From Electromagnetism to the Electromagnetic Field: The Genesis of Maxwell's Equations [Historical Corner]

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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The Electromagnetic Field and Maxwell’s Equations

Electromagnetics and Calculation of Fields ◽

10.1007/978-1-4612-0661-3_2 ◽

1997 ◽

pp. 22-46

Author(s):

Nathan Ida ◽

João P. A. Bastos

Keyword(s):

Electromagnetic Field ◽

Maxwell’S Equations ◽

Maxwell's Equations

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XI.—Electromagnetic Phenomena in a Uniform Gravitational Field

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600023014 ◽

1932 ◽

Vol 51 ◽

pp. 71-79 ◽

Cited By ~ 1

Author(s):

D. Meksyn

Keyword(s):

Electromagnetic Field ◽

Gravitational Field ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Suitable Choice ◽

Fundamental Tensor

In two recent papers Professor E. T. Whittaker has solved the electromagnetic equations for the case of a uniform gravitational field. The fundamental tensor associated with such a field makes the Riemannian tensor vanish, since such a field can be transformed away by a suitable choice of coordinates. This property enables us to find the electromagnetic field in a uniform gravitational field without solving Maxwell's equations, but by a mere transformation of co-ordinates.

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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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Origin of Gauge Theories in Electrodynamics

10.36227/techrxiv.14268602.v1 ◽

2021 ◽

Author(s):

Jay Solanki

Keyword(s):

Field Theory ◽

Electromagnetic Field ◽

Gauge Theory ◽

Gauge Invariance ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Gauge Theories ◽

Degree Of Freedom ◽

Potential Formulation ◽

Electromagnetic Field Theory

<div>The potential formulation has significant advantages over field formulation in solving complicated problems in electromagnetic field theory. One essential part of electromagnetic field theory's potential formulation is gauge invariance and gauge theories because it provides an extra degree of freedom. By using this extra degree of freedom, we can solve complicated electromagnetic problems quickly. Thus, it is necessary to include a systematic explanation of gauge theories in teaching electromagnetic theory. However, textbooks usually formulate gauge theories by using Maxwell's equations of electromagnetism, by using vector calculus identities. However, this method of formulation of gauge theories does not give a clear idea about the origin of gauge theories and gauge invariance in electromagnetism. Here the author formulates gauge theories from wave equations of the electric and magnetic fields instead of directly using Maxwell's equations. This method generalizes all gauge theories like Lorenz gauge theory, Coulomb gauge theory, Etc. Gauge theory, because of the way the author derives it, gives a distinct idea about the mathematical origin of the gauge theories and gauge invariance in electromagnetic field theory. Thus, the author reviews the origin of gauge theories in electromagnetic field theory and develops a distinct and effective method to introduce gauge theory in the teaching of electromagnetic field theory that can provide better understanding of the topic to undergraduate students.</div><div><br></div>

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Fields—Maxwell’s Equations

The Physical World ◽

10.1093/oso/9780198795933.003.0004 ◽

2017 ◽

Author(s):

Nicholas Manton ◽

Nicholas Mee

Keyword(s):

Electromagnetic Field ◽

Charged Particle ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Dipole Moments ◽

Scalar Fields ◽

Vector Calculus ◽

Klein Gordon ◽

Energy And Momentum ◽

Scalar Potentials

Chapter 3 explores the concept of the field, which is necessary to describe forces without resorting to action at a distance, and uses it to describe electromagnetism, as encapsulated by the Maxwell equations. First, scalar fields and the Klein–Gordon equation are discussed. Vector calculus is introduced. The physical meaning of Maxwell’s equations is explained. The equations are then solved for electrostatic fields. Non-uniform charge distributions and dipole moments are discussed. The vector and scalar potentials are introduced. Electromagnetic wave solutions of Maxwell’s equations are found and the Hertz experiment is described. Magnetostatics is discussed briefly. The Lorentz force is described and used to determine the motion of a charged particle in a cyclotron or synchrotron. The action principle for electromagnetism is described. The energy and momentum carried by the electromagnetic field are calculated. The reaction of a charged particle to its own electromagnetic field is considered.

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On the instability of localized EM pulses in nonlinear electrodynamics with account of temperature effects

Modern Physics Letters B ◽

10.1142/s0217984921501761 ◽

2021 ◽

Vol 35 (10) ◽

pp. 2150176

Author(s):

Mikhail B. Belonenko ◽

Natalia N. Konobeeva ◽

Alexander V. Zhukov

Keyword(s):

Electromagnetic Field ◽

Pair Production ◽

Temperature Effects ◽

Electromagnetic Pulse ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Nonlinear Electrodynamics ◽

Influence Of Temperature ◽

Schwinger Mechanism

Based on Maxwell’s equations, we study the development of electromagnetic pulse instability in nonlinear electrodynamics beyond approximation of slowly varying amplitudes and phases. The action was chosen from the Heisenberg–Euler Lagrangian based on the invariants of the electromagnetic field. We analyze the scenario for the evolution of instability, which turns out to be consistent with the earlier conclusion done within the approximation of slowly varying amplitudes and phases. In this study, we take into account the influence of temperature. The rate of pair production under the Schwinger mechanism is estimated.

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