell, J.C., “A Dynamical Theory of the Electromagnetic Field”, Philos. Trans. Roy.
Soc. London 155, pp 459-512 (1865). Abstract: Proceedings of the Royal Society of London
13, pp. 531-536 (1864).
The derivation of the electromagnetic wave equation in H begins on page 497 in the first link
below. Note how the electrostatic component, Ψ, is eliminated after equation (68), hence
leaving the elastic displacement mechanism in the wave as an effect that is connected exclu-
sively with time-varying electromagnetic induction.
http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf
http://www.zpenergy.com/downloads/Maxwell_1864_5.pdf
[6] Whittaker, E.T., “A History of the Theories of Aether and Electricity”, Chapter 4, pages
100-102, (1910)
“All space, according to the younger Bernoulli, is permeated by a fluid aether, containing an
immense number of excessively small whirlpools. The elasticity which the aether appears to
possess, and in virtue of which it is able to transmit vibrations, is really due to the presence of
these whirlpools; for, owing to centrifugal force, each whirlpool is continually striving to
dilate, and so presses against the neighbouring whirlpools.”
[7] O’Neill, John J., “PRODIGAL GENIUS, Biography of Nikola Tesla”, Long Island, New
York, 15th July 1944, Fourth Part, paragraph 23, quoting Tesla from his 1907 paper “Man’s
Greatest Achievement” which was published in 1930 in the Milwaukee Sentinel,
“Long ago he (mankind) recognized that all perceptible matter comes from a primary
substance, of a tenuity beyond conception and filling all space - the Akasha or luminiferous
ether - which is acted upon by the life-giving Prana or creative force, calling into existence, in
never ending cycles, all things and phenomena. The primary substance, thrown into
infinitesimal whirls of prodigious velocity, becomes gross matter; the force subsiding, the
motion ceases and matter disappears, reverting to the primary substance”.
http://www.rastko.rs/istorija/tesla/oniell-tesla.html
http://www.ascension-research.org/tesla.html
[8] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)
Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)
http://gsjournal.net/Science-Journals/Research%20Papers-
Mathematical%20Physics/Download/6371
See also “The Double Helix and the Electron-Positron Aether” (2017)
http://gsjournal.net/Science-Journals/Research%20Papers-
Mechanics%20/%20Electrodynamics/Download/7057
[9] The 1937 Encyclopaedia Britannica article on ‘Ether’ discusses its structure in relation to
the cause of the speed of light. It says, POSSIBLE STRUCTURE. __ The question arises as
to what that velocity can be due to. The most probable surmise or guess at present is that
the ether is a perfectly incompressible continuous fluid, in a state of fine-grained vortex
motion, circulating with that same enormous speed. For it has been partly, though as yet
incompletely, shown that such a vortex fluid would transmit waves of the same general na-
ture as light waves _i.e., periodic disturbances across the line of propagation_ and would
transmit them at a rate of the order of magnitude as the vortex or circulation speed - - - -
http://gsjournal.net/Science-Journals/Historical%20Papers-
%20Mechanics%20/%20Electrodynamics/Download/4105
8
[10] Tombe, F.D., “Displacement Current and the Electrotonic State” (2008)
http://gsjournal.net/Science-Journals/Research%20Papers-
Mechanics%20/%20Electrodynamics/Download/228
[11] Tombe, F.D., “Wireless Radiation Beyond the Near Magnetic Field” (2019)
https://www.researchgate.net/publication/335169091_Wireless_Radiation_Beyond_the_Near
_Magnetic_Field
[12] Tombe, F.D., “Radiation Pressure and E = mc2 (2018)
http://gsjournal.net/Science-Journals/Research%20Papers-
Mathematical%20Physics/Download/7324
Appendix I
(The Biot-Savart Law in the Coulomb Gauge)
The Double Helix Theory of the Magnetic Field [8], is essentially Maxwell’s sea of
aethereal vortices but with the vortices replaced by rotating electron-positron dipoles.
Within the context of a single rotating electron-positron dipole, the angular
momentum can be written as H = D×v, where D is the displacement from the centre
of the dipole and v is the circumferential velocity. When elastically bonded to other
dipoles within the wider electron-positron sea, the displacement D will be related to
the transverse elasticity through Maxwell’s fifth equation, D = εE. A full analysis can
be seen in the articles “Radiation Pressure and E = mc2 [12], and “The 1855 Weber-
Kohlrausch Experiment” [4]. If we substitute D = εE into the equation H = D×v, this
leads to,
H = εv×EC (11)
See Appendix II regarding why the magnitude of v should necessarily be equal
to the speed of light. Equation (11) would appear to be equivalent to the Biot-Savart
Law if EC were to correspond to the Coulomb electrostatic force. However, in the
context, EC will be the centrifugal force, EC = µv×H, and not the Coulomb force. If
we take the curl of equation (11) we get,
×H = ε[v(EC) EC(v) + (EC)v (v)EC] (12)
Since v is an arbitrary particle velocity and not a vector field, this reduces to,
×H = ε[v(EC) (v)EC] (13)
Since v and EC are perpendicular, the second term on the right-hand side of
equation (13) vanishes. In a rotating dipole, the aethereal flow from positron to
electron will be cut due to the vorticity, the separate flows surrounding the electron
and the positron will be passing each other in opposite directions, and so the Coulomb
force of attraction will be disengaged. Hence, the two particles will press against each
other with centrifugal force while striving to dilate, since the aether can’t pass
laterally through itself, and meanwhile the two vortex flows will be diverted up and
down into the axial direction of the double helix, [8]. Despite the absence of the
Coulomb force in the equatorial plane, EC is still nevertheless radial, and like the
9
Coulomb force, as explained in Appendix III, it still satisfies Gauss’s Law, this time
with a negative sign in the form,
EC = ρ/ε (14)
Substituting into equation (13) leaves us with,
×H = ρv = J = AC (15)
and hence since B = µH then,
×B = µJ = µAC (16)
which is Ampère’s Circuital Law in the Coulomb gauge as per equation (9).
Appendix II
(The Speed of Light)
Starting with the Biot-Savart law in the Coulomb gauge, H = εv×EC, where EC = µv×H,
means that we can then write H = εµv×(v×H). It follows therefore that the modulusHis
equal to εµv2H since v, EC, and H are mutually perpendicular within a rotating electron-
positron dipole. Hence, from the ratio εµ = 1/c2, it follows that the circumferential speed v
must be equal to c within such a rotating dipole. In other words, the ratio εµ = 1/c2 hinges on
the fact that the circumferential speed in Maxwell’s molecular vortices is equal to the speed of
light.
Appendix III
(Gauss’s Law for Centrifugal Force)
Taking the divergence of the centrifugal force, EC = µv×H, we expand as follows,
·(µv×H) = μ[H·(×v) v·(×H)] (17)
Since v refers to a point particle that is in arbitrary motion, and not to a vector field, then
×v = 0, and since ×H = J = ρv, it follows that,
·(µv×H) = μρv·v (18)
then substituting v = c as per Appendix II,
·(µv×H) = −μρc2 (19)
and substituting c2 = 1/με, this leaves us with,
·(µv×H) = −ρ/ε (20)
which is a negative version of Gauss’s law for centrifugal force.
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Dark matter and real-particle field theory" Derivation of field equations, 2. Divergence field
  • Z.-C Liang
Liang, Z.-C., "Dark matter and real-particle field theory" Derivation of field equations, 2. Divergence field, "Appendix B", (2021) https://www.researchgate.net/publication/350620139_Dark_matter_and_real-particle_field_theory