Beltrami vector fields

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

BELTRAMI VECTOR FIELDS IN ELECTRODYNAMICS—A REASON FOR REEXAMINING THE STRUCTURAL FOUNDATIONS OF CLASSICAL FIELD PHYSICS ... 

This vector field condition is sometimes referred to as Beltrami fluid flow, and was previously treated in a similar exposition by the author in 1995 [1], There it was indicated that Beltrami vector field flow is representative of a certain class of vector fields that are termed force-free. This type of field topology was first brought to prominence by Eugenio Beltrami in his 1889 paper Considerations on Hydrodynamics. [2], This type of morphology describes a regime of fluid... 

When we consider time-harmonic electrodynamics in more general media (chiral-biisotropic), A. Lakhtakia also underscored the importance of the Beltrami field condition [54], In particular, he found that time-harmonic EM fields in a homogeneous reciprocal biisotropic medium are circularly polarized, and must be described by Beltrami vector fields. 

Besides its appearance in the FFMF equation in plasma physics, as well as associated with time-harmonic fields in chiral media, the chiral Beltrami vector field reveals itself in theoretical models for classical transverse electromagnetic (TEM) waves. Specifically, the existence of a general class of TEM waves has been advanced in which the electric and magnetic field vectors are parallel [59]. Interestingly, it was found that for one representation of this wave type, the magnetic vector potential (A) satisfies a Beltrami equation ... 

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.). 

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