Maxwell—Heaviside Equations

Maxwell s famous four equations are found in many versions, for example, in differential or integral form and with different parameters involved. Equations 9.1—9.4 show one example set (the Minkowski formulation). The differential form relates the time and space derivatives at a point (in an infinitesimal small volume) to the current density at that point. The integral form relates to a defined finite volume. Ideal charge distributions are often discontinuous, and so not differentiable therefore, the integral form is a more generally applicable form. 

1 Gauss law V-E = q,/eo Flux of E through a closed surface = net enclosed charge/eo 

2 Faraday s law of induction V X E = -5B/dt Line integral of E around a loop = —rate of change of flux of B through the loop 

4 Ampere s law V X H =Jf+ dD/dt Line integral ofH around a loop = current density of free charges through the loop + rate of change of flux of D through the loop 

V is the differential vector operator called nabla or del V = id/dx + jd/dy + kd/dz, where i, j and k are unity vectors in a Cartesian coordinate system. 

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XVII. Technical Appendix B 4-Vector Maxwell-Heaviside Equations... 

A simple example in classical electrodynamics of what is now known as gauge invariance was introduced by Heaviside [3,4], who reduced the original electrodynamical equations of Maxwell to their present form. Therefore, these equations are more properly known as the Maxwell-Heaviside equations and, in the terminology of contemporary gauge field theory, are identifiable as U(l) Yang-Mills equations [15]. The subj ect of this chapter is 0(3) Yang-Mills gauge theory applied to electrodynamics and electroweak theory. 

To prove the invariance of the cyclic theorem [11-20], it is necessary only to prove the invariance of the free-space Maxwell-Heaviside equations ... 

More than a century later, Lehnert [7] introduced and developed [7-10] the concept of vacuum charge on the classical level, and showed [7-10] that this concept leads to advantages over the Maxwell-Heaviside equations in the description of empirical data, for example, the problem of an interface with a vacuum [7-10,15]. The introduction of a vacuum charge leads to axisymmetric vacuum solutions akin to the B(3> vacuum component of 0(3) electrodynamics... 

Equations (573) have overall 0(3) symmetry, and have the same structure as the Maxwell-Heaviside equations with magnetic charge and current [3,4]. From Eqs. (573), we obtain the wave equation... 

The source of Eq. (625), however, is the set of vacuum Maxwell-Heaviside equations... 

This is the same structure as the homogenous Maxwell-Heaviside equations in the vacuum, which can therefore be obtained by a consideration of relativistic helicity. 

Beltrami fields have been advanced [4] as theoretical models for astrophy-sical phenomena such as solar flares and spiral galaxies, plasma vortex filaments arising from plasma focus experiments, and superconductivity. Beltrami electrodynamic fields probably have major potential significance to theoretical and empirical science. In plasma vortex filaments, for example, energy anomalies arise that cannot be described with the Maxwell-Heaviside equations. The three magnetic components of 0(3) electrodynamics are Beltrami fields as well as being complex lamellar and solenoidal fields. The component is identically nonzero in Beltrami electrodynamics if is so. In the Beltrami... 

In this second technical appendix, it is shown that the Maxwell-Heaviside equations can be written in terms of a field 4-vector = (0, cB + iE) rather than as a tensor. Under Lorentz transformation, GM transforms as a 4-vector. This shows that the field in electromagnetic theory is not uniquely defined as a... 

Using the fact that p and J themselves form the components of a 4-vector, the Maxwell-Heaviside equations for field matter interaction can be combined into one relation between 4-vectors ... 

The results (B.7) and (B.8) are different, even though both describe a boost of the same vector equations, the Maxwell-Heaviside equations ... 

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

The result is two coupled Maxwell-Heaviside equations. Jackson shows that potentials A and in these two equations are arbitrary (i.e., yield the same force fields) [19,20]7 in a specific sense, since the A vector can be replaced with A = A + VA, where is a scalar function and VA is its gradient. The field is given by = V x A, so that the new B field becomes... 

Since the present standard U(l) electrodynamics model forbids electrical power systems with COP > 1.0, my colleagues and I also studied the derivation of that model, which is recognized to contain flaws due to its > 136-year-old basis. We particularly examined how it developed, how it was changed, and how we came to have the Lorentz-regauged Maxwell-Heaviside equations model ubiquitously used today, particularly with respect to the design, manufacture, and use of electrical power systems. 

Later H. A. Lorentz [15],5 apparently unaware of Lorenz 1867 work, independently regauged the Maxwell-Heaviside equations so that they represented a system that was in equilibrium with its active environment. This indeed simplified the mathematics, thus minimizing numerical methods. However, it also discarded all electrical windmills in a free wind —so to speak—and left only those electrical windmills in a large sealed room where there was never any net free wind. 

Such was H. A. Lorentz s prestige that, once he advanced symmetrical regauging of the Maxwell-Heaviside equations, it was rather universally adopted by electrodynamicists, who still use it today see, for example, Jackson [15]. 

Lorentz [15] (see also footnote 5, above) curtailment of the Maxwell-Heaviside equations greatly simplified the mathematics and eased the solution of the resulting equations, of course. But applied to the design of circuits, particularly during their excitation discharge, it also discarded the most interesting and useful class of Maxwellian systems, those exhibiting COP > 1.0. 

The Lehnert equations are a great improvement over the Maxwell-Heaviside equations [45,49] but are unable to describe phenomena such as the Sagnac effect and interferometry [42], for which an 0(3) internal gauge space symmetry is needed. 

If we attempt the same exercise in U(l) electrodynamics, the closed loop gives the Maxwell-Heaviside equations in the vacuum, which are invariant under T and that therefore cannot describe the Sagnac effect [44] because one loop of the Sagnac interferometer is obtained from the other loop by T symmetry. The U(l) phase factor is oof kZ + a, where a is arbitrary [44], and this phase factor is also "/ -invariant. The Maxwell-Heaviside equations in the vacuum are... 

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