volume | PIER Journals

Abstract

The equation of motion for a test particle moving in given fixed external fields is analyzed and compared to the corresponding equation of motion derived from the Darwin Lagrangian for a system of interacting charged particles. The two approaches agree as long as the part of the electric field that arises from the partial time derivative of the vector potential is taken into account. It is, however, only via the Darwin approach that the origin of this field can be understood as arising from a breakdown of the test particle approximation. Applying the formalism to an electron moving outside a long solenoid results in a classical analog of the Aharonov-Bohm effect.

1. Aharonov, Y. and D. Bohm, "Signicance of the electromagnetic potentials in the quantum theory," Phys. Rev., Vol. 115, 485-491, 1959.
doi:10.1103/PhysRev.115.485

2. Ehrenberg, W. and R. E. Siday, "The refractive index in electron optics and the principles of dynamics," Proc. Phys. Soc. B, Vol. 62, 162-173, 1949.
doi:10.1088/0370-1301/62/1/303

3. Felsager, B., Geometry, Particles, and Fields, Springer, New York, 1998.
doi:10.1007/978-1-4612-0631-6

4. Shadowitz, A., The Electromagnetic Field, Dover, New York, 1988, First published 1975.

5. Griffiths, D. J., Introduction to Quantum Mechanics, 2nd Edition, Prentice Hall, New Jersey, 2005.

6. Chambers, R. G., "Shift of an electron interference pattern by enclosed magnetic flux," Phys. Rev. Lett., Vol. 5, 3-5, 1960.
doi:10.1103/PhysRevLett.5.3

7. Batelaan, H. and A. Tonomura, "The Aharonov-Bohm effects: Variations on a subtle theme," Physics Today, 38-43, Sep. 2009.
doi:10.1063/1.3226854

8. Olariu, S. and I. Iovitzu Popescu, "The quantum effects of electromagnetic fluxes," Rev. Mod. Phys., Vol. 57, 339-436, 1985.
doi:10.1103/RevModPhys.57.339

9. Ehrlichson, H., "Aharonov-Bohm effect --- quantum effects on charged particles in field-free regions," Am. J. Phys., Vol. 38, 162-173, 1970.
doi:10.1119/1.1976266

10. Essen, H. and J. C.-E. Sten, "The magnetic interaction energy between an infinite solenoid and a passing point charge," Progress In Electromagnetics Research M, Vol. 71, 145-156, 2018.
doi:10.2528/PIERM18052908

11. Trammel, G. T., "Aharonov-Bohm paradox," Phys. Rev., Vol. 134, B1183-1184, 1964.
doi:10.1103/PhysRev.134.B1183

12. Boyer, T. H., "Misinterpretation of the Aharonov-Bohm effect," Am. J. Phys., Vol. 40, 56-59, 1972.
doi:10.1119/1.1986446

13. Boyer, T. H., "Classical electromagnetic interaction of a charged particle with a constant-current solenoid," Phys. Rev. D, Vol. 8, 1667-1679, 1973.
doi:10.1103/PhysRevD.8.1667

14. 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, Vol. 8, 1679-1693, 1973.
doi:10.1103/PhysRevD.8.1679

15. Boyer, T. H., "Proposed experimental test for the paradoxical forces associated with the Aharonov-Bohm phase shift," Foundations of Physics Letters, Vol. 19, 491-498, 2006.
doi:10.1007/s10702-006-0907-7

16. Fearn, H. and K. Nguyen, "Derivation of the Aharonov-Bohm phase shift using classical forces,", E-print arXiv:1104.1449 [quant-ph], Apr. 2011.

17. Reiss, H. R., "Physical restrictions on the choice of electromagnetic gauge and their practical consequences," J. Phys. B: At. Mol. Opt. Phys., Vol. 50, 075003-1-7, 2017.
doi:10.1088/1361-6455/aa6375

18. Xiao, G., "An interpretation for Aharonov-Bohm effect with classical electromagnetic theory,", E-print arXiv:2201.12292 [physics.class-ph], Jan. 2022.

19. Boyer, T. H., "The classical Aharonov-Bohm interaction as a relativity paradox," Eur. J. Phys., Vol. 44, 035202-1-11, 2023.

20. Boyer, T. H., "Classical electromagnetic interaction of a charge with a solenoid or toroid,", E-print arXiv:2302.01936 [physics.class-ph], Feb. 2023.

21. Darwin, C. G., "The dynamical motions of charged particles," Philos. Mag. (UK), Vol. 39, 537-551, 1920.
doi:10.1080/14786440508636066

22. Kennedy, F. J., "Approximately relativistic interactions," Am. J. Phys., Vol. 40, 63-74, 1972.
doi:10.1119/1.1986448

23. Essen, H., "Darwin magnetic interaction energy and its macroscopic consequences," Phys. Rev. E, Vol. 53, 5228-5239, 1996.
doi:10.1103/PhysRevE.53.5228

24. Essen, H., "Magnetism of matter and phase-space energy of charged particle systems," J. Phys. A: Math. Gen., Vol. 32, 2297-2314, 1999.
doi:10.1088/0305-4470/32/12/005

25. 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., Vol. 39, 3455-3477, 2006.
doi:10.1088/0305-4470/39/13/021

26. Essen, H., "Magnetic energy, superconductivity, and dark matter," Progress in Physics, Vol. 16, 29-32, 2020.

27. Landau, L. D. and E. M. Lifshitz, The Classical Theory of Fields, 4th Edition, Pergamon, Oxford, 1975.

28. Jackson, J. D., Classical Electrodynamics, 3rd Edition, John Wiley & Sons, New York, 1999.

29. Panofsky, W. K. H. and M. Phillips, Classical Electricity and Magnetism, 2nd Edition, Dover, New York, 2005, First published 1962.

30. Choudhuri, A. R., Advanced Electromagnetic Theory, Springer Nature, Singapore, 2022.
doi:10.1007/978-981-19-5944-8

31. Comay, E., "The physical meaning of gauge transformations in electrodynamics,", E-print arXiv:physics/0611148 [physics.gen-ph], Nov. 2006.

32. Konopinski, E. J., "Electromagnetic Fields and Relativistic Particles," McGraw-Hill, New York, 1981.

33. Konopinski, E. J., "What the electromagnetic vector potential describes," Am. J. Phys., Vol. 46, 499-502, 1978.
doi:10.1119/1.11298

34. Landau, L. D. and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 3rd Edition, Pergamon, Oxford, 1977.

35. Schwinger, J., L. L. DeRaad, Jr., K. A. Milton, and W. Tsai, Classical Electrodynamics, Perseus books, Reading, Massachusetts, 1998.

36. Sharma, N. L., "Field versus action-at-a-distance in a static situation," Am. J. Phys., Vol. 56, 420-423, 1988.
doi:10.1119/1.15592

37. Feynman, R. P., R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II, Mainly Electromagnetism and Matter, Addison-Wesley, Reading, Massachusetts, denitive Edition, 2006.

38. McGregor, S., R. Hotovy, A. Caprez, and H. Batelaan, "On the relation between the Feynman paradox and Aharonov-Bohm effects," New Journal of Physics, Vol. 14, 093020-1-22, 2012.
doi:10.1088/1367-2630/14/9/093020

39. Stettner, R., "Conserved quantities and radiation effects for a closed system of charged particles," Ann. Phys. (N.Y.), Vol. 67, 238-251, 1971.
doi:10.1016/0003-4916(71)90011-X

40. Kittel, C., Introduction to Solid State Physics, 7th Edition, John Wiley & Sons, New York, 1996.

41. Essen, H., "An exact formula for the electromagnetic momentum in terms of the charge density and the Coulomb gauge vector potential," Eur. J. Phys., Vol. 39, 025202-1-9, 2018.

42. Redinz, J. A., "Magnetic energy in quasistatic systems," Eur. J. Phys., Vol. 44, 015202-1-14, 2023.
doi:10.1088/1361-6404/ac9ba5

43. Singal, A. K., "Wherein lies the momentum in Aharonov-Bohm quantum interference experiment --- A classical physics perspective,", E-print arXiv: arXiv:2301.06502 [quant-ph], Jan. 2023.

44. Primakoff, H. and T. Holstein, "Many-body interactions in atomic and nuclear systems," Phys. Rev., Vol. 55, 1218-1234, 1939.
doi:10.1103/PhysRev.55.1218

45. Page, L. and N. I. Adams, Electrodynamics, Van Nostrand, New York, 1940.

46. Podolsky, B. and K. S. Kunz, Fundamentals of Electrodynamics, Marcel Dekker, New York, 1969.

47. Boyer, T. H., "Concerning classical forces, energies, and potentials for accelerated point charges," Am. J. Phys., Vol. 91, 74-78, 2023.
doi:10.1119/5.0094457

48. Griffiths, D. J., Introduction to Electrodynamics, 3rd Edition, Prentice Hall, New Jersey, 1999.

Prof. Hanno Essén