The one great contribution to Electromahnetism or even the Electra and Magnes phenomenon, was the theoretical spaciometric tool of a spherical measure.
The idea is said to have originated with Faraday, who conceived of it as a sphere of Influence. The notion underwent modification in Faradays discussions with colleagues and theoretical papers , and the notion of a Newtonian vector space was associated with it.
Under the meditations of Maxwell, the sphere of influence was" mathematically" flattened out into the notion of a "field", and the field Maxwell had in mind was a tessellated field of vortices.. By considering such fields, especially as a fluid medium, Maxwell was able to derive most of his formulae from a mechanical description of the behaviours of large vortices " lubricated" by many smaller voices in a semi rigid but dynamically responsive system.
The mathematics was horrendous, so he eagerly awaited the publication of Hamilton's Quaternions.
His formulas took on an elegant form under the quaternion notation, but the horrendous calculations soon put him and any potential readers off.
The development of other group algebras caught his attention, especially the notions of vectors . Much of the formulary was rewritten in these terms, especially on the advice of Lord Kelvin, who seeme to have taken a dislike to his fellow countryman Hamilton, and his ardent promoter Tait. In any case Maxwell substantially dropped Quaternions, complaining they were too deceptive. Neither he nor Kelvin could be bothered to learn them properly, and their basic assumptions of angles were always wrong by a factor of 2.
The field had become a quaternion field but then it became a standard in physics called a vector field. The dot and cross products of Quaternion algebra were transferred across to vector algebras eventually. Meanwhile Gibbs Baudlerized Hermann Grassmanns and Bill Clifford's work to come up with his own vector Algebra as opposed to Grassman's simple line segments or Strecken.
The normal vector for area is one of Gibbs inventions. It is a standard technique toda, but it was and is anomalous to the basic vector addition etc
The Gibbs vector field Ideas were studied by Heaviside, who contributed his own input to vector algebras and to computational calculus for transmission line phenomena. Using. His analysis, he completely rewrote Maxwells formulary. The famous 4 Maxwellian Laws are in fact a note on or a commentary on Maxwells formulary.. These Heaviide equations are based on a vector field formulation.
We have lost the spherical reference frame in which electric and magnetic behaviours were first and best understood!
We have lost sight of what these fields are, they are fields of magnetci and electric induction.
Hamilton said something very interesting to his sons, just before his death. He tried to explain the importance of Quaternions for Physics in particular, but especially for electromagnetism! For this reason I ask the reader to study this video in a new light. Notwithstanding the arcane nature of the material, a not unexpected consequence of " mathematical" chicanery, the point is clear: rotation and dilation!
Rotation is analogous to magnetic behaviour. Dilation is analogous to electric behaviour. The insight of Hamilton, approaching his death was later obscured by mathematicians, particularly Maxwell, Kelvin and Even Bill Clifford!
I returned to my study of Grassmann's Ausdehnungs Groesse, irritated by the slick presentations in the Clifford Algebras. Clifford is revered by a small and dedicated band of Mathematicians who ride roughshod over both Hamilton and Grassmann. The major part of Cliffords algebraic inspiration, like so many others after him is due to the work of Hermann Grassmann. But he is somewhat disrespected because he lacks credentials! After all he only ever worked at the primary levels of education in prussian Mathematical society. The higher levels were dominated by the eminent Gauss and his Coterie of followers.
I have remarked before how dire the situation was for all concerned, but particularly Hermann. Were it not for his brothers domineering drive to publish his Fathers works to honour him, and to prepare for his own take on scientific philosophy, his work would not have been redacted!
This was a sore point between the 2 brothers, no doubt, but the fact is it rescued Hermanns former work(1844) from obscurity! Hermann was astute enough to use this interest to reassert his ideas over those of his brothers by republishing an annotated linked version of his former treatise.
There is no doubt he was not happy with the state he was forced to publish it in in 1844, for he bemoans this in the concluding paragraphs of that Vorrede! But equally in the Vorrede to the 1862 redaction he bemoans the lack of fidelity to the original! The copious notes are intended to make a piece out of them, as originally planned.
Work done in the last year of his life in regard to the place or the presentation of Quaternions in the Ausdehnungslehre reveal his critique of Quaternions not as a competing system, but as actually a sub group within his more general treatment. This is the paper Bill Clifford used to formulate his introductory work on applying Grassmann to the undulatory or wave problem.
We know from his 1844 work that Grassmann had already success fully worked out how to apply his method to the ebb and flow of tides. His aim was to develop a complete application to corner (or joint rotations) and to Schwrnkerung( swinging arms , pendulums etc). It is therefore clear that wave mechanics would benefit from a similar treatment. Unfortunately, Bill died before he could really puzzle out how to do it.
His first attempt at it, biquaternions seemed to fail. By then the Gibbsean Vector analytical method was adopted to displace all of Hamilton's work in American physics colleges, leaving much original research unpublished as out of favour! It also successfully obscured Grassmanns contribution, with many preferring Gibbs notes to the actual Analytical methods of both Hamilton and Grassmann. Apparently he gained such favour by supporting Kelvins thermodynamical theories, who consequently promoted him and his ideas over his Nemesis Hamilton!
We have seen a steady push back to the origin of concepts created by a major problem in physical models. Dirac started the field called Quantum mechanics with a few others. At the time their was great discord in what was being discovered and how it should be interpreted. There was also a great yawning social upheaval underway, the first and second world wars. Much of what was known now became militarised, and thereby secret and subject to propagandism .
The consequence of the social upheaval was the use of science as a military and propaganda tool. Much of what was really researched and demonstrated is still top secret. Much of what the public was allowed to believe was homespun and inaccurate!
While Europe was descending into chaos, much of the best of its science etc was being collected and preserved in America. Thus American institutions come to play a powerful role in furthering the western knowledge of industrial processes and scientific theory. However, there was and is still much to recover from the libraries of the old surviving institutions.
Dirac thoroughly rewrote physics based on Grassmann algebras, and including Hamiltonian concepts. The insights and suggestions of Einstein to Schroedinger led to the simple wave equation. The solution of this equation was shown to be a probability mechanics. This again was due to Gibbs furthering statistical mechanical notions in solving physical problems, following in the footsteps of Gauss, Boltzmann and Maxwell.
However Dirac followed Grasmmann and rewrote Gibbs vector algebra into the form commonly called quantum mechanics. This was pure Grassmann and Hamiltonian algebraic formulations. This enabled him to take the time derivative of the Scroedinger wave equation, from which he derived his 4 vector equations.
If you read Hodson on the Dirac Fiasco, you will find how many scientists were emotionally not ready to give up their most treasured ideas. They fundamentally altered Dirac's Grassmann Analysis to make it conform with their ideas of reality. Of course Dirac was vindicated, but this did not change their resolve. Now Dirac's equations are are all but forgotten .
Dirac's solution were called twistors and his equations demanded anti states of twistors. By adding "mass" into the formulation this was retranslated as matter. The clear thing was that a system of equations could describe the behaviour of a substance called electrons, that is electric charge. Mass was not required to gain this insight, only rotation and translation and extension.
So the question of charge arises. Charge has a long history quite apart from electromagnetism. It is the sudden onset of force. While Faraday may initially of used the term charge he soon abandoned it in favour of electric and magnetic tension. It was Maxwell who inadvertently reintroduced the TEM, based on Coulombs work nd law, in which Coulomb drew the analogy with newtons quantity of matter. However their was no quantity of charge measurable until it was stored in Leyden jars. It being his ill defined notion: charge.
Leyden jars therefore became the quantifiers of charge by default and without ny proper definition of charge density. Thus when one attempts to understand electric measures one comes upon a tautology! The matter is not resolved until much much later when JJ Thompson claimed to have found the unit of charge called the electron.
Thus prior to this, any rational for charge was actually based on the empirical induction fields, not on any particle or quantum. The induction fields were quantified by the physical measurements of the supposed sources, that is the conductors assumed to be carrying moving "charge" , which were in fact derived from the volumes of Leyden jar battery formation.
The empirical reality was the field nd the field effects. The theoretical assumptions were some quantity of something called charge. Faraday did not subscribe to charge. He in fact increasingly talked in terms of a field.
The charge concept returns only through the proclivity to corpuscularising the active principles of phenomena. In that regard faraday was very radical in attempting to develop a spherical influence around conductors. While Maxwell took the noion of a field on board, he quantified it by measurements of the conductors. These infinitesimal measures he termed charges. But he is not clear about how they formed a " current".
In fact in the early description if phenomena he describes the induction fields as driving the 2 types of electricity apart, and in a conductor this dynamic drive disposes the electric tensions as far apart as possible. So his concept is one of field movements not charge movements. It is these field movements he considers as moving at the speed of light.
While these field movements are dynamic, he does not investigate them Rutherford, and so later slips into the error of considering them to be static in equilibrium. Charge only comes back into the picture with the Bohr and Rutherford models of the atom. Conductors thn have to be described in these terms nd electricity as the flow of Thompsons electrons.
This leads to problems that are never resolved, even though Heaviside suggested what the solutions should be: charge should be dropped in favour of a " signal" concept. As few were involved in telegraphy, this made little sense to other theoreticians, beyond Tesla, Poynting and a few others.
When Einstein proposed the massless electron, everybody was happy, few realising that this was an analogy for a field strain transmission that Heaviside called a " signal"!
Today Ivor Catt has attempted to correct this error and met with a stony silence. The few that have replied have trotted out a confused response, unaware of the flaws in their expositions, flaws only revealed by empirical data!