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An Interpretation of Faraday’s Lines of Force

Frederick David Tombe

Belfast, Northern Ireland,

United Kingdom,

sirius184@hotmail.com

7th April 2019

Abstract. The magnetic field is solenoidal, yet the Biot-Savart Law, which is the

textbook equation for the magnetic field, indicates the existence of a singularity

owing to the fact that it involves an inverse square law in distance. This dilemma is

solved within the context that an individual magnetic line of force constitutes a

double helix of sinks and sources closed on itself to form a toroidal ring vortex.

The Double Helix Theory of the Magnetic Field

I. We saw in “The Double Helix Theory of the Magnetic Field” [1], how

rotating electron-positron dipoles could be arranged in a double helix

fashion so as to account for the magnetic field. A rotating electron-

positron dipole consists of an electron and a positron undergoing a mutual

central force orbit such that the rotation axis is perpendicular to a line

joining the electron to the positron. If we stack these dipoles on top of

each other along their axes of rotation with the electrons placed

approximately above the positrons and angularly synchronized in a

twisted rope ladder fashion, we will effectively have a helical spring.

These helical springs are magnetic lines of force by virtue of the fact that

they channel a Coulomb tension along a double helix alignment within a

neutral electron-positron sea. See Figure 1 below. The angular momentum

vector H within these rotating dipoles is a measure of the magnetic

intensity of the lines of force. The rotating dipoles in one line of force

will be aligned in their mutual equatorial planes with the rotating dipoles

in an adjacent line of force and the mutual transverse velocities existing

between them will cause a centrifugal repulsion that will push them apart

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as explained in section II below. Since the magnetic lines of force

between two like poles do not join directly together, but instead spread

outwards and contact each other laterally, magnetic repulsion occurs due

to centrifugal force.

Figure 1. A close-up view of a single magnetic line of force. The electrons are

shown in red and the positrons are shown in black. The double helix is rotating

about its axis which represents the magnetic intensity vector H. The diagram is

not to scale as the relative dimensions remain uncertain. The circumferential

speed of the electrons and positrons is what determines the local speed of light.

Centrifugal Force and Magnetic Repulsion

II. Consider two electron-positron dipoles sitting side by side while rotat-

ing in the same plane and in the same direction. When the electron of one

dipole passes the positron of the other dipole in the opposite direction at

closest approach, the electrostatic field lines will connect directly be-

tween the two. According to Coulomb’s law there should be a force of

attraction acting between them as in the case of any two particles of op-

posite charge. However, in this case the two particles will possess an

enormous mutual transverse speed, and this gives us reason to believe

that the Coulomb force of attraction would be undermined. This would be

so if the electrostatic force field E is fluid based, because above a certain

threshold of mutual angular speed, the inevitable curl in the associated

velocity field would split this field between the two rotating dipoles. And

if this happens it will necessarily convert the electrostatic attraction into a

repulsion. The E field lines may remain irrotational, but their physical

cause will have changed. It will no longer be due to a tension in the fluid

but instead it will be due to side pressure from the flow lines, and so we

will now be dealing with centrifugal force as opposed to the Coulomb

force. The proof that such a fluid exists lies in the ability to explain mag-

netic repulsion and Ampère’s Circuital Law in terms of its curl. Time

varying electromagnetic induction can then be explained in terms of a

curl in the E field. We can call it the aether or the electric fluid, but it is

the primary fluid from which all matter is made. And if this fluid exists

then it should be obvious that particles are sinks or sources in it. As a

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convention, electrons will be considered to be aether sinks while posi-

trons are aether sources.

Electric current cannot be fully understood in the absence of a prima-

ry fluid flow at a deeper level than the flow of charged particles. Elec-

trons would eat their way upstream in such a fluid while positrons would

be pushed in the opposite direction, and if the fluid were inviscid, charged

particles would be accelerated by the fluid due to pressure or tension but

without taking on the fluid’s actual velocity. Electric signals in a conduct-

ing wire travel at a speed that is in the same order as the speed of light

which is probably the speed of the electric fluid.

Ampère’s Circuital Law

III. When rotating electron-positron dipoles bond together along their

rotation axes to form a double helical toroid with nothing in the toroidal

hole in the middle, the Coulomb attraction along the double helix would

tend to make the helix collapse. If the circumferential speed of each

rotating dipole is v, then ∇×v = H where H is the vorticity, and hence

∇∙H = 0 meaning that H is solenoidal. The speed v represents the vortex

flow of the electric fluid which constitutes an electric current. At the hole

in the middle of the toroid there will be a concentration of electric fluid

flowing in one direction, and the current density will be ρv = J where ρ is

the fluid density in the hole. The concentration of electric current through

the hole in the toroid prevents the toroid from collapsing into the hole.

Unlike in the case of fluid pouring down a sink, a toroid involves only

solenoidal flow and so the fluid circulates around indefinitely. The fluid

cannot pass sideways through itself inside the toroidal hole and so the

toroid cannot collapse. The double helix toroid is therefore the

fundamental basis for stability and the default alignment in the electron-

positron sea. It corresponds to a magnetic line of force and H corresponds

to magnetic intensity. And since H forms a circle around the inside of the

double helix, it follows therefore that ∇×H = J. This is Ampère’s

Circuital Law which is Maxwell’s third equation. See section V below.

Within the context that v corresponds to the circumferential current

within the individual rotating dipoles that fill all of space, a vector A will

be defined as µv, where µ, known as the magnetic permeability, is a

measure of the flux density of magnetic lines of force in the vicinity. In a

steady state magnetic field, A is a fine-grained circulating current density

(or momentum density) in the electron-positron sea. The product µH is

known as the magnetic flux density and written as B. Hence ∇×A = B and

Ampère’s Circuital Law becomes ∇×B = µJ. The equation ∇×A = B is

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Maxwell’s second equation from which ∇∙B = 0 can be derived. Maxwell

referred to the magnetic vector potential A as the electromagnetic

momentum, and although he invented displacement current, he failed to

identify it with A.

Ampère’s Circuital Law means that when a current or a particle, neu-

tral or otherwise, moves through the electron-positron sea, it causes the

electron-positron dipoles to align with their rotation axes forming sole-

noidal rings around the direction of motion. This provides a circular ener-

gy flow mechanism which replaces friction with the inertial forces and

magnetic repulsion. It’s similar in principle to the creation of smoke

rings. Maxwell explains Ampère’s Circuital Law at equation (9) in Part I

of his 1861 paper “On Physical Lines of Force” [2].

Ampère’s Circuital Law beyond atomic and molecular matter takes

the form ∇×B = µA, where A is net zero in the steady state. From this we

can then derive the electromagnetic wave equation [3], [4]. In the dynamic

state when electromagnetic radiation is passing through, the displacement

current, A, will be in an oscillating state and there will be a net flow of

electric fluid from vortex to vortex travelling at the speed of light [5], [6].

The Biot-Savart Law

IV. 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, where ε is known as

the electric permittivity (see section V below including the note on the

negative sign). A full analysis can be seen in the article “Radiation

Pressure and E = mc2” [7]. If we substitute D = −εE into H = D×v this

leads to,

H = εv×E (1)

Equation (1) would appear to be equivalent to the Biot-Savart Law

when E corresponds to the Coulomb force. However, in the context, E

will be the centrifugal force, E = v×B, and not the Coulomb force. If we

take the curl of equation (1) we get,

∇×H = ε[v(∇∙E) – E(∇∙v) + (E.∇)v – (v.∇)E] (2)

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Since v is not a vector field, this reduces to,

∇×H = ε[v(∇∙E) – (v.∇)E] (3)

Since v and E are perpendicular, the second term on the right-hand

side of equation (3) vanishes. Finally, the E field corresponds to the

centrifugal pressure in the equatorial plane within the rotating dipole.

This differs from the Coulomb force in that it arises from the curl in the

interior A field, whereas in Coulomb’s law it is the purely radial

component of A which is significant. The aethereal flow around the

positron therefore presses laterally against the aethereal flow around the

electron because the flow between the two has been cut, and in both cases

has been diverted up and down into the axial direction of the double

helix. Nevertheless, E is still radial and dependent on the magnitude of

the radial flow and so it is safe to suppose that E still satisfies Gauss’s

Law, ∇∙E = ρ/ε, and hence equation (3) becomes,

∇×H = ρv = J (4)

which is Ampère’s Circuital Law.

Unlike equation (1) above, the Biot-Savart Law is not restricted to

the context of a rotating electron-positron dipole. The Biot-Savart law is a

vector field function of general application with a polar origin located

inside a magnetic source. If we take the curl of the Biot-Savart Law, then

owing to its radial position dependence, we obtain ∇×B = 0 in the space

beyond the source. This gives the false impression that there is no curl in

a steady state magnetic field, yet the entire structure and operation of

magnetic lines of force is built upon vorticity. The vorticity in a magnetic

field is however fine-grained vorticity which is below the radar of the

Biot-Savart Law.

Another conundrum is the fact that the divergence of B is zero,

implying no sinks or sources, yet the entire B field is riddled with sinks

and sources. These two conundrums can both be resolved by realizing the

distinction between B in the context of the electron-positron sea on the

one hand, and the B(r) of the Biot-Savart Law on the other hand. B in the

electron-positron sea is a vorticity density associated with rotating

electron-positron dipoles. The curl of B in the electron-positron sea is µA

and the divergence is zero because B is solenoidal. The Biot-Savart Law

on the other hand is a position dependent function B(r) whereby the zero

divergence is based, not on the issue of B(r) being solenoidal, but rather

on the basis that it is an inverse square law position function in r.

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So, although Faraday’s lines of force are solenoidal, they are

nevertheless pierced at the sides with sources and sinks forming a double

helix pattern.

Maxwell’s Equations and Summary

V. Maxwell’s eight original equations as listed in his 1864 paper entitled

“A Dynamical Theory of the Electromagnetic Field” [4], in a section

entitled ‘General Equations of the Electromagnetic Field’, appear in

modern format as follows,

Maxwell’s Original Eight Equations 1864

(A) The Equation of Total Currents including “Displacement Current”

Jtotal = Jconduction + ∂D/∂t

(B) The Equation of Magnetic Force

∇×A = μH

(C) The Equation of Electric Current (Ampère’s Circuital Law)

∇×H = Jtotal

(D) The Equation of Electromotive Force (The Lorentz Force)

E = μv×H − ∂A/∂t − gradψ

(E) The Equation of Electric Elasticity (Hooke’s Law)

D = E

(F) The Equation of Electric Resistance (Ohm’s Law)

E = RJconduction

(G) Gauss’s Law for Free Electricity

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∇∙D = ρ

(H) The Equation of Continuity

∇∙J + ∂ρ/∂t = 0

Maxwell’s second equation, ∇×A = µH, was derived on the basis that

A is a momentum density from which we can then derive the solenoidal

equation ∇∙µH = 0 (∇∙B = 0) that is familiar in the modern listings of four

Maxwell’s equations. Modern textbooks however reverse the reasoning

and work backwards to A, giving no physical meaning to A, and in the

process adding an arbitrary constant of integration and inventing a red

herring topic known as “gauge theory”.

The Biot-Savart Law is notably absent from the list. Interestingly

there is an equation of the form H = εv×E which has a superficial

resemblance to the Biot-Savart Law and from which we can derive

Ampère’s Circuital Law (Maxwell’s third equation) simply by taking its

curl, but this equation only exists within the context of an electron-

positron sea which is not acknowledged to exist by mainstream

physicists. The velocity v term, where v determines the speed of light, is

the circumferential velocity of the individual rotating electron-positron

dipoles, while E is equal to μv×H. (See the Appendix at the end)

Modern textbooks take the curl of the actual Biot-Savart Law in

order to derive Ampère’s Circuital Law and this has caused considerable

confusion since it leads to the false conclusion that ∇×B = 0 in the

regions beyond atomic and molecular matter, where in fact we should be

using ∇×B = µA, where A is equal to zero in the steady state.

Not even Maxwell himself realized that the magnetic vector potential

A, which he termed the electromagnetic momentum, in fact corresponds

to his very own displacement current. The displacement current was

essential in Maxwell’s derivation of the electromagnetic wave equation,

but he never made the realization that the displacement current is in fact

the vector A. The general misunderstandings surrounding displacement

current is a serious problem which is fudged in the textbooks by adding a

virtual displacement current as an extra term to Ampère’s Circuital Law,

when in fact the original Ampère’s Circuital Law should simply be used

in conjunction with the actual displacement current A [3]. In a steady state

magnetic field, the displacement current A is undergoing fine-grained

circulation with no net translation, hence ∇×B = 0, while in the dynamic

state when time varying electromagnetic induction is involved, there is a

net translational flow of A which constitutes wireless electromagnetic

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radiation, and hence we have ∇×B = µA, where A satisfies the simple

harmonic equation, A = −ε∂2A/∂t2, hence A = ε∂E/∂t, where from

Faraday’s law, E = −∂A/∂t.

Maxwell’s fourth equation (D) is easily recognizable as the

Lorentz force. The three components on the right-hand side of equation

(D) appeared in equations (5) and (77) in Maxwell's 1861 paper [1]. If we

take the curl of equation (D) we end up with,

∇×E = −(v.∇)B − ∂B/∂t = −dB/dt (5)

See Appendix B in “The Double Helix Theory of the Magnetic

Field” [1] for the full analysis. Equation (5) with the convective term

(v.∇)B removed, hence leaving only the partial time derivative − ∂B/∂t

term, is one of the equations in the modern listings of four Maxwell’s

equations and it is known as Faraday’s Law. It deals only with time

varying electromagnetic induction although Faraday’s original flux law

also takes into account the convective effect E = v×B.

Maxwell’s fifth equation is a form of Hooke’s Law and it is most

relevant in relation to the elasticity of the electron-positron sea. In his

1861 paper where he first introduced the equation in the preamble of Part

III and at equation (105), Maxwell uses a negative sign even though he

derived it in conjunction with Hooke’s Law. It’s not clear why he did this,

but the negative sign is preferable when we are considering the simple

harmonic motion aspect of free oscillations [2].

References

[1] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)

http://gsjournal.net/Science-Journals/Research%20Papers-

Mathematical%20Physics/Download/6371

[2] Clerk-Maxwell, J., “On Physical Lines of Force”, Philosophical Magazine, Vol-

ume XXI, Fourth Series, London, (1861)

http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf

[3] Tombe, F.D., “Displacement Current and the Electrotonic State” (2008)

http://gsjournal.net/Science-Journals/Research%20Papers-

Mechanics%20/%20Electrodynamics/Download/228

[4] Clerk-Maxwell, J., “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 original eight Maxwell’s equations

are found in the link below in Part III entitled ‘General Equations of the Electromag-

netic Field’ which begins on page 480,

9

http://www.zpenergy.com/downloads/Maxwell_1864_3.pdf

Maxwell’s derivation of the electromagnetic wave equation is found in the link below

in Part VI entitled ‘Electromagnetic Theory of Light’ which begins on page 497,

http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf

[5] The 1937 Encyclopaedia Britannica article on ‘Ether’ discusses its structure in re-

lation 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 nature as light waves _i.e., periodic disturb-

ances across the line of propagation_ and would transmit them at a rate of the or-

der of magnitude as the vortex or circulation speed - - - -”

http://gsjournal.net/Science-Journals/Historical%20Papers-

%20Mechanics%20/%20Electrodynamics/Download/4105

[6] Tombe, F.D., “The 1856 Weber-Kohlrausch Experiment”, (2015)

http://gsjournal.net/Science-Journals/Research%20Papers-

Mathematical%20Physics/Download/6314

[7] Tombe, F.D., “Radiation Pressure and E = mc2” (2018)

http://gsjournal.net/Science-Journals/Research%20Papers-

Mathematical%20Physics/Download/7324

Appendix

H = εv×E where E = µv×H

H = εµv×(v×H)

│H│= │εµv2H│ (v, E, and H are mutually perpendicular in the rotating dipole)

εµ = 1/c2

hence v = c