An axiomatic approach to Maxwell’s equations

Heras, José A.

doi:10.1088/0143-0807/37/5/055204

Cited by 12 publications

(19 citation statements)

References 30 publications

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“…Suppose that by some means (which of course does not involve the Maxwell equations) we have found the retarded potentials (5). Differentiating these potentials one obtains their wave equations (8) and equation (6). Combining (6) and (8) one infers equations (10) and (11) which are then identified with the inhomogeneous Maxwell's equations whenever the electric and magnetic fields are defined as (12).…”

Section: Discussionmentioning

confidence: 99%

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On Feynman’s handwritten notes on electromagnetism and the idea of introducing potentials before fields

Heras¹,

Heras²

2020

Eur. J. Phys.

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In his recently discovered handwritten notes on "An alternate way to handle electrodynamics" dated on 1963, Richard P. Feynman speculated with the idea of getting the inhomogeneous Maxwell's equations for the electric and magnetic fields from the wave equation for the vector potential. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. We consider the charge conservation expressed through the continuity equation as a basic axiom and make a heuristic handle of this equation to obtain the retarded scalar and vector potentials, whose wave equations yield the homogeneous and inhomogeneous Maxwell's equations. We also show how this axiomatic-heuristic procedure to obtain Maxwell's equations can be formulated covariantly in the Minkowski spacetime.

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Section: Discussionmentioning

confidence: 99%

“…In other words: we have no gauge freedom in our approach to Maxwell's equations. equivalent to the set of equations in (8)? Let us investigate this possibility.…”

Section: Introducing the Potentials φ And A Before The Fields E And Bmentioning

confidence: 99%

On Feynman’s handwritten notes on electromagnetism and the idea of introducing potentials before fields

Heras¹,

Heras²

2020

Eur. J. Phys.

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“…Some without using the formalism of GR [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] and some using the formalism of GR in the weak field and linearized approximations. These are discussed in two separate sections below.…”

Section: Discussionmentioning

confidence: 99%

“…Jefimenko assumed c g = c and postulated a gravitoLorentz force. Recently, Heras [46], by recognizing the general validity of the axiomatic approach to Maxwell's equations of electromagnetic theory, used those axioms to derive only the field equations (leaving out gravito-Lorentz force law) of SRMG, where the in-variance of gravitational charge is considered. Other recent derivations of SRMG equations from different approaches include the works of Nyambuya [47], Sattinger [48], Vieira and Brentan [49].…”

Section: Maxwellian Gravity Of Others Without Grmentioning

confidence: 99%

Comments on gravitoelectromagnetism of Ummarino and Gallerati in “Superconductor in a weak static gravitational field” vs other versions

Behera¹

2017

Eur. Phys. J. C

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Recently reported [Eur. Phys. J. C., 77, 549 (2017). https://doi.org/10.1140/epjc/s10052-017-5116-y] gravitoelectromagnetic equations of Ummarino and Gallerati (UG) in their linearized version of general relativity (GR) are shown to match with (a) our previously reported special relativistic Maxwellian Gravity equations in the non-relativistic limit and with (b) the non-relativistic equations derived here, when the speed of gravity c g (an undetermined parameter of the theory here) is set equal to c (the speed of light in vacuum). Seen in the light of our new results, the UG equations satisfy the Correspondence Principle (cp), while many other versions of linearized GR equations that are being (or may be) used to interpret the experimental data defy the cp. Such new findings assume significance and relevance in the contexts of recent detection of gravitational waves and the gravitomagnetic field of the spinning earth and their interpretations. Being well-founded and self-consistent, the equations may be of interest and useful to researchers exploring the phenomenology of gravitomagnetism, gravitational waves and the novel interplay of gravity with different states of matter in flat space-time like UG's interesting work on superconductors in weak gravitational fields.

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“…where we have used the second equation given in (10), the property [7]: [∂F /∂t] = ∂[F ]/∂t, the Gauss theorem to transform the second volume integral of the second line into a surface integral (dS is the surface element), which is seen to vanish at infinity by assuming the boundary condition that C 2 goes to zero faster than 1/r 2 as r → ∞, and finally considering the second equation in (12). § Following a similar procedure, we can prove the second "Lorenz" condition given in (14).…”

Section: The Helmholtz Theorem For Two Retarded Fieldsmentioning

confidence: 99%

Helmholtz’s theorem for two retarded fields and its application to Maxwell’s equations

Heras¹,

Heras²

2020

Eur. J. Phys.

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An extension of the Helmholtz theorem is proved, which states that two retarded vector fields F 1 and F 2 satisfying appropriate initial and boundary conditions are uniquely determined by specifying their divergences ∇ · F 1 and ∇ · F 2 and their coupled curls −∇ × F 1 − ∂F 2 /∂t and ∇ × F 2 − (1/c 2 )∂F 1 /∂t, where c is the propagation speed of the fields. When a corollary of this theorem is applied to Maxwell's equations, the retarded electric and magnetic fields are directly obtained. The proof of the theorem relies on a novel demonstration of the uniqueness of the solutions of the vector wave equation.

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An axiomatic approach to Maxwell’s equations

Cited by 12 publications

References 30 publications

On Feynman’s handwritten notes on electromagnetism and the idea of introducing potentials before fields

On Feynman’s handwritten notes on electromagnetism and the idea of introducing potentials before fields

Comments on gravitoelectromagnetism of Ummarino and Gallerati in “Superconductor in a weak static gravitational field” vs other versions

Helmholtz’s theorem for two retarded fields and its application to Maxwell’s equations

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