Displacement Current and Maxwell’s Equations

Some remarks on the charging capacitor problem

Advanced Electromagnetics ◽

10.7716/aem.v7i2.694 ◽

2018 ◽

Vol 7 (2) ◽

pp. 10-12

Author(s):

C. J. Papachristou

Keyword(s):

Electromagnetic Field ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Displacement Current ◽

Standard Textbook ◽

Maxwell Displacement Current

The charging capacitor is the standard textbook and classroom example for explaining the concept of the so-called Maxwell displacement current. A certain aspect of the problem, however, is often overlooked. It concerns the conditions for satisfaction of the Faraday-Henry law inside the capacitor. Expressions for the electromagnetic field are derived that properly satisfy all four of Maxwell’s equations in that region.

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A Decomposition of the Electromagnetic Field — Application to the Darwin Model

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202597000542 ◽

1997 ◽

Vol 07 (08) ◽

pp. 1085-1120 ◽

Cited By ~ 19

Author(s):

P. Ciarlet ◽

E. Sonnendrücker

Keyword(s):

Electromagnetic Field ◽

Electric Field ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Computational Cost ◽

Field Application ◽

Displacement Current ◽

Computational Domain ◽

Numerical Resolution ◽

Simply Connected

In many cases, the numerical resolution of Maxwell's equations is very expensive in terms of computational cost. The Darwin model, an approximation of Maxwell's equations obtained by neglecting the divergence free part of the displacement current, can be used to compute the solution more economically. However, this model requires the electric field to be decomposed into two parts for which no straightforward boundary conditions can be derived. In this paper, we consider the case of a computational domain which is not simply connected. With the help of a functional framework, a decomposition of the fields is derived. It is then used to characterize mathematically the solutions of the Darwin model on such a domain.

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Convection Displacement Current and Generalized Form of Maxwell–Lorentz Equations

Modern Physics Letters A ◽

10.1142/s0217732397000029 ◽

1997 ◽

Vol 12 (01) ◽

pp. 1-24 ◽

Cited By ~ 11

Author(s):

Andrew E. Chubykalo ◽

Roman Smirnov-Rueda

Keyword(s):

Maxwell’S Equations ◽

Maxwell's Equations ◽

Gauge Condition ◽

Classical Electrodynamics ◽

Displacement Current ◽

Field Equations ◽

Charge System ◽

Basic Field ◽

Maxwell Displacement Current ◽

Magnetic Potentials

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 the 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 a 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 electrodynamics 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.

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