aa r X i v : . [ phy s i c s . c l a ss - ph ] S e p Concerning Hidden Momentum
Timothy H. Boyer
Department of Physics, City College of the CityUniversity of New York, New York, New York 10031
Abstract
The fact that the author of an excellent textbook on electromagnetism could be duped by ”hiddenmomentum” vividly illustrates the problematic nature of its use.
1t greatly distresses me that the ideas of ”hidden momentum,”[1][2][3] which are of du-bious value in the physics research literature, should now appear in discussions of dubiousvalue or of outright error within respectable textbooks of classical electromagnetism.[4][5]It seems that the invocation of ”hidden momentum” is simply a confusing distraction fromthe needed discussions of energy and momentum flow within systems.[6]As an example of the outright errors caused by ”hidden momentum,” consider the discus-sion where ”hidden momentum” is first invoked in an excellent undergraduate electromag-netism text.[7] In the illustrative example treated in this text, energy is transferred from abattery on the left to a resister on the right by way of a coaxial cable. The text calculatesthe Poynting vector, the total electromagnetic power flow down the cable, and the totalelectromagnetic linear momentum located in the coaxial cable. Clearly the Poynting vec-tor is associated with the transfer of energy and momentum from the battery (which losesenergy), through the coaxial cable (which does not change), to the resistor (which gains en-ergy). The battery clearly recoils on sending out the energy and becomes less massive, whilethe resistor recoils and becomes more massive on absorbing the energy. In the presentationof the textbook, it is not clear how the battery, coaxial cable, and resistor are connectedtogether. If they are all mounted on a board with wheels, then (in response to the couplingof the recoils) we would expect the board would roll to the left so as to maintain the systemcenter of energy at a constant location in an inertial frame where the system was initiallyat rest.[8]However, rather than discussing these aspects of energy flow, the text invokes the myster-ies of ”hidden momentum.” The text states: ”In this case it turns out that there is ’hidden’mechanical momentum associated with the flow of current, and this exactly cancels themomentum in the fields.” The text also makes an appeal to an irrelevant later example of”hidden momentum” where no battery and resistor are present. Thus the text makes theerroneous suggestion that the electrical currents in the cable carry a ”hidden mechanicalmomentum” in the opposite direction from the electromagnetic energy flow. This is not thecase.How do we know that that the currents in the coaxial cable do not carry the mysterious”hidden mechanical momentum” invoked by the text? I believe that we can make theessential physics of the situation clearer by making the current carriers less obscure. Let’sreplace the current-carrying coaxial cable by two frictionless parallel filaments on which2harged beads of negligible mass are allowed to slide freely. There are positively chargedbeads lined up along the top filament and negatively charged beads lined up on the bottomfilament, leading to an electrostatic potential between the filaments. A source of energyat the left takes a pair of beads (one positive and one negative) and moves them apart,pushing the positively charged bead onto the top filament and the negatively charged beadonto the bottom filament. On doing so, the source of energy both does work and also recoilsto the left. The energy is absorbed at the right-hand end of the filaments where a positivebead is removed from the top filament while a negative bead is removed from the bottomfilament, and energy is transferred as the beads are removed and brought together. Theenergy and momentum flow down the pair of filaments is purely electromagnetic and isgiven by Poynting’s vector, and there is electromagnetic momentum stored in the regionof the filaments. However, there is nothing even remotely resembling ”hidden mechanicalmomentum” in the currents on the filaments. This example suggests that the comments inthe text claiming the existence of ”hidden mechanical momentum” associated with the flowof current are simply wrong.Perhaps the situation is even more transparent if we consider electromagnetic energytransmitted through vacuum from the energy source on the left to an energy receiver on theright in the form of electromagnetic waves. Once again the energy source on the left will loseenergy while the energy sink on the right will gain energy, and the space in between sourceand sink will contain both electromagnetic energy and momentum. In this case there areno electrical currents for which one can invoke ”hidden mechanical momentum.” Of course,in all these cases, if the energy source and energy sink are mounted on a board on wheels,the board will roll to the left due to the recoils associated with energy transmission andreception.[8]The fact that the author of an excellent textbook on electromagnetism could be dupedby ”hidden momentum” vividly illustrates the problematic nature of its use. [1] S. Coleman and J. H. Van Vleck, ”Origin of ’hidden momentum forces’ on magnets,” Phys.Rev. , 1370-1375 (1968).[2] Y. Aharonov, P. Pearle, and L. Vaidman, ”Comment on ’proposed Aharonov-Casher effect: nother example of an Aharonov-Bohm effect arising from a classical lag,’” Phys. Rev. ,485-491 (1988).[3] L. Vaidman, ”Torque and force on a magnetic dipole,” Am. J. Phys. Introduction to Electrodynamics
Classical Electrodynamics
Spacetime Physics (W. H. Freeman, San Francisco 1963), p. 147-148.(W. H. Freeman, San Francisco 1963), p. 147-148.