Archive for the 'Classical Electrodynamics' Category

Jan 29 2023

Magnetic Fields as Effecting Coulomb Groups

An electric field of an electromagnetic wave does the work to extend the magnetic field of the same wave. What makes the electric field turn around must have something to do with running out of energy to extend the magnetic field further. Griffiths says: “Magnetic forces do no work” ([1], pg. 207], and that is why it is said that transmission of the Coulomb field is “a diffeomorphism on the electric fields of the gamma rays”:

Magnetic fields can act as guides however, and can help hold together a groupoid in the gamma ray field so it can act transitively. There is “energy stored in the magnetic field” ([1], pg. 317] and “Magnetic forces may alter the direction in which a particle moves, but they cannot speed it up or slow it down.” ([1], pg. 207) It is the same in Coulomb groups, spherical or concentrated, that carry the Coulomb field, – there are electric currents that are altered in direction by magnetic fields. Another example of this is gravitational lensing.

An involution may be a charged particle, or nucleus, with mass, as it absorbs gravitons for the energy to send out Coulomb groups, or it may be a Coulomb group itself in an open field. As a spherical group travels, for example, it takes on new gamma rays and leaves some behind, and the new gamma rays may be called an involution as they become part of the Coulomb group.

When it is said that with Coulomb phonon transmission, the gamma rays are “frozen in time” up to “10 meters at least”: ,

it is in relation to travel, though they may travel a miniscule amount. It is torsion that transmits the Coulomb field, and the angular velocity, ω, is higher the stronger the field.

In a Cartan decomposition, “g1 = t1 + p1 and g2 = t2 + p2“ ([2], pg.517), p is the peak point of the electric field of a graviton. In a Riemannian globally symmetric space of type I, p follows the peak of a sine wave, and it also follows the peak in a Riemannian globally symmetric space of type II.

[1] Griffiths, David J., “Introduction to Electrodynamics”, Prentice Hall, 1999

[2] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012

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Jan 17 2023

Action of the Electric Field

When a molecule is formed, each nucleus senses the one(s) closest by its spherical pulses. Then each nucleus starts sending out alternating concentrated groupoids toward the nearest nuclei in the molecule.

In a Coulomb attraction, the groupoid decides how to bisect by the spin of a target. The two brackets then compress against other gamma rays and subsequentially spring back and squeegee along the backside of the target in what is called a pullback. Past the target, the brackets “re-emerge as action morphisms of Lie algebroids” ([1], pg. 152), and join a spherical group.

The scalar potential has units of J/s, which is energy per time. The electric field has units of N/C, and Force = mass x acceleration per Newton’s second law. The acceleration is less for a larger mass of charge, and there are neutrons in most nuclei which makes the effect greater. The electric field travels faster the denser a gravitational field is, though the speed difference may not be discernable.

We can have “a π-saturated open set” ([1], pg. 97) with “saturated local flow”, though the gravitons will be at various phases on sine waves when an electric field comes through. Thus, in terms of analytic coordinates, “such coordinates do not usually exist for Lie groupoids.” ([1], pg.  pg. 142) What we have is an infinitesimal zigzag pattern, though when we back out to the classical level, it does not matter for any application.

As said earlier, Coulomb repulsion acts on the frontside of another charge. The electric field travels much faster than the charged mass it pushes, in part due to inertia, so likewise, after the push, the brackets join another spherical group behind the target. A nuclear concentrated groupoid may join a spherical groupoid once it passes a target.

In both cases, Coulomb attraction or repulsion, the spherical group from which the brackets came mends itself.

[1] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

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Nov 17 2022

The Coulomb Gauge    

There is another name for a free graviton, – it is “the identity isomorphism idEx, here denoted 1x, and the elements 1x, x ϵ M, act as unities for any multiplication in which they can take part” ([1], pg. 4). We see that unlike π (pi), idEx has some degree of circular polarization and/or skewed sine waves. In some writing instances π is the same as idEx and I am not trying to dictate how they should be used.

In the “Coulomb, radiation, or transverse gauge. This is the gauge in which ∇ · A = 0” ([2], pg. 241), we have a classical description. In the tensor sense, we have the forms Χij. The direction we choose for Χ is always transverse to the radial electric field at a chosen point, and the coordinate frame Ui is picked centered on the same point, creating a k-plane. We have that “The forms Χij are the transition forms for the Lie algebroid atlas {Ui, ψi, Θi}” ([1], pg. 206), and Θi varies with the density of the gamma ray field:

Considering the transition form TP/G [1], we may here call G the density of the gravitational field. It is seen that as the density goes up the transition angle Θi decreases for a given charge and distance from the charge.

In Jackson’s problem 6.19 (b), “the original and space-inverted vector potential differ by a gauge transformation” ([2], pg. 291). Though the earth catches some of the sun’s gravitons all the time, the sun’s gravitons during the day are greater at the face of the earth than at night, and inverted, changing the Coulomb gauge.

With the “Lorenz condition (1867), ∇ · A + (1/c2) ẟφ/dt = 0” ([2], pg. 240), it is mathematically shown that the system {Ui, ψi, Θi} acts fast compared to the gradient of A, and
           ιX (φ ˄ ψ) = ιX(φ) ˄ ψ + (-1)i φ ˄ ιX(ψ)              ([1], pg. 306)
Also, as small as gravitons are, we may as well call the k-planes “flat connections Θi“ ([1], pg. 206).

Since we have “t the fixed point set of θ” ([3], pg. 401), t is on the center line of a gamma ray, and “g0 = t0 + p0 is a Cartan decomposition of g0“ ([3], pg. 184). In certain situations the center can shift as well, in which case “c0 is the center of t0” ([3], pg. 452) as t0 moves back and forth.

With the polarization factor, it is interesting to call h the vector summation of two gamma ray electric fields. When a gravitational field is yet more compact, h is the summation of more than 2 electric fields, so that “f: MH be a smooth map” ([1], pg. 183), and “Let h be a proper subalgebra of g of maximum dimension” ([3], pg. 160).

Incidentally, the identity isomorphism reminds us of quantum 1:

[1] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society
[2] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc
[3] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012

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Oct 05 2022

The Vector Potential

In electrodynamics we find that “A quantum-mechanical description of photons necessitates quantization of only the vector potential” ([1], pg. 242), as in the summation of all the manifolds of gravitational fields at a given location. In a more densely packed summation of manifolds, the action of an electric charge will have a lesser rotational effect on the electric fields of the gamma rays than on a less dense field. The power of the rotation is the same in either field however, as long as we are referring to a gravitational field that is not too sparse for electric fields to propagate.

“The definition of B = ∇ x A specifies the curl of A, but it doesn’t say anything about the divergence – we are at liberty to pick that as we see fit, and zero is ordinarily the simplest choice.” ([2], pg. 235) The reason we may pick the divergence as zero is that the manifolds “are frozen in time for phonon transmission”:

As far as group action, Mackenzie [3] calls these “groupoids”, such as an ellipsoid, a spheroid, or another 3-dimensional shape. The definition of a spheroid I find is that it is like a sphere, but not a perfect sphere, and in the present case we have “oscillations and accordion motion in multiple axes”:

On a side note, though related to manifolds of gravitational fields, the Nobel Prize in Physics is being given this year for essentially this:

[1] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc
[2] Griffiths, David J., “Introduction to Electrodynamics, Third Edition”, c. 1999, Prentice-Hall, Inc.
[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

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Feb 11 2022

Isotropic Property of the Coulomb Potential

In the vicinity of where our machines have been, we know that electric current will flow in any designated direction and is not particular to the direction of the highest flux density of gravitons.

For various reasons, we cannot have protons and electrons continuously flipping, – the Stern-Gerlach experiment proves that they do not. There must be internal processes of the proton and electron which produce isotropic electric fields. Some of this was previously addressed in two blog entries:

It is possible that not all conjugate wave gravitons pass straight through a proton or electron, or that even with a free proton or electron that the gravitons leaving have just entered. Some may make a horseshoe pattern and come out near the same point entered. They may be able to come back out at any angle. As compressed as the gravitons become inside a particle, almost any shape can occur. Gravitational pressure dictates a consistent size of a free proton or electron.

With the flux density coming out of the face of the earth, we seem to have a conundrum with the idea of gravitational pressure, one side having much greater pressure than the other. Why do gravitons not burst out the top, resulting in particle collapse? It also begs the question as to why electrons are perfectly round, and not teardrop shaped:

Possibly, branes form at the top of an electron and reform in a spin flip.  These branes would be linked inside the particle so that they do not bust out, and may deflect some exiting gravitons at various angles. These branes may also help keep the electron round. Here we are designating “top” as away from the highest flux density of gravitons.

As far as isotropic fields, at this point we must say that it is designed internal to the proton or electron and is of consistent pattern.  The open field starts just outside the particle, so it is maintained that electric and magnetic fields transmit openly by “phase shift and chirality” or “phase shift and parity”.  The Coulomb force is considered instantaneous at reasonable distances:

It appears as though this is necessary, because then the speed that free gravitons travel at, the speed of light in a vacuum, does not effect the electric and magnetic fields generated.

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Feb 09 2022

Core of an Electron or Proton

We can see from the calculation of the diameter of a free electron that as the density of the gravitational field goes down, the diameter increases.  This would be because of less gravitational pressure on the outside of the electron.

As gravitons enter a proton, electron, neutron, or nucleus, as conjugate waves or to take residence, the buildup takes on a fuzzy look that makes them look larger.  If we take a core diameter of 1.3335 x 10-15 m, the part that produces the fundamental charge, and add one graviton wavelength, we arrive at 5.30 x 10-15 m diameter, which is close to the classical diameter of the electron, 5.64 x 10-15 m *.  One graviton wavelength is used because one-half wavelength is on one side of the electron and one-half wavelength of a different graviton is on the other side.

We may call these outer layer gravitons tentacles or strings.  When nuclear fission occurs, the de Broglie wavelength of a neutron can come in at an angle where the strings on each entity hook and help pull the neutron into the nucleus. The cross section for this process is larger for slow neutrons vs fast neutrons in part because of the longer de Broglie wavelength.

* Jackson, J. D., Classical Electrodynamics, Third Edition, c. 1999 John David Jackson, John Wiley & Sons, Inc., p. 695

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Nov 14 2021

Greatest Lower Bound

One would guess that the particle physicists and quantum field theorists may like a 1.3335 x 10-15 m diameter of a free electron, because it is closer to a point particle than many estimates of the diameter.  It is possible that 1.3335 x 10-15 m is also the limit inferior of the sequence S137 to Sn in an atomic orbital.

The maximum diameter, on the other hand, will depend on the element and on the orbital.  At a spin flip, electrons in all orbitals may reduce to 1.3335 x 10-15 m, before taking off on a new trajectory and increasing in diameter again.  We cannot speak of a limit superior of the sequence of diameters of the electron in an atomic orbital nevertheless.  That will depend on the direction of electron travel, and on whether the atom is at the surface of the earth, or at some other planet.  For the latter, it depends on the density of the gravitational field.

We may also ask whether 1.3335 x 10-15 m is the greatest lower bound at all locations in the universe.  This raises the question of whether the fine structure constant is a universal constant, or whether or not the Coulomb gauge is the same everywhere.

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Jan 25 2021

Final Mediator

Going back to the April 2007 paper once again: “… and the same Coulomb force being the final mediator of the gravitational force as proposed, the mass energy of the proton appears to be integral to gravitational field action.”, we can go further back in history to examine the path of how we arrived at this place.

In the scientific community, the cause of gravity is by many still considered as unknown.  As Hughes-Hallett et al. puts it in relation to the gravitational force: “How does acceleration come about?  How does the velocity change?  Through the action of forces.  Newton placed a new emphasis on the importance of forces.  Newton’s laws of motion do not say what a force is, they say how it acts.” 1

Newton himself says in The Principia: “…considering in this treatise not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions.” 2  In Definition 8 Newton says: “This concept is purely mathematical, for I am not now considering the physical causes and sites of forces.” 3

Roger Cotes, Plumian Professor of Astronomy and Experimental Philosophy at Cambridge University wrote in his Editor’s Preface to the Second Edition of The Principia: “But will gravity be called an occult cause and be cast out of natural philosophy on the grounds that the cause of gravity itself is occult and not yet found?” 4  Electromagnetic waves were not yet discovered in 1713 when Cotes wrote this, so it is difficult to imagine how anyone could have discovered the cause of gravity back then.

1  Hughes-Hallet, Gleason, McCallum et al., Calculus, John Wiley & Sons, Inc., 2005, p.304

2  Cohen, Whitman, Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy, University of California Press, 1999, p. 588

3  Ibid., p. 407

4  Ibid,. p. 392

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Dec 04 2019

Electric and Magnetic Fields

Jon Rogawski has a cool diagram and calculation using Faraday’s Law on page 979 of his Multivariable Calculus book*. The magnetic field around the straight wire with an alternating current flowing in it produces a voltage in an adjacent looped wire with no conventional energy applied except that from the other wire.

Of course this magnetic field, and likewise for electric fields, must have a pervasive field in which to transmit. One may say it transmits through the air, however Faraday’s law also applies in space.

These E and B fields transmit by turning and bending the E and B fields of the dense gamma rays.

* Rogawski, Jon, “Multivariable Calculus”, W. H. Freeman and Company, c. 2008

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Sep 09 2019

Atomic Stability

Published by under Classical Electrodynamics

Rutherford states that “Larmor has shown that the condition for no loss of energy by radiation is that the vector sum of the accelerations of all the component charged particles shall be permanently zero.  If this condition is not fulfilled there will be a steady drain of internal energy from the atom in the form of electromagnetic radiation, and unless this is balanced in some way by the absorption of energy from outside, the atom must ultimately become unstable and break up into a new system.” * (Rutherford references “Larmor:  Aether and Matter, p. 233”)

This is interesting in that by 1906 it was realized that radiation energy would need to be exchanged in order that atoms and molecules not collapse.  Even within the atom, by stating “vector sum” it is alluded that energy is exchanged.  Keep in mind that this was before the Bohr model of the atom.

In the next paragraph, Rutherford has:  “The positively and negatively charged particles constituting the atom must be so arranged as to form a stable aggregate under their forces of attraction and repulsion, and at the same time their arrangement and motions must be such that no energy is radiated from the atom.”

* E. Rutherford, Radioactive Transformations, Yale University, 1906, p. 262

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