The European Physical Journal H

, Volume 39, Issue 3, pp 283–302 | Cite as

Classical formulations of the electromagnetic self-force of extended charged bodies

  • P.W. Smorenburg
  • L.P.J. Kamp
  • O.J. Luiten


Several formulations of the classical electrodynamics of charged particles, as have been developed in the course of the twentieth century, are compared. The mathematical equivalence of the various dissimilar expressions for the electromagnetic self-force is demonstrated explicitly by deriving these expressions directly from one another. The new connections that are established present the previously published results on a common basis, thereby contributing to a coherent historical picture of the development of charged particle models.


Casimir Force Classical Electrodynamic Radiation Reaction Coulomb Gauge Generalize Hypergeometric Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, M. 1902. Principien der Dynamik des Elektrons. Ann. Phys. (Leipzig) 315: 105-179CrossRefADSGoogle Scholar
  2. 2.
    Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover, New YorkGoogle Scholar
  3. 3.
    Ackerhalt, J.R., P.L. Knight and J.R. Eberly. 1973. Radiation reaction and radiative frequency-shifts. Phys. Rev. Lett. 30: 456-460CrossRefADSGoogle Scholar
  4. 4.
    Aguirregabiria, J.M., J. Llosa and A. Molina. 2006. Motion of a classical particle. Phys. Rev. D 73: 125015MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Ares de Parga, G. 2006. A physical deduction of an equivalent Landau-Lifshitz equation of motion in classical electrodynamics. A new expression for the large distance radiation rate of energy. Found. Phys. 36: 1474-1510MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. 6.
    Barut, A.O. and J.P. Dowling. 1987. Quantum electrodynamics based on self-energy: spontaneous emission in cavities. Phys. Rev. A 36: 649-654CrossRefADSGoogle Scholar
  7. 7.
    Barut, A.O. and J.P. Dowling. 1989. QED based on self-fields: a relativistic calculation of g-2. Z. Naturforsch. A 44: 1051-1055Google Scholar
  8. 8.
    Barut, A.O., J. Kraus, Y. Salamin and N. Ünal. 1992. Relativistic theory of the Lamb shift in self-field quantum electrodynamics. Phys. Rev. A 45: 7740-7745CrossRefADSGoogle Scholar
  9. 9.
    Bohm, D. and M. Weinstein. 1948. The self-oscillations of a charged particle. Phys. Rev. 74: 1789-1798CrossRefzbMATHADSGoogle Scholar
  10. 10.
    Born, M. 1909. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. Ann. Phys. (Leipzig) 335: 1-56CrossRefADSGoogle Scholar
  11. 11.
    Bosanac, S.D. 2001. General classical solution for the dynamics of charges with radiation reaction. J. Phys. A 34: 473-490MathSciNetCrossRefzbMATHADSGoogle Scholar
  12. 12.
    Boyer, T.H. 1968. Quantum electromagnetic zero-point energy of a conducting spherical shell and the Casimir model for a charged particle. Phys. Rev. 174: 1764-1776CrossRefADSGoogle Scholar
  13. 13.
    Casimir, H.B.G. 1953. Introductory remarks on quantum electrodynamics. Physica 19: 846-849CrossRefADSGoogle Scholar
  14. 14.
    Compagno, G. and F. Persico. 2002. Self-dressing and radiation reaction in classical electrodynamics. J. Phys. A 35: 3629-3645MathSciNetCrossRefzbMATHADSGoogle Scholar
  15. 15.
    Dirac, P.A.M. 1938. Classical theory of radiating electrons. Proc. R. Soc. London A 167: 148-169CrossRefADSGoogle Scholar
  16. 16.
    Epp, R.J., R.B. Mann and P.L. McGrath. 2009. Rigid motion revisited: rigid quasilocal frames. Classical Quant. Grav. 26: 035015MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Erber, T. 1961. The classical theories of radiation reaction. Fortschr. Phys. 9: 343-392MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fermi, E. 1921. Sulla dinamica di un sistema rigido di cariche elettriche in moto traslatorio. Nuovo Cimento 22: 199-207CrossRefGoogle Scholar
  19. 19.
    Fermi, E. 1923. Correzione di una contraddizione tra la teoria elettrodinamica e quella relativistica delle masse elettromagnetiche. Nuovo Cimento 25: 159-170CrossRefGoogle Scholar
  20. 20.
    Fermi, E. 1927. Sul meccanismo dell’emissione nella meccanica ondulatoria. Rend. Lincei 5: 795-800zbMATHGoogle Scholar
  21. 21.
    Ford, G.W. and R.F. O’Connell. 1991. Radiation reaction in electrodynamics and the elimination of runaway solutions. Phys. Lett. A 157: 217-220CrossRefADSGoogle Scholar
  22. 22.
    Gill, T.L., W.W. Zachary and J. Lindesay. 2001. The classical electron problem. Found. Phys. 31: 1299-1355MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gralla, S.E., A.I. Harte and R.M. Wald. 2009. Rigourous derivation of electromagnetic self-force. Phys. Rev. D 80: 024031CrossRefADSGoogle Scholar
  24. 24.
    Hammond, R.T. 2013. Electrodynamics and radiation reaction. Found. Phys. 43: 201-209MathSciNetCrossRefzbMATHADSGoogle Scholar
  25. 25.
    Hansen, E.R. 1975. A Table of Series and Products. Prentice-Hall, Englewood Cliffs. pp. 5 and 124Google Scholar
  26. 26.
    Herglotz, G. 1903. Zur Elektronentheorie. Nachr. Ges. Wiss. Göttingen 1903: 357-382zbMATHGoogle Scholar
  27. 27.
    Hnizdo, V. 2000. The electromagnetic self-force on a charged spherical body slowly undergoing a small, temporary displacement from a position of rest. J. Phys. A 33: 4095-4103MathSciNetCrossRefzbMATHADSGoogle Scholar
  28. 28.
    Jackson, J.D. 1999. Classical Electrodynamics, 3rd edn. Wiley, New YorkGoogle Scholar
  29. 29.
    Janssen, M. and M. Mecklenburg. 2006. From classical to relativistic mechanics: electromagnetic models of the electron. In: V.F. Hendricks, K.F. Jørgenson, J. Lützen and S.A. Pedersen (eds.) Interactions: Mathematics, Physics and Philosophy, 1860-1930, Springer, Dordrecht, pp. 65-134Google Scholar
  30. 30.
    Jiménez, J.L. and I. Campos. 1999. Models of the classical electron after a century. Found. Phys. Lett. 12: 127-146CrossRefGoogle Scholar
  31. 31.
    Kholmetskii, A.L. 2006. On “gauge renormalization” in classical electrodynamics. Found. Phys. 36: 715-744MathSciNetCrossRefzbMATHADSGoogle Scholar
  32. 32.
    Landau, L.D. and E.M. Lifshitz. 1975. Classical Theory of Fields, 4th rev. edn. Pergamon, Oxford. Section 76Google Scholar
  33. 33.
    Leonardt, U. and W.M.R. Simpson. 2011. Exact solution for the Casimir stress in a spherically symmetric medium. Phys. Rev. D 84: 081701(R)CrossRefADSGoogle Scholar
  34. 34.
    Lorentz, H.A. 1916. The Theory of Electrons, 2nd edn. Teubner, LeipzigGoogle Scholar
  35. 35.
    Luke, Y.L. 1962. Integrals of Bessel Functions. McGraw-Hill, London. p. 28Google Scholar
  36. 36.
    Lyle, S.N. 2010. Self-Force and Inertia. Springer, Berlin. Chap. 12Google Scholar
  37. 37.
    Martins, A.A. and M.J. Pinheiro. 2008. On the electromagnetic origin of inertia and inertial mass. Int. J. Theor. Phys. 47: 2706-2715MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Medina, R. 2006. Radiation reaction of a classical quasi-rigid extended particle. J. Phys. A 39: 3801-3816MathSciNetCrossRefzbMATHADSGoogle Scholar
  39. 39.
    Milonni, P.W., J.R. Ackerhalt and W.A. Smith. 1973. Interpretation of radiative corrections in spontaneous emission. Phys. Rev. Lett. 31: 958-960CrossRefADSGoogle Scholar
  40. 40.
    Milton, K.A. 1980. Semiclassical electron models: Casimir self-stress in dielectric and conducting balls. Ann. Phys. (New York) 127: 49-61MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Milton, K.A., L.L. DeRaad Jr., and J. Schwinger. 1978. Casimir self-stress on a perfectly conducting spherical shell. Ann. Phys. (New York) 115: 388-403MathSciNetCrossRefADSGoogle Scholar
  42. 42.
    Moniz, E.J. and D.H. Sharp. 1977. Radiation reaction in nonrelativistic quantum mechanics. Phys. Rev. D 15: 2850-2865CrossRefADSGoogle Scholar
  43. 43.
    Morse, P.M. and H. Feshbach. 1953. Methods of Theoretical Physics. McGraw-Hill, New York. Part 2, p. 1255Google Scholar
  44. 44.
    Noja, D. and A. Posilicano. 1999. On the point limit of the Pauli-Fierz model. Ann. Inst. Henri Poincaré A 71: 425-457MathSciNetzbMATHGoogle Scholar
  45. 45.
    Oliver, M.A. 1998. Classical electrodynamics of a point particle. Found. Phys. Lett. 11: 61-82MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ori, A. and E. Rosenthal. 2004. Calculation of the self force using the extended-object approach. J. Math. Phys. 45: 2347-2364MathSciNetCrossRefzbMATHADSGoogle Scholar
  47. 47.
    Panofsky, W.K.H. and M. Phillips. 2005. Classical Electricity and Magnetism, 2nd edn. Dover, MineolaGoogle Scholar
  48. 48.
    Pierce, E. 2007. The lock and key paradox and the limits of rigidity in special relativity. Am. J. Phys. 75: 610-614CrossRefADSGoogle Scholar
  49. 49.
    Poincaré, M.H. 1906. On the dynamics of the electron. Rend. Circ. Mat. Palermo 21: 129-176CrossRefzbMATHGoogle Scholar
  50. 50.
    Prigogine, I. and F. Henin. 1962. Motion of a relativistic charged particle. Physica 28: 667-688MathSciNetCrossRefzbMATHADSGoogle Scholar
  51. 51.
    Puthoff, H.E. 2007. Casimir vacuum energy and the semiclassical electron. Int. J. Theor. Phys. 46: 3005-3008CrossRefzbMATHGoogle Scholar
  52. 52.
    Roa-Neri, J.A.E. and J.L. Jiménez. 1993. On the classical dynamics of non-rotating extended charges. Nuovo Cimento B 108: 853-869CrossRefADSGoogle Scholar
  53. 53.
    Roa-Neri, J.A.E. and J.L. Jiménez. 2002. An alternative approach to the classical dynamics of an extended charged particle. Found. Phys. 32: 1617-1634MathSciNetCrossRefGoogle Scholar
  54. 54.
    Rohrlich, F. 2002. Dynamics of a classical quasi-point charge. Phys. Lett. A 303: 307-310MathSciNetCrossRefzbMATHADSGoogle Scholar
  55. 55.
    Rohrlich, F. 2007. Classical Charged Particles, 3rd edn. World Scientific, New Jersey. Section 6-3Google Scholar
  56. 56.
    Schott, G.A. 1908. Über den Einfluß von Unstetigkeiten bei der Bewegung von Elektronen. Ann. Phys. (Leipzig) 330: 63-91CrossRefADSGoogle Scholar
  57. 57.
    Senitzky, I.R. 1973. Radiation-reaction and vacuum-field effects in Heisenberg-picture quantum electrodynamics. Phys. Rev. Lett. 31: 955-958CrossRefADSGoogle Scholar
  58. 58.
    Slater, L.J. 1966. Generalized Hypergeometric Functions. Cambridge University Press, CambridgeGoogle Scholar
  59. 59.
    Smorenburg, P.W., L.P.J. Kamp, G.A. Geloni and O.J. Luiten. 2010. Coherently enhanced radiation reaction effects in laser-vacuum acceleration of electron bunches. Laser Part. Beams 28: 553-562CrossRefADSGoogle Scholar
  60. 60.
    Smorenburg, P.W., L.P.J. Kamp and O.J. Luiten. 2013. Ponderomotive manipulation of cold subwavelength plasmas. Phys. Rev. E 87: 023101CrossRefADSGoogle Scholar
  61. 61.
    Sommerfeld, A. 1904a. Simplified deduction of the field and forces of an electron moving in any given way. Proc. K. Akad. Wet. Amsterdam, Sect. Sci. 7: 346-367. (English translation)ADSGoogle Scholar
  62. 62.
    Sommerfeld, A. 1904b. Zur Elektronentheorie. Nachr. Ges. Wiss. Göttingen 1904: 99-439zbMATHGoogle Scholar
  63. 63.
    Stratton, J.A. 1941. Electromagnetic Theory. McGraw-Hill, London. Section 7.8Google Scholar
  64. 64.
    Villarroel, D. 2006. Enlarged Lorentz-Dirac equations. J. Phys. A 39: 8543-8556MathSciNetCrossRefzbMATHADSGoogle Scholar
  65. 65.
    Watson, G.N. 1966. A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge. p. 528Google Scholar
  66. 66.
    Whittaker, E.T. and G.N. Watson. 1962. A Course of Modern Analysis, 4th edn. Cambridge University Press, CambridgeGoogle Scholar
  67. 67.
    Wildermuth, K. 1955. Zur physikalischen Interpretation der Elektronenselbstbeschleunigung. Z. Naturforsch. A 10: 450-459zbMATHADSGoogle Scholar
  68. 68.
  69. 69.
  70. 70.
    Yaghjian, A.D. 2006. Relativistic Dynamics of a Charged Sphere, 2nd edn. Springer, New YorkGoogle Scholar
  71. 71.
    Zygmund, A. 1968. Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge. Vol. 2, p. 243Google Scholar

Copyright information

© EDP Sciences and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Coherence and Quantum Technology (CQT), Eindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations