Lorentz’s equation

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 KG equation is Lorentz invariant, as required, but presents some other problems. Unlike Schrodinger s equation the KG equation is a second order differential equation with respect to time. This means that its solutions are specified only after an initial condition on bothand d /dt has been given. However, in contrast to k, d /dt has no direct physical interpretation [61]. Should the KG equation be used to define an equation of continuity, as was done with Schrodinger s equation (4), it is found to be satisfied by... 

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... 

I returned to the University of Toronto in the summer of 1940, having completed a Master s degree at Princeton, to enroll in a Ph.D. program under Leopold Infeld for which I wrote a thesis entitled A Study in Relativistic Quantum Mechanics Based on Sir A.S. Eddington s Relativity Theory of Protons and Electrons. This book summarized his thought about the constants of Nature to which he had been led by his shock that Dirac s equation demonstrated that a theory which was invariant under Lorentz transformation need not be expressed in terms of tensors. 

The refractive index is an important quantity for characterizing the structure of polymers. This is because it depends sensitively on the chemical composition, on the tacticity, and - for oligomeric samples - also on the molecular weight of a macromolecular substance. The refractive indices (determined using the sodium D line) of many polymers are collected in the literature. In order to characterize a molecule s constitution one requires knowledge of the mole refraction, Rg. For isotropic samples, it can be calculated in good approximation by the Lorentz-Lorenz equation ... 

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]. 

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 ... 

One of the most widely used steric parameters is molar refraction (MR), which has been aptly described as a "chameleon" parameter by Tute (160). Although it is generally considered to be a crude measure of overall bulk, it does incorporate a polarizability component that may describe cohesion and is related to London dispersion forces as follows MR = 4TrNa/By where N is Avogadro s number and a is the polarizability of the molecule. It contains no information on shape. MR is also defined by the Lorentz-Lorenz equation ... 

Figure 47 shows taken from Equation 20 versus Vj. It shows that S. is quite sensitive to Vp and is therefore a good means to evaluate v, with the numerical values of Fig. 47. It can be estimated that the tensile modulus E of the bulk PMMA is not affected by the very low pressure toluene gas environment during the short duration of the experiment. The optical craze index in PMMA in air without load is known as n = 1.32, which corresponds to v = 0.6. From the optical interferometry, it is known that the craze just before breakage is twice as thick as unloaded, (v, = 0.3) and hence using Lorentz-Lorenz equation its optical index is n = 1.15. From Figs. 46 and 47 it can be concluded that the bulk modulus around the propagating crack is about 4400 MPa, which is a somewhat high value, in view of the strain rates at a propagating crack tip (10 to s" ). Using the scatter displayed in Fig. 46, it can be concluded from Fig. 47 that the fibril volume fraction is constant, v = 0.3, within a scatter band of 0.08, and is therefore not sensitive to the toluene gas.

Pressure-induced changes in the refractive index and the thickness are theoretically rationalized by Tait s and Lorentz-Lorenze s equations. Studies... 

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]. 

The atomic theory of electricity simplifies the field (Mjuations, by considering only fields in a vacuum the units can thivn Ih ( >]iosen so that the field equations contain no material consta,nts (or at most, tlie velocity of light c = 3-10 cm./sec., which is retained so tliat we may be able to use the ordinary units of kinematics, centimetres a,nd seconds). This is the standpoint of Lorentz s electron theory (1897), which dominated physics abont the turn of the century. At that time we were... 

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 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]. 

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