I got the idea from a Phipps after investigating Nikolova/Zimmerman’s technical report and being referred to “Heretical Verities”* by a Princeton plasma physicist who had to validate computational models against their experiments. I believe Phipps may have gotten it from Hertz although there was probably some contribution from J. P. Wesley.
* It turns out the author of the preface was someone I knew from SLAC, who knew Phipps because they both went to UofIL HS thence to Harvard, but I didn’t know anything about that association or that book until the replication of Nikolova/Zimmerman experiments began in Urbana of all places, in 2014. Nor was the plasma physicist associated with those authors. He just had problems believing the issue of “unobservable outside of QM” had been sufficiently verified given his experience with plasma simulations.
Those experiments were initiated when a long-time friend of Carver Mead (who had a hand in encouraging Mead to complete the book “Collective Electrodynamics”) became aware of the Nikolova/Zimmerman patent – in part because of his skepticism of received wisdom regarding the vector potential’s interpretation. That’s the contact with the plasma physicist’s skepticism.
Without getting into the specifics of the current discussion - shrot on time - the Geometric Algebra (GA) / Clifford Algebra formulation of EM is the best, I believe. Essentially, it augments the algebra of vectors with higher-dimension entities such as planes of rotation (bivectors formed through the outer product of orthogonal vectors). See John Denker’s: Electromagnetism using Geometric Algebra versus Components.
I first got into GA through researching MaxEnt; David Hestenes was active in that area, but his main work was reintroducing the use of Clifford Algebras as a lingua franca for physics, for which he was awarded the Oersted Medal.
Edit: one more GA link, relevant to the original topic:" Can physics laws be derived from monogenic functions"" Jose Almeida wrote several papers that deserve more attention on the optical interpretation of relativity. “the author intends to show that GTR and Quantum Mechanics (QM) can be seen as originating from monogenic functions in the algebra of the 5-dimensional spacetime G(4,1). … Euclidean relativistic dynamics resembles Fermat’s principle extended to 4 dimensions and is thus designated as 4-Dimensional Optics (4DO).”
You appear to agree that the Lorentz force equation is no different from the other expressions that you listed above. Thus, I’m perplexed why we are having this discussion. On the other hand, you wrote:
Yet it is not unlike the Lorentz force at all; it is the same equation written in a different way. It is not like the Lorentz force and it’s the equivalent to the Lorentz force.
You seem to be in a logical cleft stick with no way out. Perhaps Bertrand Russell can help:
You belittled me over this subexpression. When I pointed out your inappropriate behavior, you doubled down by saying: “Expect some sarcasm in the meantime.”
When I pointed out that one of the most highly cited papers on the magnetic vector potential used the same subexpression to again point out your inappropriate behavior, you, instead, read me as saying that I thought that paper’s equation was the same as my original equation (despite my saying the exact opposite).
On that basis, you tripled down and then quadrupled down on your inappropriate behavior.
How much further are you willing to go before thanking me for retaining some semblance of civility?
So, I guess Mr. Russell was of no help. Better luck next time. On the other hand, Beyond the Fringe guys are always good for some laughs. Surely, you must have enjoyed that. You’re welcome.
(1) E = -∇Φ - dA/dt
(2) dA/dt = ∂A/∂t dt/dt + ∂A/∂x dx/dt + ∂A/∂y dy/dt + ∂A/∂z dz/dt
(3) dA/dt = ∂A/∂t 1 + ∂A/∂x dx/dt + ∂A/∂y dy/dt + ∂A/∂z dz/dt
(4) dA/dt = ∂A/∂t +∂A/∂x dx/dt + ∂A/∂y dy/dt + ∂A/∂z dz/dt
(5) vx = dx/dt, vy = dy/dt, vz = dz/dt
(6) dA/dt = ∂A/∂t + ∂A/∂x vx + ∂A/∂y vy + ∂A/∂z vz
(7) (v· ∇)A = ∂A/∂x vx + ∂A/∂y vy + ∂A/∂z vz
(8) dA/dt = ∂A/∂t + (v· ∇)A
(9) E = -∇Φ - ∂A/∂t - (v· ∇)A
(10) ∇(v· A) = (v· ∇)A + (A · ∇)v+ v× (∇ × A) + A × (∇ × v)
(11) ∇ × v= 0
(12) (A · ∇)v= 0
(13) ∇(v· A) = (v· ∇)A + 0 + v× (∇ × A) + A × (0)
(14) ∇(v· A) = (v· ∇)A + v× (∇ × A)
(15) (v· ∇)A + v× (∇ × A) = ∇(v· A)
(16) (v· ∇)A = - v× (∇ × A) + ∇(v· A)
(17) E = -∇Φ - ∂A/∂t + v× (∇ × A) - ∇(v· A)
(18) F = q E
(19) F = q (-∇Φ - ∂A/∂t + v× (∇ × A) - ∇(v· A))
(1) Postulate (And yes I know defining “E” this way is “wrong”.)
(2) Definition of total derivative
(3&4) Multiplicative identity
(5) Definition of velocity vector components
(6) Substitute (5) in (4)
(7) Advection identity
(8) Substitute (7) in (6)
(9) Substitute (8) in (1)
(10) Vector identity
(11) Experimental condition (direction of v not dependent on position)
(12) Experimental condition (v not dependent on position)
(13) Substitute (11 & 12) in (10)
(14) Multiplicative identity
(15) Commutativity of equality
(16) Subtract v × (∇ × A) from both sides
(17) Substitute (16) in (9)
(18) Electromotive force definition
(19) Substitute (17) in (18)
I just realized that, in addition to my being a bit too-generous in thanking Dr. Lorentz for “correcting” me for using the same expression, ∇(v ·A ), that Konopinski used, another source of misunderstanding may have been the quasi-static condition Konopinski imposed on equation (2) which makes it the equivalent of the Lorentz Force law. The quasistatic condition does render ∇(v ·A ) “unmeasurable” except in its quantum mechanical effect on the phase of the charged particle. But the reason I have, from the outset, been trying to get across to our good Dr. another interpretation of ∇(v ·A ) is that the condition is far from quasi-static within a receiving antenna consisting of relativistic plasma electrons aligned with the curl-free A region of a Hertzian dipole.
A “Heretical Verities” quote from Phipps seems quite proper:
“My message in this book, which now draws to its close, is that if you believe the experts when they tell you your native wit and critical sense are worthless then in your own case you prove them right…but if you resist their browbeating their is hope in your case for individual salvation—though Barnum be right about the public.”
These University of Illinois at Urbana physicists are nothing but trouble with their penchant for actually conducting experiments. Aside from Dr. Phipps there is this other UofIL physicst Chalmers W. Sherwin who disagreed with Phipps regarding Phipps’s refusal (to his dying day) to believe that Lorentz length contraction (as opposed to a larger time dilation than that predicted by SRT) had been observed. However, Dr. Sherwin actually ran an experiment in which he failed to find a contraction effect:
While it may be the case that there are others who have managed to poke holes in Sherwin’s experiment, I haven’t found them. And, although one shouldn’t take GPT4’s word for anything, it is generally quite willing to enforce the scientific consensus when asked questions such as the one I just did:
Just so everyone knows how great the contribution of “immigration” has been to western civilization: It terminated my authority to hire the folks I needed to pursue this:
Maybe Dr. Lorentz can explain why it is that the imperial “we” of Wolfram’s community, believes it is appropriate to drop physical dimensions in the case of a differential of a constant function with a physical dimension. Perhaps he thinks that instead of me merely committing career suicide to try to get physicists to be serious about the foundation of mathematical physics, I should have gone to the vestibule of the NSF and set myself on fire.
After wrangling Mathematica far more than should be necessary, I got it to finally behave itself and produce the simulate scope traces (composite Weber Electrodynamics Force and Rx charge velocity below it) that qualitatively matched what we saw coming from the neon indicator bulb receiving antenna with relativistic electrons with bias current qualitatively simulated by the velocity graph. The Mathematica notebook is at a repo I just created with some additional comments at the top:
I can’t tell you how frustrating it is to find myself watching the world catching up to where I was about a decade ago and being deprived of the resources to catch the wave.
. Hop on the bus with the rest of us.