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Maxwell's Equations

New -- March 20, 2004

James Clerk Maxwell is routinely considered one of the greatest physicists to have ever lived – viewed by most modern day physicists as being on the same rarified level as Sir Isaac Newton and Albert Einstein. Einstein himself described Maxwell's work as the “most profound and the most fruitful that physics has experienced since the time of Newton .” This is pretty fair praise.

Fundamental to Maxwell's fame is a set of equations involving electromagnetism, which have come to be known collectively as “Maxwell's Equations”. [2] These equations are discussed in numerous websites, but Wikipedia [1] has one of the best presentations and explanations of the traditional four equations. (4/1/07) Of particular note is the fact that the Wikipedia article includes eight of the original Maxwell's Euqations. The fact there may have been twenty is, alas, not discussed. (See below.) Alternatively, the Hyperphysics site is also excellent, and notes in its introduction the following:

“Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field.

“Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships.” [3]

In truth the four equations traditionally included in the teaching of physics are more aptly named the “Maxwell-Heaviside Equations” inasmuch as Oliver Heaviside reformulated Maxwell's original equations from a quaternion format into a simple vector format. Maxwell's original paper [4] consisted of 20 equations with 20 unknowns. According to Tom Bearden [5], Maxwell's 1865 paper had its quaternion equations reduced to vector notation -- after a comparatively limited debate among some 30 scientists – a notation advocated by Heaviside, Gibbs, et al – after Maxwell was already dead.

“After publication of the first edition of his Treatise, Maxwell of course also caught strong pressure from his own publisher to get rid of the quaternions (which few persons understood). Maxwell thus rewrote and simplified about 80% of his own 1873 Treatise before he died of stomach cancer in 1879.  The second edition of that treatise was later published with that 80% revision done by Maxwell himself under strong pressure, and with a guest editor. But the 1865 Maxwell paper shows the real Maxwell theory, with 20 equations in 20 unknowns (they are explicitly listed in the paper).  The equations taught today in universities as ‘Maxwell's equations' are actually Heaviside's equations, with a further truncation via the symmetrical regauging performed by Lorentz.” [5]

Bearden [6] goes on to point out that:

“A higher group symmetry algebra such as quaternions will contain and allow many more operations than a lower algebra such as tensors, which itself contains more than an even lower algebra such as vectors.” [6]

In effect, the reduction of Maxwell's original theory from 20 equations to 4 – purely in order to make the mathematics a bit easier for the poor physicists – severely limits the capabilities of the original theory. This shows up dramatically in the Second Law of Thermodynamics, where the original equations were effectively “regauged” in order to force the theory to obey the law of conservation of energy. In all respects a return to the quaternion format in Maxwell's original equations seems likely to yield astounding results. Quaternions cannot simply be ignored any longer.

[4/1/05] The other equations of the 20 equation set of quaternion equations may have boggled many a mind when they were first introduced, but as others have shown in modern times, the remaining equations may be very useful in engineering the exctraction of energy from the Quantum vacuum.

If in fact the theory is left to its own merits, all manner of possibilities exist – effectively the same potentiality as displayed in Connective Physics. The key is in avoiding limiting assumptions whose only merits are those of mathematical simplicity.

A corollary to this avoidance of blinders has been addressed at the University of Virginia , where an apparent exception to Ampere's Law was discussed – “When does Ampere's Law go wrong?” The critical factor was in the dynamic nature of the system.

“Ampere's law was established as the result of large numbers of careful experiments on all kinds of current distributions. So how could it be that something of the kind we describe above was overlooked? The reason is really similar to why electromagnetic induction was missed for so long. No-one thought about looking at changing fields, all the experiments were done on steady situations .” [7] [ Emphasis added]

The great flaw in mainstream physics today is that no one seems willing to look at the effects of accelerating fields. Adding a fourth term to the electromagnetic equations in fact yields such a condition, and lo and behold, conservation laws are redefined in a wholly connected and virtually unlimited universe. It's just the kind of thing Maxwell's Demon (and/or Tom Bearden [8]) could appreciate.



[1] http://en.wikipedia.org/wiki/Maxwell's_equations

[2] Technically, the four equations are separately entitled: “Gauss' law for electricity,” 2) “Gauss' law for magnetism,” 3) “Faraday's law of induction” and 4) “Ampere's law.”

[3] http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html

[4] James Clerk Maxwell, “A Dynamical Theory of the Electromagnetic Field,” Philosophical Transactions of the Royal Society of London , 155 , 459-512 (1865).

[5] http://www.cheniere.org/correspondence/120403a.htm

[6] http://www.cheniere.org/correspondence/011203.htm

[7] http://www.phys.virginia.edu/classes/109N/more_stuff/Maxwell_Eq.html

[8] Tom Bearden's website is: http://www.cheniere.org/ , where we find, among other thingshttp://www.cheniere.org/references/maxwell.htm

“See James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field," Roy . Soc. Trans., Vol. CLV, 1865, p 459. Read Dec. 8, 1864. Also in The Scientific Papers of James Clerk Maxwell, 2 vols. bound as one, ed. 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. 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.”

“Quaterions have a vector and a scalar part, and have a higher topology than vector and tensor analysis. Maxwell has 20 quaterion equations with 20 unknowns! See Bearden's book: Energy from the Vacuum; Concepts and Principles.”

http://www.cheniere.org/books/efv/toc.htm provides a glimpse (especially 1.5.2; pages 52-54 in Adobe format) of what Bearden thinks is wrong in electrodynamics. 


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