The force law of classical electrodynamics: Lorentz versus Einstein and Laub

The classical theory of electrodynamics is built upon Maxwell’s equations and the concepts of electromagnetic field, force, energy, and momentum, which are intimately tied together by Poynting’s theorem and the Lorentz force law. Whereas Maxwell’s macroscopic equations relate the electric and magnetic fields to their material sources (i.e., charge, current, polarization and magnetization), Poynting’s theorem governs the flow of electromagnetic energy and its exchange between fields and material media, while the Lorentz law regulates the backand- forth transfer of momentum between the media and the fields. As it turns out, an alternative force law, first proposed in 1908 by Einstein and Laub, exists that is consistent with Maxwell’s macroscopic equations and complies with the conservation laws as well as with the requirements of special relativity. While the Lorentz law requires the introduction of hidden energy and hidden momentum in situations where an electric field acts on a magnetic material, the Einstein-Laub formulation of electromagnetic force and torque does not invoke hidden entities under such circumstances. Moreover, the total force and the total torque exerted by electromagnetic fields on any given object turn out to be independent of whether force and torque densities are evaluated using the Lorentz law or in accordance with the Einstein-Laub formulas. Hidden entities aside, the two formulations differ only in their predicted force and torque distributions throughout material media. Such differences in distribution are occasionally measurable, and could serve as a guide in deciding which formulation, if either, corresponds to physical reality.

[1]  R. Boyd,et al.  Momentum of Light in a Dielectric Medium , 2010 .

[2]  D. Griffiths.  Introduction to Electrodynamics , 2017 .

[3]  Comment on ‘‘Torque and force on a magnetic dipole,’’ by L. Vaidman [Am. J. Phys. 58, 978–983 (1990)] , 1992 .

[4]  G. Spavieri.  Proposal for experiments to detect the missing torque in special relativity , 1990 .

[5]  Y. Shen.  Electrostriction, optical Kerr effect and self-focusing of laser beams , 1966 .

[6]  M. Mansuripur.  Computation of electron diffraction patterns in Lorentz electron microscopy of thin magnetic films , 1991 .

[7]  J. Jackson.  Classical Electrodynamics, 3rd Edition , 1998 .

[8]  W. Shockley,et al.  "Try Simplest Cases" Discovery of "Hidden Momentum" Forces on "Magnetic Currents" , 1967 .

[9]  H. Haus.  Momentum, energy and power densities of TEM wave packet , 1969 .

[10]  Hermann A. Haus,et al.  Electrodynamics of Moving Media , 1968 .

[11]  Masud Mansuripur.  On the foundational equations of the classical theory of electrodynamics , 2013 .

[12]  R. Loudon.  Radiation pressure and momentum in dielectrics , 2004 .

[13]  A. Smolyakov,et al.  Electromagnetic forces and internal stresses in dielectric media. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  W. Shockley.  A "try simplest cases" resolution of the abraham-minkowski controversy on electromagnetic momentum in matter. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[15]  L. Vaidman.  Torque and force on a magnetic dipole , 1990 .

[16]  S. Reynolds,et al.  Hidden momentum, field momentum, and electromagnetic impulse , 2009 .

[17]  Masud Mansuripur,et al.  Radiation pressure and the distribution of electromagnetic force in dielectric media. , 2005 .

[18]  M. Mansuripur.  Resolution of the Abraham-Minkowski controversy , 2010, 1208.0872.

[19]  K. Young,et al.  Microscopic Derivation of the Helmholtz Force Density , 1981 .

[20]  Edwin L. Kerr.  Filamentary Tracks Formed in Transparent Optical Glass by Laser-Beam Self-Focusing. III. Filament Formation , 1972 .

[21]  H. Haus,et al.  Force on a conducting current loop , 1969 .

[22]  R. Boyd,et al.  Electromagnetic momenta and forces in dispersive dielectric media , 2009, 0910.0770.

[23]  M. Khorrami.  Comment on "Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation". , 2013, Physical review letters.

[24]  D. Griffiths.  Resource Letter EM-1: Electromagnetic Momentum , 2012 .

[25]  Masud Mansuripur,et al.  Trouble with the Lorentz law of force: response to critics , 2012, Other Conferences.

[26]  J. Shapiro.  Torques without Rotation: the Right-Angle Lever , 2012, 1206.4487.

[27]  P. Saldanha.  Division of the energy and of the momentum of electromagnetic waves in linear media into electromagnetic and material parts , 2011, 1102.0491.

[28]  Concerning “hidden momentum” , 2007, 0709.3457.

[29]  B. Kemp.  Resolution of the Abraham-Minkowski debate: Implications for the electromagnetic wave theory of light in matter , 2011 .

[30]  Timothy H. Boyer,et al.  Examples and Comments Related to Relativity Controversies , 2012, 1206.5322.

[31]  K. Webb.  Dependence of the radiation pressure on the background refractive index. , 2013, Physical Review Letters.

[32]  A. Einstein.  On the Electrodynamics of Moving Bodies , 2005 .

[33]  S. S. Hakim,et al.  An Experimental Determination of the Excess Pressure produced in a Liquid Dielectric by an Electric Field , 1962 .

[34]  Boyer.  Proposed Aharonov-Casher effect: Another example of an Aharonov-Bohm effect arising from a classical lag. , 1987, Physical review. A, General physics.

[35]  Iver Brevik,et al.  Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor , 1979 .

[36]  M. Mansuripur,et al.  Maxwell's macroscopic equations, the energy-momentum postulates, and the Lorentz law of force. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  P. Krumm,et al.  On the origin of magnetic dynamics , 1986 .

[38]  Electromagnetic-force distribution inside matter , 2013, 1308.3511.

[39]  N. Kroll,et al.  Temporal and Spatial Gain in Stimulated Light Scattering , 1971 .

[40]  J. V. Vleck,et al.  Origin of "Hidden Momentum Forces" on Magnets , 1968 .

[41]  J. Sinova,et al.  Anomalous hall effect , 2009, 0904.4154.

[42]  D. Griffiths.  Dipoles at rest , 1992 .

[43]  M. Mansuripur.  Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation. , 2012, Physical review letters.

[44]  V. Namias.  Electrodynamics of moving dipoles: The case of the missing torque , 1989 .

[45]  Über die im elektromagnetischen Felde auf ruhende Körper ausgeübten ponderomotorischen Kräfte , 1908 .

[46]  V. Hnizdo,et al.  Mansuripur's paradox , 2013, 1303.0732.

[47]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[48]  Masud Mansuripur,et al.  Whence the Minkowski momentum? , 2009, NanoScience + Engineering.

[49]  Vaidman,et al.  Comment on "Proposed Aharonov-Casher effect: Another example of an Aharonov-Bohm effect arising from a classical lag" , 1988, Physical review. A, General physics.

[50]  W. Shockley.  "Hidden Linear Momentum" Related to theα→·E→Term for a Dirac-Electron Wave Packet in an Electric Field , 1968 .

[51]  Electromagnetic force and torque in ponderable media. , 2008, Optics express.

[52]  Arthur Ashkin,et al.  Radiation Pressure on a Free Liquid Surface , 1973 .

[53]  Daniel J. Cross,et al.  Resolution of the Mansuripur Paradox , 2012, 1205.5451.

[54]  C. D. Wright,et al.  The physical principles of magneto-optical recording , 1996 .

[55]  F. Rasetti,et al.  Deflection of Mesons in Magnetized Iron , 1944 .