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arXiv:physics/0506181 · Journal of Physics A Mathematical and General

Darwin–Lagrangian analysis for the interaction of a point charge and a magnet: considerations related to the controversy regarding the Aharonov–Bohm and Aharonov–Casher phase shifts

Timothy H Boyer

Published Mar 15, 2006• 24 citations
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Abstract

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.

DOI
10.1088/0305-4470/39/13/021

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Darwin-Lagrangian Analysis for the Interaction of a Point Charge and a Magnet: Considerations Related to the Controversy Regarding the Aharonov-Bohm and Aharonov-Casher Phase Shifts

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The classical electromagnetic interaction of a point charge and a magnet is discussed by first calculating the interaction of point charge with a simple model magnetic moment and then suggesting a multiparticle limit. The Darwin Lagrangian is used to analyze the electromagnetic behavior of the model magnetic moment (composed of two oppositely charged particles of different mass in an initially circular orbit) interacting with a passing point charge. The changing mangetic 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. It is suggested that in the limit of a multiparticle magnetic toroid, the electric fields of the passing charge are screened out of the body of the magnet while the magnetic fields penetrate into the magnet. This is consistent with our understanding of the penetration of electromagnetic velocity fields into ohmic conductors. Conservation laws are discussed. 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.

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Physical Review A•Nov 1, 1987•110 citations
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.
The classical electromagnetic interaction of a point charge and a magnet is discussed by first calculating the interaction of point charge with a simple model magnetic moment and then suggesting a multiparticle limit. The Darwin Lagrangian is used to analyze the electromagnetic behavior of the model magnetic moment (composed of two oppositely charged particles of different mass in an initially circular orbit) interacting with a passing point charge. The changing mangetic 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. It is suggested that in the limit of a multiparticle magnetic toroid, the electric fields of the passing charge are screened out of the body of the magnet while the magnetic fields penetrate into the magnet. This is consistent with our understanding of the penetration of electromagnetic velocity fields into ohmic conductors. Conservation laws are discussed. 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.
Abstract The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov–Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism.
The magnetic Aharonov–Bohm (A–B) effect occurs when a point charge interacts with a line of magnetic flux, while its reciprocal, the Aharonov–Casher (A–C) effect, occurs when a magnetic moment interacts with a line of charge. For the two interacting parts of these physical systems, the equations of motion are discussed in this paper. The generally accepted claim is that both parts of these systems do not accelerate, while Boyer has claimed that both parts of these systems do accelerate. Using the Euler–Lagrange equations we predict that in the case of unconstrained motion, only one part of each system accelerates, while momentum remains conserved. This prediction requires a time-dependent electromagnetic momentum. For our analysis of unconstrained motion, the A–B effects are then examples of the Feynman paradox. In the case of constrained motion, the Euler–Lagrange equations give no forces, in agreement with the generally accepted analysis. The quantum mechanical A–B and A–C phase shifts are independent of the treatment of constraint. Nevertheless, experimental testing of the above ideas and further understanding of the A–B effects that are central to both quantum mechanics and electromagnetism could be possible.
The magnetic Aharonov–Bohm (A–B) effect occurs when a point charge interacts with a line of magnetic flux, while its reciprocal, the Aharonov–Casher (A–C) effect, occurs when a magnetic moment interacts with a line of charge. For the two interacting parts of these physical systems, the equations of motion are discussed in this paper. The generally accepted claim is that both parts of these systems do not accelerate, while Boyer has claimed that both parts of these systems do accelerate. Using the Euler–Lagrange equations we predict that in the case of unconstrained motion, only one part of each system accelerates, while momentum remains conserved. This prediction requires a time-dependent electromagnetic momentum. For our analysis of unconstrained motion, the A–B effects are then examples of the Feynman paradox. In the case of constrained motion, the Euler–Lagrange equations give no forces, in agreement with the generally accepted analysis. The quantum mechanical A–B and A–C phase shifts are independent of the treatment of constraint. Nevertheless, experimental testing of the above ideas and further understanding of the A–B effects that are central to both quantum mechanics and electromagnetism could be possible.
In 1984 Aharonov and Casher proposed that a beam of neutral particles with magnetic dipole moments passing around opposite sides of a line charge will undergo a relative quantum phase shift. This is currently under experimental investigation. Aharonov and Casher claimed that the magnetic dipole particles should undergo the phase shift despite experiencing no classical force. They suggested that this new effect has a ``duality'' with the solenoid Aharonov-Bohm effect where charged particles passing around a magnetic solenoid experience a phase shift despite, it is claimed, experiencing no classical force. Here it is pointed out that a magnetic dipole particle passing a line charge does indeed experience a classical electromagnetic force in the usual electric-current model for a magnetic dipole. This force will produce a relative lag between dipoles passing on opposite sides of the line charge, and the classical lag then leads to a quantum phase shift in exact agreement with that calculated by Aharonov and Casher. Thus actually the proposed Aharonov-Casher effect has a transparent explanation as a classical lag effect. It is emphasized that the solenoid Aharonov-Bohm effect is currently the only phase-shift effect which cannot be explained in an obvious fashion as arising from a classical lag and it is again proposed that also this shift may actually involve a classical electromagnetic lag effect. A natural experimental consequence suggested by the lag point of view is the breakdown of the interference pattern when the lag becomes comparable to the wave-packet coherence length.