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# An Interpretation of Faraday's Lines of Force

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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.
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1
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
2
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
3
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
4
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
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)
5
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.
6
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
7
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
8
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
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-
[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-
[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
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,
[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-
[6] Tombe, F.D., “The 1856 Weber-Kohlrausch Experiment”, (2015)
http://gsjournal.net/Science-Journals/Research%20Papers-
[7] Tombe, F.D., “Radiation Pressure and E = mc2 (2018)
http://gsjournal.net/Science-Journals/Research%20Papers-
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
... Maxwell identified A with Faraday's electrotonic state. See "An Interpretation of Faraday's Lines of Force", [8]. ...
... The velocity field is more correctly the momentum field A known as the magnetic vector potential [8]. It is Maxwell's displacement current. ...
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