Free-space Maxwell equations electromagnetic field

Up to now, we have examined how the Beltrami vector field relation surfaces in many electromagnetic contexts, featuring predominantly plane-wave solutions (PWSs) to the free-space Maxwell equations in conjunction with biisotropic media (Lakhtakia-Bohren), in homogeneous isotropic vacua (Hillion/Quinnez), or in the magnetostatic context exemplified by FFMFs associated with plasmas (Bostick, etc.). 

Consequently, much like the Beltrami vortex filaments discussed earlier in conjunction with the magnetostatic FFMF, the Beltrami vector relations associated with nonluminal solutions to the free space Maxwell equations, are directly related to physical classical field phenomena currently unexplainable by accepted scientific paradigms. For instance, such non-PWS of the free-space Maxwell equations are direct violations of the sacrosanct principle of special relativity [72], as well as exhibit other counterintuitive properties. Yet, even more extraordinary, these non-PWS are not only theoretical possibilities, but have been demonstrated to exist empirically in the form of the so-called evanescent mode propagation of electromagnetic energy [72-76]. 

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

In order to rmderstand how light can be controlled, we must first review some of tire basic properties of tire electromagnetic field [8], The electromagnetic tlieory of light is governed by tire equations of James Clerk Maxwell. The field phenomena in free space with no sources are described by tire basic set of relationships below ... 

The electromagnetic field in free space is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell s equations... 

For the time being, let us consider the conventional view wave-particle duality. Then, propogation of photon is the same as propagation of electromagnetic field E, B. In free space the charge density is null everywhere, except possibly at the source. The photon is chargeless hence, if Maxwell s equations are applicable to a photon in vacuum, pe = 0 everywhere. This leads to some contradiction. 

In Section III we reviewed our own work on the solutions of Maxwell s equations, which hint to the existence of non-conventional magnetic scalar potentials in free space. The symmetrized set of Maxwell s equations [87] suggests the existence of two novel electromagnetic fields P, N, that lead to the conventional fields E, B. 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

Electromagnetic wave — Oscillatory propagating electromagnetic field. Maxwell s equations, for free space, can be manipulated into the form of two extremely concise vector equations, V2 = eoEof f and V2E =... 

Suppose the waveguide is composed of an unbounded uniform medium of refractive index i.e. effectively free space . The modal fields are then found from Maxwell s equations to be the fields of transverse electromagnetic, or TEM, waves propagating in the z-direction parallel to the waveguide axis. Thus, the propagation constant )S = n k, the longitudinal components satisfy e = h = 0, and the transverse electric and magnetic fields are related by... 

The potentials for a uniform electric field corresponds to those of the electric dipole approximation, except that in the latter case the electric field is uniform only over the molecular volume and not over the entire space. Uniform fields extending over all space must be considered a mathematically convenient idealization they are allowed by Maxwell s equations, but not realizable experimentally. The same holds true for the source-free electromagnetic waves discussed above. It is also important to note that uniform fields do not form the static limit of an electromagnetic wave, in the limit % — 0 both the electric and magnetic fields go to zero. 

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