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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 + ε∂ES/∂t) (1)

where the electrostatic term, ES, satisfies Gauss’s Law,

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∇·ES = ρ/ε (2)

The extra ε∂ES/∂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,

ε∂ES/∂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. Liang’s 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

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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

ε∂ES/∂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,

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A = μ/4π∫V (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, ε∂EK/∂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) = ε∂ES/∂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 = με∂EK/∂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 −ε∂2AC/∂t2, and from the oscillatory nature we know that,

AC = −ε∂2AC/∂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

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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

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[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

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[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

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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 modulus│H│is

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.

8th February 2022 amendment