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Progress In Electromagnetics Research M

THE MAGNETIC INTERACTION ENERGY BETWEEN AN INFINITE SOLENOID AND A PASSING POINT CHARGE

Hanno Essén, Johan C.-E. Stén

Published Jan 1, 2018• 2 citations
Publisher

Abstract

The standard expression for the magnetic interaction energy used in the study of the Aharonov-Bohm effect is investigated. We calculate the magnetic interaction energy between a point charge and an infinite solenoid from first principles. Two alternative expressions are used: the scalar products of the currents with the vector potentials and the scalar product of the magnetic fields. The alternatives are seen to agree. The latter approach also involves taking into account surface integrals at infinity, which are shown to be zero. Our model problem indicates no classical Aharonov-Bohm effect, but we also discuss the normally neglected fact of energy non-conservation. The problem is treated from the point of view of Lagrangian and Hamiltonian mechanics.

DOI
10.2528/pierm18052908

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The standard expression for the magnetic interaction energy used in the study of the Aharonov-Bohm effect is investigated. We calculate the magnetic interaction energy between a point charge and an infinite solenoid from first principles. Two alternative expressions are used: the scalar products of the currents with the vector potentials and the scalar product of the magnetic fields. The alternatives are seen to agree. The latter approach also involves taking into account surface integrals at infinity, which are shown to be zero. Our model problem indicates no classical Aharonov-Bohm effect, but we also discuss the normally neglected fact of energy non-conservation. The problem is treated from the point of view of Lagrangian and Hamiltonian mechanics.
The classical electromagnetic interaction for a charged particle with a long, constant-current solenoid is investigated in the context of classical electromagnetism. The conservation laws connected with energy, linear momentum, and angular momentum are first verified. Then the changes in the electromagnetic field quantities are evaluated explicitly, and the values are seen to have interesting connections with the solenoid vector potential. The calculations are of interest in connection with the Aharonov-Bohm effect involving the passage of electrons past a solenoid, and the classical expressions for energy and momentum are reminiscent of terms in the quantum-mechanical calculations.
"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."
The classical electromagnetic interaction of a point charge and a magnet is discussed by first calculating the interaction of a point charge with a simple model magnetic moment and then suggesting a multiparticle limit. The Darwin–Lagrangian is used to analyse the electromagnetic behaviour of the model magnetic moment (composed of two oppositely charged particles of different masses in an initially circular Coulomb orbit) interacting with a passing point charge. Considerations of force, energy, momentum and centre of energy are treated through second order in 1/ c . The changing magnetic moment is found to put a force back on a passing charge; this force is of order 1/ c 2 and depends upon the magnitude of the magnetic moment. The limit of a many-particle magnet arranged as a toroid is discussed. It is suggested that in the multiparticle limit, the electric fields of the passing charge are screened out of the body of the magnet while the magnetic fields of the passing charge penetrate into the body of the magnet. This is consistent with our understanding of the penetration of electromagnetic velocity fields into ohmic conductors. The proposed multiparticle limit is consistent with the conservation laws for energy and momentum, as well as constant motion of the centre of energy, and Newton's third law for the net Lorentz forces on the magnet and on the point charge. The work corresponds to a classical electromagnetic analysis of the interaction which is basic to understanding the controversy over the Aharonov–Bohm and Aharonov–Casher phase shifts and represents a refutation of the suggestions of Aharonov, Pearle and Vaidman.
A new classical electromagnetic analysis is presented suggesting that the Aharonov-Bohm phase shift is overwhelmingly likely to arise from a classical lag effect based upon classical electromagnetic forces. The analysis makes use of several aspects of classical electromagnetic theory which are unfamiliar to most physicists, including the Darwin Lagrangian, acceleration-based electric fields, internal electromagnetic momentum in a magnet, and a magnet model involving at least three mutually-interacting particles. Only when the acceleration-based electric forces acting on the passing charge are included do we find consistency with all the relativistic conservation laws: energy, linear momentum, angular momentum, and constant center-of-mass velocity. The electric forces on the passing charge lead to a lag effect which accounts quantitatively for the Aharonov-Bohm phase shift. Thus the classical analysis strongly suggests that the Aharonov-Bohm phase shift (observed when electrons pass a long solenoid which corresponds to a line of magnetic dipoles) is the analogue of the Matteucci-Pozzi phase shift (observed when electrons pass a line of electric dipoles). The classical electromagnetic analysis suggests experiments to distinguish the proposed classical-based lag effect from the presently accepted view that the Aharonov-Bohm phase shift is a quantum topological effect arising from magnetic fluxes in the absence of classical electromagnetic forces.
We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase shift involving three entities: the electron, the charges constituting the solenoid current, and the vector potential. The usual calculation supposes that the solenoid's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized electron moving in this vector potential. Recently, Vaidman presented a semiclassical calculation [L. Vaidman, Phys. Rev. A 86, 040101 (2012)], later confirmed by a fully quantum calculation of Pearle and Rizzi [preceding paper, Phys. Rev. A 95, 052123 (2017)], where it is supposed that the electron's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized solenoid charges moving in this vector potential. Here we present a third calculation, which supposes that the electron and solenoid currents may be well approximated as classical sources. The AB phase shift is then shown to be acquired by the quantized vector potential. We next show these are three equivalent alternative ways of calculating the AB shift. We consider the exact problem where all three entities are quantized. We approximate the wave function as the product of three wave functions: a vector potential wave function, an electron wave function, and a solenoid wave function. We apply the variational principle for the exact Schr\"odinger equation to this approximate form of solution. This leads to three Schr\"odinger equations, one each for vector potential, electron, and solenoid, each with classical sources for the other two entities. However, each Schr\"odinger equation contains an additional real ccc-number term, the time derivative of an extra phase. We show that these extra phases are such that the phase of the total wave function produces the AB shift. Since none of the three entities requires different treatment from any of the others, this leads to three alternative views of the physical cause of the AB magnetic effect.