The Maxwell-Faraday Equation Extended for Convection

Abstract

To show that the curl of E = v×B is an additional convective component to the Maxwell-Faraday equation.

Full text

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The Maxwell-Faraday Equation Extended for Convection
Frederick David Tombe,
Belfast, Northern Ireland,
United Kingdom,
sirius184@hotmail.com
17th July 2023
Abstract. To show that the curl of E = v×B is an additional convective component to the
Maxwell-Faraday equation.
The Electromagnetic Forces
I. When a charged particle is placed in a magnetic field, it experiences a force
under two conditions. If the magnetic field is varying in time, the particle
experiences a force,
E = −∂A/∂t (1)
where E is the force per unit charge, and where,
B = ∇×A (2)
When the charged particle is moving with velocity, v, it experiences a
deflecting force in the form,
E = v×B (3)
We will now take the curl of the total electromagnetic force, as in,
∇×E = ∇×[−∂A/∂t + v×B] (4)
From equation (2), this leads to the Maxwell-Faraday equation, but with an
additional convective term, ∇×(v×B), on the right-hand-side, as in,
∇×E = −∂B/∂t + ∇×(v×B) (5)
If we can show that,
∇×(v×B) = −(v·∇)B (6)
then we will have shown that ∇×(v×B) is the convective component of the
total time derivative,

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d/dt = ∂/∂t + (v·∇) (7)
where the partial time derivative, ∂/∂t, is known as the local time derivative,
and where (v·∇) is known as the convective derivative. Hence,
∇×E = −dB/dt (8)
Unlike in the 1982 derivation, [1], this will now be demonstrated by
multiplying out the components in full.
The Analysis
II. The analysis begins by multiplying out the components of v×B. These are,
(vyBz – vzBy)i (9)
(vzBx – vxBz)j (10)
(vxBy – vyBx)k (11)
We then take the curl of equations (9), (10), and (11), and at this stage it is
important to bear in mind that v is the velocity of a charged particle and that it is
not therefore a vector field. Hence, the spatial derivatives of v will all be zero.
The result is,
[vx∂By/∂y + vx∂Bz/∂z]i (12)
[vy∂Bz/∂z + vy∂Bx/∂x]j (13)
[vz∂Bx/∂x + vz∂By/∂y]k (14)
Now we need to take account of the fact that magnetic fields are
solenoidal, and that from equation (2), we know that,
∇·B = 0 (15)
and hence,
∂Bx/∂x + ∂By/∂y + ∂Bz/∂z = 0 (16)

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As pointed out by Professor Halim Boutayeb, Professor at Université du
Québec en Outaouais, in a private correspondence on ResearchGate, we can
isolate any of the three terms on the left-hand-side of equation (16) and equate it
with the negative of the sum of the other two terms. As such, substituting
equation (16) into (12), (13), and (14), we obtain,
−[vx∂Bx/∂x]i (17)
−[vy∂By/∂y]j (18)
−[vz∂Bz/∂z]k (19)
These are the equivalent of −(v·∇)B which is what we set out to
demonstrate. ∇×(v×B) is therefore the additional convective component
required to make the Maxwell-Faraday equation into a total time derivative
equation as per equation (8).
The Galilean Transformation
III. The convective term, v×B, in electromagnetic induction is not the
consequence of a Galilean transformation. The Galilean transformation ignores
the existence of the physical medium through which motion is defined and
which is responsible for generating both the inertial forces and the
electromagnetic forces. This medium is a dense sea of rotating electron-positron
dipoles that fills all of space, [2], [3], [4], [5], [6].
Conclusion
IV. The term, v×B, which first appeared in the electromotive force equation
(equation (77)) in Maxwell’s 1861 paper, “On Physical Lines of Force”, [7], is
the convective term which would complete the Maxwell-Faraday equation, that
being the curl of equation (77). This is the deflecting force which acts on a
moving charged particle in an already existing background magnetic field, and
which is due to the asymmetrical contact with the field. The magnetic field is
constructed from tiny rotating electron-positron dipoles, mutually aligned along
their rotation axes, and so the component of the particle’s motion at right-angles
to the field lines, will generate a differential centrifugal pressure on either side
of the particle, at right-angles to that component of its motion. This will cause
the particle to deflect away from its straight-line path. Meanwhile, the moving
particle itself also generates its own superimposed magnetic field which causes
an additional v×B force to press inwards evenly all around it, at right-angles to
its direction of motion. This latter force is also accompanied by an asymptotic

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factor which predicts a terminal speed and hence betrays the inertial resistance
that arises due to the presence of the all-pervading electron-positron sea. In
order to derive v×B in this latter context, we need to use a Lorentz
transformation and we need to also involve the electrostatic force, −∇ψ, since it
is involved in the construction of the magnetic field lines, [8].
Finally, it’s important to note that the Lorentz transformation is not actually
a coordinate frame transformation at all, but rather it is an analysis which
describes the manner in which a rotating dipole precesses and tends towards a
terminal speed as it is linearly accelerated through the electron-positron sea, [8].
References
[1] Tombe, F.D., “Maxwell’s Equations and Galilean Relativity”, (1984)
https://www.gsjournal.net/Science-Journals/Journal%20Reprints-
Mechanics%20/%20Electrodynamics/Download/3757
[2] Tombe, F.D., “The Double Helix Theory of the Magnetic Field”, (2006)
Galilean Electrodynamics, vol. 24, number 2, p.34, (March/April 2013)
https://www.researchgate.net/publication/295010637_The_Double_Helix_Theory_of_the_M
agnetic_Field
[3] Tombe, F.D., “Atomic Clocks and Relativity”, (2023)
https://www.researchgate.net/publication/372165749_Atomic_Clocks_and_Relativity
[4] Tombe, F.D., “The Positronium Orbit in the Electron-Positron Sea”, (2020)
https://www.researchgate.net/publication/338816847_The_Positronium_Orbit_in_the_Electr
on-Positron_Sea
[5] Tombe, F.D., “The Double Helix and the Electron-Positron Aether”, (2017)
https://www.researchgate.net/publication/319914395_The_Double_Helix_and_the_Electron-
Positron_Aether
[6] Tombe, F.D., “The Electron-Positron Sea”, (2014)
https://www.researchgate.net/publication/320173646_The_Electron-
Positron_Sea_The_Electric_Sea
[7] Clerk-Maxwell, J., “On Physical Lines of Force”, Philosophical Magazine, Volume
XXI, Fourth Series, London, (1861)
http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf
[8] Tombe, F.D., “The Lorentz Aether Theory”, (2020)
https://www.researchgate.net/publication/339696770_The_Lorentz_Aether_Theory