Resource Letter EM-1: Electromagnetic Momentum David J. Griffiths Citation: American Journal of Physics 80, 7 (2012); doi: 10.1119/1.3641979 View online: http://dx.doi.org/10.1119/1.3641979 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/80/1?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in The electromagnetic momentum of static charge-current distributions Am. J. Phys. 82, 869 (2014); 10.1119/1.4879539 Resource Letter N-1: Nanotechnology Am. J. Phys. 82, 8 (2014); 10.1119/1.4827826 Self-dual electromagnetic fields Am. J. Phys. 78, 858 (2010); 10.1119/1.3379299 Hidden momentum, field momentum, and electromagnetic impulse Am. J. Phys. 77, 826 (2009); 10.1119/1.3152712 Electromagnetic waves in dissipative media revisited Am. J. Phys. 72, 393 (2004); 10.1119/1.1639010 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.248.29.220 On: Wed, 27 Aug 2014 03:24:59 RESOURCE LETTER Resource Letters are guides for college and university physicists, astronomers, and other scientists to literature, websites, and other teaching aids. Each Resource Letter focuses on a particular topic and is intended to help teachers improve course content in a specific field of physics or to introduce nonspecialists to this field. The Resource Letters Editorial Board meets at the AAPT Winter Meeting to choose topics for which Resource Letters will be commissioned during the ensuing year. Items in the Resource Letter below are labeled with the letter E to indicate elementary level or material of general interest to persons seeking to become informed in the field, the letter I to indicate intermediate level or somewhat specialized material, or the letter A to indicate advanced or specialized material. No Resource Letter is meant to be exhaustive and complete; in time there may be more than one Resource Letter on a given subject. A complete list by field of all Resource Letters published to date is at the website http:// ajp.dickinson.edu/Readers/resLetters.html. Suggestions for future Resource Letters, including those of high pedagogical value, are welcome and should be sent to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455; e-mail: [email protected] Resource Letter EM-1: Electromagnetic Momentum David J. Griffiths Department of Physics, Reed College, Portland, Oregon 97202 (Received 29 July 2011; accepted 1 September 2011) This Resource Letter surveys the literature on momentum in electromagnetic fields, including the general theory, the relation between electromagnetic momentum and vector potential, “hidden” momentum, the 4/3 problem for electromagnetic mass, and the Abraham–Minkowski controversy regarding the field momentum in polarizable and magnetizable media. VC 2012 American Association of Physics Teachers. [DOI: 10.1119/1.3641979] I. INTRODUCTION But locating this “hidden momentum” can be subtle and difficult. According to classical electrodynamics, electric and mag- (3) A moving charge drags around the momentum in its netic fields (E and B) store linear momentum, which must fields, which means (in effect) that it has “extra” mass. be included if the total momentum of a system is to be con- But this “electromagnetic mass” is inconsistent with what served. Specifically, the electromagnetic momentum per unit you get from the energy in the fields (using Einstein’s for- volume is mula E ¼ mc2)—by a notorious factor of 4/3, in the case of a spherical shell. Which mass (if either) is “correct”? g ¼ ðE  BÞ; (1) 0 (4) Inside matter, which is subject to polarization and mag- as first proposed by Poynting (Refs. 30–32). Field momen- netization, the effective field momentum is modified. tum is most dramatically demonstrated in the laboratory by Minkowski proposed the pressure of light on an absorbing or reflecting surface. [In 1619 Kepler suggested that the pressure of light explains gM ¼ðD  BÞ; (2) why comet tails point away from the sun (Ref. 29). The theory was developed by Maxwell (Ref. 10) and confirmed Abraham advocated experimentally by Lebedew (Ref. 25) and Nichols and Hull 1 (Ref. 28). Some introductory textbooks offer a quick qualita- g ¼ ðE  HÞ: (3) A c2 tive explanation for the pressure exerted on a perfect conduc- tor: E drives charge in (say) the x direction and B (in the y For over a century a debate has raged: which expression is direction) then exerts a force in the z direction. This naive right? Or are they perhaps both right, and simply describe argument is faulty (Refs. 27, 33, and 16).] different things? How can the question be settled, theoreti- But the notion that fields carry momentum leads to several cally and experimentally? Although many distinguished intriguing problems, some of which are not entirely resolved authors claim to have resolved the issue, the dispute contin- after more than a century of debate. ues to this day. (1) For a point charge q in an external field represented by In Section II, I summarize the theory. I then survey each the vector potential A, the electromagnetic momentum is of the four controversies described qualitatively above. In qA. This suggests that A can be thought of as “potential the final section, I briefly consider electromagnetic momen- momentum per unit charge,” just as the scalar potential tum in quantum mechanics, where the photon makes the V is “potential energy per unit charge.” But this interpre- story in some respects more concrete and intuitive. tation raises questions of its own, and it has never been universally accepted. II. THEORY (2)AccordingtoEq.(1), even purely static fields can A. Nonrelativistic store momentum. How can a system at rest carry mo- mentum? It cannot … there must be some compensat- Electrodynamics (Refs. 2 and 6) is based on Maxwell’s ing non-electromagnetic momentum in such systems. equations, which tell us how the sources (charge density q 7 Am. J. Phys. 80 (1), January 2012 http://aapt.org/ajp VC 2012 American Association of Physics Teachers 7 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.248.29.220 On: Wed, 27 Aug 2014 03:24:59 and current density J) generate electric and magnetic fields The corresponding statement for electromagnetic energy is (E and B): @u 1 @B þrS ¼ðE Á JÞ: (14) rE ¼ q; rE ¼ ; @t 0 @t This is Poynting’s theorem; E Á J is the power per unit vol- @E rB ¼ 0; rB ¼ l0J þ l00 ; (4) ume delivered by the fields to the electric charges. Except in @t regions where E Á J ¼ 0 (empty space, for example) the elec- and the Lorentz force law, which tells us the force exerted by tromagnetic energy by itself is not conserved, because the the fields on a point charge q moving with velocity v: fields do work on the charges. Similarly, @g $ F ¼ qðE þ v  BÞ: (5) r>¼ðqE þ J  BÞ: (15) @t The homogeneous Maxwell equations (the two that do not involve q or J) allow us to express the fields in terms of sca- Here (qE þ J  B) is the force per unit volume exerted by lar and vector potentials the fields on the electric charges. Except (for example) in empty space, electromagnetic momentum by itself is not @A conserved. [Nor, therefore, is mechanical momentum sepa- E ¼rV À ; B ¼rÂA: (6) @t rately conserved. This means that Newton’s Third Law (although it holds in electrostatics and magnetostatics)isnot Electromagnetic fields store energy and momentum (and obeyed in electrodynamics (Refs. 22 and 2).] for that matter also angular momentum). The energy per unit As we shall see, it is no accident that the same quantity volume in the fields is (E  B) appears in the Poynting vector and in the momentum density (Ref. 1), 1 1 2 2 2 u ¼ 0E þ B ; (7) S c g (16) 2 l ¼ 0 $ (or that the same quantity > plays a dual role as force-per- and the momentum density is unit-area and momentum flux). g ¼ 0ðE  BÞ: (8) B. Relativistic The fields also transport energy and momentum from one place to another. The energy flux (energy per unit time, per 1. Notation unit area) is given by the Poynting vector, The theory is more elegant in covariant (relativistic) nota- l 1 tion. The (Cartesian) space-time coordinates are x ¼ (ct, x, S ¼ ðE  BÞ (9) y, z), Greek indices run from 0—the “temporal” coordi- l0 nate—to 3, while Roman indices go from 1 to 3—the “spatial” coordinates. We use the metric (S Á da is the energy per unit time transported through a 0 1 “window” of area da). The momentum flux is related to the 10 0 0 Maxwell stress tensor: B 0 À10 0C gl ¼ B C (17) @ 00À10A 1 2 1 1 2 Tij ¼ 0 EiEj À dijE þ BiBj À dijB (10) 00 0À1 2 l0 2 (Specifically, the momentum per unit time transported and the Einstein convention (sum repeated indices). The $ energy density u, the energy flux S, the momentum density through a window da is > Á da ). For example, the energy $ and momentum per unit time radiated (to infinity) by a non- g, and the stress tensor > go together to make the stress- relativistic point charge q are energy tensor: 2 dE l0q 2 l u ½S=c ¼ a ; (11) H ¼ $ (18) dt 6pc ½cg½À> dp l q2 $ 0 2 This is entirely general—in other contexts u, S, g, and > will ¼ 3 a v; (12) dt 6pc not have their electromagnetic form [Eqs.