We have recently 1 argued that classical electrodynamics can predict nonlocal effects by showing ... more We have recently 1 argued that classical electrodynamics can predict nonlocal effects by showing an example of a topological and nonlocal electromagnetic angular momentum. In this paper we discuss the dual of this angular momentum which is also topological and nonlocal. We then unify both angular momenta by means of the electromagnetic angular momentum arising in the configuration formed by a dyon encircling an infinitely-long dual solenoid enclosing uniform electric and magnetic fluxes and show that this electromagnetic angular momentum is topological because it depends on a winding number, is nonlocal because the electric and magnetic fields of this dual solenoid act on the dyon in regions for which these fields are excluded and is invariant under electromagnetic duality transformations. We explicitly verify that this duality-invariant electromagnetic angular momentum is insensitive to the radiative effects of the Liénard-Wiechert fields of the encircling dyon. We also show how duality symmetry of this angular momentum suggests different physical interpretations for the corresponding angular momenta that it unifies.
Classical electrodynamics is a local theory describing local interactions between charges and ele... more Classical electrodynamics is a local theory describing local interactions between charges and electromagnetic fields and therefore one would not expect that this theory could predict nonlocal effects. But this perception implicitly assumes that the electromagnetic configurations lie in simply connected regions. In this paper, we consider an electromagnetic configuration lying in a non-simply connected region, which consists of a charged particle encircling an infinitely long solenoid enclosing a uniform magnetic flux, and show that the electromagnetic angular momentum of this configuration describes a nonlocal interaction between the encircling charge outside the solenoid and the magnetic flux confined inside the solenoid. We argue that the nonlocality of this interaction is of topological nature by showing that the electromagnetic angular momentum of the configuration is proportional to a winding number. The magnitude of this electromagnetic angular momentum may be interpreted as t...
We reply to some comments made by Davis (2020 Eur. J. Phys. 40 018001) on our paper (2020 Eur. J.... more We reply to some comments made by Davis (2020 Eur. J. Phys. 40 018001) on our paper (2020 Eur. J. Phys. 41 035202), by arguing that Davis’s assertions are unsupported in some cases and are unsatisfactory in other cases.
In his recent paper (2020 Eur. J. Phys. 41 045202), Davis makes the claim that potentials and fie... more In his recent paper (2020 Eur. J. Phys. 41 045202), Davis makes the claim that potentials and fields are ill-defined in the conventional treatment of electromagnetism. He argues that ‘the usual treatment is ambiguous, with that ambiguity being reflected in the gauge transformation equations’. He then proposes an approach based on two operational versions of Helmholtz’s theorem and claims that his approach does not exhibit gauge freedom and allows a rigourous definition of electromagnetic potentials. Here I argue that Davis’s approach does not provide a more rigours definition of potentials than that provided by the standard approach. Apparently, Davis does not realize that when applying an operational version of Helmholtz’s theorem to Maxwell’s equations, he is not avoiding gauge invariance but tacitly applying it by choosing the particular gauge-condition related to this version of the theorem. The application of the instantaneous Helmholtz’s theorem to Maxwell’s equations is equiv...
In his recently discovered handwritten notes on 'An alternate way to handle electrodynamics' date... more 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. 'He (Feynman) said that he would start with the vector and scalar potentials, then everything would be much simpler and more transparent.' M A Gottlieb-M Sands Conversation. 4
An extension of the Helmholtz theorem is proved, which states that two retarded vector fields F 1... more 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.
It is well-known that the speed c u = 1/ √ ǫ 0 µ 0 is obtained in the process of defining SI unit... more It is well-known that the speed c u = 1/ √ ǫ 0 µ 0 is obtained in the process of defining SI units via action-at-a-distance forces, like the force between two static charges and the force between two long and parallel currents. The speed c u is then physically different from the observed speed of propagation c associated with electromagnetic waves in vacuum. However, repeated experiments have led to the numerical equality c u = c, which we have called the c equivalence principle. In this paper we point out that ∇ × E = −[1/(ǫ 0 µ 0 c 2)]∂B/∂t is the correct form of writing Faraday's law when the c equivalence principle is not assumed. We also discuss the covariant form of Maxwell's equations without assuming the c equivalence principle.
This paper presents a simple and systematic method to show how the potentials in the Lorentz, Cou... more This paper presents a simple and systematic method to show how the potentials in the Lorentz, Coulomb, Kirchhoff, velocity and temporal gauges yield the same retarded electric and magnetic fields. The method appropriately uses the dynamical equations for the scalar and vector potentials to obtain two wave equations, whose retarded solutions lead to the electric and magnetic fields. The advantage of this method is that it does not use explicit expressions for the potentials in the above gauges, which are generally simple to obtain for the scalar potential but generally difficult to calculate for the vector potential. The spurious character of the term generated by the scalar potential in the Coulomb, Kirchhoff and velocity gauges is noted. The non spurious character of the term generated by the scalar potential in the Lorenz gauge is emphasized.
In a recent paper Davis formulated a generalized Helmholtz theorem for a time-varying vector fiel... more In a recent paper Davis formulated a generalized Helmholtz theorem for a time-varying vector field in terms of the Lorenz gauge retarded potentials. The purposes of this comment are to point out that Davis's generalization of the theorem is a version of the extension of the Helmholtz theorem formulated some years ago by McQuistan and also by Jefimenko and more recently by the present author and to show that Davis's expression for the time-dependent vector field is also valid for potentials in gauges other than the Lorenz gaug
It is well-known that the speed $c_u=1/\sqrt{\epsilon_0\mu_0}$ is obtained in the process of defi... more It is well-known that the speed $c_u=1/\sqrt{\epsilon_0\mu_0}$ is obtained in the process of defining SI units via action-at-a-distance forces, like the force between two static charges and the force between two long and parallel currents. The speed $c_u$ is then physically different from the observed speed of propagation $c$ associated with electromagnetic waves in vacuum. However, repeated experiments have led to the numerical equality $c_u=c,$ which we have called the $c$ equivalence principle. In this paper we point out that $\nabla\times{\bf E}=-[1/(\epsilon_0\mu_0 c^2)]\partial{\bf B}/\partial t$ is the correct form of writing Faraday's law when the $c$ equivalence principle is not assumed. We also discuss the covariant form of Maxwell's equations without assuming the $c$ equivalence principle.
We have recently 1 argued that classical electrodynamics can predict nonlocal effects by showing ... more We have recently 1 argued that classical electrodynamics can predict nonlocal effects by showing an example of a topological and nonlocal electromagnetic angular momentum. In this paper we discuss the dual of this angular momentum which is also topological and nonlocal. We then unify both angular momenta by means of the electromagnetic angular momentum arising in the configuration formed by a dyon encircling an infinitely-long dual solenoid enclosing uniform electric and magnetic fluxes and show that this electromagnetic angular momentum is topological because it depends on a winding number, is nonlocal because the electric and magnetic fields of this dual solenoid act on the dyon in regions for which these fields are excluded and is invariant under electromagnetic duality transformations. We explicitly verify that this duality-invariant electromagnetic angular momentum is insensitive to the radiative effects of the Liénard-Wiechert fields of the encircling dyon. We also show how duality symmetry of this angular momentum suggests different physical interpretations for the corresponding angular momenta that it unifies.
Classical electrodynamics is a local theory describing local interactions between charges and ele... more Classical electrodynamics is a local theory describing local interactions between charges and electromagnetic fields and therefore one would not expect that this theory could predict nonlocal effects. But this perception implicitly assumes that the electromagnetic configurations lie in simply connected regions. In this paper, we consider an electromagnetic configuration lying in a non-simply connected region, which consists of a charged particle encircling an infinitely long solenoid enclosing a uniform magnetic flux, and show that the electromagnetic angular momentum of this configuration describes a nonlocal interaction between the encircling charge outside the solenoid and the magnetic flux confined inside the solenoid. We argue that the nonlocality of this interaction is of topological nature by showing that the electromagnetic angular momentum of the configuration is proportional to a winding number. The magnitude of this electromagnetic angular momentum may be interpreted as t...
We reply to some comments made by Davis (2020 Eur. J. Phys. 40 018001) on our paper (2020 Eur. J.... more We reply to some comments made by Davis (2020 Eur. J. Phys. 40 018001) on our paper (2020 Eur. J. Phys. 41 035202), by arguing that Davis’s assertions are unsupported in some cases and are unsatisfactory in other cases.
In his recent paper (2020 Eur. J. Phys. 41 045202), Davis makes the claim that potentials and fie... more In his recent paper (2020 Eur. J. Phys. 41 045202), Davis makes the claim that potentials and fields are ill-defined in the conventional treatment of electromagnetism. He argues that ‘the usual treatment is ambiguous, with that ambiguity being reflected in the gauge transformation equations’. He then proposes an approach based on two operational versions of Helmholtz’s theorem and claims that his approach does not exhibit gauge freedom and allows a rigourous definition of electromagnetic potentials. Here I argue that Davis’s approach does not provide a more rigours definition of potentials than that provided by the standard approach. Apparently, Davis does not realize that when applying an operational version of Helmholtz’s theorem to Maxwell’s equations, he is not avoiding gauge invariance but tacitly applying it by choosing the particular gauge-condition related to this version of the theorem. The application of the instantaneous Helmholtz’s theorem to Maxwell’s equations is equiv...
In his recently discovered handwritten notes on 'An alternate way to handle electrodynamics' date... more 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. 'He (Feynman) said that he would start with the vector and scalar potentials, then everything would be much simpler and more transparent.' M A Gottlieb-M Sands Conversation. 4
An extension of the Helmholtz theorem is proved, which states that two retarded vector fields F 1... more 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.
It is well-known that the speed c u = 1/ √ ǫ 0 µ 0 is obtained in the process of defining SI unit... more It is well-known that the speed c u = 1/ √ ǫ 0 µ 0 is obtained in the process of defining SI units via action-at-a-distance forces, like the force between two static charges and the force between two long and parallel currents. The speed c u is then physically different from the observed speed of propagation c associated with electromagnetic waves in vacuum. However, repeated experiments have led to the numerical equality c u = c, which we have called the c equivalence principle. In this paper we point out that ∇ × E = −[1/(ǫ 0 µ 0 c 2)]∂B/∂t is the correct form of writing Faraday's law when the c equivalence principle is not assumed. We also discuss the covariant form of Maxwell's equations without assuming the c equivalence principle.
This paper presents a simple and systematic method to show how the potentials in the Lorentz, Cou... more This paper presents a simple and systematic method to show how the potentials in the Lorentz, Coulomb, Kirchhoff, velocity and temporal gauges yield the same retarded electric and magnetic fields. The method appropriately uses the dynamical equations for the scalar and vector potentials to obtain two wave equations, whose retarded solutions lead to the electric and magnetic fields. The advantage of this method is that it does not use explicit expressions for the potentials in the above gauges, which are generally simple to obtain for the scalar potential but generally difficult to calculate for the vector potential. The spurious character of the term generated by the scalar potential in the Coulomb, Kirchhoff and velocity gauges is noted. The non spurious character of the term generated by the scalar potential in the Lorenz gauge is emphasized.
In a recent paper Davis formulated a generalized Helmholtz theorem for a time-varying vector fiel... more In a recent paper Davis formulated a generalized Helmholtz theorem for a time-varying vector field in terms of the Lorenz gauge retarded potentials. The purposes of this comment are to point out that Davis's generalization of the theorem is a version of the extension of the Helmholtz theorem formulated some years ago by McQuistan and also by Jefimenko and more recently by the present author and to show that Davis's expression for the time-dependent vector field is also valid for potentials in gauges other than the Lorenz gaug
It is well-known that the speed $c_u=1/\sqrt{\epsilon_0\mu_0}$ is obtained in the process of defi... more It is well-known that the speed $c_u=1/\sqrt{\epsilon_0\mu_0}$ is obtained in the process of defining SI units via action-at-a-distance forces, like the force between two static charges and the force between two long and parallel currents. The speed $c_u$ is then physically different from the observed speed of propagation $c$ associated with electromagnetic waves in vacuum. However, repeated experiments have led to the numerical equality $c_u=c,$ which we have called the $c$ equivalence principle. In this paper we point out that $\nabla\times{\bf E}=-[1/(\epsilon_0\mu_0 c^2)]\partial{\bf B}/\partial t$ is the correct form of writing Faraday's law when the $c$ equivalence principle is not assumed. We also discuss the covariant form of Maxwell's equations without assuming the $c$ equivalence principle.
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