Classical Electromagnetic Interaction of a Charge with a Solenoid or Toroid

TL;DR

This paper provides a classical electromagnetic analysis of the interaction between a charged particle and a solenoid or toroid, highlighting the Lorentz force effects and comparing classical predictions with quantum results, especially regarding the Aharonov-Bohm effect.

Contribution

It offers a detailed classical electromagnetic perspective on charge-solenoid interactions using the Darwin Lagrangian, clarifying force directions and deflections without quantum assumptions.

Findings

Classical analysis predicts charge deflections matching quantum results in magnitude.

The Lorentz force on the solenoid or toroid is analyzed in different reference frames.

Deflection directions are unambiguous and independent of Planck's constant.

Abstract

The Aharonov-Bohm phase shift in a particle interference pattern when electrons pass a long solenoid is identical in form with the optical interference pattern shift when a piece of retarding glass is introduced into one path of a two-beam optical interference pattern. The particle interference-pattern deflection is a relativistic effect of order 1/c^2, though this relativity aspect is rarely mentioned in the literature. Here we give a thorough analysis of the classical electromagnetic aspects of the interaction between a solenoid or toroid and a charged particle. We point out the magnetic Lorentz force which the solenoid or toroid experiences due to a passing charge. Although analysis in the rest frame of the solenoid or toroid will involve back Faraday fields on the charge, the analysis in the inertial frame in which the charge is initially at rest involves forces due to only electric fields where forces are equal in magnitude and opposite in direction. The classical analysis is made using the Darwin Lagrangian. We point out that the classical analysis suggests an angular deflection independent of Planck's constant where the deflection magnitude is identical with that given by the traditional quantum analysis, but where the deflection direction is unambiguous.