Convection Displacement Current and Generalized Form of Maxwell–Lorentz Equations

Abstract

Some mathematical inconsistencies in the conventional form of Maxwell's equations extended by Lorentz for a single charge system are discussed. To surmount these in framework of Maxwellian theory, a novel convection displacement current is considered as additional and complementary to the famous Maxwell displacement current. It is shown that this form of the Maxwell-Lorentz equations is similar to that proposed by Hertz for electrodynamics of bodies in motion. Original Maxwell's equations can be considered as a valid approximation for a continuous and closed (or going to infinity) conduction current. It is also proved that our novel form of the Maxwell-Lorentz equations is relativistically invariant. In particular, a relativistically invariant gauge for quasistatic fields has been found to replace the non-invariant Coulomb gauge. The new gauge condition contains the famous relationship between electric and magnetic potentials for one uniformly moving charge that is usually attributed to the Lorentz transformations. Thus, for the first time, using the convection displacement current, a physical interpretation is given to the relationship between the components of the four-vector of quasistatic potentials. A rigorous application of the new gauge transformation with the Lorentz gauge transforms the basic field equations into an independent pair of differential equations responsible for longitudinal and transverse fields, respectively. The longitudinal components can be interpreted exclusively from the standpoint of the instantaneous "action at a distance" concept and leads to necessary conceptual revision of the conventional Faraday-Maxwell field. The concept of electrodynamic dualism is proposed for self-consistent classical electrodynamics. It implies simultaneous coexistence of instantaneous long-range (longitudinal) and Faraday-Maxwell short-range (transverse) interactions that resembles in this aspect the basic idea of Helmholtz's electrodynamics.

On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell’s Equations

“…Therefore, the denial of the exclusive use of partial time derivatives to describe time variation for an observer at rest and a recognition of a deeper underlying meaning of the total time derivative in Eulerian description imply inevitable changes in the structure of mathematical solutions to Maxwell's equations (as regards this aspect, some alternative frameworks for classical electrodynamics were recently discussed in [6], [12]− [15] ). However, a wider analysis of the integral form of Maxwell's equations on basis of the local Convection Theorem does not look tractable at present stage.…”

Untitled

“…tend to zero in (69) leaving only partial time derivatives in agreement with (10)- (13). The difference of the present approach [15] with Hertz's covering theory (and with Phipps' neo-Hertzian approach [13,14]) consists in the definition of the total time derivative (66) for a medium at rest (not in motion with the possible implication of Galilean invariance). Below we shall demonstrate that the set (65)-(68) possesses invariance properties in any inertial frame of reference.…”

“…F or some purposes, it is convenient to decompose (15), (16) into two pairs of second order differential equations for each component of general solution of (15), (16): and with initial and boundary conditions given, for instance, in the case of electric potential. The equation (20), apart from (iii), is supplemented by Whereas (22) has to be added with In the theory of differential equations any complete solution of (15), (16) consists of a general solution of homogeneous D' Alembert's equation plus some particular solution of the inhomogeneous one. Thus, we can assume that the same procedure can be applied to its equivalent formulation in form (20)- (23).…”

“…Thus, we can assume that the same procedure can be applied to its equivalent formulation in form (20)- (23). On one hand, a complete solution should be formed by two independent general solutions satisfying homogeneous Poisson's and homogeneous wave equations, respectively, and, on the other hand, it has to include one particular solution (as a linear combination of non-reducible components (18), (19), satisfying inhomogeneous D' Alembert's equations (15,16). Relationship between both components (longitudinal and transverse) of electromagnetic field is guided by (25) and (26) and is contained in the particular solution of inhomogeneous D' Alembert's equations.…”

“…Before going on to a more general consideration of a large number of sources, it is worth to draw attention that we arrived to the most compact differential form of Maxwell-Hertz equations in the reference system at rest [15].…”