Charge density, 0 electrodynamics

Poisson equation — In mathematics, the Poisson equation is a partial differential equation with broad utility in electrostatics, mechanical engineering, and theoretical physics. It is named after the French mathematician and physicist Simoon-Denis Poisson (1781-1840). In classical electrodynamics the Poisson equation describes the relationship between (electric) charge density and electrostatic potential, while in classical mechanics it describes the relationship between mass density and gravitational field. The Poisson equation in classical electrodynamics is not a basic equation, but follows directly from the Maxwell equations if all time derivatives are zero, i.e., for electrostatic conditions. The corresponding ( first ) Maxwell equation [i] for the electrical field strength E under these conditions is... 

It is known from the classical electrodynamics that the electrostatic potential energy of charge densities can be calculated by using the multipole expansion of the charge distributions (see above). The electrostatic interaction energy between charge distributions has the following form ... 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

The original formal theory is expressed in terms of quanttun electrodynamics with the continuum mediwn characterized by its spectnun of complex dielectric frequencies. A more recent formulation, derived from this theory, is based on the extension of the reaction field concept to a dipole subject to fluctuations exclusively electric in origin. Another procedme has been formulated starting, as for the repulsion contribution, from the theory of intermolecular forces. Following the scheme commonly exploited to derive the electrostatic contribution to the interaction energy, the molecule B is substituted by a continuum medium, the solvent S, described by a surface charge density as induced by the solute transition densities of M (the equivalent of A) and spreading on the cavity surface. 

Two general theoretical approaches have been applied in the analysis of heterogeneous materials. The macroscopic approach, in terms of classical electrodynamics, and the statistical mechanics approach, in terms of charge-density calculations. The first is based on the application of the Laplace equation to calculate the electric potential inside and outside a dispersed spherical particle (11, 12). The same result can be obtained by considering the relationship between the electric displacement D and the macroscopic electric field Em a disperse system (12,13). The second approach takes into account the coordinate-dependent concentration of counterions in the diffuse double layer, regarding the self-consistent electrostatic poton tial of counterions via Poisson s equation (5, 16, 17). Let us consider these approaches briefly. 

Focusing of ions in curved FAIMS (4.3.1) means a pseudopotential bottoming near the gap median. Devices using such wells to guide or trap ions (e.g., quadrupole filters or traps and electrodynamic funnels) have finite charge capacity or saturation current (/sat) the Coulomb potential scales as the charge density squared and, above some density, exceeds the well depth and expels excess ions from the device. Simulations... 

The magnetic resjwnse of diamagnetic atoms, molecules and clusters, i.e., typical quantum mechanical systems, can be effectively interpreted and visualized via the laws of classical electrodynamics, allowing for functions of position r which describe the electronic charge density p(r), a scalar property, and the electronic current density J(r), a vector field, evaluated by quantum mechanical methods. 

The molecule is a classical oscillating charge density (usually a point dipole) and the metal nanoparticle is a continuous body characterized by the frequency-dependent dielectric function (see Chapter 1). This is by far the most common description of the metal-molecule electrodynamic coupling problem in the literature. Notably, sometimes even the metal nanoparticle is reduced to a polarizable dipole. Depending on the phenomenon under study, this may be acceptable or results in an oversimplification [50]. 

This result, as well as the form of expressions (23) and (24), shows that the charge and current density relations (3), (4), and (8) of the present extended theory become consistent with and related to the Dirac theory. It also implies that this extended theory can be developed in harmony with the basis of quantum electrodynamics. 

Therefore, the vacuum charge and current densities of Panofsky and Phillips [86], or of Lehnert and Roy [10], are given a topological meaning in 0(3) electrodynamics. In this condensed notation, the vacuum 0(3) field tensor is given by... 

The Lagrangian (850) shows that 0(3) electrodynamics is consistent with the Proca equation. The inhomogeneous field equation (32) of 0(3) electrodynamics is a form of the Proca equation where the photon mass is identified with a vacuum charge-current density. To see this, rewrite the Lagrangian (850) in vector form as follows ... 

Equation (C.5) means that there are no magnetic charge or current densities in 0(3) electrodynamics. 

It is emphasized, however, that there is no reason to assume plane waves. These are used as an illustration only, and in general the vacuum charge current densities of 0(3) electrodynamics are richly structured, far more so than in U(l) electrodynamics, where vacuum charge current densities also exist from the first principles of gauge theory as discussed already. 

Therefore, a check for self-consistency has been carried out for indices p 2 and v = 1. It has been shown, therefore, that in pure gauge theory applied to electrodynamics without a Higgs mechanism, a richly structured vacuum charge current density emerges that serves as the source of energy latent in the vacuum through the following equation ... 

It thus becomes clear that the vacuum charge current density introduced by Lehnert is an excitation above the true vacuum in classical electrodynamics. The true vacuum is defined by Eq. (337). It follows that in the true classical vacuum, the electromagnetic field also disappears. 

We have established that, in 0(3) electrodynamics, the vacuum charge current densities first proposed by Lehnert [42,45,49] take the form... 

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