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Revista Facultad de Ingeniería Universidad de Antioquia

The Influence of the Electric Field Outside a Resistive Solenoid on The Aharonov-Bohm Effect

S Héctor Torres, André Koch Torres Assis

Published Jan 1, 2001• 0 citations
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Abstract

"It is a known fact that there is no magnetic field outside an infinite solenoid carrying a constant current, althoughthere is magnetic vector potential non-null outside it. The existence of the Aharonov-Bohm effect (AB) is usuallyconsidered as proof of the relevance of the vector potential for quantum mechanics. In this paper we will see that thereis a non null electric field outside an infinite resistive solenoid carrying a constant current and its possible relevanceto the analysis of the Aharonov-Bohm effect. When calculating, we introduce the Galilean invariance of Maxwell´sequations and then we obtain the contribution to the AB effect due to electric potential."

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In this paper it is presented a manifestly covariant formulation of the Aharonov-Bohm (AB) phase difference for the magnetic AB effect. This covariant AB phase is written in terms of the Faraday 2-form F and using the decomposition of F in terms of the electric and magnetic fields as four-dimensional (4D) geometric quantities. It is shown that there is a static electric field outside a stationary solenoid with resistive conductor carrying steady current, which causes that the AB phase difference in the magnetic AB effect may be determined by the electric part of the covariant expression, i.e., by the local influence of the 4D electric field and not, as generally accepted, in terms of nonzero vector potential.

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In classical electrodynamics the vacuum is defined as a region where there are no electric or magnetic fields. In such a region, a charged particle (such as an electron) will feel no effect — the Lorentz force is zero. The space ex-ternal to a perfect (i.e., infinite) solenoid can be considered an electromagnetic vacuum, since ~E and ~B vanish there. While a non-zero vector potential ~A does exist outside the solenoid, it can exert no influence on the particle, and as such cannot be directly detected or quantified classically. However, in 1959 Aharonov and Bohm predicted that a vector field would exert a purely quantum-mechanical effect on the phase of the particle’s wave function, which in principle should be detectable. The predicted phase shift was not observed experimentally until 1986, when Tonomura brilliantly verified the effect using a microscopic solenoid. This paper provides a simplified explanation of the Aharonov-Bohm prediction using a path-integral approach that is suitable for the advanced undergraduate. 1.

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We present a review and discussion of the physical meaning of the vector potential in electromagnetism by means of a classical experiment in which a long solenoid, connected to a sinusoidal voltage supply, produces an electric field and a vector potential in a circuit exterior to the solenoid itself. Some basic considerations are recalled concerning the meaning of action-at-a-distance and local description, in terms of 'real fields' of the induced electromotive force on the circuit outside the solenoid. From this viewpoint the roles played by the magnetic field, the electric field and the vector potential to account for the physical effects are reviewed. Finally, the connections between these considerations and the magnetic Aharonov–Bohm effect are recalled in the light of recent theoretical work aimed at explaining the Aharonov–Bohm phase shift in terms of the magnetic field without resorting to the vector potential.

European Journal of Physics•Jan 6, 2004•28 citations
"It is a known fact that there is no magnetic field outside an infinite solenoid carrying a constant current, althoughthere is magnetic vector potential non-null outside it. The existence of the Aharonov-Bohm effect (AB) is usuallyconsidered as proof of the relevance of the vector potential for quantum mechanics. In this paper we will see that thereis a non null electric field outside an infinite resistive solenoid carrying a constant current and its possible relevanceto the analysis of the Aharonov-Bohm effect. When calculating, we introduce the Galilean invariance of Maxwell´sequations and then we obtain the contribution to the AB effect due to electric potential."
In this paper it is presented a manifestly covariant formulation of the Aharonov-Bohm (AB) phase difference for the magnetic AB effect. This covariant AB phase is written in terms of the Faraday 2-form F and using the decomposition of F in terms of the electric and magnetic fields as four-dimensional (4D) geometric quantities. It is shown that there is a static electric field outside a stationary solenoid with resistive conductor carrying steady current, which causes that the AB phase difference in the magnetic AB effect may be determined by the electric part of the covariant expression, i.e., by the local influence of the 4D electric field and not, as generally accepted, in terms of nonzero vector potential.
Whether the time-dependent Aharonov-Bohm (AB) effect even exists or not has been the subject of long-standing debate. There are two factors complicating the problem. First, in the closed spacetime line integral of the vector potential that is thought to give the AB-phase shift, how to treat the time-varying vector potential is highly nontrivial. Second, the time-varying magnetic flux generates induced electric field even outside the solenoid. In the present paper, motivated by a recent work by Gao, we re-investigate the role of the induced electric field with the utmost care. This analysis reveals a highly nontrivial effect of the induced electric field, which turns out to be useful for verifying the very existence of the time-dependent AB-effect.
In classical electrodynamics the vacuum is defined as a region where there are no electric or magnetic fields. In such a region, a charged particle (such as an electron) will feel no effect — the Lorentz force is zero. The space ex-ternal to a perfect (i.e., infinite) solenoid can be considered an electromagnetic vacuum, since ~E and ~B vanish there. While a non-zero vector potential ~A does exist outside the solenoid, it can exert no influence on the particle, and as such cannot be directly detected or quantified classically. However, in 1959 Aharonov and Bohm predicted that a vector field would exert a purely quantum-mechanical effect on the phase of the particle’s wave function, which in principle should be detectable. The predicted phase shift was not observed experimentally until 1986, when Tonomura brilliantly verified the effect using a microscopic solenoid. This paper provides a simplified explanation of the Aharonov-Bohm prediction using a path-integral approach that is suitable for the advanced undergraduate. 1.
We discuss two possible covariant generalizations of the Aharonov–Bohm effect – one expression in terms of the space–time line integral of the four-vector potential and the other expression in terms of the space–time “area” integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov–Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov–Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, E=−∂tA, outside the solenoid. This result may already have been confirmed experimentally.
In this Comment it is shown that it cannot be argued that in the magnetic AB effect there is no force acting on the particle, i.e., that the observed phase shift is entirely due to nonzero vector potential. In stationary resistive conductors carrying constant currents there are quasistatic surface charges, which generate not only the electric field inside the wire driving the current, but also a static electric field outside it. These external static electric fields have nothing to do with Boyer's force picture and with his result for the existence of a time delay.
We present a review and discussion of the physical meaning of the vector potential in electromagnetism by means of a classical experiment in which a long solenoid, connected to a sinusoidal voltage supply, produces an electric field and a vector potential in a circuit exterior to the solenoid itself. Some basic considerations are recalled concerning the meaning of action-at-a-distance and local description, in terms of 'real fields' of the induced electromotive force on the circuit outside the solenoid. From this viewpoint the roles played by the magnetic field, the electric field and the vector potential to account for the physical effects are reviewed. Finally, the connections between these considerations and the magnetic Aharonov–Bohm effect are recalled in the light of recent theoretical work aimed at explaining the Aharonov–Bohm phase shift in terms of the magnetic field without resorting to the vector potential.