The concept of hidden momentum is reviewed, and the first rigorous derivation from Maxwell's equations is provided for the electromagnetic force on electrically small perfect electric conductors of arbitrary shape in bandlimited but otherwise arbitrarily time-varying fields. It is proven for the Amperian magnetic dipoles of these perfect conductors that a "hidden-momentum" electromagnetic force exists that makes the force on these time varying Amperian magnetic dipoles equal to the force on magnetic-charge magnetic dipoles with the same time varying magnetic dipole moment in the same time varying externally applied fields. The exact Mie solution to the perfectly conducting sphere under plane-wave illumination is used to prove that the expressions for the total and hidden-momentum forces on the arbitrarily shaped electrically small perfect conductors correctly predict the forces on perfectly conducting spheres. Remarkably, it is found that the quadrupolar fields at the surface of the sphere are required to obtain the correct total force on the sphere even though the quadrupolar moments are negligible compared to the dipole moments as the electrical size of the sphere approaches zero.
2. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941.
3. 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
4. Calkin, M. G., "Linear momentum of the source of a static electromagnetic field," Am. J. Phys., Vol. 39, 513-516, May 1971.
doi:10.1119/1.1986204
5. Vaidman, L., "Torque and force on a magnetic dipole," Am. J. Phys., Vol. 58, 978-983, October 1990.
doi:10.1119/1.16260
6. Coleman, S. and J. H. Van Vleck, "Origin of `hidden momentum forces' on magnets," Phys. Rev., Vol. 171, 1370-1375, July 1968.
doi:10.1103/PhysRev.171.1370
7. Penfield, Jr., P. and H. A. Haus, Electrodynamics of Moving Media, M.I.T. Press, Cambridge, MA, 1967.
8. Griffiths, D. J., Introduction to Electrodynamics, 4th Ed., Cambridge University Press, 2017.
9. Boyer, T. H., "Classical interaction of a magnet and a point charge: The Shockley-James paradox," Phys. Rev. E, Vol. 91, 013201(1-11), 2015.
doi:10.1103/PhysRevE.91.013201
10. Boyer, T. H., "Interaction of a magnet and a point charge: Unrecognized internal electromagnetic momentum," Am. J. Phys., Vol. 83, 433-442, 2015.
doi:10.1119/1.4904040
11. De Groot, S. R. and L. G. Suttorp, Foundations of Electrodynamics, Amsterdam, North-Holland, 1972.
12. Maxwell, J. C., A Treatise on Electricity and Magnetism, Unabridged, 3rd Ed., Dover, New York, 1954; The Dover edition is an unabridged, slightly altered, republication of the third edition, published by the Clarendon Press in 1891.
13. Yaghjian, A. D., "Reflections on Maxwell’s treatise," Progress In Electromagnetics Research, Vol. 149, 217-249, November 2014; see also A.D. Yaghjian, ``An overview of Maxwell's Treatise,'' FERMAT Multimedia, Vol. 11, 2015.
doi:10.2528/PIER14092503
14. 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
15. Hnizdo, V., "Comment on `Torque and force on a magnetic dipole'," Am. J. Phys., Vol. 60, 279-281, March 1992.
doi:10.1119/1.16912
16. Furry, W. E., "Examples of momentum distributions in the electromagnetic field and in matter," Am. J. Phys., Vol. 37, 621-636, June 1969.
doi:10.1119/1.1975729
17. Yaghjian, A. D., Relativistic Dynamics of a Charged Sphere: Updating the Lorentz-Abraham Model, 2nd Ed., Springer, New York, 2006.
doi:10.1007/b98846
18. Phillips, H. B., Vector Analysis, John Wiley & Sons, New York, 1933.
19. Raab, R. E. and O. L. De Lange, Multipole Theory in Electromagnetism, Clarendon Press, Oxford, 2005.
20. Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed., Butterworth Heinemann, Oxford, UK, 1984.
21. Yaghjian, A. D., "Electric dyadic Green’s functions in the source region," Proc. IEEE, Vol. 68 & 69, 248–263 & 282–285, February 1980 & February 1981.
22. Haus, H. A. and J. R. Melcher, Electromagnetic Fields and Energy, Prentice Hall, Englewood Cliffs, NJ, 1989.
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