[Paper Review] Classical Electromagnetic Interaction of a Point Charge and a Magnetic Moment: Considerations Related to the Aharonov-Bohm Phase Shift

Timothy H. Boyer|ArXiv.org|Jul 4, 2001

Quantum and Classical Electrodynamics29 references3 citations

TL;DR

This paper proposes a relativistic classical model explaining the Aharonov-Bohm and Aharonov-Casher phase shifts via electromagnetic forces, showing that a moving point charge induces a $1/c^2$-order electric force on the charge due to a changing magnetic moment in a relativistic hydrogen-atom-based magnetic dipole model. The key result is that Lorentz forces on the charge and magnet are equal and opposite, restoring Newton's second law and classically explaining the phase shifts without invoking quantum topology.

ABSTRACT

A fundamentally new understanding of the classical electromagetic interaction of a point charge and a magnetic moment through order second order in 1/c is suggested. This relativistic analysis connects together hidden momentum in magnets, Solem's strange polarization of the classical hydrogen atom, and the Aharonov-Bohm Phase shift. We use a relativistic magnetic moment model consisting of many superimposed classical hydrogen atoms interacting through the Darwin Lagrangian with an external charge but not with each other. The analysis of Solem regarding the strange polarization of the classical hydrogen atom is seen to give a fundamentally different mechanism for the electric field of the passing charge to change the magnetic moment. The changing magnetic moment leads to an electric force back at the point charge which i)is second order in 1/c, ii)depends upon the magnetic dipole moment, changing sign with the dipole moment, iii)is odd in the charge q of the passing charge, and iv)reverses sign for charges passing on opposite sides of the magnetic moment. We suggest that a realistic multi-particle magnetic moment involves a changing magnetic moment which keeps the electromagnetic field momentum constant. This means also that the magnetic moment does not allow a significant shift in its internal center of energy. This criterion also implies that the Lorentz forces on the charged particle and on the point charge are equal and opposite and that the center of energy of each moves according to Newton's second law where the force is exactly the Lorentz force. Finally, we note that the results and suggestion given here are precisely what are needed to explain both the Aharonov-Bohm phase shift and the Aharonov-Casher phase shift as arising from classical electromagnetic forces.

Motivation & Objective

• To resolve long-standing ambiguities in the classical electromagnetic interaction between a point charge and a magnetic dipole moment, particularly regarding momentum and force balance.
• To address the missing role of the electric field in altering the magnetic moment of a relativistic magnetic dipole, which affects the force back on the passing charge.
• To provide a classical explanation for the Aharonov-Bohm and Aharonov-Casher phase shifts by showing they arise from electromagnetic forces, not quantum topology.
• To restore the semiclassical connection between classical and quantum physics by demonstrating that Lorentz forces on both charge and magnet are equal and opposite under a new criterion of constant electromagnetic field momentum.

Proposed method

• Uses a non-relativistic rigid-ring model to show negligible $1/c^4$ back-force, proving its inadequacy for $1/c^2$ analysis.
• Applies a relativistic model of superimposed classical hydrogen atoms (and anti-atoms) interacting via the Darwin Lagrangian with an external charge, not with each other.
• Incorporates Solem’s analysis of the 'strange polarization' of the classical hydrogen atom to show that an external electric field induces a change in the magnetic moment.
• Derives the resulting electric force on the passing charge as $\sim q^3 e_b^3 \omega / c^2$, which is $1/c^2$-order and odd in $q$, reversing sign on opposite sides of the magnet.
• Introduces a criterion that a realistic magnet must maintain constant electromagnetic field momentum, implying controlled magnetic moment changes to preserve force balance.
• Uses analogy with conductors to argue that a real magnet adjusts its internal dynamics to prevent net change in field momentum, ensuring Newton’s second law holds for both objects via Lorentz forces.

Experimental results

Research questions

• RQ1 Why has the classical electromagnetic interaction between a point charge and a magnetic dipole moment been poorly understood despite its foundational role in electromagnetism?
• RQ2 What is the role of the electric field from a passing charge in altering the magnetic moment of a relativistic magnetic dipole, and how does this affect the force on the charge?
• RQ3 How can the Aharonov-Bohm and Aharonov-Casher phase shifts be explained classically through electromagnetic forces rather than as purely quantum topological effects?
• RQ4 What conditions must a realistic magnetic dipole satisfy to ensure that the Lorentz forces on the point charge and the magnet are equal and opposite, preserving Newton’s second law?
• RQ5 How does the Darwin Lagrangian and the concept of hidden momentum in relativistic systems contribute to a consistent classical explanation of the phase shifts?

Key findings

• The rigid-ring model fails to produce a significant $1/c^2$-order force because the back-force from Faraday induction is $\sim 1/c^4$, making it irrelevant at this order.
• In the relativistic hydrogen-atom model, Solem’s 'strange polarization' leads to a $1/c^2$-order change in magnetic moment due to the external charge’s electric field.
• The resulting force on the passing charge is of order $q^3 e_b^3 \omega / c^2$, odd in $q$, linear in the magnetic moment, and reverses sign on opposite sides of the magnet.
• This force mechanism satisfies all qualitative requirements for a classical explanation of the Aharonov-Bohm phase shift.
• The criterion that the electromagnetic field momentum remains constant during interaction ensures equal and opposite Lorentz forces on the charge and magnet, restoring Newton’s second law.
• The model reinstates the semiclassical connection between classical and quantum physics by showing the phase shifts arise from classical electromagnetic lag effects, not from non-local quantum topology.