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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Classical Electromagnetic Interaction of a Charged Particle with a Constant-Current Solenoid

Timothy H. Boyer

Published Sep 15, 1973• 59 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.8.1667

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•Mar 1, 2014•0 citations
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.
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.
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.
In the context of the Aharonov-Bohm effect, it is well-known that the local action of the vector potential or the non-local action of the magnetic field are properties attributable to quantum physics, which can explain the observed Aharonov-Bohm phase. Here we point out that, surprisingly, these properties are also attributable to classical physics and that this result follows from examining the electromagnetic angular momentum of a static electric charge outside an infinitely-long solenoid of finite radius carrying a uniform magnetic flux. This angular momentum may be expressed in three equivalent forms. The first form shows that there is an electromagnetic angular momentum even in regions for which the magnetic field vanishes but the potential does not, which stresses the physical nature of the vector potential in classical physics. The second form shows that there is an electromagnetic angular momentum produced by the non-local action of the magnetic field. The third form shows that the Aharonov-Bohm phase is proportional to the magnitude of the electromagnetic angular momentum, which sheds light on the physical meaning of this phase.
In the Aharonov–Bohm setup, a double-slit experiment, when a long but thin solenoid of current is introduced between the two coherent beams of electrons behind the slits, an extra phase difference between the interfering beams appears, as shown by a shift in the interference pattern. This mysterious effect purportedly arises owing to an electromagnetic momentum, attributed to the presence of a vector potential at the location of either beam, due to the solenoid of current even when the magnetic field is zero outside the solenoid. It has remained a puzzle, how mere potential, thought to be just a mathematical tool for calculating electromagnetic field, can give rise to electromagnetic momentum in a system. Experimentally, the effect has been amply verified, with hardly any doubts that the observed effect is real. A satisfactory physical explanation of the existence of momentum, at least under the aegis of classical electromagnetism, is still missing since inception of the idea more than half a century back. We show here the presence of electromagnetic momentum in the product of the drift velocities of the current-carrying charges within the solenoid and the mass equivalent of their potential energies in the electric field of the external charges.
The magnetic Aharonov-Bohm effect shows that charged particles may be affected by the vector potential in regions without any electric or magnetic fields [1]. The Aharonov-Bohm effect was experimentally confirmed [2-3] and has been found in many situations [4-6]. A common explanation is based on quantum mechanics, which states that the wavefunctions associated with the charges will accumulate a phase shift due to the vector potential. However, consensus about its nature and interpretation has not been achieved [7-14]. We here propose a simple but reasonable interpretation based on the theory for electromagnetic radiation and couplings [15]. The energy associated with a pulse radiator is divided into a Coulomb-velocity energy and a radiative energy, together with a macroscopic Schott energy accounting for the energy exchange between them. All these energies are expressed with terms including the potentials, so are the mutual coupling energies. There exists a force acting on the moving charges even though the fields completely vanish. This force makes the charges pass through the magnetic solenoid in different velocity with different path length, causing a phase shift the same as that obtained with quantum mechanics. The theory is originally aimed for providing an interpretation for electromagnetic radiation and mutual coupling. It is derived directly from the Maxwell theory with no modification but only substitution and reorganization.
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.