Hidden mechanical momentum and the field momentum in stationary electromagnetic and gravitational systems

American Journal of Physics

Volumn 65, Issue 6, 1997, Pages 515-518

  Hnizdo, V ^a  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031479134     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18581     Document Type: Article

References (30)

1. 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. A 26, 412-413 (1968).
2. 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. A 26, 412-413 (1968).
3. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990); V. Hnizdo, "Comment on 'Torque and force on a magnetic dipole,'" ibid. 60, 279-280 (1992).
4. L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990); V. Hnizdo, "Comment on 'Torque and force on a magnetic dipole,'" ibid. 60, 279-280 (1992).
5. V. Hnizdo, "Hidden momentum and the electromagnetic mass of a charge and current carrying body," Am. J. Phys. 65, 55-65 (1997).
6. L. Vaidman, Ref. 2, discusses briefly the hidden momentum in gravitational systems.
7. An ideal fluid has a negligible viscosity and thermal conductivity, and it cannot support any shear stresses.
8. ij.
9. ij.
10. The pressure in an ideal fluid is an invariant scalar, see C. Møller, The Theory of Relativity (Clarendon, Oxford, 1972), 2nd ed., pp. 191-192.
11. ij, in a violation of Pascal's law (see the first footnote in Sec. 35 of L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Ref. 6).
12. Strictly speaking, the force density f acting on a given element of the body should include the contributions from the "body" (as opposed to "surface") forces due to all the other elements of the body, but we shall assume that such contributions can be neglected in comparison with the external forces (see Ref. 12, however, for a case where such an assumption cannot be made).
13. i and the fact that ▽·(ah) =h·▽a for a divergenceless h.
14. Following the already accepted terminology, by "hidden momentum" is thus meant a nonzero mechanical momentum of a body the center of mass of which is at rest; however, in a recent paper of E. Comay, "Exposing 'hidden momentum,'" Am. J. Phys. 64, 1028-1034 (1996), the term "hidden momentum" is used in a rather less restricted way.
15. 3r=0 for a divergenceless j.
16. 2)m×E.
17. A strictly straight tube cannot confine the fluid to a finite region of space; the tube has to have bends that fashion it into a closed loop. One can assume, however, that the bends are negligible in size compared to the straight sections of the tube.
18. That this is true for v=ω×r with ω constant ("constant angular velocity") is seen most easily by using cylindrical coordinates ρ,φ: v·[(v·▽)(γv)]=v·[v∂(γv)/ρ∂ φ]=0, as putting the z axis along ω, v =(-v sin φ,v cos φ,0) with v=|v|=ωρ.
19. This equation can be obtained also by transforming Euler's equation from the local rest frame of a constant-velocity fluid to the laboratory frame, see, in the electromagnetic setting, V. Hnizdo, "Hidden momentum of a relativistic fluid carrying current in an external electric field," Am. J. Phys. 65, 92-94 (1997).
20. 2=0, while the divergence of the velocity itself ▽·v=0.
21. 0, the hidden momentum of Eq. (13) vanishes as the particle current density nv is divergenceless in the stationary situation.
22. Vaidman, Ref. 2.
23. 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, 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); L. Vaidman, Ref. 2.
24. 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, 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); L. Vaidman, Ref. 2.
25. 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, 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); L. Vaidman, Ref. 2.
26. P. Lorrain and D. R. Corson, Electromagnetic Fields and Waves (Freeman, San Francisco, 1970), 2nd ed., p. 251; D. Bedford and P. Krumm, "On relativistic gravitation," Am. J. Phys. 53, 889-890 (1985); P. Krumm and D. Bedicid, "The gravitational Poynting vector and energy transfer," Am. J. Phys. 55, 362-363 (1987).
27. P. Lorrain and D. R. Corson, Electromagnetic Fields and Waves (Freeman, San Francisco, 1970), 2nd ed., p. 251; D. Bedford and P. Krumm, "On relativistic gravitation," Am. J. Phys. 53, 889-890 (1985); P. Krumm and D. Bedicid, "The gravitational Poynting vector and energy transfer," Am. J. Phys. 55, 362-363 (1987).
28. P. Lorrain and D. R. Corson, Electromagnetic Fields and Waves (Freeman, San Francisco, 1970), 2nd ed., p. 251; D. Bedford and P. Krumm, "On relativistic gravitation," Am. J. Phys. 53, 889-890 (1985); P. Krumm and D. Bedicid, "The gravitational Poynting vector and energy transfer," Am. J. Phys. 55, 362-363 (1987).
29. D. Bedford and P. Krumm, Ref. 21.
30. 2/4πG)b×e, as proposed by P. Krumm and D. Bedford, Ref. 21.