Skip to main content
SpringerLink
Log in

Classical Electromagnetic Interaction of a Charge with a Solenoid or Toroid

  • Published:
Foundations of Physics Aims and scope Submit manuscript

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 \(\hbar \), where the deflection magnitude is identical with that given by the traditional quantum analysis, but where the deflection direction is unambiguous.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

All the data involved in this manuscript are included within the manuscript.

References

  1. Moellenstedt, G., Bayh, W.: Messung der kontinuierlichen phasenschiebung von Elecktronenwellen in kraftfeldfreien Raum durch das magnetische Vektorpotential einer Luftspule. Naturwissenschaften 49, 81–82 (1962)

    ADS  Google Scholar 

  2. Bayh, W.: Messung der kontinuierlichen Phasenschiebung von Elektronenwellen im kraftfeldfreien Raum durch das magnetische Vektorpotential einer Wolfram-Wendel. Z. Phys. 169, 492–510 (1962)

    ADS  Google Scholar 

  3. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)

    ADS  MathSciNet  MATH  Google Scholar 

  4. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, Vol. II, Sect. 15-5. Addison-Wesley, Reading (1964)

  5. Shadowitz, A.: The Electromagnetic Field, pp. 197, 208–209, 517–522. Dover, New York (1988)

    Google Scholar 

  6. Garg, A.: Classical Electromagnetism in a Nutshell, pp. 107–108. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  7. Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn., pp. 384–391. Pearson Prentice Hall, Upper Saddle River (2005)

    Google Scholar 

  8. Balentine, L.E.: Quantum Mechanics, pp. 220–223. Prentice Hall, Englewood Cliffs (1990)

    Google Scholar 

  9. Boyer, T.H.: Concerning classical forces, energies, and potentials for accelerated point charges. Am. J. Phys. 91, 74–78 (2023)

    ADS  Google Scholar 

  10. Boyer, T.H.: The classical Aharonov-Bohm interaction as a relativity paradox. Eur. J. Phys. 44, 035202 (2023)

    Google Scholar 

  11. Boyer, T.H.: Concerning the direction of the Aharonov–Bohm deflection, submitted for publication

  12. Jackson, J.D.: Classical Electrodynamics, 2nd edn., pp. 593–595. John Wiley & Sons, New York (1975)

    MATH  Google Scholar 

  13. Boyer, T.H.: Classical electromagnetic deflections and lag effects associated with quantum interference pattern shifts: considerations related to the Aharonov–Bohm effect. Phys. Rev. D 8, 1679–1693 (1973)

    ADS  Google Scholar 

  14. These acceleration terms appear in the work of Page, L., Adams, N.I.: Electrodynamics. Van Nostrand, New York (1940), p. 175; Action and reaction between moving charges. Am. J. Phys. 13, 141–147 (1945)

  15. Boyer, T.H.: The Aharonov–Bohm effect as a classical electromagnetic lag effect: an electrostatic analogue and possible experimental test. Il Nuovo Cimento 100, 685–701 (1987)

    MathSciNet  Google Scholar 

  16. Jackson, J.D.: Classical Electrodynamics, p. 389. John Wiley & Sons, New York (1962)

    Google Scholar 

  17. Griffiths, D.J.: Introduction to Electrodynamics, 4th edn., pp. 571–572. Pearson, New York (2013)

    Google Scholar 

  18. Boyer, T.H.: Misinterpretation of the Aharonov–Bohm Effect. Am. J. Phys. 40, 56–59 (1972)

    ADS  Google Scholar 

  19. Aharonov, Y., Casher, A.: Topological quantum effects for neutral particles. Phys. Rev. Lett. 53, 319–321 (1984)

    ADS  MathSciNet  Google Scholar 

  20. Shockley, W., James, R.P.: Try simplest cases discovery of hidden momentum forces on magnetic currents. Phys. Rev. Lett. 18, 876–879 (1967)

    ADS  Google Scholar 

  21. Griffiths, D.J.: Introduction to Electrodynamics, 3rd edn., pp. 357, 361, 520–521. Prentice-Hall, Upper Saddle River (1999)

    Google Scholar 

  22. Jackson, J.D.: Classical Electrodynamics, 3rd edn., pp. 189, 618. John Wiley & Sons, New York (1999)

    Google Scholar 

  23. Zangwill, A.: Modern Electrodynamics, pp. 521–522. Cambridge University Press (2013)

    MATH  Google Scholar 

  24. Mansuripur, M.: Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation. Phys. Rev. Lett. 108, 193901 (2012)

    ADS  Google Scholar 

  25. Chambers, R.G.: A shift of an electron interference pattern by enclosed magnetic flux. Phys. Rev. Lett. 5, 3–5 (1960)

    ADS  Google Scholar 

  26. Olariu, S., Iovitzu Popescu, I.: The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57, 339–436 (1985)

    ADS  Google Scholar 

  27. Batelaan, H., Tonomura, A.: The Aharonov–Bohm effect: variations on a subtle theme.’ Phys. Today 38-43 (September 2009)

  28. Liebowitz, B.: Significance of the Aharonov-Bohm effect. Nuovo Cim. 38, 932–950 (1965)

    Google Scholar 

  29. Trammel, G.T.: Aharonov–Bohm paradox. Phys. Rev. 134, B1183 (1964)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Boyer, T.H.: Classical electromagnetic interaction of a charged particle with a constant-current solenoid. Phys. Rev. D 8, 1667–1679 (1973)

    ADS  Google Scholar 

  31. Boyer, T.H.: Penetration of electromagnetic velocity fields through a conducting wall of finite thickness. Phys. Rev. E 53, 6450–6459 (1996)

    ADS  Google Scholar 

  32. Boyer, T.H.: Understanding the penetration of electromagnetic velocity fields into conductors. Am. J. Phys. 67, 954–958 (1999)

    ADS  Google Scholar 

  33. Boyer, T.H.: Does the Aharonov–Bohm effect exist? Found. Phys. 30, 893–905 (2000)

    MathSciNet  Google Scholar 

  34. Boyer, T.H.: Classical electromagnetism and the Aharonov–Bohm phase shift. Found. Phys. 30, 907–932 (2000)

    MathSciNet  Google Scholar 

  35. Matteucci, G., Pozzi, G.: New diffraction experiment on the electrostatic Aharonov–Bohm effect. Phys. Rev. Lett. 54, 2469–2470 (1985)

    ADS  Google Scholar 

  36. Boyer, T.H.: 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. J. Phys. A:Math. Gen. 39, 3455–3477 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  37. Caprez, A., Barwick, B., Batelaan, H.: A macroscopic test of the Aharonov–Bohm effect. Phys. Rev. Letters 99, 210401(1–4) (2007)

    ADS  Google Scholar 

  38. Boyer, T.H.: Proposed experimental test for the paradoxical forces associated with the Aharonov–Bohm phase shift. Found. Phys. Lett. 19, 491–498 (2006)

    MATH  Google Scholar 

  39. Coleman, S., Van Vleck, J.H.: Origin of hidden momentum forces on magnets. Phys. Rev. 171, 1370–1375 (1968)

    ADS  Google Scholar 

  40. Boyer, T.H.: Classical interaction of a magnet and a point charge: the Shockley–James paradox. Phys. Rev. E 91(11), 013201 (2015)

    ADS  MathSciNet  Google Scholar 

  41. Boyer, T.H.: Interaction of a magnet and a point charge: unrecognized internal electromagnetic momentum. Am. J. Phys. 83, 433–442 (2015)

    ADS  Google Scholar 

  42. Cimmino, A., Opat, G.I., Klein, A.G., Kaiser, H., Werner, S.A., Arif, M., Clothier, R.: Observation of the topological Aharonov–Casher phase shift by neutron interferometry. Phys. Rev. Lett. 63, 380–383 (1989)

    ADS  Google Scholar 

  43. Boyer, T.H.: Proposed Aharonov–Casher effect: another example of an Aharonov–Bohm effect arising from a classical lag. Phys. Rev. A 36, 5083–5086 (1987)

    ADS  Google Scholar 

  44. Aharonov, Y., Pearle, P., Vaidman, L.: Comment on proposed Aharonov–Casher effect: another example of an Aharonov–Bohm effect arising from a classical lag. Phys. Rev. 115, 485–491 (1988)

    Google Scholar 

  45. Boyer, T.H.: Examples and comments related to relativity controversies. Am. J. Phys. 80, 962–971 (2012)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, City College of the City University of New York, New York, New York, 10031, USA

    Timothy H. Boyer

Authors
  1. Timothy H. Boyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The sole author of this manuscript made all contributions to it.

Corresponding author

Correspondence to Timothy H. Boyer.

Ethics declarations

Competing interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boyer, T.H. Classical Electromagnetic Interaction of a Charge with a Solenoid or Toroid. Found Phys 53, 71 (2023). https://doi.org/10.1007/s10701-023-00712-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-023-00712-y

Keywords

Search

Navigation