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European Journal of Physics

In nature, there are fluctuating electric fields outside an infinite solenoid

Timothy H. Boyer

Published Feb 14, 2025• 1 citation
Publisher

Abstract

Abstract Because the relativistic transformation of time involves spatial coordinates, multipole moments can take on very different values in relatively-moving inertial frames. As a related aspect, an infinite solenoid of finite cross-section will have non-zero time-varying electric fields outside its winding arising from the discrete charges carrying the solenoid currents. In the rest frame of the solenoid, the time averages of these time variations vanish and may be ignored. However, these time-varying electric fields have non-zero time averages in an inertial frame where the solenoid is moving perpendicular to its axis, and this non-zero time average plays an important role in the classical electromagnetic interpretation of the Aharonov–Bohm phase shift, which was treated in earlier work. Here we illustrate these ideas for a circular current loop, and then note their applicability to solenoids and toroids. The ‘ideal magnetic moment’ with no charges and no spatial extent is completely unphysical, and is seen to lead, in a moving inertial frame, to a different and incorrect average electric field from that where discrete charges are used and the radius of the magnetic moment is finite.

DOI
10.1088/1361-6404/adb660

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Abstract Because the relativistic transformation of time involves spatial coordinates, multipole moments can take on very different values in relatively-moving inertial frames. As a related aspect, an infinite solenoid of finite cross-section will have non-zero time-varying electric fields outside its winding arising from the discrete charges carrying the solenoid currents. In the rest frame of the solenoid, the time averages of these time variations vanish and may be ignored. However, these time-varying electric fields have non-zero time averages in an inertial frame where the solenoid is moving perpendicular to its axis, and this non-zero time average plays an important role in the classical electromagnetic interpretation of the Aharonov–Bohm phase shift, which was treated in earlier work. Here we illustrate these ideas for a circular current loop, and then note their applicability to solenoids and toroids. The ‘ideal magnetic moment’ with no charges and no spatial extent is completely unphysical, and is seen to lead, in a moving inertial frame, to a different and incorrect average electric field from that where discrete charges are used and the radius of the magnetic moment is finite.
"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."
We consider a quantum charged particle moving in the x y plane under the action of a time-dependent magnetic field described by means of the linear vector potential of the form A = B ( t ) − y ( 1 + β ) , x ( 1 − β ) / 2 . Such potentials with β ≠ 0 exist inside infinite solenoids with non-circular cross sections. The systems with different values of β are not equivalent for nonstationary magnetic fields or time-dependent parameters β ( t ) , due to different structures of induced electric fields. Using the approximation of the stepwise variations of parameters, we obtain explicit formulas describing the change of the mean energy and magnetic moment. The generation of squeezing with respect to the relative and guiding center coordinates is also studied. The change of magnetic moment can be twice bigger for the Landau gauge than for the circular gauge, and this change can happen without any change of the angular momentum. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields.
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
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.
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.
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.