Hidden momentum and the electromagnetic mass of a charge and current carrying body

American Journal of Physics

Volumn 65, Issue 1, 1997, Pages 55-65

  Hnizdo, V ^a  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

Author keywords

[No Author keywords available]

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EID: 0031538225     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18789     Document Type: Article

References (53)

1. A. Pais, "The early history of the theory of the electron: 1897-1947," in Aspects of Quantum Theory, edited by A. Salam and E. P. Wigner (Cambridge U. P., Cambridge, 1972), pp. 79-93.
2. 1/2m v, see Pais, Ref. 1. An interesting episode in the early history of special relativity concerning experimental tests of the differing Abraham and Lorentz-Einstein predictions for the variation of the electron mass with velocity is described in great detail by J. T. Cushing, "Electromagnetic mass, relativity, and the Kaufmann experiments," Am. J. Phys. 49(12), 1113-1149 (1981).
3. See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), 2nd ed., Sec. 14.1.
4. 2.
5. See Pais, Ref. 1.
6. A relatively recent exchange is between T. H. Boyer, "Classical model of the electron and the definition of electromagnetic field momentum," Phys. Rev. D 25, 3246-3250 (1982); and F. Rohrlich, "Comment on preceding paper by T. H. Boyer," Ibid., 3251-3255 (1982).
7. A relatively recent exchange is between T. H. Boyer, "Classical model of the electron and the definition of electromagnetic field momentum," Phys. Rev. D 25, 3246-3250 (1982); and F. Rohrlich, "Comment on preceding paper by T. H. Boyer," Ibid., 3251-3255 (1982).
8. e in terms of the radius a [see our Eq. (3)], without affecting any of the points under the discussion.
9. Similar calculations, using the device of Lorentz transforming into the rest frame, will be carried out in more detail in Sec. II of this paper. A simple nonrelativistic calculation displaying the factor of 4/3 discrepancy has been given recently by P. Moylan, "An elementary account of the factor of 4/3 in the electromagnetic mass," Am. J. Phys. 63(9), 818-820 (1995).
10. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
11. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
12. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
13. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
14. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
15. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
16. The original papers of H. Poincaré, Co. R. Acad. Sci. Paris 140, 1504 (1905), and "Sur la dynamique de l'électron," Rend. Cire. Mat. Palermo 21, 129 (1906), are not easily accessible; however, there is an English, "modernized" presentation of the latter paper by H. M. Schwartz, "Poincaré's Rendiconti paper on relativity. Part I," Am. J. Phys. 39(11), 1287-1294 (1971); "Part II," ibid. 40(6), 862-872 (1972); and "Part III," ibid. 40(9), 1282-1287 (1972); an early discussion of his ideas is by H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952), 2nd ed., pp. 213-215; see also Jackson, Ref. 3, pp. 792-793.
17. See, for example, L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1975), 4th ed., Secs. 33 and 34.
18. An authoritative, detailed history of QED has appeared recently: S. S. Schweber, QED and the Men Who Made It (Princeton U.P., Princeton, NJ, 1994).
19. F. Rohrlich, "Self-energy and stability of the classical electron," Am. J. Phys. 28, 639-643 (1960); "Electromagnetic momentum, energy, and mass," ibid. 38(11), 1310-1316 (1970); Classical Charged Particles (Addison-Wesley, Reading, MA, 1965 and 1990).
20. F. Rohrlich, "Self-energy and stability of the classical electron," Am. J. Phys. 28, 639-643 (1960); "Electromagnetic momentum, energy, and mass," ibid. 38(11), 1310-1316 (1970); Classical Charged Particles (Addison-Wesley, Reading, MA, 1965 and 1990).
21. F. Rohrlich, "Self-energy and stability of the classical electron," Am. J. Phys. 28, 639-643 (1960); "Electromagnetic momentum, energy, and mass," ibid. 38(11), 1310-1316 (1970); Classical Charged Particles (Addison-Wesley, Reading, MA, 1965 and 1990).
22. E. Fermi, "Über einen Widerspruch zwischen der elektrodynamischen und der relativistischen Theorie der elektromagnetische Masse," Z. Phys. 24, 340-346 (1922); B. Kwal, "Les expressions de l'energie et l'impulsion du champ électromagnetique propre de l'electron en mouvement," J. Phys. Radium 10, 103-104 (1949).
23. E. Fermi, "Über einen Widerspruch zwischen der elektrodynamischen und der relativistischen Theorie der elektromagnetische Masse," Z. Phys. 24, 340-346 (1922); B. Kwal, "Les expressions de l'energie et l'impulsion du champ électromagnetique propre de l'electron en mouvement," J. Phys. Radium 10, 103-104 (1949).
24. μν).
25. E. J. Konopinski, Electromagnetic Fields and Relativistic Particles (McGraw-Hill, New York, 1981), pp. 432-436.
26. Rohrlich, Ref. 12; see also Jackson, Ref. 3, Sec. 17.5, and Konopinski, Ref. 15.
27. D. J. Griffiths and R. E. Owen, "Mass renormalization in classical electrodynamics," Am. J. Phys. 51(12), 1120-1126 (1983). These authors also consider, and analyze in detail on the example of a "dumbbell" configuration of charge, a dynamic mechanism for the generation of electromagnetic mass in the self-force on an accelerating extended charged body, i.e., in the force that arises from the retarded electromagnetic field of the body itself. In the classical electron theory, the inclusion of the self-force led to the Abraham-Lorentz equation of motion, which includes the radiative reaction on a charged particle, see Jackson, Ref. 3, Sec. 17.3.
28. This viewpoint has been expressed again very recently by K. McDonald, "Answer to Question #26," Am. J. Phys. 64(1), 15-16 (1996), in an answer to R. H. Romer, "Question #26. Electromagnetic field momentum," ibid. 63(9), 777-779 (1995), who posed the question whether the usual definition of electromagnetic field momentum, which may yield a nonzero momentum for a static field, is appropriate in the case of a field "bound" to its sources, i.e., a self-field. Romer, loc. cit. gives several references to the literature on the electromagnetic momentum of static fields.
29. This viewpoint has been expressed again very recently by K. McDonald, "Answer to Question #26," Am. J. Phys. 64(1), 15-16 (1996), in an answer to R. H. Romer, "Question #26. Electromagnetic field momentum," ibid. 63(9), 777-779 (1995), who posed the question whether the usual definition of electromagnetic field momentum, which may yield a nonzero momentum for a static field, is appropriate in the case of a field "bound" to its sources, i.e., a self-field. Romer, loc. cit. gives several references to the literature on the electromagnetic momentum of static fields.
30. W. Shockley and R. P. James, " 'Try simplest cases' discovery of 'hidden momentum' forces on 'magnetic currents,' " Phys. Rev. Lett. 18, 876-879 (1967); H. A. Haus and P. Penfield, "Force on a current loop," Phys. Lett. 26A, 412-413 (1968).
31. W. Shockley and R. P. James, " 'Try simplest cases' discovery of 'hidden momentum' forces on 'magnetic currents,' " Phys. Rev. Lett. 18, 876-879 (1967); H. A. Haus and P. Penfield, "Force on a current loop," Phys. Lett. 26A, 412-413 (1968).
32. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58(10), 978-983 (1990), and references therein; V. Hnizdo, "Hidden momentum of a relativistic fluid carrying current in an external electric field," Am. J. Phys. 65(1), 92-94 (1997).
33. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58(10), 978-983 (1990), and references therein; V. Hnizdo, "Hidden momentum of a relativistic fluid carrying current in an external electric field," Am. J. Phys. 65(1), 92-94 (1997).
34. S. Coleman and J. H. Van Vleck, "Origin of 'hidden momentum forces' on magnets," Phys. Rev. 171, 1370-1375 (1968); M. G. Calkin, "Linear momentum of the source of a static electromagnetic field," Am. J. Phys. 39(5), 513-516 (1971); Y. Aharanov, P. Pearle, and L. Vaidman, "Comment on 'Proposed Aharanov-Casher effect: Another example of Aharanov-Bohm effect arising from a classical lag,' " Phys. Rev. A 37, 4052-4055 (1988); Vaidman, Ref. 20; E. Comay, "Exposing 'hidden momentum,' " Am. J. Phys. 64, 1028-1034 (1996).
35. S. Coleman and J. H. Van Vleck, "Origin of 'hidden momentum forces' on magnets," Phys. Rev. 171, 1370-1375 (1968); M. G. Calkin, "Linear momentum of the source of a static electromagnetic field," Am. J. Phys. 39(5), 513-516 (1971); Y. Aharanov, P. Pearle, and L. Vaidman, "Comment on 'Proposed Aharanov-Casher effect: Another example of Aharanov-Bohm effect arising from a classical lag,' " Phys. Rev. A 37, 4052-4055 (1988); Vaidman, Ref. 20; E. Comay, "Exposing 'hidden momentum,' " Am. J. Phys. 64, 1028-1034 (1996).
36. S. Coleman and J. H. Van Vleck, "Origin of 'hidden momentum forces' on magnets," Phys. Rev. 171, 1370-1375 (1968); M. G. Calkin, "Linear momentum of the source of a static electromagnetic field," Am. J. Phys. 39(5), 513-516 (1971); Y. Aharanov, P. Pearle, and L. Vaidman, "Comment on 'Proposed Aharanov-Casher effect: Another example of Aharanov-Bohm effect arising from a classical lag,' " Phys. Rev. A 37, 4052-4055 (1988); Vaidman, Ref. 20; E. Comay, "Exposing 'hidden momentum,' " Am. J. Phys. 64, 1028-1034 (1996).
37. S. Coleman and J. H. Van Vleck, "Origin of 'hidden momentum forces' on magnets," Phys. Rev. 171, 1370-1375 (1968); M. G. Calkin, "Linear momentum of the source of a static electromagnetic field," Am. J. Phys. 39(5), 513-516 (1971); Y. Aharanov, P. Pearle, and L. Vaidman, "Comment on 'Proposed Aharanov-Casher effect: Another example of Aharanov-Bohm effect arising from a classical lag,' " Phys. Rev. A 37, 4052-4055 (1988); Vaidman, Ref. 20; E. Comay, "Exposing 'hidden momentum,' " Am. J. Phys. 64, 1028-1034 (1996).
38. S. Coleman and J. H. Van Vleck, "Origin of 'hidden momentum forces' on magnets," Phys. Rev. 171, 1370-1375 (1968); M. G. Calkin, "Linear momentum of the source of a static electromagnetic field," Am. J. Phys. 39(5), 513-516 (1971); Y. Aharanov, P. Pearle, and L. Vaidman, "Comment on 'Proposed Aharanov-Casher effect: Another example of Aharanov-Bohm effect arising from a classical lag,' " Phys. Rev. A 37, 4052-4055 (1988); Vaidman, Ref. 20; E. Comay, "Exposing 'hidden momentum,' " Am. J. Phys. 64, 1028-1034 (1996).
39. Vaidman, Ref. 20; V. Hnizdo, "Comment on 'Torque and force on a magnetic dipole,' " Am. J. Phys. 60(3), 279-280 (1992).
40. Vaidman, Ref. 20; V. Hnizdo, "Comment on 'Torque and force on a magnetic dipole,' " Am. J. Phys. 60(3), 279-280 (1992).
41. One consequence of hidden momentum is that the forces acting between a current-carrying body and a moving charged particle satisfy Newton's third law in the nonrelativistic limit of slow motion, see W. H. Furry, "Examples of momentum distributions in the electromagnetic field and in matter," Am. J. Phys. 37(6), 621-636 (1969), and V. Hnizdo, "Conservation of linear and angular momentum and the interaction of a moving charge with a magnetic dipole," ibid. 60(3), 242-246 (1992).
42. One consequence of hidden momentum is that the forces acting between a current-carrying body and a moving charged particle satisfy Newton's third law in the nonrelativistic limit of slow motion, see W. H. Furry, "Examples of momentum distributions in the electromagnetic field and in matter," Am. J. Phys. 37(6), 621-636 (1969), and V. Hnizdo, "Conservation of linear and angular momentum and the interaction of a moving charge with a magnetic dipole," ibid. 60(3), 242-246 (1992).
43. Romer, Ref. 18. The shells are assumed to be massive, i.e., not massless, of small but macroscopic thickness, and, if needed, with thin struts that would keep them fixed to the common center, independently from each other (the struts of the larger shell would pass through holes in the smaller shell); the charges/currents are on the outer surfaces of the shells and occupy layers of negligible thickness. The electric field is that of a charge distribution that would be induced on the surface of a conducting sphere by an external uniform electric field, but there is no external field here and the charges are assumed to be fastened on the shell; similarly, the currents are assumed to be "fastened" on the second shell, where the macroscopic charge density vanishes. The shells must be made, therefore, of a nonconducting material.
44. See, for example, Calkin, Ref. 21.
45. Another way to calculate this quantity is as the force on a surface element of charge due to the arithmetic mean of the electric fields inside and outside the surface of the shell.
46. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, Oxford, 1986), 3rd ed., Sec. 2.
47. See, for example, Jackson, Ref. 3, p. 552.
48. Landau and Lifshitz, Ref. 10, Sec. 35.
49. αβ the matrix of the Lorentz transformation.
50. Landau and Lifshitz, Ref. 10, Sec. 6, Problem 1.
51. e+m as the charge density, and hence also the electric-field force, are assumed to vanish on the surface of the current-carrying shell and so will make no contribution to the volume-averaged stress tensor in a surface integral like that of Eq. (26).
52. See, for example, Jackson, Ref. 3, p. 518.
53. Jackson, Ref. 3, pp. 791-796.