RE: Electrodynamic equations
Date: Sat, 4 May 2002 16:12:38 -0500
Yes, the history gets muddled a bit.
Maxwell's seminal theory is contained in James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field," Royal Society Transactions, Vol. CLV, 1865, p 459. Read Dec. 8, 1864. Also in The Scientific Papers of James Clerk Maxwell, 2 vols. bound as one, edited by W. D. Niven, Dover, New York, 1952, Vol. 1, p. 526-597. Two errata are given on the unnumbered page prior to page 1 of Vol. 1. In this paper Maxwell presents his seminal theory of electromagnetism, containing 20 equations in 20 unknowns. His general equations of the electromagnetic field are given in Part III, General Equations of the Electromagnetic Field, p. 554-564. On p. 561, he lists his 20 variables. On p. 562, he summarizes the different subjects of the 20 equations, being three equations each for magnetic force, electric currents, electromotive force, electric elasticity, electric resistance, total currents; and one equation each for free electricity and continuity. In the paper, Maxwell adopts the approach of first arriving at the laws of induction and then deducing the mechanical attractions and repulsions.
The notation is different from the modern quaternion notation, and so perhaps should be referred to as "quaternion-like" in the modern sense.
So resistant to quaternions were the times, that Maxwell himself began simplifying his theory farther away from quaternions. As an example, after the first edition of his Treatise (book) in 1873, the outcry was so great that he began rewriting and simplifying his own theory. He died before the second edition could be finished and published, but he had changed more than half the chapters. The second edition therefore differs fairly significantly from the first edition, and particularly from the 1865 paper.
Also, in 1967 Ludwig Lorenz developed a parallel theory of electrodynamics very similar to Maxwell's, and he also symmetrically regauged his equations (which is a dramatic simplification, indeed one which arbitrarily discards any net interaction between an active environment and the physical system being modeled). Lorenz got very shabby treatment at the hands of the scientific community, and some years later when H.A. Lorentz published his version of Maxwell's theory with symmetrical regauging of the equations, credit was erroneously given to Lorentz (and still is) for that action.
Our wry comment is that this symmetrical regauging --- whether of Maxwell's equations or Heaviside's truncated version --- discards all permissible Maxwellian systems that are open systems far from equilibrium in an active environmental energy exchange. In modern terms, this discards the local curved spacetime and also the local active vacuum, insofar as any net usable exchange with the system is concerned.
Yet the symmetrical regauging assumes two separate and simultaneous free changes of the potential energy of the system. Unless one wishes to discard the conservation of energy law, that in turn implicitly assumes (in modern terms) that energy from the vacuum has entered the system in two ways (changing two potentials) but very selectively and just so that the resulting two new and free force fields that result are everywhere equal and opposite.
Interestingly, that assumes that a continuous exchange of energy between the active vacuum and the system is ongoing, but all the net energy entering the system is "locked up" as a stress potential with a net translation vector summation of zero.
But a stress potential change means that one has assumed INTERNAL work continuously being performed on the system, by its peculiar exchange of energy with the external active environment. In short, that continually produces and maintains the assumed increased stress.
Further, the increase in potential energy of the system is also an increase in the local energy density of the vacuum (i.e., the local energy density of spacetime). That represents at least a rotation of the system frame out of the laboratory frame of the observer.
So the standard statement in all the EM textbooks that the symmetrically regauged Maxwell-Heaviside equations describe exactly the same system as the unregauged equations described, is quite a non sequitur. It's like saying that a system operating between two elephants pushing against it in opposite directions, is identically the same system without the elephants.
Anyway, that little trick --- the Lorenz/Lorentz symmetrical regauging -- ARBITRARILY discarded the entire permissible class of EM systems that are far from equilibrium in their active exchange with (1) the local vacuum, and (2) the local curvatures of spacetime (every energy density change in the system produces a curvature of local spacetime, negating the Lorentz regauging assumption of a flat local spacetime.
So electrical engineers do not perform a thorough analysis on their power systems at all. To do that, one has to analyze the supersystem, consisting of (1) the physical system and its dynamics (normal sense), (2) the active vacuum and its dynamics, and (3) the local curvatures of spacetime and their dynamics. All three components of the supersystem interact with each other, so it's highly nonlinear.
To do such an analysis, one has to discard the standard U(1) classical EM model used by electrical engineers, and utilize an EM model in a higher group symmetry algebra. O(3) EM advanced by Evans and Vigier is really good, as are quaternions. To see what Tesla's patented circuits are doing, one has to analyze them in such a higher group symmetry electrodynamics. The novel actions do not even show to a tensor analysis (as shown by Barrett).
Tesla was unique in his approach to electrodynamics. He was a tireless and highly innovative experimenter, and thought in terms of material fluid theory as did just about everyone else (Maxwell's theory is indeed a material fluid dynamics theory, and still assumes a material ether in its equations which were never changed after the destruction of the material luminiferous ether by the Michelson-Morley experiments.). But Tesla had a photographic mind, and also incredible powers of visualization (as he wrote, until he was 12 he visualized things in his mind so vividly that he could not differentiate between physical objects observed and his own thought images. He had to find a "trick" that allowed him to tell the difference).
So he used simple algebra, but also used what we can only call "super simulation" in his own head. It's as if he had a great simulation program going on continually in his head, in a very large supercomputer. So he would do an experiment 100 times in excruciating detail in his mind/vision/simulator, and adjust the variables and components used in his mind-simulation until he obtained the results he sought.
So this was the real secret of Tesla's great discoveries. He did so many experiments, and fitted his mind-simulation so accurately, that he was the equivalent of a group of researchers using modern highly sophisticated computer simulation tools.
And quite a bit of what he "saw" is still outside electrodynamics. But he saw it the way nature did it, because he fitted that simulation to thousands of experiments. He was correct that there was no transverse EM wave in vacuum, but it will take probably another two centuries before the scientific community will give up their love of substituting the effect for cause (something absolutely rampant in physics, from mechanics to electrodynamics to particle physics). Our instruments measure a transverse EM wave in a receiving antenna, sure, but they are actually measuring the electron precession waves (accounting for the fact that electrons move longitudinally down a wire with great difficulty, with only the Drude drift velocity of, say, a few inches per hour). Instead, since an electron in the Drude gas is spinning in its 3-d aspects and thus acts partly as a gyro due to the longitudinal constraints, the electron precesses laterally in the wire. The old guys thought that a "shaking" material ether had entered the wire and "perturbed" or entrained the "material electric fluid", so that the shaking of the material electric fluid in the wire was exactly the same as the shaking of the material ether fluid outside the wire. (the electron and atom and nucleus had not been discovered, there was no particle physics to speak of, no special or general relativity, no quantum mechanics, and no quantum electrodynamics --- and in fact, no Drude gas theory as yet).
And that, together with Faraday's notion that his field lines were physical, like taut strings in space, meant that perturbation of the "field lines" in space gave a set of "plucked string" or transverse wave oscillations. And that seemed to them to be completely consistent with what they measured in their circuits! They never even thought to differentiate electron precession waves from longitudinal forces pressing on the gyroscopic electrons --- because there were no electrons (as far as anyone knew at the time).
So the foundations of classical EM are very old (at least 137 years or more) and have not been updated to reflect lots of physics discovered since Faraday's researchers and Maxwell's 1865 paper (and the various truncations of that theory).
Anyway, that's my take on all of it.