Maxwell-Lorentz equation

The Equations of Oscillation.—Let now X, Y, Z denote the electrical forces in the directions of the x, y, z axes, respectively, while L, M> N denote the corresponding magnetic forces. If we eliminate L, M, N in the Maxwell-Lorentz equations we obtain relations of the familiar type... 

In the Lorentz gauge, Maxwell s equations when expressed in terms of the potentials assume the following form... 

However, in Maxwell s days everyone assumed that there had to be a mechanical underpinning for the theory of EM. Many researchers worked on very detailed hidden variable theories for the EM field, in an attempt to prove that the laws of EM were in fact a theorem in NM, just like Kepler s laws are a theorem in NM. No one noticed that it was impossible to do this, since Maxwell s equations are not Galilei invariant and Newton s laws are. That includes Lorentz who discovered around 1900 that the Maxwell equations are invariant under another transformation that now bears his name. 

The Ether is not useful to teach MT. The EM field is most effectively viewed as an irreducible entity completely defined by Maxwell s equations. (If one wants to make the interaction with point charges in N.M or QM explicit, one can add the Lorentz force or the minimal coupling.) All physical properties of th EM field and its interaction with matter follow from Maxwell s equations and the matter equations. 

The basic mechanism of spin-orbit coupling is magnetic induction. It is therefore a truly relativistic effect, as will be discussed shortly. The potentials of a moving charge can be found from Maxwell s equations, as direct solutions or from Lorentz transformations of potentials of a static charge to a moving frame. Maxwell s equations can be divided into the homogeneous parr... 

Maxwell s equations, as well as the Lorentz force, can be derived from the Lagrangian density... 

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... 

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. 

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. 

This finding may be related to similar remarks of Ritz [30] regarding Lorentz electron theory [45]. Ritz concluded that the solutions to the wave equations were more fundamental than Maxwell s equations. In his words [30, p. 172] on voit qu en derniere analyse c est laformule des actions elementaires, et non le systeme de equations aux derivees partielles, qui est Vexpression exacte et complete de la theorie de Lorentz (emphasis in original). 

This form corresponds to a breakdown of Lorentz invariance induced by the propagation of the photon. Suppose that the light travels a distance L in time t e 2 in the presence of a metric fluctuation h . The effects of such a field in Maxwell s equations may be considered as follows ... 

This is incompatible with Maxwell s equations, as shown below by using Gauss s law, Eq. (2.7.16), and the Lorentz force, Eq. (2.7.24). Assume that the two systems S and S move at velocities v and v and relative velocity V= v — v. If we use the Galileian transformation and assume that the charge q and the electric displacement D is the same in the two systems ... 

We have performed numerical experiments using a three dimensional relativistic kinetic electromagnetic particle-in-cell code The code works from first principles by solving the Lorentz force equation for the particles and the Maxwell s equations for the electromagnetic fields. 

James Clerk Maxwell died in 1879, the same year that Albert Einstein was born. Sixteen years later Einstein recognized that Maxwell s equations are covariant with respect to the Lorentz transformations between relatively moving inertial frames of reference, that is, reference frames that are in constant relative motion in a straight line. Thus, Einstein recognized in 1895 that the laws of electrodynamics, expressed with Maxwell s held equations, must be in one-to-one correspondence in all possible inertial frames of reference, from the view of any one of them [1]. 

According to Lorentz, the electric field E acting on a molecule in the isotropic medium (local field) and causing its polarization is not equal to the mean (macroscopic) field E which satisfies the phenomenological Maxwell s equations, but is determined by... 

The conditions of propagation of an electromagnetic wave in the ionospheric medium are determined by the Maxwell and Lorentz equations. These equations determine the properties of the electric (E) and magnetic (H) fields, as well as the displacement (D) and the induction (B) as a function of the electric charge and current densities (J) (see, for example, Budden, 1961 Davies, 1965) ... 

The theoretical content of classical electrodynamics can be summarized by the Lorentz force law and Maxwell s equations. The Lorentz force law describes the force on a charge q moving with velocity v in the presence of an electric field E and a magnetic field B ... 

The name of this gauge refers to the Dutch physicist Hendrik Antoon Lorentz (1853-1928), whereas the gauge was really introduced by the Danish physicist Ludvig Valentin Lorenz who not only introduced this gauge as well as their retarded solutions (91), but also independently developed Maxwell s equations [39]. 

There are a number of approximations required to obtain a description such as (16). Maxwell s equations are invariant to Lorentz transformations and so must be the particle hamiltonian. For the free particle, it is assumed an infinite-dimensional Hilbert space. Here, there is a problem since the momentum range is continuous. The common practice is to define a volume V, e.g. a cubic box, and allow only wave functions fulfilling periodic boundary conditions. The problem is that a square box of volume V is not Lorentz invariant. The reader will have an excellent discussion on this and other related matters in Veltman s book [20]. Here, Eq. (16) is retained as a useful ansatz [21]. 

Maxwell s equations, which were first presented in 1864 and published in 1865 [40], completely describe the classical behavior of electric and magnetic fields and — supported by the Lorentz force law — their interaction with charged particles and currents. In Gaussian units their differential form is given by... 

Attempts were first made to cast Maxwell s equations in a Galileo-invariant form. However, all such attempts resulted in prediction of effects that contradicted experiment or eluded detection. Instead, Lorentz noticed that these equations were invariant under the following transformation ... 

Furthermore, the key factor in successful HH cells (pot operations) is to predict and maintain MHD stability. This leads to the characterization of the internal MHD flow using Navier-Stokes equation of motion and Maxwell s equations, including of charge distribution, Faraday s law of induction. Ohm s law, Lorentz force law, Pcasson s equation, and even LaHace s equation. The set of these laws and equations constitute the MHD expressions. 

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