Electric/magnetic dipole in an electromagnetic field: force, torque and energy

Transcription

1 Eur. Phys. J. Plus (2014) 129: 215 DOI /epjp/i y Regular Artile THE EUROPEAN PHYSICAL JOURNAL PLUS Eletri/magneti dipole in an eletromagneti field: fore, torque and energy Alexander Kholmetskii 1,2,a, Oleg Missevith 3, and T. Yarman 3,4 1 Belarusian State University, Minsk, Belarus 2 Institute for Nulear Problems, Belarusian State University, Minsk, Belarus 3 Okan University, Akfirat, Istanbul, Turkey 4 Savronik, Eskisehir, Turkey Reeived: 16 June 2014 Published online: 9 Otober 2014 Soietà Italiana di Fisia / Springer-erlag 2014 Abstrat. In this paper we ollet the relativisti expressions for the fore, torque and energy of a small eletri/magneti dipole in an eletromagneti field, whih we reently obtained (A.L. Kholmetskii et al., Eur. J. Phys. 33, L7 (2011), Prog. Eletromagn. Res. B 45, 83 (2012), Can. J. Phys. 9, 576 (2013)) and onsider a number of subtle effets, haraterized the behavior of the dipole in an external field, whih seem interesting from the pratial viewpoint. 1 Introdution The problem of derivation of relativisti expressions for the fore and torque experiened by a small eletri/magneti dipole in an external eletromagneti field was the subjet of essential interest of researhers at the seond half of the 20th entury, and ontinues to attrat attention up to date. The essential progress in this area was ahieved with the derivation of the Bargmann-Mihel-Telegdi (BMT) equation (1), whih desribes the time evolution of the partile s spin with the inlusion of Thomas preession [2]. At the same time, the approah used in the derivation of the BMT equation (the introdution of the spin four-vetor with the vanishing time omponent in the proper frame of the partile [1,3]) is onvenient to determine the motion of the partile s spin in its proper frame in terms of the external eletromagneti (EM) field measured in the laboratory. However, we remind that the transformation of the torque omponents is guided by the orresponding torque four-tensor [4], and one an hek that its appliation does not allow to obtain an expliit analytial expression for the torque in the laboratory frame. At the same time, the derivation of the fore and torque expressions, where all quantities are measured in the laboratory, has a prinipal importane in the onsistent investigation of relativisti effets, related to the motion of an eletri/magneti dipole in an EM field. In addition, before the disovery of the hidden momentum of magneti dipoles [5], its ontribution to the fore and torque was not taken into aount; moreover, it ontinues to be ignored in some modern publiations, inluding textbooks (e.g., [6]). One should mention that even in the past years, the problem of deriving the orret expression for the fore and torque on a ompat dipole faed a number of errors made by various authors (e.g., [7 11]). Nevertheless, the orret Lorentz-invariant expression for the fore on a moving eletri/magneti dipole, seemingly for the first time, has been ahieved in our reent ontribution [12], whih opened the way for the derivation of the relativisti equations for the desription of a torque on a moving dipole [13], and its eletromagneti energy [14]. Now it seems useful from the pratial viewpoint to ollet these equations altogether in a single paper and to disuss some subtle points, related to their physial meaning, whih were not speially ommented in the mentioned papers [12 14]. The obtained equations for the fore F, torque T and energy E of a small eletrially neutral dipole with the proper eletri p 0 and magneti m 0 dipole moments, moving at veloity ν in the external eletri E and magneti B a khol123@yahoo.om

2 Page 2 of 13 Eur. Phys. J. Plus (2014) 129: 215 fields, are as follows [12 14]: F = (p E)+ (m B)+ 1 d dt (p B) 1 d (m E), dt (1) T = p E + m B + 1 ν (p B) 1 ν (m E), (2) E = (p E) (m B) 1 (p B) ν + 1 (m E) ν, (3) where ( 1) p = p 0 ν 2 (p 0 ν)ν + ν m 0, (4) ( 1) m = m 0 ν 2 (m 0 ν)ν + p 0 ν (5) are, respetively, the eletri and magneti dipole moments of a moving dipole in the frame of observation [6, 15, 16], and =(1 ν 2 / 2 ) 1/2 is the Lorentz fator. Here we stress that eq. (3) determines the eletromagneti energy of the dipole only, and does not inlude the frations of energy absorbed (extrated) in the power supply and in a soure of external magneti field [14]. We point out that the quantities entering into eqs. (1)-(3) are defined in the frame of observation (laboratory frame), and these equations also an be expressed either via the quantities measured in the rest frame of a dipole, or via some ombination of these quantities (e.g., the proper eletri and magneti dipole moments, measured in the rest frame of a dipole, and eletromagneti fields measured in the laboratory. As we will see below, the latter ombination is onvenient for the analysis of the energy of the dipole). In set. 2 we disuss the physial meaning of eqs. (1) (3), fousing mainly our attention on the expressions for torque (2) and energy (3) for a moving dipole, insofar as the expression for the fore experiened by a dipole has been already disussed in the related publiations [9, 11, 12]. Finally, we onlude in set Expressions for the fore, torque and energy of a moving dipole: Origin and impliations 2.1 Fore on a moving eletri/magneti dipole in an eletromagneti field In this subsetion we address eq. (1) and remind that it originates from the Lorentz fore law, applied to the eletrially neutral ompat bunh of harges [9] and also inludes the fore omponent due to time variation of the hidden momentum of the magneti dipole (see, e.g. [3,5,17 19]). More speifially, the sum of the terms ontaning p (i.e. the first and third terms) desribes the Coulomb interation of the harges of the eletri dipole with the eletri field, and the interation of the onvetive urrents of the dipole with the magneti field. The sum of the terms ontaning m (i.e. the seond and fourth terms) is responsible for the interation of the proper (losed) urrent of the magneti dipole with the magneti field, and also inludes the hidden-momentum ontribution. A real situation might be more ompliated, beause the moving eletri dipole develops the magneti dipole moment, and the moving magneti dipole possesses an eletri dipole moment, see eqs. (4), (5). However, we skip the detailed analysis of the various terms in eq. (1), whih an be found in the above-mentioned publiations [9, 11, 12], and would like to give answer to the question, whih we ask time by time: why the Lorentz fore law alone fails to desribe the hidden-momentum ontribution to the resultant fore on a magneti dipole, and the latter should be exogeneously added to the motional equation for a bunh of harges in an eletromagneti field? The general answer to this question is straightforward: the Lorentz fore law is relevant only for point-like harges, whereas an eletri/magneti dipole always represents some finite-size distribution of the harges. If so, an external eletromagneti field should indue some hanges in its inner volume (polarization, magnetization, the appearane of mehanial stresses, et.). The mentioned hanges an be lassified as some seondary effets emerging in the bulk of the dipole due to the external eletromagneti field. It was pointed out for the first time in ref. [5] that suh seondary effets an be attributed to the appearane of the momentum omponent of the magneti dipole, even if the dipole, as the whole, is at rest in the frame of observation; that is why this omponent was named as the hidden momentum. Moreover, it was further proven in ref. [17] that for a magneti dipole, resting in an eletri field, being reated by a stati-harge distribution, the hidden momentum of the dipole is exatly equal with the reverse sign to the interational eletromagneti momentum of this onfiguration, and the time variation of the hidden momentum (whih indues the appearane of fore on the dipole) is exatly equal with the reverse sign to time variation of the field momentum. Therefore, the hidden momentum plays an important role in the energy-momentum balane for isolated systems, whih inlude harges and magneti dipoles.

3 Eur. Phys. J. Plus (2014) 129: 215 Page 3 of 13 F hidden F p q m. v F in -F p y x Fig. 1. Interation of a resting harge q with a moving magneti dipole m. The dipole is moving without frition along the x-axis inside a thin insulating tube (not shown in the figure), and the harge is rigidly attahed to the tube. The tube has a single degree of freedom to move along/opposite to the y-axis. To demonstrate the validity of this statement, let us onsider the following problem (see fig. 1). Let a small magneti dipole with the proper magneti dipole moment m{0, 0, m} move inside a thin isolated tube oriented along the x-axis. Let a point-like harge q be rigidly fixed inside the same tube at the origin of the oordinates. We further asume that at the initial time moment t = 0 the magneti dipole has the x-oordinate X 1, and analyze the implementation of the energy-momentum onservation law, onsidering another time moment t = T, when the x-oordinate of the dipole beomes equal to X 2. First onsider the fore, ating on the resting harge due to the moving magneti dipole, whih has two omponents: the interation of the harge q with the indued eletri field E in = A t, where A is the vetor potential of the dipole; the Coulomb interation of the harge q with the eletri dipole moment, developed by the moving magneti dipole. Assuming for simpliity a weak relativisti limit to the auray 2, whih is suffiient for further analysis, we straightforwardly alulate both fore omponents. The vetor potential of the magneti dipole at the loation of the harge has only the y-omponent A y = m x 2, and the orresponding fore on the harge due to the indued eletri field (F in ) y = q A y t = 2qmv x 3. (6) Further, for the problem in question the eletri dipole moment of the moving magneti dipole lies in the negative y-diretion, and is equal to p y = vm. (7) Hene the fore on the harge due to eletri dipole (7) reads as (F p ) y = qmv x 3. (8) Considering now the reative fore on the magneti dipole on behalf of the resting harge, and ignoring (for a moment) the fore due to the time variation of the hidden momentum (i.e. the last term in eq. (1)), we get the single fore omponent, desribing the Coulomb fore on the eletri dipole (7) due to the resting harge. This fore omponent lies in the negative y-diretion and has the opposite sign with respet to the fore (8). As a result, we have the exat balane of the Coulomb fores between harge and dipole, so that the non-ompensated fore (6) remains. Hene, while the dipole moves along the tube, the latter experienes the net fore (6) and does aelerate in the negative y-diretion during the motion of the magneti dipole from the point X 1 to the point X 2. This result already looks ounter-intuitive. What is more, we an show that it violates the law of onservation of total energy (partiles and fields). Indeed, the momentum transmitted to the tube during the time T is equal to (P tube ) y = T 0 T (F in ) y dt = 0 2qmv X2 x 3 dt = 2qm X 1 x 3 dx = qm ( 1 X ) X2 2, (9)

4 Page 4 of 13 Eur. Phys. J. Plus (2014) 129: 215 and the additional kineti energy of the system magneti dipole, harge and tube in the non-relativisti limit reads as E k = (P tube) 2 y 2M, (10) where M is the total rest mass of this onfiguration. The hange of kineti energy (10) an be ounterated only by the orresponding hange of the eletromagneti interational energy between magneti dipole and harge. This energy has only the eletri omponent, aused by the interation of the eletri dipole moment (7) with the harge q. However, one an see that this energy is equal to zero at any time moment 0 <t<t. Indeed, the salar potential of the eletri dipole moment (7) ϕ = p x/x 3 is equal to zero at the loation of the harge q at any time moment. Thus, the energy (10) is taken from nothing. This paradox is immediately resolved, when the hidden-momentum ontribution (the last term of eq. (1)) is taken into aount. Indeed, the eletri field produed by the harge at the loation of the magneti dipole is equal to E x = q/x 2,and (F hidden ) y = mv E x x = 2qmv x 3. (11) Comparing now eqs. (6) and (11), we see that their sum is equal to zero and thus, no net fore ats on the tube, so that it remains at rest during the entire time interval 0 <t<t. Thus, the total energy-momentum onservation law is perfetly fulfilled for the problem in fig. 1, when the fore omponent due to the time variation of the hidden momentum is taken into aount. 2.2 Torque on a moving eletri/magneti dipole in an eletromagneti field The total torque on an eletri/magneti dipole an be presented in the form T = (r f total )d, (12) where f total is the total fore density, r is for the position vetor of any point inside the dipole (where the enter of mass of the dipole is loated in the origin of the oordinates), and stands for the volume of the dipole. Like for the total fore on a dipole, the fore density f total should be presented as the sum where f L is the density of the Lorentz fore f total = f L + f h, (13) f L = ρ total E + 1 j total B (14) (see, e.g., refs. [3,9]), and f h = 1 (M E) (15) t is the density of the fore due to the hidden-momentum ontribution [13]. Here ρ total, j total are the total harge density and urrent density, respetively, and M is the magnetization. We point out that the use of the partial time derivative in eq. (15) instead of the total time derivative for the orresponding fore due to the time variation of the hidden momentum (the last term in eq. (1)) is related to the fat (usually omitted in the literature) that the measurement of magnetization (polarization) is arried out at a fixed point in the laboratory, whereas the measurement of the magneti (eletri) dipole moment of a moving bunh of harges is arried out for a volume o-moving with this bunh. Below we onsider the ase, when the free harges are absent, so that for a material medium we have the known relationships [3, 9]: ρ total = P, (16) j total = P + M, (17) t where P is the medium polarization. Thus, ombining eqs. (13) (17), we derive the total fore density in the form f total = ( P )E +( M) B + 1 P t B 1 (M E). (18) t

5 Eur. Phys. J. Plus (2014) 129: 215 Page 5 of 13 Substituting further eq. (18) into eq. (12), we get the expression for the total torque exerted on an eletri/magneti dipole in an eletromagneti field: [ ( T total = d r ( P )E +( M) B + 1 P t B 1 )] (M E). (19) t In order to larify the physial meaning of the torque (19) and its omponents, we an simplify it in the ase of a small dipole. This approximation implies, first of all, that the spatial variation of eletri and magneti fields at the loation of the dipole an be pratially ignored. This statement is expressed in the form of inequalities [11]: E i r j Δr j E i, B i r j Δr j B i (i, j =1...3), (20) where Δr j is the typial size of the dipole along the dimension j. Thus in the evaluation of the torque ontributions of eq. (19) for small dipole, we an take the fields E(r), B(r), as funtions of spatial oordinates, to be onstant within the volume of suh a dipole, beause any terms, whih inlude partial spatial derivatives of the eletri and magneti field beome negligible due to the inequalities (20). We notie that the approximation of onstant fields annot be adopted in the alulation of the fore (1) ating on a small dipole, beause some of the fore omponents are vanishing at E(r), B(r) = onst., and we have to involve the terms, ontaining their spatial derivatives (see, e.g., [9,11]). At the same time, when the motion of dipoles is onsidered, the variation of eletri and magneti fields along a trajetory of a moving dipole, in general, annot be ignored, so that the approximation of a onstant (as a funtion of spatial oordinates) field is relevant for a small volume of dipole only, and annot be applied to the entire spae. So, we onsider a small eletri/magneti dipole, moving at veloity ν in the external eletri E and magneti B fields of a laboratory frame. For simpliity we assume that the eletri p = P d and magneti m = Md dipole moments are onstant values in the rest frame of the dipole. With these limitations we obtain for the first term in the integrand of eq. (19) r ( P)Ed = p E, (21) whih desribes the ontribution to the torque due to the Coulomb interation. The derivation of this equation is presented in ref. [13]. Next we evaluate the torque omponent due to interation of the proper urrent of a dipole with a magneti field (seond term in the integrand of eq. (19)). Using the vetor identities a b = b a and a ( b) = (a b) b ( a) (a )b (b )a, this term an be presented in the form r (( M) B)d = r (B ( M))d = r ( (B M) (B )M (M )B M ( B))d = [r (B )M]d. (22) Here we have taken into aount that the integral [r (B M)]d an be transformed into the surfae integral, where the magnetization M is vanishing; also we notie that in the adopted approximation (B onst. inside the volume of the dipole), the terms (M )B and M ( B) are vanishing too. Integrating by parts the remaining integral in eq. (22), we obtain [13] [r (B )M]d = μ B. (23) Further, we evaluate the term responsible for the interation of the onvetive urrents of a dipole with a magneti field (third term in the integrand of eq. (19)). Here we notie that due to the adopted onstany of the proper eletri dipole moment, we get P / t = 0 in the rest frame of the dipole. However, for a moving dipole the stationary distribution of its harges yields dp /dt = 0, and hene P / t = (ν )P. With the latter equality we derive 1 r ((ν )P B)d = 1 r ((ν )(P B))d = 1 ν (p B). (24) The proof of eq. (24) an be found in ref. [13]. Addressing now the last term of the integrand of eq. (19) (the hidden-momentum ontribution), and taking into aount that for the stationary magnetization t (M E) = (ν )(M E), we obtain by analogy with eq. (24) T h = 1 (r t ) (M E) d = 1 (r (ν )(M E))d = 1 ν (m E). (25)

6 Page 6 of 13 Eur. Phys. J. Plus (2014) 129: 215 Finally, substituting eqs. (21), (23)-(25) into eq. (19), we derive the expression for the total torque exerted on a small dipole in an eletromagneti field [13]: T total = p E + m B + 1 ν (p B) 1 ν (m E), (26) where all quantities are evaluated in the laboratory frame. The first and seond terms on the rhs of eq. (26) look similar to the orresponding known terms for the torque on a resting dipole (see, e.g. [15]), though now they inlude the eletri p and magneti μ dipole moments of a moving dipole. The third term is responsible for the interation of the onvetive urrents of a dipole with the magneti field, while the fourth term stands for the hidden-momentum ontribution to the torque on a dipole. To our reolletion, the last two terms in eq. (26) seem not to have been reported before publiation of our paper [13]. Now it is worth omparing our equation (26) with the results obtained earlier by Namias. In ref. [7] he onsidered separately a torque on an eletri dipole (p 0 0,m 0 = 0) and a torque on a magneti dipole (p 0 =0,m 0 0). Applying the eletri-harge model and magneti-harge model, he subsequently derived the expressions for the torque as follows (in Gaussian units): T p = p E + 1 p (ν B)(p 0 0, m 0 =0), (27a) and T m = m B 1 m (ν E)(p 0 =0, m 0 0). (27b) We stress that the summation of eqs. (27a) and (27b) (to try to get the general expression for the torque at p 0 0, μ 0 0) is in general inorret, beause in this ase we fall outside the sope of validity of eah of these equations. Thus, in order to make a omparison of eq. (26) with the results of ref. [7], we an apply eq. (26) to the partiular ases p 0 0,m 0 =0andp 0 =0,m 0 0. In the first ase eq. (26) yields T total = p E + 1 (p 0 ν) B + 1 ν (p B) 1 2 ν ((p 0 ν) E) (p 0 0, m 0 =0), (28) where we have taken into aount that the moving eletri dipole develops the magneti dipole moment (the last term of eq. (5)). We see that, in general, our equation (28) differs from the Namias equation (27a) in terms of the order (ν/) 2.In partiular, the last term of eq. (28) indiates that the hidden momentum should be also assoiated with the magneti dipole moment, arising due to the motion of the eletri dipole. The physial meaning of this term, whih is missing both in the eletri-harge model and in the magneti-harge model, is disussed in more detail in ref. [13] (see also footnote 1 below). Next we apply eq. (26) to the analysis of some physial problems, whih ould be interesting to the readers. First onsider a resolution of the paradox by Mansuripur via a diret appliation of this equation. In a reent letter [10] Mansuripur questioned the orretness of the Lorentz fore law in material media, arguing that the fore law by Einstein and Laub [20,21] must be adopted instead, insofar as only the latter law provides the implementation of the momentum onservation law and exhibits ompatibility with speial relativity. As a entral point of his argumentation, the author of [10] onsiders the interation between a point-like harge q and a point-like magneti dipole m, whih are both at rest with respet to eah other in a referene frame K, and the diretion of m is orthogonal to the line joining harge and dipole (the z-axis). In this frame the mutual fore between harge and magneti dipole is equal to zero, and no torque is exerted on the dipole by the harge at rest. Then he onsiders the situation, where the frame K is moving at onstant veloity along the z-axis of the laboratory frame K, and shows that in this frame the fores ating on harge and dipole remain zero. However, one an hek that the Lorentz fore law (14) yields a non-vanishing torque exerted on the moving magneti dipole by the moving harge. This obviously represents a non-adequate result, sine in the proper frame of harge and dipole K, the torque is equal to zero. On the other hand, when the Einstein-Laub formula is applied, both the fore and the torque are equal to zero in the frames K and K. Based on this result, Mansuripur onluded that for material media the Lorentz fore law must be abandoned in favor of the Einstein-Laub law. The subsequent omments on this paper [22 25] invalidated the onlusion by Mansuripur and showed that the approah based on the Lorentz fore law with taking into aount the hidden momentum of the magneti dipole yields the vanishing torque in both K and K frames, whih resolves the paradox. Here we mention that Mansuripur was well aware of the hidden-momentum ontribution to the fore on a magneti dipole (the last term of eq. (1)), but deided, perhaps, that this ontribution ould be omitted in the problem he onsidered, beause in his onfiguration the fore omponent due to the time variation of the hidden momentum is exatly equal to zero. However, he missed the fat that the torque on the magneti dipole due to the hidden-momentum ontribution is not zero. In general, the aspet of non-vanishing torque on some body with zero total fore on this body is trivial in lassial mehanis. However,

7 Eur. Phys. J. Plus (2014) 129: 215 Page 7 of 13 y p α p K p x p E y y osα K x K x Fig. 2. Small eletri dipole with its proper eletri dipole moment p 0, oriented along the x p-axis of its rest frame K p, is moving in the plane xy of the laboratory frame K at onstant veloity, onstituting an angle α with the x-axis. We want to alulate the torque exerted on the dipole due to the onstant eletri field E, lying in the positive x-diretion, in the frame K and in another inertial frame K, moving along the x-axis with onstant veloity os α. this fat was overlooked by the author of [10] and this fat was not stressed in the omments [22 25] too, with respet to the fore and torque due to the hidden momentum ontribution. Indeed, applying eq. (26) to the mentioned problem of ref. [10] about the interation of a point-like harge and a magneti dipole, we obtain in the laboratory frame K T total = p E 1 ν (m E), (29) where the eletri dipole moment p = 1 (ν m) appears due to the relativisti polarization of the moving magneti dipole. Further on, the term p E an be presented in the form p E = 1 (ν m) E = 1 E (ν m) =1 m (E ν)+1 ν (E m) =1 ν (E m), (30) where we have used the Jaobi identity and taken into aount that for the problem in question the vetors ν, E are parallel to eah other at the loation of the point-like dipole, and E ν = 0. Hene, substituting eq. (30) into eq. (29), we get T total = 0 in the laboratory frame K, as it should be aording to the relativisti requirements. Thus, in ontrast to the laim made in ref. [10], we onlude that the approah based on the Lorentz fore law omplemented by the hidden-momentum ontribution, yielding eq. (26), is fully ompatible with the relativisti requirements and the momentum onservation law. Here we again highlight the Lorentz invariane of the novel expression (26), whih thus inorporates all the relativisti effets of speial relativity. For example, let us show that in suessive spae-time transformations, eq. (26) along with the relativisti transformations for the eletri and magneti dipole moments (4), (5) yields relativistially adequate results with the inlusion of the Thomas-Wigner rotation [26]. Consider the following problem (see fig. 2). The eletri dipole with the proper eletri dipole moment p 0, oriented along the x p -axis of its rest frame K p, is moving at onstant veloity, onstituting an angle α with the x-axis of the laboratory frame K. The stati eletri field E, as measured in the laboratory frame, lies along the x-axis. We want to alulate the torque exerted on the dipole in the frame K, and also in another inertial referene frame K,whih moves along the x-axis of K with onstant veloity os α. First of all, we observe that the motion of the frame K along the lines of the onstant eletri field E signifies that this field is the same in both K and K frames, and no magneti field emerges in these frames. Hene, aording to eq. (26), the expression for the torque on the dipole has the same form in K and K frames: T total = p E 1 ν (m E), (31) where, however, the parameters ν, p and m are different for observers in these frames.

8 Page 8 of 13 Eur. Phys. J. Plus (2014) 129: 215 In partiular, for a laboratory observer we have ν = { os α, sin α, 0}. Thus, substituting this veloity into eqs. (4), (5), and taking into aount that p 0 {p 0, 0, 0}, m 0 = 0, we obtain p x = p 0 [ ] ( 1) 2 (p ( 1) 0 x ) x = p 0 1 os 2 α, (32a) ( 1) p y = 2 (p ( 1) 0 x ) y = p 0 sin α os α, m z = p 0 y = p 0 sin α (32b), (32) where =(1 2 / 2 ) 1/2. Equation (32b) shows the appearane of the non-vanising y-omponent of the eletri dipole moment on the y-axis of the frame K, so that the vetor p is no longer parallel to the x-axis of this frame. From the physial viewpoint, this result is explained by the relativisti ontration of the moving sale along the diretion of motion. Indeed, the omponent of the length of the dipole on vetor is ontrated 1/ times, while the omponent of the length orthogonal to remains unhanged. Hene the dipole as a whole is turned with respet to the x-axis at a negative angle ϕ, defined via the equation tan ϕ = p ( 1) y sin α os α = p x 1 ( 1) os 2 α. To the auray of alulations 2, whih is suffiient for further analysis, this equation yields ϕ sin α os α = x y 2 2. (33) Here the negative sign of the angle ϕ orresponds to the lokwise rotation. Correspondingly, the vetor p is no longer parallel to the eletri field E for a laboratory observer, and the Coulomb fore indues the non-vanishing torque omponent (T Coulomb ) z =(p E) z = p y E p 0 E x y 2 2 (34) in the positive z-diretion. At the same time, the Coulomb ontribution (34) does not yet determine the total torque on the dipole. The moving eletri dipole develops the magneti dipole moment, whih in our ase is determined by eq. (32). Aording to eq. (31), it gives one more omponent of the torque: Combining eqs. (35) and (32), we derive T hidden = 1 ν (m E). (35) (T hidden ) z = 1 (ν (m E)) z = x m ze = p 0 E x y 2, (36) and this torque omponent lies in the negative z-diretion. Thus, the total torque on the moving eletri dipole is equal to the sum of eqs. (34), (36), and lies in the negative z-diretion: (T total ) z p 0 E x y 2 2. (37) Here it is interesting to note that a laboratory observer sees the moving eletri dipole to be turned in the lokwise diretion with respet to the lines of eletri field E, and intuitively he may expet the torque on the dipole in the pozitive z-diretion. However, the moving eletri dipole has the magneti dipole moment and, like an atual magneti dipole, possesses the hidden momentum, interating with the eletri field and induing the torque omponent (36) in the negative z-diretion. As a result, the total torque on the dipole appears to be negative too. This result looks somewhat paradoxial. However, its adequay from the relativisti viewpoint an be demonstrated via alulation of the torque on the dipole for an observer in the frame K.

9 Eur. Phys. J. Plus (2014) 129: 215 Page 9 of 13 In this frame the veloity of dipole is parallel to the y -axis, and the eletri field E = E is parallel to the x -axis. In these onditions the seond term on the rhs of eq. (31) (the hidden-momentum ontribution) is vanishing: T hidden = 1 (m E )= 1 m ( E ) 1 E (m )=0, where we have taken into aount the orthogonality of vetors, E and m (whih lie along the z -axis). Hene, only the Coulomb ontribution to the torque (the first term on the rhs of eq. (31)) is fixed in the frame K. In order to determine its value, we have to find the omponents of the eletri dipole moment p for an observer in K.Using eqs. (32a-), whih give the omponents of the eletri and magneti dipole moments in K, and taking into aount that the frame K is moving in K at the veloity os α along the x -axis, we derive via eq. (4) p x = p x ( 1) ( os α) 2 (p x os α) os α = p x [ 1 ( 1) ( p x 1 2 os 2 ) ( α 2 2 p os 2 ) α 2, (38a) p y = p y + 1 (m z os α) = p 0 ( 1) p 0 2 sin α os α p 0 2 sin α os α 2 sin α os α + p 0 2 sin α os α 2 = p 0 2 sin α os α 2 2 where we designated =(1 2 os 2 α ) 1/ os 2 α 2 2. Thus, the angle ϕ between the vetor p and x -axis 2 of the frame K is equal to ] = p 0 x y 2 2, (38b) ϕ p y p = x y x 2 2, (39) and happens to be positive (the rotation is in the ounterlokwise diretion). Correspondingly, the Coulomb omponent of the torque on the dipole (whih represents the total torque in the frame K ), is equal to (T total) z =(p E ) z = p y E p 0 E x y 2 2 (40) and lies in the negative z -diretion. Thus, we find that the torque on the moving eletri dipole, as seen in the frames K and K, is direted opposite to the z-axis; it is the same in the adopted auray of alulations 2 (ompare eqs. (37) and (40)), and indues a lokwise rotation of the dipole for the observers in these frames. We see a full adequay of this result from the relativisti viewpoint 1. It is also important to notie that the rotation of the vetor p at the angle ϕ with respet to the x -axis onurrently means the same rotation of the x p -axis of K p with respet to the x -axis of K. Furthermore, onsidering the eletri dipole p 0, originally oriented along the y p -axis of its rest frame K p, and ating in a similar way, we an show that the y p -axis of K p is seen in the frame K to be turned at the same angle ϕ with respet to the y -axis. Thus we onlude that for the motion diagram of fig. 2, the axes of the frames K p and K are no longer parallel to eah other and experiene a ommon rotation at the angle ϕ. This is the known relativisti effet of the Thomas-Wigner rotation, appearing in suessive spae-time transformations with non-ollinear veloities. 2.3 Energy of eletri/magneti dipole in an eletromagneti field Deriving an expression for the energy of eletrially neutral eletri/magneti dipole, we assume for simpliity the onstany of its proper eletri p 0 and magneti m 0 dipole moments. Furthermore, at the initial time moment t =0 let the dipole move with onstant veloity ν, as measured in the laboratory frame, being far from the region with the eletri E and magneti B fields. When the dipole is approahing the field region, it begins to experiene the 1 Here we mention that the appliation of Namias equations (27a), (27b) to the alulation of a torque on a moving eletri dipole in the frame K gives a positive value of the z-omponent of this torque (induing a ounterlokwise rotation of the dipole), whih thus is relativistially non-adequate.

10 Page 10 of 13 Eur. Phys. J. Plus (2014) 129: 215 eletromagneti fore F aording to (1). In these onditions, we assume the presene of some external fore F ext = F, whih exatly ounterats the fore (1), so that the veloity of the dipole remains onstant in the entire region with eletromagneti field, and its non-eletromagneti energy M 0 2 (where M 0 is the rest mass of the dipole) does not hange with time. Hene, the work done to the eletri/magneti dipole in the external eletri and magneti fields is equal to the eletromagneti energy of the dipole in this field. At any time moment t, this work is equal to E = t 0 F ext νdt = Substituting eq. (1) into eq. (41), we get the eletromagneti energy of the dipole: t F νdt = (p E) ds t 0 F νdt. (41) t d dt (p B) νdt + 1 t d (m E) νdt, 0 dt E EM = (m B) ds or E EM = (p E) (m B) 1 (p B) ν + 1 (m E) ν, (42) where we designated ds = νdt the element of the path of the dipole. As we have mentioned above, due to the onstany of ν, eq. (42) determines the eletromagneti energy of the eletri/magneti dipole in an external eletromagneti field, though we point out that this energy still does not inlude the fration of energy absorbed (or extrated) in a power supply, maintaining the onstany of m 0 for the dipole in question. One should add that this equation also does not inlude the energy absorbed (or extrated) in a soure of external magneti field, if we assume that the motion of the dipole does not ause any variation of the external field. We see that at v = 0, eq. (42) is transformed into the known expression for the eletromagneti energy of a resting eletri/magneti dipole, i.e. (E EM ) 0 = (p 0 E) (m 0 B). (43) For a moving dipole, in addition to the replaements p 0 p, m 0 m in the firsts two terms of eq. (42) in omparison with eq. (43), two new terms with expliit dependene on ν, i.e. 1 (p B) ν and 1 (m E) ν, do emerge. The energy term 1 (m E) ν, being related to the hidden-momentum ontribution to the energy of a moving dipole, for the first time has been introdued by Hnizdo [27] based on the requirement of Lorentz invariane of lassial eletrodynamis with the inlusion of hidden momentum. To our reolletion, the energy term 1 (p B) ν was not reported earlier and it is diretly related to the time variation of the ross produt 1 (p B), whih we suggested to name as latent momentum of eletri dipole [14]. For further progress in the analysis of the energy of the eletri/magneti dipole in an eletromagneti field, we write eq. (42) via the proper eletri and magneti dipole moments, using eqs. (4) and (5). Substituting these equations into equation (42), we derive after straightforward manipulations [14] E EM = (p 0 E) 2 (m 0 B) 2 (p 0 ν)( 1) 2 v 2 (ν E) (m 0 ν)( 1) 2 v 2 (ν B). (44) Finally, the energy (44) an be also expressed through the eletri E 0 and magneti B 0 fields in the rest frame of the dipole, when we use the relativisti transformations of the fields (see, e.g. [3]): ( 1) E = E 0 v 2 (E 0 ν)ν ν B 0, ( 1) B = B 0 v 2 (B 0 ν)ν + ν E 0. (45) Substituting eqs. (45) into eq. (44), we obtain (for the details of alulations see [14]) E EM = 1 ( p 0 E 0 + m 0 B 0 + (p 0 B 0 ) ν (m ) 0 E 0 ) ν. (46) The obtained eq. (46) an be onsidered as the relativisti transfomation of the eletromagneti energy of the eletri/magneti dipole, whih an be rewritten in the symboli form as follows: E EM = 1 ((E EM) 0 +(p l0 + p h0 ) ν), (47) where the proper eletromagneti energy of the dipole (E EM ) 0 is defined by eq. (43), p l0 = p 0 B 0 / is the proper latent momentum, and p h0 = m 0 E 0 / is the proper hidden momentum of the dipole. Here it is worth stressing that

11 Eur. Phys. J. Plus (2014) 129: 215 Page 11 of 13 eq. (47) does not oinide with the relativisti transformation of energy-momentum, beause E EM does not determine the total energy of the eletri/magneti dipole, as we mentioned above. For pratial appliations, eq. (44) is the most onvenient. Indeed, this equation operates with the eletri E and magneti B fields diretly measured in a laboratory, as well as with the proper eletri p 0 and magneti m 0 dipole moments, usually onsidered as known quantities for a given bunh of harges. Analyzing the struture of eq. (44), we reveal its important property: the energy of the eletri dipole depends solely on the presene of an eletri field E, and does not depend on the magneti field B. Analogously, the energy of the magneti dipole depends solely on the presene of a magneti field B, and does not depend on the eletri field E. In partiular, we observe that for an eletri dipole, moving in a time-independent magneti field with vanishing eletri field (E = 0), the energy of the dipole is equal to zero at any time moment. This result reflets the known fat that the energy of any bunh of harges, moving in a magneti field, remains unhanged, sine the magneti Lorentz fore omponent is always orthogonal to the veloity of the harges. It is interesting to notie that the symmetri situation ours with respet to a magneti dipole, moving in a time-independent eletri field with vanishing magneti field (B = 0): the energy of suh a magneti dipole is equal to zero too. From the physial viewpoint, the onstany of energy of a magneti dipole in a time-independent eletri field an be explained by the fat that its hidden energy (the last term in eq. (3)) (m E) ν E h = (48) is exatly balaned by the eletri energy U rel = (ν m 0) E, (49) assoiated with the relativisti polarization of a moving magneti dipole, whih leads to the appearane of the eletri dipole moment p = ν m0. Indeed, summing up eqs. (48) and (49), we have E h + U rel = = (m E) ν (ν m) E = (ν (m m 0)) E (ν m 0) E (ν m 0) E ( 1) (ν ν) E = v 2 (m 0 ν) =0, where we have used the vetor identity (a b) =( a) b, as well as the transformation (5) at p 0 =0. Here we should like to point out that in the analysis performed just above, the assumption about the independene of eletri and magneti fields on time is important: otherwise, an eletri field varying with time indues a magneti field, and vie versa, whih makes the situation more ompliated. Finally, it is worth emplasizing that in the ultra-relativisti ase, the energy (44) essentially depends on the mutual orientation of vetors p 0, E, ν and m 0, B, ν. First of all, we onsider the fration of energy (44), whih inludes the eletri dipole moment, i.e. E p = (p 0 E) 2 (p 0 ν)( 1) 2 v 2 (ν E). (50) We see that in the partiular ase, where vetors p 0, E, ν are ollinear to eah other, this energy is equal to (E p ) // = (p 0 E). (51a) Furthermore, in the ase, where one of the vetors (p 0 or E) is orthogonal to ν, the energy beomes (E p ) = (p 0 E) 2. (51b) The differene between eqs. (51a) and (51b) is atually ruial at 1, and it should be taken into aount in pratial appliations.

12 Page 12 of 13 Eur. Phys. J. Plus (2014) 129: 215 From the physial viewpoint, the appearane of fator 1/ in eq. (51a) is explained by the ontration of the length of the eletri dipole along the veloity ν, whih indues the proportional derease of its energy. The additional suppression of the energy (51b) 1/ times in omparison with eq. (51a) in both possible ases (p 0 ν, ande is not ollinear to ν; E ν, andp 0 is not ollinear to ν) is aused by the interation of the magneti dipole moment p0 ν with the eletri field (i.e. the ontribution of the hidden momentum of the magneti dipole, emerging due to the motion of the eletri dipole), whih thus redues the value of the eletromagneti energy of the entire bunh of harges in omparison with energy (51a). For more explanation of this effet see ref. [14]. The energy of a magneti dipole in a magneti field is defined by the expression E m = (m 0 B) 2 (m 0 ν)( 1) 2 v 2 (ν B), (52) whih is symmetri to eq. (50) with respet to the replaements p 0 m 0, E B. Correspondingly, all the omments made above on eq. (50) remain true with respet to eq. (52), too. In partiular, the energy of the magneti dipole is equal to (E m ) // = (m 0 B) (53a) in the ase m 0 //B//ν, and (E m ) = (m 0 B) 2, (53b) when one of the vetors (m 0 or B) is orthogonal to ν. Again, the differene between eqs. (53a) and (53b) beomes ruial at 1 (the ultrarelativiasti ase), and this irumstane should be taken into aount in pratial appliations, dealing with the relativisti motion of magneti dipoles in an eletromagneti fields. From the physial viewpoint, the appearane of fator 1/ in eq. (53a) (when vetors m 0 and ν are ollinear to eah other) is explained by the time dilation effet for the moving magneti dipole, whih dereases the urrent of the dipole and its magneti dipole moment by 1/ times. The additional suppression of the energy (53b) 1/ times in omparison with eq. (53a) in the ase m 0 ν is aused by the ontration of the area of the dipole along the diretion of ν. In another ase of validity of eq. (53b) (B ν, andm 0 not ollinear to ν) the mehanism of suppression of energy is more ompliated and is skipped in this paper. We only mention that it ombines the effets of relativisti ontration of the area of a magneti dipole, as well as the interation of the latent momentum of the eletri dipole (emerging due to the motion of the magneti dipole) with the magneti field B. 3 Conlusion In this paper we olleted altogrther the relativisti expressions for the fore, torque and energy of a small eletri/magneti dipole in an eletromagneti field, whih we reently obtained, and analyzed a number of subtle effets, related to the relativisti motion of dipoles. In partiular, we presented the solution of the paradox by Mansuripur with the expliit explanation of its origin. We have shown that the obtained expressions for the fore and torque on a moving dipole, along with the relativisti transformation of the eletri/magneti dipole moment, do inlude all of the known relativisti effets (in partiular, the Thomas-Wigner rotation in suessive spae-time transformations) and provide adequate results from the relativisti point of view. Finally, we have shown that the energy of the ultrarelativisti eletri/magneti dipole essentially depends on the mutual orientation of veloity, eletri (magneti) dipole moment and eletri (magneti) field. Referenes 1.. Bargmann, L. Mihel,.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959). 2. L.H. Thomas, Nature (London) 117, 514 (1926). 3. J.D. Jakson, Classial Eletrodynamis, 3rd edition (Wiley, New York, 1998). 4. J.D. Jakson, Am. J. Phys. 72, 1484 (2004). 5. W. Shokley, R.P. James, Phys. Rev. Lett. 18, 876 (1967). 6. A.O. Barut, Eletrodynamis and lassial theory of fields & partiles (Dover, New York, 1980). 7.. Namias, Am. J. Phys. 57, 171 (1989). 8. G.P. Fisher, Am. J. Phys. 39, 1528 (1971). 9. J.E. erstein, Eur. J. Phys. 18, 113 (1997).

13 Eur. Phys. J. Plus (2014) 129: 215 Page 13 of M. Mansuripur, Phys. Rev. Lett. 108, (2012). 11. A.L. Kholmetskii, O.. Missevith, T. Yarman, Eur. J. Phys. 32, 873 (2011). 12. A.L. Kholmetskii, O.. Missevith, T. Yarman, Eur. J. Phys. 33, L7 (2012). 13. A.L. Kholmetskii, O.. Missevith, T. Yarman, Prog. Eletromagn. Res. B 45, 83 (2012). 14. A.L. Kholmetskii, O.. Missevith, T. Yarman, Can. J. Phys. 9, 576 (2013). 15. W.K.H. Panofsky, M. Phillips, Classial Eletriity and Magnetism, seond edition (Addison-Wesley, Reading, MA, 1969). 16. A.L. Kholmetskii, O.. Missevith, T. Yarman, Prog. Eletromagn. Res. B 47, 263 (2013). 17. Y. Aharonov, P. Pearle, L. aidman, Phys. Rev. A 37, 4052 (1988). 18. L. aidman, Am. J. Phys. 58, 978 (1990) Hnizdo, Eur. J. Phys. 33, L7 (2012). 20. A. Einstein, J. Laub, Ann. Phys. (Leipzig) 331, 532 (1908). 21. A. Einstein, J. Laub, Ann. Phys. (Leipzig) 331, 541 (1908). 22. D.A. anzella, Phys. Rev. Lett. 110, (2013). 23. S.M. Barnett, Phys. Rev. Lett. 110, (2013). 24. P.L. Saldanha, Phys. Rev. Lett. 110, (2013). 25. M. Khorrami, Phys. Rev. Lett. 110, (2013). 26. C. Møller, The Theory of Relativity (Clarendon Press, Oxford, 1973) Hnizdo, Magn. Eletr. Sep. 3, 259 (1992).