Comay s Paradox: Do Magneti Charges Conserve Energy? 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (June 1, 2015; updated July 16, 2015) The interation energy of an eletri harge q and a point eletri dipole with moment p of fixed magnitude, both at rest, is, in Gaussian units, Ep E q dvol = 1 ( ) p r E r 3 q dvol = 1 ( ) p r r E 3 q dvol + 1 p r r E 3 q dvol = 1 p r p r Sat r E 3 q darea + r 3 qδ3 (r r q ) dvol = p qr q rq 3 = p E q (eletri harge and permanent eletri dipole), (1) taking the eletri dipole to be at the origin, suh that the field at the dipole (i.e., at the origin) due to harge q at loation r q is E q = qr q /r 3 q with E q =q δ 3 (r r q ), while the field of the eletri dipole an be related to the gradient of a salar potential as E p = (p r/r 3 ). The fore on the eletri dipole p is F p =(p p )E q = p (p E q )= p U int, (2) where p is the gradient at the loation of the dipole p, and we note that E q =0for harge q at rest. The fore on the eletri harge q is F q = q E p = q q p r q r 3 q = q (p E q )= q p F p, (3) where q is the gradient at the loation of the harge q. Similarly, interation energy of a (Gilbertian) magneti harge p (aka magneti monopole) and a point magneti dipole m G onsisting of a pair of opposite magneti harges with fixed separation, all at rest, is BmG B p dvol = m G B p (Gilbertian magneti harge and Gilbertian magneti dipole),(4) taking the magneti dipole to be at the origin, suh that the field at the dipole (i.e., atthe origin) due to magneti harge p at loation r p is B p = p r p /r 3 p with B p =pδ 3 (r r p ), while the field of the magneti dipole an be related to the gradient of a salar potential as B mg = (m G r/r 3 ). The fore on the Gilbertian magneti dipole m G is F mg =(m G m )B p = m (m G B p )= m U int, (5) 1
where m is the gradient at the loation of the dipole m, and we note that B p =0for magneti harge p at rest. The fore on the magneti harge p (in vauum) is F p = p B mg = p p m G r p r 3 p = p (m G B p )= p m F mg, (6) where p is the gradient at the loation of the magneti harge p. However, if the point magneti dipole m A is Ampèrian, suh as that of an eletron, proton or neutron, for whih B ma =0, 1 its interation energy with a Gilbertian magneti pole appears to vanish, BmA B p dvol = 1 B ma p r dvol = 1 = 1 ( ) p r B m A dvol + 1 p r B ma dvol p Sat r B m A darea = 0 (Gilbertian magneti harge and Ampèrian magneti dipole), (7) taking the magneti harge to be at the origin, and noting that the field of a magneti dipole falls off as 1/r 3 at large distanes. 2 The fore on the Ampèrian magneti dipole m A is 3 while the fore on the magneti harge p (in vauum) is F ma = m (m A B p ), (8) F p = p B ma = p p m A r p r 3 p = p (m A B p )= m (m A B p )= F ma, (9) It is agreeable that F p = F ma, but neither of these fores is assoiated with a onserved field energy unless the interation energy were m A B p rather than zero as found in eq. (7). The impliation is that the interation of a Gilbertian magneti harge with a point (permanent) Ampèrian magneti dipole does not onserve energy. 4 Can this be so? A possible resolution of this paradox is that Gilbertian magneti harges annot exist (along with eletri harges and urrents), as holds in Nature as far was we know. This paradox is due to Comay [2, 3], who did not offer a resolution. 1 For disussion of how we know that the magneti moment of a neutron is Ampèrian, see [1]. 2 The argument of eq. (7) does not depend on the Ampèrian urrent loop being pointlike, but only that the magneti field B ma of the loop obeys B ma = 0 and that this field falls off as 1/r 3 at large distanes. 3 See, for example, se. 5.7 of [4]. 4 TheaseofanAmpérian magneti moment of finite size, whose steady urrent is maintained by a battery, is onsidered in Appendix A. 2
2 Solution 2.1 Delta Funtions Assoiated with Point Dipoles As disussed, for example, in se. 4.1 of [4], the field of a point dipole m at the origin onsisting of a pair of opposite (Gilbertian) harges an be written as 3(m ˆr) ˆr m B m,g = r 3 3 m δ3 (r), (10) where the delta funtion at the origin desribes the large field between the two harges that point from the positive to the negative harge, i.e., opposite to the diretion of the momentum m. In ontrast, as disussed in se. 5.6 of [4], if the magneti dipole is due to an (Ampèrian) loop of eletri urrent, the field at the origin points in the same diretion as the moment m, with Hene, B m,a = Bm,Amperian B p 3(m ˆr) ˆr m r 3 + 8π 3 m δ3 (r) =B m,g +m δ 3 (r). (11) Bm,Gilbertian B p dvol = dvol + m δ 3 (r) B p dvol = m B p + m B p =0. (12) That is, keeping trak of the delta funtion at the enter of a point dipole does not resolve the paradox, as laimed in [5], but reinfores it. 2.2 Interation Energy of a Dira String Paradoxes similar to the above involving magneti harges were disussed by Lipkin and Peshkin [6, 7], who suggested that they should be onsidered in the ontext of Dira s theory of magneti harges as being at the end of strings of magneti flux [8, 9]. 5 However, Lipkin and Peshkin did not provide a lear resolution of these paradoxes. We now transribe an argument by Getino, Rojo and Rubio [10] that the interation energy between an Ampèrian magneti dipole and the Dira string assoiated with a Gilbertian magneti harge equals m A B p, 6 whih resolves Comay s paradox to the extent that suh strings are physial. However, the field energy assoiated with a pair of Gilbertian magneti harges then beomes doubtful. 2.2.1 Gilbertian Magneti Charge + Ampèrian Magneti Moment Sine quantum theories of partiles are based on Hamiltonians in whih the anonial momentum of a partile with eletrial harge q is p anonial = p meh +qa/, wherea is the vetor potential of the eletromagneti field ating on the harge, it is desirable that the magneti 5 A lassial presentation of Dira strings is given in se. 6.12 of [4]. 6 Suh interation energy eah goes against Dira s view [9], supposing eah pole to be at the end of an unobservable string. 3
field of magneti harge p be expressible in terms of a vetor potential. However, sine the magneti field B p = p (r r p )/ r r p 3 of a magneti harge obeys B p =pδ 3 (r r p ), we annot write B p = A p. Dira s suggestion [8] was that a magneti harge p is at the end of an infinite string, here labeled s, and the interior of this string arries magneti flux p. That is, denoting ŝ as the unit vetor tangent to the string, pointing to the end where the magneti harge resides, the magneti field of the string an be written as B s =pδ s, (13) where the vetor delta funtion δ s is parallel to ŝ, andobeysδ s = 0 for points not on the string. Furthermore, δ s ˆn darea = sign(ŝ ˆn), (14) for an integral over a surfae piered by the string at a point where ˆn is the unit vetor normal to the surfae. Then, the total magneti flux aross a surfae surrounding the magneti harge is zero, B =0whereB = B p + B s, and we an now assoiate a vetor potential A p with the field B of a magneti harge. For example, the vetor potential of a magneti harge p at the origin, with Dira string along the negative-z axis, is A p = p 1 os θ ˆφ, (15) r sin θ in spherial oordinates (r, θ, φ). Turning to the issue of the interation energy between a Gilbertian magneti harge p, taken to be at the origin, and an Ampèrian magneti dipole m A at r m, we write the total field of the magneti harge as B p + B s, suh that BmA (B p + B s ) BmA B s dvol = dvol = p B ma δ s dvol = p (B ma δ s )(ds darea) s surfaes to ŝ = p (B ma ds)(δ s darea) =p B ma ds s surfaes to ŝ s = p ϕ ma ds = pϕ ma (0) = p m A r m = m s rm 3 A B p, (16) wherewehaveexpressedthevolumeintegralofb ma δ s as an integral along the Dira string times integrals over surfaes penetrated by the string; then sine δ s and ds are parallel, they an be exhanged in the integrands of the seond and third lines; and we note that the magneti field of the Ampèrian magneti dipole an be expressed in terms of a magneti salar potential ϕ ma = m (r r m )/ r r m 3 for points outside the dipole urrent, with B ma = ϕ ma. Hene, the interation energy of the Ampèrian magneti dipole with the Dira string assoiated with the magneti harge p has the desired value m A B p needed to 4
restore onservation of energy, so long as the Dira string does not pass through the urrent loop of the Ampèrian dipole. 7 2.2.2 Two Gilbertian Magneti Charges The magneti field energy of two Gilbertian magneti hagres, eah with an assoiated Dira string, would be (Bp1 + B s1 ) (B p2 + B s2 ) dvol Bp1 B p2 Bp1 B s2 Bp2 B s1 Bs1 B s2 = dvol + dvol + dvol + dvol = p 1p 2 + p 2 ϕ r p1 (r p2 )+p 1 ϕ p2 (r p1 )=3 p 1p 2, (17) 12 r 12 assuming that the two Dira strings do not interset, so B s1 B s2 = 0, and noting that in eq. (16) the field B ma ould be replaed by the field B p of a Gilbertian magneti harge whose magneti salar potential is ϕ p (r) =p/ r r p (outside the Dira string). However, sine the fore between two Gilbertian magneti harges is F = p 1 p 2 r 12 /r12, 3 we expet their interation energy to be p 1 p 2 /r 12. Hene, the introdution of Dira strings does not appear to resolve Comay s paradox in the larger sense of aounting for field energy in systems involving Gilbertian magneti harges (as well as eletri urrents). It ontinues to seem that an eletromagneti field theory in whih the field energy is (E 2 + B 2 ) dvol/8π in not ompatible with the existene of both eletri and magneti harges, and that Comay s paradox remains unresolved, at least in a lassial ontext. 2.3 Quantum Analyses Comay s paradox suggests the despite their appeal, magneti harges are not ompatible with lassial eletrodynamis. Present enthusiasm of magneti harges is in the quantum ontext, starting with the landmark papers of Dira [8, 9], and extended to gauge theories by thooft [11] and Polyakov [12]. As remarked in se. 3.1.7 of [13], There is no lassial Hamiltonian theory of magneti harge. Appendix A: Current Loop Maintained by a Battery If the urrent of an Ampèrian dipole is maintained by a onstant-urrent soure ( battery of appropriately variable voltage V ), the latter an ontribute to the energy stored in the system, whih is therefore not neessarily equal to the work done by the eletromagneti fores. 7 Lipkin and Peshkin [6, 7] noted that in ase the magneti harge moves through the urrent loop, possibly on a trajetory that passes through the loop several times, the Dira string must beome wound around the urrent loop. 5
Current Loop Brought in from Infinity For example, suppose the system of Gilbertian magneti harge and Ampèrian magneti dipole, both at rest, is reated by first bringing the magneti harge in from infinity to its final position, and then bringing the magneti dipole in from infinity. The field of the magneti harge does work W p = rm F ma dx m = rm m (m A B p ) dx m = m A B p, (18) realling eq. (7) and noting that the displaement dx m is opposite to the fore F ma on the magneti dipole as it moves in from infinity. In addition, the battery does work to maintain onstant urrent I in the Ampèrian loop of (onstant) Area, W battery = V battery Idt= I dt d ϕ dt = IΔϕ = B p IArea = B p m A, (19) noting that to keep the urrent onstant the battery must provide a voltage equal and opposite to the EMF indued in the urrent loop due to the hanging magneti flux ϕ = Bp darea aording to Faraday s law, EMF ind = (d/dt)ϕ/. 8 Thus, zero total work is required to assemble the Gilbertian magneti harge and the Ampèrian magneti dipole in the above senario, so it is agreeable that the magneti field interation energy (7) is zero in this ase. 9 Current Raised from Zero Another senario for assembly of the Gilbertian magneti harge and Ampèrian magneti dipole is that initially the urrent is zero. Then, the harge and loop are brought to their final positions, and the urrent in loop is raised until its magneti moment is the desired m A. Sine the Lorentz fore on the urrent is perpendiular to the latter, no work is done by the field of the magneti harge as the urrent is raised. The only work done is that by the battery to overome the bak EMF due to the self indutane of the urrent loop as the urrent rises. This results in energy stored in the field of the urrent loop, whih is onsidered as a self energy, and not part of the possible interation energy with the magneti harge. Hene, also in this senario, zero work is done while assembling the system that ontributes to an interation field energy. Thus, Comay s paradox does not apply to Ampèrian magneti moments that are loops of urrent maintained by batteries. The paradox exists only if Nature inludes inlude Gilbertian magneti harges as well as permanent Ampèrian magneti moments, suh as those of eletrons, protons and neutrons. 10 Suh permanent moments are not well explained in lassial eletrodynamis, and are a feature of quantum eletrodynamis. 8 In ase the field of the magneti harge flips the diretion of the moment m A from antiparallel to parallel B p, the work done by the field is 2m A B p, and this energy omes from the battery, as aording to eq. (19), W battery = IΔϕ/ =2IB p Area/ =2B p m A. The interation field energy remains zero during this proess. 9 This argument is given in se. 5.7 of [4] and on p. 986 of [14]. 10 If one assoiates Dira strings with magneti harges to resolve Comay s paradox for permanent Ampèrian magneti dipoles, then the interation energy (16) between the dipole and the string violates the onservation of energy that holds in the absene of suh strings. 6
Hene, Comay s paradox is an aspet of the protrusion of quantum physis into the lassial realm. Appendix B: Field Momentum and Angular Momentum In 1904, J.J. Thomson [15, 16] showed that the field momentum of a magneti harge and eletri harge, both at rest, is zero, P EM = p EM dvol = E B dvol = 0, (20) supposing that the field of the magneti harge is given by B p = p (r r p )/ r r p 3. He also showed that the field angular momentum of this system is L EM = r p EM dvol = r E B dvol = qp ˆR, (21) where unit vetor ˆR points from the eletri harge to the magneti harge. 11 For systems at rest with fields that fall off suffiiently quikly at large distanes, and for whih the magneti field an be dedued from a vetor potential, the field momentum and angular momentum an be omputed in other ways [20], inluding P EM = ρa (C) dvol, L EM = r ρa(c) dvol, (22) where ρ is the eletri harge density and A (C) is the vetor potential in the Coulomb gauge. However, the forms (22) appear to be problemati for the vetor potential A p assoiated with the Dira string of a magneti harge p, as this would imply P EM = qa(c) p (r q ), L EM = r q qa(c) p (r q ). (23) suh that P EM would be nonzero in general, while L EM would not point along ˆR = ˆr q when the magneti harge is at the origin. If the field momentum of a system at rest is nonzero, that system must also ontain an equal and opposite hidden momentum, suh that the total momentum of the system is zero. 12 A system of strutureless eletri and magneti harges (at rest) annot have any hidden (internal) momentum, so it is agreeable that the field momentum of this system is zero aording to eq. (10). However, if we onsider that the magneti harge is assoiated with a Dira string, there is some kind of hidden struture to the system, whih ould then have nonzero field momentum (with equal and opposite hidden momentum residing in the string). 11 The result (21) was antiipated by Darboux in 1878 [17] and by Poinaré in 1896 [18], but without interpretation of the vetor qp ˆR/ as the field angular momentum [19]. 12 For a review of the onept of hidden momentum, see [21]. 7
Referenes [1] K.T. MDonald, Fores on Magneti Dipoles (Ot. 26, 2014), http://physis.prineton.edu/~mdonald/examples/neutron.pdf [2] E. Comay, Interations of Stati Eletri and Magneti Fields, Lett. Nuovo Cim. 38, 421 (1983), http://physis.prineton.edu/~mdonald/examples/em/omay_ln_38_421_83.pdf [3] E. Comay, A Differene between Solenoid and Magneti Spin, J. Mag. Mag. Mat. 43, 59 (1984), http://physis.prineton.edu/~mdonald/examples/em/omay_jmmm_43_59_84.pdf [4] J.D. Jakson, Classial Eletrodynamis, 3rd ed. (Wiley, New York, 1999). [5] M. Tejedor and H. Rubio, Interations of Stati Eletri and Magneti Fields, Nuovo Cim. 103B, 669 (1989), http://physis.prineton.edu/~mdonald/examples/em/tejedor_n_103b_669_89.pdf [6] H.J. Lipkin and M. Peshkin, Magneti Monopoles, Eletri Currents, and Dira Strings, Phys. Lett. 179B, 109 (1986), http://physis.prineton.edu/~mdonald/examples/em/lipkin_pl_179b_109_86.pdf [7] H.J. Lipkin and M. Peshkin, Magneti Monopoles and Dipoles in Quantum Mehanis, Ann. N.Y. Aad. Si. 480, 210 (1986), http://physis.prineton.edu/~mdonald/examples/qed/lipkin_anyas_480_210_86.pdf [8] P.A.M. Dira, Quantised Singularities in the Eletromagneti Field, Pro. Roy. So. London A 133, 60 (1931), http://physis.prineton.edu/~mdonald/examples/qed/dira_prsla_133_60_31.pdf [9] P.A.M. Dira, The Theory of Magneti Poles, Phys. Rev. 74, 817 (1948), http://physis.prineton.edu/~mdonald/examples/qed/dira_pr_74_817_48.pdf [10] J.M. Getino, O. Rojo and H. Rubio, Interation between Eletri Currents and Magneti Monopoles, Europhys. Lett. 15, 821 (1991), http://physis.prineton.edu/~mdonald/examples/em/getino_epl_15_821_91.pdf [11] G. thooft, Magneti Monopoles in Unified Gauge Theories, Nul. Phys. B279, 276 (1974), http://physis.prineton.edu/~mdonald/examples/ep/thooft_np_b79_276_74.pdf [12] A.M. Polyakov, Partile Spetrum in Quantum Field Theory, JETP Lett. 20, 194 (1975), http://physis.prineton.edu/~mdonald/examples/ep/polyakov_jetpl_20_194_75.pdf [13] K.A. Milton, Theoretial and experimental status of magneti monopoles, Rep. Prog. Phys. 69, 1637 (2006), http://physis.prineton.edu/~mdonald/examples/ep/milton_rpp_69_1637_06.pdf [14] D.J. Griffiths, Dipoles at rest, Am. J. Phys. 60, 979 (1992), http://physis.prineton.edu/~mdonald/examples/em/griffiths_ajp_60_979_92.pdf [15] J.J. Thomson, On Momentum in the Eletri Field, Phil. Mag. 8, 331 (1904), http://physis.prineton.edu/~mdonald/examples/em/thomson_pm_8_331_04.pdf 8
[16] K.T. MDonald, J.J. Thomson and Hidden Momentum (Apr. 30, 2014), http://physis.prineton.edu/~mdonald/examples/thomson.pdf [17] G. Darboux, ProblèmedeMéanique, Bull. Si. Math. Astro. 2, 433 (1878), http://physis.prineton.edu/~mdonald/examples/em/darboux_bsma_2_433_78.pdf [18] H. Poinaré, Remarques sur une expèriene de M. Birkeland, Comptes Rendus Aad. Si. 123, 530 (1896), http://physis.prineton.edu/~mdonald/examples/em/poinare_ras_123_530_96.pdf http://physis.prineton.edu/~mdonald/examples/em/poinare_ras_123_530_96_english.pdf [19] K.T. MDonald, Birkeland and Poinaré: Motion of an Eletri Charge in the Field of a Magneti Pole (Apr. 15, 2015), http://physis.prineton.edu/~mdonald/examples/birkeland.pdf [20] K.T. MDonald, Four Expressions for Eletromagneti Field Momentum (April 10, 2006), http://physis.prineton.edu/~mdonald/examples/pem_forms.pdf [21] K.T. MDonald, On the Definition of Hidden Momentum (July 9, 2012), http://physis.prineton.edu/~mdonald/examples/hiddendef.pdf 9