E x B is mathematically defined and derived, some say from Maxwells equations by Poynting. However it is physically ambiguous without the Fleming Right or Left hand rules of thumb! This makes it a mathematical oddity, and worse still physically untrustworthy.
Maxwell was an early adopter of Hamilton's Quaternions. He was a brilliant mathematical thinker and so was not unwilling to plough through the pages of algebraic manipulation. However, many of his contemporaries, including Lord Krlvin were not so willing. Minkoeski was not willing to rewrite his work in terms of the new notations. Thus Maxwell found himself up a creek without a paddle! He did resent this and expressed it in a famous quote, but this was not because he could not do Quaternions, but because his colleagues were reluctant to do so and criticised him for wandering off into obscurity!
However Maxwell had benefitted from his course in Quaternions and he laboured to rewrite his conclusions in the new, less subtle Gibbs and Kelvin vector notation. He found it lead to slippery issues with the sign of terms which was annoying, and which did not occur in Quaternions. Eventually Pauli got some consensus around an exclusion principle which it was hoped fixed the problem. It was a most strange and disagreeable state of affairs in mathematics, especially when the Maths was being corrected by the Physicists and engineers! Lewis carol wrote Alice in wonderland in protest at this situation.
Later Bill Clifford got hold of the algebras by the scruff of the neck and realising the superiority of the Grassmann lineal Algebra as well as the Grassmann method of Analysis nd synthesis, he combined the Quaternions into the Grassmann algebra by a note Grassmann had made on that very subject, before dying. Clifford clearly loved both men's insight, and he lovingly showed how they could be represented in a simpler notation.. In addition he was able to show haw Grassmanns algebraic methods of synthesis could model any form of measurement and calculation system, neatly and succinctly.
Clifford generalised the Grassmann product into the lineal combination of 3 different types of product. The scalar product, the dot product, and the cross product. He showed how these 3 products were distributed in a combinatorial table, and because of that others soon twigged on to the relationship with Cayley's 4 x4 matrix algebrs.
E x B arises from this mathematical history. However, in applying it to physics, in particular the production of electrical current and Electro Motive Force it became necessary to tie it to the existing rules of thumb, the Fleming – Faraday right and left hand rules. The careless application of this rule to this product has led to confusion over the phase or phase angle of EMF production.
We have been incorrectly taught that their is an electromagnetic field, and we have been led to see a magnetic phenomenon as a so called electric phenomenon. Instead of progressing from electra to Magnes we have combined the 2 in an odd way that frankly still leaves us puzzled!
The dynamic magnetic phenomenon in the rocks from Magnesia have always been considered as related to the phenomenon in rubbed amber, but it is not obvious how they are connected.
At the same time as Oersted noticed a magnetic compass pointing to a wire through or by which a charge was being discharged from a Leyden jar or a "current" from a voltaic cell, Arago's was showing that rotating discs of metal produced a coupling magnetic field.. History shows how going with Oersted has led to some interesting difficulties, whereas going with Arago's would have introduced the concept of a rotating magnetic field transiting along the wire.
The voltaic current was also a mistake, as it soon became apprent the so called electricity defined as the so called movement of charge could not propagate through a wire at all. It was then confined to the surface where it's movement was further misunderstood.
In fact I do not need to concern myself with these details as23 now know the charge model to be fictitious.
E and B were introduced as arbitrary vectors supposedly giving the orientation of the electric and magnetic fields. Their is no such component, rather their is a magnetic fid and a twisting klop of wire. It is this loop that holds the fictitious vectors as measures of through flux and a through current!
I now know that electricity and magnetism have so much more to do with the surface and boundary conditions of moving material in. A dynamic rotating magnetic field. The orthogonality is not just axial, it is phase and also surface orthogonality.
The behaviour of rotating projectile fluids at the boundary surface of a stationary
Y or rotating form, particularly with sharp edges is now od interest..
To say that light or EM energies are polarised longitudinally and to ignore transverse polarisation along a longitudinal axis is to skew the empirical data what may well be called polarisation of propagated energy in the direction of propagation, may also be confused ith polarisation of the energy transverse to the direction of propagation. In fact and in general polarisation may be the nature of the action of pressure in a surface and normal to a surface.
It is known that fluids have no form except the boundary of their container. It is proposed that energy has this fluid property, and that in fact Newtonian motive has such an identical property to fluid that it will serve as the concept of energy at least initially.
The Newtonian fluid motive has Newyonian measures such as acceleration and instantaneous or sequential velocity change. The notion of celerity represents Newtonin motive in equilibrium. Newtonin motive was defined to exist within bounded forms and to transmit by and through the contact of these boundaries.
I will free Newtonian motive to roam arbitrarily in space and to by its rotational and trochoidal nature ,form regional boundaries in the wada basin sense. In this way boundaries are not impervious to motive but are in fact regions where different laws of motive hold with some measure of stability or endurance.
This will allow what Newton could never apprehend, how force could be transmitted in a rarefied medium at a distance. The principal mechnism for this will be the shock wave or the Dirac delta function
http://en.wikipedia.org/wiki/Magnus_effect this is a kind of refraction.
http://www.dtic.mil/dtic/tr/fulltext/u2/a125332.pdf this is a kind of liquid projectile spinning in a ballistic trajectory.
A spinning projectile http://www.scielo.org.ar/pdf/laar/v38n3/v38n3a07.pdf
Spinning waves sought