volume | PIER Journals

Abstract

Using realistic classical models of microscopic electric-charge electric dipoles and electric-current (Amperian) magnetic dipoles, it is proven that the Einstein-Laub macroscopic electromagnetic force on a macroscopic-continuum volume of these classical dipoles equals the sum of the microscopic electromagnetic forces on the discrete classical dipoles in that volume. The internal (hidden) momentum of the discrete Amperian magnetic dipoles is rigorously derived and properly included in the determination of the macroscopic force from the spatial averaging of the microscopic forces. Consequently, the Abraham/Einstein-Laub rather than the Minkowski macroscopic electromagnetic-field momentum density gives the total microscopic electromagnetic-field momentum in that volume. The kinetic momentum is found for the volume of the macroscopic continuum from Newton's relativistic equation of motion. It is shown that the difference between the kinetic and canonical momenta in a volume of the macroscopic continuum is equal to the sum of the ``hidden electromagnetic momenta'' within the electric-current magnetic dipoles and within hypothetical magnetic-current electric dipoles replacing the electric-charge electric dipoles in the classical macroscopic continuum. To obtain the correct unambiguous value of the force on a volume inside the continuum from the force-momentum expression, it is mandatory that the surface of that volume be hypothetically separated from the rest of the continuum by a thin free-space shell. Two definitive experiments performed in the past with time varying fields and forces are shown to conclusively confirm the Einstein-Laub/Abraham formulation over the Minkowski formulation.

1. Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed., Butterworth Heinemann, Oxford, UK, 1984.

2. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941.

3. Panofsky, W. K. H. and M. Phillips, Classical Electricity and Magnetism, 2nd Ed., Addison-Wesley, Reading, MA, 1962.

4. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, New York, 1999.

5. Van Bladel, J. G., Electromagnetic Fields, 2nd Ed., Wiley/IEEE, Hoboken, NJ, 2007.
doi:10.1002/047012458X

6. Abraham, M., "Zur elektrodynamik bewegter Korper," Rend. Circ. Mat. Palermo, Vol. 28, 1-28, 1909.
doi:10.1007/BF03018208

7. Abraham, M., "Sull’elettrodinamica di Minkowski," Rend. Circ. Mat. Palermo, Vol. 30, 30-46, 1910.
doi:10.1007/BF03014862

8. Einstein, A. and J. Laub, "On the Pondermotive forces exerted on bodies at rest in the electromagnetic field," The Collected Papers of Albert Einstein, Vol. 2: The Swiss Years: Writings, 1900–1909, 339-348, Princeton Univ., 1990.

9. Minkowski, H., "Die Grundgleichungen fur die elektromagnetischen Vorgange in bewegten Korpern," Nachr. Ges. Wiss. Gottingen, 53-111, 1908, reprinted in Math. Annaln., Vol. 68, 472–575, 1910.

10. Griffiths, D. J., "Resource letter EM-1: Electromagnetic momentum," Am. J. Phys., Vol. 80, 7-19, 2012.
doi:10.1119/1.3641979

11. McDonald, K. T., "Bibliography on the Abraham-Minkowski debate (2017),", http://kirkmcd.princeton.edu/examples/.

12. Mansuripur, M., "Force, torque, linear momentum, and angular momentum in classical electrodynamics," Applied Physics A, Vol. 123, No. 653, 1-11, 2017.

13. Silveirinha, M. G., "Reexamination of the Abraham-Minkowski dilemma," Phys. Rev. A, Vol. 96, 033831, 2017.
doi:10.1103/PhysRevA.96.033831

14. Maxwell, J. C., A Treatise on Electricity and Magnetism, Unabridged 3rd Ed., Dover, New York, 1954.

15. Yaghjian, A. D., "Reflections on Maxwell’s Treatise," Progress In Electromagnetics Research, Vol. 149, 217-249, 2014.
doi:10.2528/PIER14092503

16. Liu, H., L. Rondi, and J. Xiao, "Mosco convergence for HH (curl) spaces, higher integrability for Maxwell’s equations, and stability in direct and inverse EM scattering problems," J. Eur. Math. Soc., Vol. 21, 2945-2993, October 2019.

17. Yaghjian, A. D., "Maxwell’s definition of electric polarization as displacement," Progress In Electromagnetics Research M, Vol. 88, 65-71, 2020.
doi:10.2528/PIERM19090802

18. Yaghjian, A. D., "Classical power and energy relations for macroscopic dipolar continua derived from the microscopic Maxwell equations," Progress In Electromagnetics Research B, Vol. 71, 1-37, 2016.
doi:10.2528/PIERB16081901

19. Hansen, T. B. and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications, Wiley/IEEE Press, New York, 1999.
doi:10.1109/9780470545522

20. Raab, R. E. and O. L. DeLange, Multipole Theory in Electromagnetism, Clarendon Press, Oxford, 2005.

21. De Groot, S. R. and L. G. Suttorp, "Foundations of Electrodynamics," North-Holland, 1972.

22. Fano, R. M., L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces, MIT Press, Cambridge, MA, 1960.

23. Kinsler, P., A. Favaro, and M. W. McCall, "Four Poynting theorems," European J. Phys., Vol. 30, 983-993, July 2009.

24. Lorentz, H. A., "Weiterbildung der Maxwellischen Theorie: Elektronentheorie," Encyklopadie der Mathematischen Wissenschaften, Vol. 5, Part 2, 1904-1922, December 1903.

25. Einstein, A., "To Walter Dallenbach (1918)," The Collected Papers of Albert Einstein, Vol. 8: The Berlin Years: Correspondence, 1914–1918 (English Translation Supplement), item 565, Princeton Univ., 1999.

26. Yaghjian, A. D., "Extreme electromagnetic boundary conditions and their manifestation at the inner surfaces of spherical and cylindrical cloaks," Metamaterials, Vol. 4, 70-76, August–September 2010.
doi:10.1016/j.metmat.2010.03.006

27. Silveirinha, M. G., "Poynting vector, heating rate, and stored energy in structured materials: A first-principles derivation," Phys. Rev. B, Vol. 8, 235120, December 2009.
doi:10.1103/PhysRevB.80.235120

28. Yaghjian, A. D., "Power flow, energy density, and group/energy transport velocities in spatially dispersive media," Radio Science, Vol. 53, 303-313, March 2018.
doi:10.1002/2017RS006489

29. Yaghjian, A. D., "Force and hidden momentum for classical microscopic dipoles," Progress In Electromagnetics Research B, Vol. 82, 165-188, 2018.
doi:10.2528/PIERB18092007

30. Boyer, T. H., "Classical interaction of a magnet and a point charge: The Shockley-James paradox," Phys. Rev. E, Vol. 91, No. 013201, 1-11, January 2015.

31. Boyer, T. H., "Interaction of a magnet and a point charge: Unrecognized internal electromagnetic momentum," Am. J. Phys., Vol. 83, 433-442, May 2015.
doi:10.1119/1.4904040

32. Nieto-Vesperinas, M., J. J. Saenz, R. Gomez-Medina, and L. Chantada, "Optical forces on small magnetodielectric particles," Optics Express, Vol. 18, No. 11428, 1-16, May 2010.

33. Chaumet, P. and A. Rahmani, "Electromagnetic force and torque on magnetic and negative-index scatterers," Optics Express, Vol. 17, No. 2224, 1-11, February 2009.

34. Yaghjian, A. D., "Electric dyadic Green’s function in the source region," Proc. IEEE, Vol. 68, 248-263, February 1980; also Vol. 69, 282–285, February 1981.
doi:10.1109/PROC.1980.11620

35. Yaghjian, A. D., Relativistic Dynamics of a Charged Sphere: Updating the Lorentz-Abraham Model, 3rd Ed., Springer, New York, 2022.
doi:10.1007/978-3-031-06067-0

36. Lembessis, V. E., M. Babiker, C. Baxter, and R. Loudon, "Theory of radiation forces and momenta for mobile atoms in light fields," Phys. Rev. A, Vol. 48, 1594-1603, August 1993.
doi:10.1103/PhysRevA.48.1594

37. Baxter, C., M. Babiker, and R. Loudon, "Canonical approach to photon pressure," Phys. Rev. A, Vol. 47, 1278-1287, February 1993.
doi:10.1103/PhysRevA.47.1278

38. Barnett, S. M., "Resolution of the Abraham-Minkowski dilemma," Phys. Rev. Lett., Vol. 104, 070401, February 2010.
doi:10.1103/PhysRevLett.104.070401

39. Barnett, S. M. and R. Loudon, "The enigma of optical momentum in a medium," Phil. Trans. R. Soc. A., Vol. 368, 927-939, March 2010.
doi:10.1098/rsta.2009.0207

40. James, R. P., "Force on permeable matter in time-varying fields,", Ph.D. thesis, Stanford University, December 1968.

41. Walker, G. B., D. G. Lahoz, and G. Walker, "Measurement of the Abraham force in a barium titanate specimen," Can. J. Phys., Vol. 53, 2577-2586, December 1975.
doi:10.1139/p75-313

42. Marx, G. and G. Gyorgyi, "Der Energie-Impuls-Tensor des elektromagnetischen Feldes und die ponderomotorischen Krafte in Dielektrika," Acta Physica Academiae Scientiarum Hungaricae, Vol. 3, 213-242, April 1954.

Dr. Arthur D. Yaghjian