Mansuripur s Paradox

Transcription

1 1 Problem Mansuripur s Paradox Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ (May 2, 2012; updated Marh 10, 2015) An eletrially neutral urrent-loop, 1 with magneti dipole m 0,thatisatrestinastati, uniform eletri field E 0 experienes no fore or torque. However, if that system is observed in the lab frame where the loop has veloity v parallel to E 0,andv, where is the speed of light, then there appears to be an eletri dipole moment, 2 p = v/ m 0 assoiated with the loop (in Gaussian units). Naïvely, the torque on this moment due to the eletri field (whih has strength E = E 0 + O(v 2 / 2 )inthelabframe)isτ = p E E 0 vm 0 /. Can/should the torque be different in different frames of referene? This paradox was first posed by Spavieri [3] 3 and more reently by Mansuripur [7]. 4,5 It is a oneptual variant of a famous problem by Shokley [16] that introdued the onept of hidden mehanial momentum. 6 The paradox is ompounded by supposing the stati, uniform eletri field is due to a single eletri harge q at large, fixed distane from the magneti moment in the rest frame of the latter, and the lab-frame veloity v is along the line of enters of the harge and moment. Disuss the fore on harge q in the lab frame. 7 1 This problem avoids the deliate issue of whether a urrent loop atually is eletrially neutral, naïvely assuming this to be so. For disussion of the small departures from this assumption required so that the transverse fore on the ondution harges equal the entripetal fore, see [1]. 2 See, for example, eq. (2) of [2]. 3 Spavieri built on earlier examples of Bedford and Krumm [4] that were also disussed by Namias [5] and by Vaidman [6]. 4 Mansuripur s main onern seems to be with an old debate about the eletrodynamis of moving media, and the validity of the Lorentz fore law in suh media, being unaware that there is no generally valid form of the Lorentz fore law that uses the total E, D, B and H fields of marosopi eletrodynamis. Among the extensive literature on this subjet, see, for example [8]. See also [9] by the author. Mansuripur laims to resolve the paradox by use of the so-alled Einstein-Laub fore density [10, 11], whih was derived from an invalid form of the Lorentz fore law and leads to nonzero self fores, as disussed in [12] and in se of [13]. Mansuripur also beomes enmeshed in the so-alled Abraham-Minkowski debate onerning the nature of eletromagneti momentum in media at rest (see, for example, [14], where Table 1 summarizes five variants of the eletromagneti fore law). Mansuripur onsiders that magneti dipoles based on ondutionurrent loops are equivalent to (quantum) magnetization, and notes that if one uses the Abraham field momentum rather than the Minkowski momentum, then there is no field momentum for magnetization in an eletri field, and no hidden mehanial momentum. However, this argument does not apply to a magneti dipole based on a loop of ondution urrent, whih appears to be the subjet of [7]. 5 A related issue is that the torque on a pair of harged partiles with relative motion is nonzero in general, but is zero if one partile is fixed in a partiular inertial frame [15]. 6 See [17] for ommentary on the relation between hidden mehanial momentum and the Abraham- Minkowski debate. 7 This onfiguration is that of the Aharonov-Bohm effet [18], whih has the lassial paradox that if the harge is in motion in the rest frame of the magneti moment, then the magneti moment, but not the harge, is subjet to a fore [19]. 1

2 2 Solution The magneti moment of a loop of urrent I of radius a has the magnitude m 0 = πa 2 I/, so the supposed lab-frame torque, of magnitude τ = E 0 vm 0 / = πa 2 IE 0 v/ 2, is an effet of order 1/ 2. Hene, disussion of the problem should inlude all relevant effets at order 1/ 2, even though the restrition that v suggests that terms of order 1/ 2 ould be ignored. 8 The presene of a torque τ = p E would suggest that the lab-frame eletri dipole moment p will preess about m 0 so as to bring it into alignment with the eletri field E. However, the apparent eletri dipole moment p = v/ m 0 is independent of E, andso annot move into alignment with that field. 9 It must be that the torque, if it exists, has no effet on the mehanial onfiguration of the system. A key to the resolution of the paradox is that moving dipoles do not have the properties that one might naïvely assoiate with them, based on understanding in their rest frame (se. 2.1). Analysis of a moving dipole in the lab frame is best obtained via transformation from its rest frame, whih implies that the torque is nonzero in the lab frame in the present example. Before detailed analysis of the lab-frame torque in se. 2.3, we first disuss a fore paradox (se. 2.2) whih illustrates that the lab-frame fields of a moving dipole are not those naïvely assoiated with the lab-frame moments p and m. The need for a nonzero torque in the lab frame is related to the existene of time-dependent hidden angular momentum in the lab frame, are disussed in ses The omplementary problem of an eletri dipole in an external magneti field is onsidered in se Finally, in se. 2.7 we note that the present example annot be realized in systems with ordinary (or superonduting) urrents, but ould only arise in rather aademi thought experiments if ondution-urrent loops are involved. 2.1 Ambiguity as to the Meaning of Moving Dipole Moments The onepts of dipole moments arose in examples of systems with harges at rest, and/or where all urrent densities J are divergene free (i.e., J = 0). In partiular, the definition of the magneti moment m as r J m = dvol (1) 2 assumes that J = 0, whih does not hold for the present example in the lab frame where the urrent density has bulk motion. One an define eletri and magneti dipole moments p and m in any frame in terms of volume integrals of the densities P and M of eletri and magneti polarization (in that frame), respetively, p = P dvol, m = M dvol. (2) 8 This point was emphasized by Coleman and Van Vlek [20] in their ommentary on [16]. 9 When v is parallel to E (and B = 0), there is no preession of the magneti moment m 0. See, for example, [21, 22]. This result applies also to spin-1/2 elementary partiles, whose magneti moment is the quantum equivalent of urrent loops, rather than pairs of equal and opposite magneti harges, as first noted by Fermi [23] based on onsiderations of the hyperfine interation. 2

3 The relativisti transformations of densities P and M were first disussed by Lorentz [24], who noted that they follow the same transformations as do the magneti and eletri fields B = H +4πM and E = D 4πP, respetively, P = γ ( P 0 + v M 0 ) (γ 1)(ˆv P 0 )ˆv, M = γ ( M 0 v P 0 ) (γ 1)(ˆv M 0 )ˆv, (3) where the inertial lab frame moves with veloity v with respet to the (inertial) rest frame of the polarization densities, and γ =1/ 1 v 2 / 2. Considerations of the fields moving eletri dipoles perhaps first arose in the ontext of Čerenkov radiation when the dipole veloity v exeeds the speed of light /n in a medium of index of refration n [25]. Later disussions by Frank of his pioneering work are given in [26, 27]. 10 As noted by Frank [27], 11 speialization of eq. (3) to point eletri and magneti dipole moments p and m, and to low veloities, leads to the forms p p 0 + v m 0, m m 0 v p 0, p 0 p v m, m 0 m + v p, (4) where p 0 and m 0 and the moments in the rest frame of the point partile, while p and m are the moments when the partile has veloity v in the lab frame. However, the fields assoiated with an eletri and/or magneti dipole moving at low veloity are not simply the instantaneous fields of the moments p and m, whih leads to ambiguities in interpretations of the physial signifiane of the dipole moments p and m in the lab frame. In se. III of [32] it is laimed that the magneti moment of a point eletri dipole p 0 whih moves with veloity v is m = v 2 p 0, (5) whih differs from eq. (4) by a fator of 2. Additional support for this form appears to be given in probs. 6.21, 6.22 and of [33]. The relation (5) (and an infinite set of other definitions) an be used (with are) in the lab frame, but we use the onvention of eq. (4) in the following. As disussed in se. 2.3, the fores and torques experiened by moving moments in the lab frame are not equal to those that might be expeted on the lab-frame values of moments p and m. This situation is somewhat anomalous in the theory of relativity, and it is therefore not surprising that it has led to various onfusions over the years, apparently beginning with the Trouton-Noble experiment [34]. The approah taken in the rest of this note is that onsistent results in the lab frame are best obtained by transformation of the orresponding results in the rest frame of the system. 2.2 The External Field is Due to a Single Distant Charge Suppose that the external field E 0 = E 0 ˆx near the dipole is due to a single harge q at x = d 0 for large d 0 (in the rest frame). 10 The relation that p v/ m 0 when p 0 = 0 appears in eq. (13) of [28], whih was inspired by se. 600 of [29]. See also [30]. That a moving urrent leads to an apparent harge separation was noted in [31]. 11 Ref. [27] reounts a past ontroversy that these transformations might involve the index n if the magneti dipole were not equivalent to an Ampèrian urrent loop. 3

4 In the lab frame we might argue that the fore on q is due to both the eletri field from the apparent eletri dipole p v/ m 0 and the magneti field of the magneti moment m m 0, F q ( =? q E p + v ) ( B m q p + v d 3 0 m ) 0 = 2q v d 3 0 m 0 d 3 0 = 2qvm 0 d 3 0 ŷ. (6) How an this be, as the fore on the harge is zero in the rest frame, and 3-fore is invariant under low-veloity transformations? Is the Lorentz fore law inompatible with relativity? The issue is that the fields of a dipole moving at low veloity are not the same as the (instantaneous) fields of the moments obtained by transformation of the moments from their rest frame. That is, the meaning of a moving dipole must be onsidered with are. The proper alulation is that F q = q (E + v ) B, (7) where E and B are the Lorentz transformations of the fields of the magneti moment m 0 in its rest frame, where E m =0, B m = m 0, (8) d 3 0 at harge q. The (low-veloity) transforms of these to the lab frame are E E m v B m = v B m, B B m + v E m = B m. (9) Using these in eq. (7) we find F q = 0 as expeted (and that the Lorentz fore law works fine). It remains disonerting that the eletri field in the lab frame at harge q is the negative of that inferred from the relation p v/ m 0. For additional disussion of these matters, see [2], partiularly se Torque in the Lab Frame A Naïve Analysis To alulate the torque on the dipole in the lab frame, we might presume that the apparent eletri dipole moment p v/ m 0 an be assoiated with harges ±Q with small separation, suh that p = Q(r + r ). In this ase, we ould write (following [5]) τ? p = r + Q (E + v ) B + r Q (E + v ) B = Q(r + r ) (E + v ) B ( v ) = p E + p B, (10) in the limit of a point dipole. Similarly, if the moving partile has magneti dipole m in the lab frame, it seems reasonable (again following [5]) that the partial torque τ p due to external E and B fields an be dedued by supposing that the dipole onsists of a pair of magneti 4

5 harges ±Q m subjet to the Lorentz fore Q m (B v/ E), whih leads (in the limit of a point dipole) to ( τ? m = r + Q m B v ) ( E + r Q m B v ) E = Q m (r + r ) (B v ) E ( v ) = m B m E. (11) The total torque (in ase of a Gilbert magneti dipole) is then τ = τ p + τ m? p E + m B + p ( v B ) m ( v E ). (12) In the present example, the external eletri field in the lab frame is just E E 0 to order v/, and the external magneti field is B v/ E 0 v/ E. Also, the magneti moment is m 0 and the eletri dipole moment is p v/ m 0 in this frame, to order v/. Then, to this order, the total torque on the moving dipole is ( τ? v ) p E m E. (13) While this omputation of the torque is not equal to p E in general, it does equal this if v is parallel to E (or if m 0 is parallel to v E) Analysis via Transformation of Torque from the Rest Frame However, a lesson of ses is that omputations in the lab frame involving the apparent dipole moments p and m (in that frame) may give invalid results. Furthermore, we desire an analysis for an Ampèrian magneti dipole. A better proedure is to transform omputations in the rest frame to the lab frame. For this, it is insightful to onsider the (antisymmetri) torque tensor τ μν = (r μ f ν r ν f μ ) dvol = r μ F ν r ν F μ, (14) where r μ =(t, r), the Lorentz 4-fore density is f μ =(J E/, ρe + u/ B) forasystem with harge-urrent-density 4-vetor (ρ, J), and the Lorentz 4-fore is F μ = γ u (F u/, F) for a system of partiles with veloities u and γ u =1/ 1 u 2 / 2 Thetorque3-vetorτ 0 =(τ 23, τ 13,τ 12 ) on a general dipole with eletri-harge dipole moment p 0 and Ampèrian magneti moment (due to eletrial urrents) m 0 in its rest frame is ( τ 0 = r 0 ρ 0 E 0 + J ) 0 B 0 dvol 0 (r0 B 0 )J 0 (r 0 J 0 )B 0 = ρ 0 r 0 dvol 0 E 0 + dvol 0 = p 0 E 0 + m 0 B 0, (15) noting that for steady urrent J 0, r 0 J 0 dvol 0 = n q n r 0n u 0n = 1 d q n r0n 2 =0, (16) 2 dt 5 n

6 and (r0 B 0 )J 0 dvol 0 = n = n = n q n (r 0n B 0 )u 0n q n (r 0n B 0 )u 0n 2 q n (r 0n u 0i ) 2 n q n (u 0n B 0 )r 0n d q n (r 0n B 0 )r 0n 4 dt B 0 = m 0 B 0, (17) n where the magneti moment in the rest frame is m 0 = n q n (r 0n u 0n ) 2 r0 J 0 = dvol 0. (18) 2 The rest-frame torque tensor also has omponents r0,i (J 0 E 0 ) τ 0,0i = τ 0,i0 = t 0 f 0,i dvol dvol = [m 0 E 0 ] i, (19) for a system subjet to zero total Lorentz fore, f 0 dvol = 0, noting that r0 (J 0 E 0 ) dvol 0 = q n (u 0n E 0 )r 0n n = q n (u 0n E 0 )r 0n q n (r 0n E 0 )u 0n + 1 d q n (r 0n E 0 )r 0n dt n n n = q n (r 0n u 0n ) E 0 = m 0 E 0. (20) 2 n The result (19) may be surprising, in that for a ase like the present example, in whih the rest-frame 3-torque (15) is zero, the 4-torque is non-zero, and onsequently the 3-torque is nonzero in other frames. 12 This peuliar result holds only for a partile with an (Ampèrian) magneti dipole moment due to eletri urrents. If the magneti dipole onsisted of a pair of opposite magneti harges (Gilbert dipole), τ 0,0i would be zero (assuming that the system does not have a magneti urrent loop and assoiated Gilbertian eletri dipole moment). Hene, the torque on a moving magneti dipole in an external eletri fields provides a lassial distintion between Ampèrian and Gilbertian moments (whih distintion is often onsidered to be relevant only in the quantum domain). For low veloities of the lab frame we have that r r 0 + vt, dvol dvol 0, (21) ρ ρ 0 + J 0 v, J J ρ 0 v, (22) E E 0 v B 0, B B 0 + v E 0. (23) 12 This effet invalidates a laim in [35] that if the 3-torque τ is zero in one (inertial) frame it is zero in all (inertial) frames. 6

7 Hene, the lab-frame Lorentz fore density is related to that in the rest frame by ρe + J B ρ 0E 0 + J 0 B 0 + (J 0 E 0 )v 2, (24) whih an be onfirmed expliitly using eqs. (22)-(23). 13 The lab-frame 3-torque at time t =0isthus, ( τ = r ρe + J B ) dvol τ 0 + r (J 0 E 0 )v r0 (J 0 E 0 ) dvol τ dvol 0 v = p 0 E 0 + m 0 B 0 + v (m 0 E 0 ) (26) realling eqs. (15) and (20). We an express the lab-frame torque in terms of lab-frame quantities using eq. (4), τ p 0 E 0 + m 0 B 0 + v (m 0 E 0 ) (p v ) m (E + v ) B + (m + v ) p + v [( m + v ) p (E + v )] B ( v ) ( v ) p E + m B + p B B p (B v ) E ( v ) ( v ) + E m m E + v (m E) = p E + m B + v (p B) (Ampèrian magneti dipole), (27) noting the identity that a (b ) =b (a ) (a b). This result differs somewhat from that of eq. (12). 14 In the rest frame of the present example the 3-vetor τ 0, eq. (15), is zero, while to order 1/ 2 the lab-frame 3-torque is τ p E (present example, Ampèrian magneti dipole), (28) 13 We ould also represent the Lorentz fore density in terms of polarization densities P and M, replaing ρ by P and J by P/ t + M, where we an assoiate the eletri and magneti dipole moments p and m of a point partile at the origin with free polarization densities P and M aording to P = p δ 3 (r) and M = m δ 3 (r). These relations are best established first in the rest frame, where P 0 = p 0 δ 3 (r 0 )and M 0 = m 0 δ 3 (r 0 ). and then transforming to the lab frame via eq. (3), with the result for low veloities, P P 0 + v M 0 = ( p 0 + v ) m 0 δ 3 (r), M = M 0 v P 0 = ( m 0 v ) p 0 δ 3 (r). (25) Then, the inverse relations p = P dvol and m = M dvol are onsistent with eq. (4). See [36, 37] for use of this approah for the present example. 14 Mansuripur states without apparent derivation in either [7] or [38] that τ = p E+m B for a system on whih the total fore F is zero. 7

8 sine B = 0 here. This onfirms the existene of a nonzero lab-frame torque in the present example, so paradox remains as its physial signifiane in ase of an Ampèrian magneti dipole. If instead the magneti dipole m 0 were made from opposite magneti harges (and the eletri dipole p 0 is made from opposite eletri harges), then τ 0,0i = 0, and the lab-frame 3-torque is given by τ = τ 0 = p 0 E 0 + m 0 B 0 p E + m B + v (p B m E) (Gilbert magneti dipole). (29) Thus, the naïve omputation (12) is slightly wrong in general, whih further illustrates the diffiulty in interpreting lab-frame dipole moments. In the rest frame of present example the torque 4-tensor is zero (for a Gilbert magneti dipole), so this tensor is also zero in the lab frame, as is the lab-frame 3-torque, 15 τ = 0 (present example, Gilbert magneti dipole). (30) Mansuripur s paradox vanishes for the ase of a Gilbert magneti dipole, but remains for an Ampèrian one Transformation in Two Steps An analysis that uses rest-frame and lab-frame quantities in the same steps an be given by supposing that the moments in eqs. (10)-(11) are those in the rest frame (and that the magneti moment is Gilbertian), while the positions, veloities and fields are those in the lab frame [5]. This analysis an be regarded as a partial transformation of the analysis in the rest frame (ignoring possible nonzero omponents τ 0i in that frame), in whih the positions and fields are transformed to the lab frame, but the moments are not transformed. In this hybrid view, the lab-frame torques are τ p = p 0 E + p 0 ( v B ), τ m = m 0 B m 0 ( v E ). (31) In the present example, p 0 and B are zero, while v E, so the total torque τ = τ p + τ m in the lab frame is zero, aording to eq. (31). We an omplete the partial transformations of eq. (31), realling eq. (4), τ p (p v ) ( v ) m E + p B, (32) τ m = (m + v ) p ( v ) B m E. (33) 15 In Mansuripur s example, B = 0 in the lab frame where p = v/ m, so eq. (29) beomes τ = p E v/ (m E) =(v m)e/ (E m)v/ =0,sineE and v are parallel to one another and perpendiular to m. 8

9 Using the identity that a (b ) =b (a ) (a b) the total torque in the lab frame an be written as τ = τ p + τ m p E + m B + v (p B m E) (Gilbert magneti dipole).(34) as previously found in eq. (29), where all quantities in this expression are in the lab frame. 16 The torque (34) vanishes in the present example where B =0,m m 0, p v/ m 0 and v E. 2.4 Field Momentum and Hidden Mehanial Momentum in the Rest Frame The example of Mansuripur ontains an additional subtlety that deserves omment. Namely, even in its rest frame the system possesses nonzero eletromagneti field momentum P EM. 17 For systems in whih effets of radiation and of retardation an be ignored, the eletromagneti momentum an be alulated in various equivalent ways [40], ϱa E B V J P EM = dvol = dvol = dvol, (35) 4π 2 where ϱ is the eletri harge density, A is the magneti vetor potential (in the Coulomb gauge where A =0),E is the eletri field, B is the magneti field, V is the eletri (salar) potential, and J is the eletri urrent density. The first form is due to Faraday [41] and Maxwell [42], the seond form is due to Poynting [43], Poinaré [44] and Abraham [45], and the third form was introdued by Furry [46]. Sine a system at rest must have zero total momentum [20], it must also possess a hidden mehanial momentum P hidden equal and opposite to the field momentum. 18 This hidden momentum is a relativisti effet of order 1/ 2. 19,20,21 We evaluate P EM for a magneti moment m 0 = πa 2 I ẑ/ due to urrent I whih flows in a irular loop of radius a subjet to external eletri field E 0 that makes angle α to m, i.e., 16 Thus, the argument in [5] applies to a Gilbert dipole but not to an Ampèrian dipole, as noted in [39]. This argument is not self-evidently orret to this author, so this setion provided the validation needed (in my view). 17 This is also true for the onfiguration of the Aharonov-Bohm effet [18]. 18 For ommentary on hidden momentum by the author, see [47]. 19 The eletromagneti field momentum P EM is also an effet of order 1/ 2 in that the vetor potential A and the magneti field B are of order 1/, so all three forms of eq. (35) are of order 1/ As pointed out by Griffiths [39], and earlier by Haus [48], lassial systems with nonzero hidden mehanial momentum must have moving parts. If magneti harges existed, a system of stati eletri and magneti harges would have no hidden mehanial momentum, and its total field momentum must also be zero, as expliitly verified in [39] for simple systems inluding dipoles. For example, a Gilbertian magneti dipole involves no eletri urrent, so eq. (36) would give zero field momentum, and hene zero hidden mehanial momentum. 21 Mansuripur [38] denies the existene of hidden mehanial momentum, while apparently aepting the usual relations (35) for eletromagneti field momentum. This view seems to be based on the use of the Abraham rather than Minkowski field momentum, and the mistaken belief that all magneti dipoles an be represented via a (quantum) magnetization density. Mansuripur may have been led to this view by his emphasis of the so-alled Einstein-Laub fore density [10] (whih is more onsistent with the Gilbertian result (34), as shown by Cross [37]). 9

10 E 0 = E 0 (sin α ˆx +osα ẑ). The largest magneti field is inside the loop, in the ẑ diretion, so the seond form of eq. (35) indiates that P EM will be in the ŷ diretion. This result is ounterintuitive in that the diretion of the momentum is not related to the diretion of the veloity (if any). In the present problem P EM is perpendiular to v, so the field angular momentum (39) is nonzero and position/time dependent for motion along the x-axis. We use the third form of eq. (35) to ompute the field momentum. The external eletri field an be derived from the salar potential V = E 0 (x sin α+z os α),and the y-omponent of J dvol is Iaos φdφ in ylindrial oordinates (ρ, φ, z) entered on the moment. Then, noting that x = a os φ and z = 0 on the loop, we find P EM,y = 2π VJy ( E 0 a os φ sin α)(iaos φ) dvol = dφ = πa2 IE 0 sin α = m 0E 0 sin α. (36) That is, 22 P EM = E 0 m 0 (Ampèrian magneti dipole). (37) As the total momentum of the system at rest must be zero, we infer that there exists hidden mehanial momentum given by P hidden = P EM = E 0 m 0. (38) The momenta (37)-(38) are effets of order 1/ 2. See, for example, [49] for a lassial model of the hidden momentum (38). 2.5 Torque and Changing Hidden Angular Momentum A lassial (Ampèrian) magneti moment m 0 has intrinsi mehanial angular momentum L 0 =2Mm 0 /Q where M and Q are the mass and harge of the partiles whose motion generates the moment. In addition, the moment is assoiated with hidden mehanial angular momentum given by L hidden = r P hidden, (39) where r is the position of the enter of the moment. In the inertial frame where the magneti moment has position r = vt = vt ˆx, withv (suh that the eletri field and the moment have the same values as in the moment s rest frame to order v/, and the field momentum and the hidden mehanial momentum have their rest-frame values to order 1/ 2 ), the mehanial angular momentum of the system is 23 L meh = L 0 + L hidden L 0 vt E m 0. (40) 22 The result (37) first appeared in eq. (37) of [46]. 23 The intrinsi mehanial angular momentum L 0 has orretions at order v 2 / 2, but these are timeindependent in the lab frame. 10

11 To support this time-varying mehanial angular momentum, we expet from lassial mehanis that the system must be subjet to a torque, τ = dl meh dt v E m 0 When E and v are parallel, we an rewrite eq. (41) as. (41) τ = E v m 0 = p E (present example, Ampèrian magneti dipole). (42) This equals the lab-frame torque (28) (for v E) omputed in se via transformation from the rest frame, and the paradoxial torque is needed to ause the hanges in the lab-frame hidden mehanial angular momentum. 24 There also exists the eletromagneti field angular momentum L EM = r P EM = L hidden, (43) whose time rate of hange is equal and opposite to that of the hidden mehanial angular momentum, dl EM = dl hidden. (44) dt dt We an say that the torque dl hidden /dt auses the hange in L EM Flow of Field Momentum and Angular Momentum As first noted by Poinaré [44], an eletromehanial system ontains densities p meh of mehanial momentum as well as p EM = S = E B (45) 2 4π of eletromagneti field momentum (in media where E = D and B = H as in the present example). Changes in these momentum densities an be related to a fore density f whih is the divergene of a stress tensor T, f = T = p EM t + p meh t, (46) where the stress tensor an be written as the sum of mehanial and eletromagneti stress tensors, T = T meh + T EM, (47) with T EM ij = E ie j + B i B j 4π δ ij E 2 + B 2 8π, (48) 24 Essentially similar arguments have been given by Griffiths and Hnizdo [36], by Cross [37], by Vanzella [50], and by Saldanha [51]. Sine eq. (28) was dedued from the Lorentz fore law, this law is not brought into doubt by the present example, ontrary to the laim in [7]. 11

12 whih is often alled the Maxwell stress tensor. Outside of matter, where the density p meh and the tensor T meh are zero, eq. (46) an be written as p EM t T EM = 0 (outside matter), (49) whih indiates that the (tensor) flux of eletromagneti momentum p EM is given by T EM in empty spae. 25 We define the densities of eletromagneti and mehanial angular momenta as l EM = r p EM, l meh = r p meh. (50) Then, the vetor moment of eq. (46) leads to the relation r f = r T = r T = l EM t + l meh, (51) t where the ross produt r T involves only the seond index of T, andthei omponent of r T is, noting that the tensor T is symmetri, r T i = ɛ ijk x j T lk x l = x l ɛ ijk x j T lk = r T i. (52) See, for example, [53]. Outside of matter the density l meh is zero, and eq. (51) an be written as l EM r T EM = 0 (outside matter), (53) t whih indiates that the tensor r T (whose omponents are ɛ ikl x k T jl ) equals the (tensor) flux of the density l EM of eletromagneti angular momentum in empty spae. We an regard the hanges in the lab-frame eletromagneti-angular-momentum density as due to the flux of angular momentum, desribed by the tensor r T EM,outfromthetimedependent hidden mehanial angular momentum in the urrents of the moving magneti moment Eletri Dipole in an External Magneti Field In the omplementary example of a point partile with eletri dipole moment p 0 at rest in a onstant, uniform magneti field B 0 there is no torque on the partile in either the rest frameorinalabframethathaslowveloityv parallel to B 0, realling eq. (26). If the magneti field were due to a distant magneti monopole, there would be no hidden mehanial momentum in the system, as noted in [39]. Hene there would be no hanging 25 Another interpretation of eq. (49) is that the fore density T EM auses thetimerateofhangeofthe eletromagneti-momentum density p EM. Suh interpretation is deliate in that T EM = (E B)/4π t, so eq. (49) is more of a tautology than a ause/effet relation. 26 Thetorquedensityr f an be said to ause the hanges in the densities l EM and l meh of eletromagneti and mehanial angular momenta. However, this interpretation has the same deliay disussed in the previous footnote. 12

13 hidden mehanial angular momentum in the lab frame, and no torque would be needed there. In a more realisti example the magneti field is due to eletrial urrents in, say, an Ampèrian magneti dipole. In this ase the system at rest possesses nonzero eletromagneti field momentum P EM = B 0 p 0 /2 [52]. The hidden mehanial momentum of the system is the negative of this, and in the lab frame the time rate of hange of the orresponding hidden mehanial angular momentum is dl hidden /dt = v (B 0 p 0 /2) = vp 0 B 0 ẑ/2, for moment p 0 in the z-diretion and B 0 and v in the x-diretion. It may appear paradoxial that the hidden mehanial angular momentum is time dependent in the lab frame, but there is no torque there on the dipole. However, we should also onsider the effet of the fields of the eletri dipole on the soure of the external magneti field. For example, suppose the magneti field B 0 = B 0 ˆx at the origin is due to a urrent loop at x = d 0 on the x-axis with (Ampèrian) magneti moment m 0 = B 0 d 3 0 ˆx/2. If the eletri dipole at the origin has moment p 0 = p 0 ẑ, then the eletri field on the magneti moment is E = p 0 /d 3 0. The torque on the magneti moment in the lab frame follows from eq. (26) as τ v/ (m 0 E) = vm 0 E ẑ/ = vp 0 B 0 ẑ/2. This torque equals the time rate of hange of the hidden mehanial angular momentum found above, so the absene of a lab-frame torque on the eletri dipole at the origin (at time t = 0) in this example is not paradoxial Physial Realizations of Magneti Moments The behavior of a moving urrent loop in an external eletri field depends on the physial nature of the urrent. If the urrent flows in a resistive ondutor, that ondutor would shield the urrent from a onstant, uniform external eletri field E if the ondutor is at rest or in uniform motion with respet to the field. In this ase there would be no Lorentz fore on the urrent due to the external field, and no torque in the frame where the urrent loop has veloity v. Similarly, if the urrent loop is a superondutor, the superurrent is shielded from the external field, and there is no torque. A model of a neutral urrent loop that ould realize Mansuripur s paradox is a pair of nononduting, oaxial disks with positive harge fixed to the rim of one and negative harge on the other, with the disks rotating in opposite senses with the same magnitude of angular veloity. The paradox applies also to models in whih the urrent is a harged, ompressible gas or liquid that flow inside a nononduting tube (models i and iii of [6]). 28 In sum, the present example an be realized only in rather aademi thought experiments if the magneti momenent is due to ondution urrent loops. The most pratial realization of the present example would involve magneti fields due 27 A deliate point is that the eletromagneti field momentum of the magneti dipole in the eletri field oftheeletridipoleisp EM = E m 0 /, whih equals field momentum B 0 p 0 /2 oftheeletridipole in the field of the magneti dipole. These are not different momenta, but two omputations of the field momentum of the system. See [52] for a omputation in whih the magneti field is due to a long solenoid magnet. 28 To have an eletrially neutral urrent loop, one must postulate a pair of suh tubes that ontaining opposite harged gas/liquid flowing in opposite diretions. 13

14 to intrinsi (Ampèrian) magneti momentums, suh as assoiated with a nononduting permanent magnet, or a neutron. Aknowledgment The author thanks Daniel Cross, David Griffiths, Vladimir Hnizdo, Zurab Silagadze, Lev Vaidman and Daniel Vanzella for e-disussions that resulted in substantial hanges from earlier versions of this note. Referenes [1] K.T. MDonald, Charge Density in a Current-Carrying Wire (De. 23, 2010), [2] V. Hnizdo and K.T. MDonald, Fields and Moments of a Moving Eletri Dipole (Nov. 29, 2011), [3] G. Spavieri, Proposal for Experiments to Detet the Missing Torque in Speial Relativity, Found. Phys. Lett. 3, 291 (1990), [4] D. BedfordandP. Krumm, On the origin of magneti dynamis, Am. J. Phys. 54, 1036 (1986), [5] V. Namias, Eletrodynamis of moving dipoles: The ase of the missing torque, Am.J. Phys. 57, 171 (1989), [6] L. Vaidman, Torque and fore on a magneti dipole, Am. J. Phys. 58, 978 (1990), [7] M. Mansuripur, Trouble with the Lorentz Law of Fore: Inompatibility with Speial Relativity and Momentum Conservation, Phys. Rev. Lett. 108, (2012), [8] D. Shieber, Some Remarks on the Classial Eletrodynamis of Moving Media, Appl. Phys. 14, 327 (1977), [9] K.T. MDonald, Methods of Calulating Fores on Rigid Magneti Media (Marh 18, 2002), [10] A. Einstein and J. Laub, Über die elektromagnetishen Grundgleihungen für bewegte Köper, Ann. Phys. 26, 532 (1908), Über die im elektromagnetishen Felde auf ruhende Köper ausgeübten ponderomotorishen Kräfte, Ann. Phys. 26, 541 (1908), 14

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17 [38] M. Mansuripur, Maxwell s marosopi equations, the energy-momentum postulates, and the Lorentz law of fore, Phys. Rev. E 79, (2009), Nature of eletri and magneti dipoles gleaned from the Poynting theorem and the Lorentz fore law of lassial eletrodynamis, Opt. Comm. 283, 4594 (2010), [39] D.J. Griffiths, Dipoles at rest, Am. J. Phys. 60, 979 (1992), [40] K.T. MDonald, Three Expressions for Eletromagneti Field Momentum (Apr. 10, 2006), [41] Part I, se. IX of M. Faraday, Experimental Researhes in Eletriity (Dover Publiations, New York, 2004; reprint of the 1839 edition). [42] Ses and 57 of J.C. Maxwell, A Dynamial Theory of the Eletromagneti Field, Phil. Trans. Roy. So. London 155, 459 (1865), [43] J.H. Poynting, On the Transfer of Energy in the Eletromagneti Field, Phil. Trans. Roy. So. London 175, 343 (1884), [44] H. Poinaré, La Théorie de Lorentz et la Prinipe de Réation, Arh. Neer. 5, 252 (1900), Translation: The Theory of Lorentz and the Priniple of Reation, [45] M. Abraham, Prinzipien der Dynamik des Elektrons, Ann. Phys. 10, 105 (1903), [46] W.H. Furry, Examples of Momentum Distributions in the Eletromagneti Field and in Matter, Am. J. Phys. 37, 621 (1969), [47] K.T. MDonald, On the Definition of Hidden Momentum (July 9, 2012), [48] H.A. Haus, Eletrodynamis of Moving Media and the Fore on a Current Loop, Appl. Phys. A 27, 99 (1982), [49] K.T. MDonald, Hidden Momentum in a Current Loop (June 30, 2012), [50] D.A.T. Vanzella, Comment on Trouble with the Lorentz law of fore: Inompatibility with speial relativity and momentum onservation, (May 4, 2012), 17

18 [51] P. Saldanha, Comment on Trouble with the Lorentz Law of Fore: Inompatibility with Speial Relativity and Momentum Conservation, [52] K.T. MDonald, Eletromagneti Momentum of a Capaitor in a Uniform Magneti Field (June 18, 2006), [53] L. Levitov, MIT Problem Set #2 (Feb. 19, 2004), Levitov s prob. 1 is taken from prob. 5, hap. 3 of J. Shwinger, L.L. DeRaad, Jr, K.A. Milton and W.-Y. Tsai, Classial Eletrodynamis (Perseus Books, Reading, MA, 1998). 18