Charge-current density

A high rate discharge led to the recombination of isolated lithium which resulted in an increase in cycle life, and the cycle life decreased with an increase in the charge current density [115],... 

A further condition on j x) can be obtained as follows since jM(x) is the source of the electromagnetic fields—it is the conserved (dltju(x) = 0) charge-current density—the operator... 

Charge conjugate operator, 545 Charge-current density renormalized, 597... 

Relativistic charge-current densities expressed in terms of G-spinor basis sets for stable and economical numerical calculations [2]. 

The G-spinor representation (12), when substituted into (5), results in a time-dependent charge-current density... 

Both the conductivity and the susceptibility contribute to the imaginary part of the permittivity Im e) = Im x) + Re a/to. A nonzero value for Im e) manifests itself physically by absorption of electromagnetic energy in the medium. We may associate Im(x) with the bound charge current density and Re charge current density. Absorption is determined by the sum of these two quantities, however, and it is not possible to determine by absorption measurements their relative contributions. This underscores our assertion that there is no clearly defined distinction between free and bound charges. 

The present chapter is devoted mainly to one of these new theories, in particular to its possible applications to photon physics and optics. This theory is based on the hypothesis of a nonzero divergence of the electric field in vacuo, in combination with the condition of Lorentz invariance. The nonzero electric field divergence, with an associated space-charge current density, introduces an extra degree of freedom that leads to new possible states of the electromagnetic field. This concept originated from some ideas by the author in the late 1960s, the first of which was published in a series of separate papers [10,12], and later in more complete forms and in reviews [13-20]. 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

Returning to the form (3) of the space-charge current density, and observing that (j, ) is a 4-vector, the Lorentz invariance thus leads to... 

The Lagrangian (824), which is the same as the Lagrangian (839), gives the inhomogeneous equation (826) using the same Euler-Lagrange equation (843). Therefore the photon mass can be identified with the vacuum charge-current density as follows (in SI units) ... 

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 ... 

The individual terms of the charge current density (Jv) in the vacuum are Noether currents of the type (101)—(106) and we have the following identifications under all conditions ... 

In the case of reactant adsorption (see Sect. 1.1.3), the charging current is given by an expression similar to eqn. (7) (with TA replaced by T0 in the example discussed) and it is to be noted that T0 plays a role in determining both the faradaic currrent density, jF, and the charging current density, c. 

The current density, will be the sum of the faradaic current density, jF, and the charging current density, c, cf. eqn. (8). The latter is related to the interfacial potential indicated in an implicit way by eqn. (20). The theory of the electrical double layer provides no analytical expression for the relation between E and qM and so, rigorously, this part of the problem would have to be solved numerically using the empirical relationship, which is known for many commonly used indifferent electrolytes. If Cd = dqM/dE is the differential double-layer capacity, we have... 

Both the double-layer charging and the faradaic charge transfer are non-linear processes, i.e. the charging current density, jc, and the faradaic... 

Although being of great fundamental importance, it should not be ignored that practical application of the semi-integral analysis requires separation of the faradaic current density jF, i.e. subtraction of the charging current density jc, from the overall current density, j, as well as perfect instrumental compensation or numerical subtraction of the ohmic potential drop jARn in order to obtain the interfacial potential E. 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

III. Classical Lehnert and Proca Vacuum Charge Current Density... 

It is customary to develop the Proca equation in terms of the vacuum charge current density... 

III. CLASSICAL LEHNERT AND PROCA VACUUM CHARGE CURRENT DENSITY... 

Therefore, /M ( vac) is a covariant conserved charge current density in the vacuum. The coefficient g of the covariant derivative has the units [47-61] of k/A in the vacuum. Using... 

So it is also possible to use the form (139) for the vacuum charge current density, a form that eliminates any geometric unit such as Ar that is not fully relativistic. However, A is, strictly speaking, a potential energy difference and not a field. 

The term —g2m2A /X implies that the electromagnetic 4-potential has acquired mass. Simultaneously there appear two other terms. All four vacuum charge current densities produce vacuum energy through the equation... 

The structure of these vacuum charge current densities can be developed as follows in terms of time-like, longitudinal and transverse components. In this... 

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 ... 

Therefore the Lehnert equation (253) correctly conserves action under a local U(l) gauge transformation in the vacuum. Such a transformation leads to a vacuum charge current density as the result of gauge theory itself, because U(l) gauge theory has a scalar internal space that supports A and A. These must be complex in order to define the globally conserved charge ... 

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