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Nature of the electromagnetic between classical magnetic dipoles Masud Mansuripur College of Optical Sciences, The University of Arizona, Tucson [Published in Spintronics X, edited by H.-J. Drouhin, J.E. Wegrowe, M. Razeghi, and H. Jaffrès, Proceedings of SPIE Vol. 10357, 103570R-1:6 (2017). doi: 10.1117/12.2273216] Abstract. The Lorentz force law of classical electrodynamics states that the force exerted by the magnetic induction on a particle of charge moving with velocity is given by = × . Since this force is orthogonal to the direction of motion, the is said to be incapable 𝑭𝑭of performing mechanical work. Yet𝑩𝑩 there is no denying that𝑞𝑞 a permanent 𝑽𝑽 can readily 𝑭𝑭perform𝑞𝑞𝑽𝑽 mechanical𝑩𝑩 work by pushing/pulling on another permanent magnet — or by attracting pieces of magnetizable material such as scrap iron or iron filings. We explain this apparent contradiction by examining the magnetic Lorentz force acting on an Amperian current loop, which is the model for a . We then extend the discussion by analyzing the Einstein-Laub model of magnetic dipoles in the presence of external magnetic fields. 1. Introduction. The force exerted by one magnetic dipole on another is capable of performing mechanical work, which appears to be in violation of the Lorentz force law of classical electrodynamics. The discrepancy is resolved if one examines the model of a magnetic dipole as an Amperian current loop, which allows for the balancing of the translational kinetic energy gained (or lost) by the moving current loop against its rotational kinetic energy. This problem will be analyzed in some detail in Sec.2, where we show that the change in the loop’s rotational kinetic energy is also related to its changing magnetic dipole in accordance with Larmor’s diamagnetic susceptibility of the loop. We proceed to address, in Sec.3, the question of what would happen if a magnetic dipole failed to behave as an Amperian current loop. This brings up an alternative model of and magnetic dipoles, which will be treated in the context of the Einstein-Laub formulation of the classical theory. In Sec.4 we examine the electromagnetic (EM) energy of an immobile magnetic dipole in the presence of an externally-applied static magnetic field. Two approaches to EM energy, namely, the Lorentz approach (which considers magnetic dipoles as Amperian current loops), and the Einstein-Laub approach (which treats such dipoles as pairs of north-south magnetic monopoles) will be considered. Finally, in Sec.5, we extend the discussion of Sec.4 to the case of moving magnetic dipoles within a static magnetic field. The predictions of the Lorentz and Einstein-Laub formulations turn out to be in complete agreement in all the cases examined in Secs.4 and 5. The paper closes with a few concluding remarks in Sec.6. 2. Interaction between two magnetic dipoles. Consider a magnetic point-dipole , sitting at the origin of a spherical coordinate system ( , , ), as shown in Fig.1. The magnetic field of the dipole in 0 the surrounding free space is1,2 𝑚𝑚 𝒛𝒛� 𝑟𝑟 𝜃𝜃 𝜑𝜑 ( , , ) = (2 cos + sin ). (1) 𝑚𝑚0 3 The reason for the free-𝑯𝑯space𝑟𝑟 𝜃𝜃 per𝜑𝜑meability4𝜋𝜋𝜇𝜇0𝑟𝑟 appearing𝜃𝜃 𝒓𝒓� in Eq.(1)𝜃𝜃 𝜽𝜽� is that we are using the notation in which the magnetic induction is defined as = 0 + . According to this definition, the magnetization 2 𝜇𝜇 has the units of the -field (i.e., weber/m or0 in , the international system of units), and the magnetic dipole moment of a small current 𝑩𝑩loop 𝜇𝜇of 𝑯𝑯area 𝑴𝑴 carrying the current is = . 𝑴𝑴 Let a rigid circular𝐵𝐵 ring of radius and negligible 𝑆𝑆𝑆𝑆thickness be z0 centered on the -axis at = . The ring, which is uniformly𝐴𝐴 charged 𝐼𝐼 𝑚𝑚 𝜇𝜇 𝐼𝐼𝐼𝐼 𝑅𝑅 around its circumference, has total charge , total mass , and rotates R 0 z0 × at a constant angular𝑧𝑧 velocity𝑧𝑧 𝑧𝑧 around the -axis. The Ω circulating around the ring in the azimuthal 𝑞𝑞direction is𝓂𝓂 thus given by Ω 𝑧𝑧 Fig.1. The magnetic dipole sits at the origin of the coordinate system. A 𝝋𝝋� circular ring of radius and negligible thickness is placed parallel to the - 𝜃𝜃 0 y plane at = . The ring, which𝑚𝑚 𝒛𝒛� is uniformly charged around its circumference, has a total charge , total𝑅𝑅 mass , and rotates with a constant angular velocity𝑥𝑥𝑥𝑥 𝑚𝑚0𝒛𝒛� 0 around𝑧𝑧 the 𝑧𝑧 -axis. 𝑞𝑞 𝓂𝓂 x Ω 𝑧𝑧 1

= (2 ), the mechanical angular momentum of the ring is = ( ) , and the magnetic dipole moment of the spinning ring is = ½( ) = ½( ) . 2 𝐼𝐼 𝑞𝑞IΩn ⁄the𝜋𝜋 spherical coordinate system of Fig.1,2 the angle 𝑳𝑳subtended𝓂𝓂𝑅𝑅 Ωby𝒛𝒛� the ring at the origin is 1 0 0 = tan ( ). The Lorentz𝒎𝒎 force-density𝜇𝜇 𝑞𝑞𝑅𝑅 (i.e.,Ω 𝒛𝒛� force per𝜇𝜇 𝑞𝑞 unit⁄𝓂𝓂 length𝑳𝑳 of the ring) produced by the - field of the−1 dipole acting on the spinning loop is readily found𝜃𝜃 to be 𝜃𝜃 𝑅𝑅⁄𝑧𝑧0 𝐵𝐵 0 ( ) ( ) 𝑚𝑚 𝒛𝒛� = × = · (2) 𝑚𝑚0 2 cos 𝜃𝜃𝒓𝒓� + sin 𝜃𝜃𝜽𝜽� 𝐼𝐼𝑚𝑚0 2 cos 𝜃𝜃 𝜽𝜽� − sin 𝜃𝜃𝒓𝒓� ⁄ ⁄ 1 2 2 3 2 2 2 3 2 Upon integrating the𝒇𝒇 above𝐼𝐼𝝋𝝋� force-density4𝜋𝜋�𝑅𝑅 + around𝑧𝑧0 � the perimeter4𝜋𝜋�𝑅𝑅 +of𝑧𝑧 0the� ring, we find that only the - component of the Lorentz force survives. The net force is thus given by 𝑧𝑧 ( ) = 2 = = · (3) 2 𝐼𝐼𝑚𝑚0 −2 cos 𝜃𝜃 sin 𝜃𝜃 −sin 𝜃𝜃 cos 𝜃𝜃 3𝐼𝐼𝑚𝑚0𝑅𝑅 sin 𝜃𝜃 cos 𝜃𝜃 3𝐼𝐼𝑚𝑚0𝑅𝑅 𝑧𝑧0 ⁄ ⁄ ⁄ 𝑧𝑧 2 2 3 2 2 2 3 2 2 2 5 2 Incidentally,𝐹𝐹 in𝜋𝜋 𝜋𝜋the limit 4when𝜋𝜋�𝑅𝑅 + 𝑧𝑧0 � 0, the force− 2 �𝑅𝑅is +equivalent𝑧𝑧0 � to− 2(�𝑅𝑅 +𝑧𝑧0 �) , where = and = (2 | | ) along the -axis; see Eq.(1). The bottom line is that the dipole 𝑧𝑧 1 1 exerts2 an attractive force on 3the 𝑅𝑅ring→ that is centered𝐹𝐹 𝒛𝒛� on the -axis at =𝒎𝒎 ∙ 𝜵𝜵while𝑯𝑯 carrying𝒎𝒎 the 0 0 0 0 constant𝜇𝜇 𝜋𝜋𝑅𝑅 𝐼𝐼𝒛𝒛� current𝑯𝑯 along𝑚𝑚 𝒛𝒛�⁄ . 𝜋𝜋When𝜇𝜇 𝑧𝑧 the radius 𝑧𝑧of the ring is sufficiently small, one may suppose that𝑚𝑚 the𝒛𝒛� 𝑧𝑧 0 spinning ring is a magnetic𝐹𝐹 point𝒛𝒛� -dipole , in which case it is easy 𝑧𝑧to see how𝑧𝑧 the𝑧𝑧 opposing poles of the two dipoles tend 𝐼𝐼to attract𝝋𝝋� each other. So long𝑅𝑅 as the dipoles are pinned down at their respective locations, there will be no movement and, therefore,𝑚𝑚1 no𝒛𝒛� mechanical work is performed by one dipole on the other. However, if the dipole at the origin remains fixed in place, but the dipole at = is allowed to move along the -axis, the force of Eq.(3) will perform mechanical work, which appears to contradict the well-known assertion that magnetic fields acting on electric charges cannot do any𝑧𝑧 work.𝑧𝑧0 To resolve𝑧𝑧 the contradiction,𝐹𝐹𝑧𝑧𝒛𝒛� let us now assume that the ring spinning around the -axis at = is also travelling along at a constant velocity . The rotating charges are thus subjected to a new Lorentz force, whose linear density around the circumference of the ring is given by 𝑧𝑧 𝑧𝑧 𝑧𝑧0 𝑧𝑧 𝑣𝑣𝒛𝒛�

= × = = . (4) 𝑞𝑞𝑞𝑞𝒛𝒛� 𝑚𝑚0�2 cos 𝜃𝜃𝒓𝒓� +sin 𝜃𝜃𝜽𝜽�� 3𝑞𝑞𝑞𝑞𝑚𝑚0 sin 𝜃𝜃 cos 𝜃𝜃 3𝑞𝑞𝑞𝑞𝑚𝑚0𝑧𝑧0 ⁄ ⁄ ⁄ 2 2 2 3 2 2 2 2 3 2 2 2 2 5 2 The force-density𝒇𝒇 2𝜋𝜋𝜋𝜋 performs4𝜋𝜋� 𝑅𝑅mechanical+𝑧𝑧0 � work8 𝜋𝜋 𝑅𝑅 on�𝑅𝑅 the+𝑧𝑧0 �rotating𝝋𝝋� charges8𝜋𝜋 �𝑅𝑅 +of𝑧𝑧 0the� ring𝝋𝝋� at a rate given by the dot-product of and the azimuthal velocity of the charges. The rate at which work is done on the entire ring is subsequently𝒇𝒇2 obtained by integrating𝑊𝑊 around the perimeter of the ring, that is, 𝒇𝒇2 𝑅𝑅Ω𝝋𝝋� ( )( ) = ( ) = = · (5) 2 𝑑𝑑𝑑𝑑 2𝜋𝜋 3𝑞𝑞𝑞𝑞𝑚𝑚0𝑧𝑧0𝑅𝑅Ω 2𝜋𝜋𝜋𝜋 3𝐼𝐼𝑚𝑚0𝑅𝑅 𝑧𝑧0𝑣𝑣 ⁄ ⁄ 2 2 2 2 5 2 2 2 5 2 A comparison𝑑𝑑𝑑𝑑 of Eq.(5)∫0 𝒇𝒇 with∙ 𝑅𝑅 ΩEq.(3)𝝋𝝋� 𝑅𝑅𝑅𝑅 𝑅𝑅reveals8 𝜋𝜋that�𝑅𝑅 +the𝑧𝑧0 �energy delivered2�𝑅𝑅 +𝑧𝑧0 � to the current loop in accordance with Eq.(5) is taken away from the translational kinetic energy of the traveling loop along the -axis at the rate of . [Conversely, if happens to be negative, the loop gains translational kinetic energy as it moves down toward the dipole , while simultaneously losing rotational kinetic energy in 𝑧𝑧accordance with Eq.(5)𝐹𝐹𝑧𝑧.𝑣𝑣] We have thus confirmed𝑣𝑣 that a magnetic field cannot perform any (net) work on moving charged particles — again for the simple𝑚𝑚0𝒛𝒛� reason that the Lorentz force of the -field is always orthogonal to the direction of motion of each such charge. The mechanical (2 ) acting on the ring is, of course, responsible for chang𝐵𝐵 ing the ring’s angular momentum and, consequently,2 its dipole moment = ½( ) , at the following rate: 𝜋𝜋𝑅𝑅 𝑓𝑓2 𝒛𝒛� 1 0 = 𝑳𝑳 = (2 ) = 𝒎𝒎 =𝜇𝜇 𝑞𝑞⁄𝓂𝓂 𝑳𝑳 . (6) 2 2 2 2 𝑑𝑑𝒎𝒎1 𝜇𝜇0𝑞𝑞 𝑑𝑑𝑳𝑳 𝜇𝜇0𝑞𝑞 3𝜇𝜇0𝑞𝑞 𝑅𝑅 𝑚𝑚0𝑧𝑧0𝑣𝑣 𝜇𝜇0𝑞𝑞 𝑅𝑅 3𝑚𝑚0𝑧𝑧0𝑣𝑣 2 ⁄ ⁄ 2 2 2 5 2 2 2 5 2 𝑑𝑑𝑑𝑑 �2𝓂𝓂� 𝑑𝑑𝑑𝑑 �2𝓂𝓂� 𝜋𝜋𝑅𝑅 𝑓𝑓 𝒛𝒛� 8𝜋𝜋𝓂𝓂�𝑅𝑅 +𝑧𝑧0 � 𝒛𝒛� � 4𝓂𝓂 � 2𝜋𝜋�𝑅𝑅 +𝑧𝑧0 � 𝒛𝒛�

2

The bracketed term on the right-hand-side of Eq.(6) is Larmor’s diamagnetic susceptibility3 associated with the magnetic dipole . For sufficiently small , the remaining term on the right-hand- side of Eq.(6), namely, 3 (2 ), is the time-rate-of-change of the -field as seen by the traveling 1 point-dipole while passing through𝑚𝑚4 𝒛𝒛� = at the constant velocity𝑅𝑅 ; see Eq.(1). It is seen that the 0 dipole moment does in𝑚𝑚 fact𝑣𝑣𝒛𝒛� ⁄change𝜋𝜋𝑧𝑧0 in precisely the way it is expected to𝐵𝐵 change (given its diamagnetic susceptibility)𝑚𝑚 1when𝒛𝒛� passing through the𝑧𝑧 nonuniform𝑧𝑧0 magnetic field produced𝑣𝑣𝒛𝒛� by the stationary dipole . The changin𝑚𝑚1 g magnitude of the dipole moment guarantees that any gain (or loss) in its translational kinetic energy is balanced by the concomitant loss (or gain) in the rotational kinetic energy 𝑚𝑚of 0the𝒛𝒛� revolving charge(s) that constitute the dipole moment𝑚𝑚 1𝒛𝒛� .

3. Beyond the Amperian current loop model. The analysis𝑚𝑚 of1𝒛𝒛� the preceding section applies to magnetic dipoles that originate in the orbital motion of around atomic/molecular nuclei. It is not clear if -based magnetic dipoles would behave in similar fashion. In the Einstein-Laub formulation of EM force and torque,4,5 magnetic dipoles are treated as primordial sources of the EM field (along with electric dipoles, in addition to free electric charges and currents). The Einstein-Laub force and torque 4-6 exerted by the fields ( , ) and ( , ) on a magnetic dipole are EL EL 𝑭𝑭 𝑻𝑻 ( , ) = ( ) ( ) × 𝑬𝑬 𝒓𝒓 𝑡𝑡 𝑯𝑯 𝒓𝒓 𝑡𝑡 𝒎𝒎 , (7a) EL ( , ) = × + × 0 𝑭𝑭 𝒓𝒓 𝑡𝑡 𝒎𝒎 ∙ 𝜵𝜵 𝑯𝑯 − 𝜕𝜕𝒎𝒎⁄𝜕𝜕𝜕𝜕 . 𝜀𝜀 𝑬𝑬 (7b) EL EL Any changes that might occur 𝑻𝑻in the𝒓𝒓 𝑡𝑡dipole𝒓𝒓 moment𝑭𝑭 𝒎𝒎 itself𝑯𝑯 in the presence of external EM fields are unspecified. Moreover, with the defined as ( , ) = ( , ) × ( , ) in the Einstein-Laub formulation,5,6 the energy exchange rate between𝒎𝒎 the EM field and a magnetic dipole is given by ( ). A point-dipole such as ( ) ( ) ( 𝑺𝑺) 𝒓𝒓, whose𝑡𝑡 𝑬𝑬 magnitude𝒓𝒓 𝑡𝑡 𝑯𝑯 𝒓𝒓 𝑡𝑡 remains constant while traveling along the -axis, will have the following energy-exchange-rate with the magnetic𝒎𝒎 1 1 field ( )𝑯𝑯 of∙ Eq.(1):𝜕𝜕𝒎𝒎⁄𝜕𝜕 𝜕𝜕 𝑚𝑚 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑦𝑦 𝛿𝛿 𝑧𝑧 − 𝑣𝑣𝑣𝑣 𝒛𝒛� 𝑚𝑚 𝑧𝑧 𝑯𝑯 𝒓𝒓 ( ) = ( ) ( ) ( ) = ∞ . (8) ∞ 𝑚𝑚0𝑚𝑚1𝑣𝑣 ′ 3𝑚𝑚0𝑚𝑚1𝑣𝑣 3 4 −∞ 2𝜋𝜋𝜇𝜇0𝑧𝑧 2𝜋𝜋𝜇𝜇0𝑧𝑧0 ∭Equation𝑯𝑯 ∙ (8)𝜕𝜕𝒎𝒎 is⁄ 𝜕𝜕in𝜕𝜕 complete𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 accord− � with−∞ Eq.(3), 𝛿𝛿as 𝑥𝑥it 𝛿𝛿shows𝑦𝑦 𝛿𝛿 that𝑧𝑧 − translational𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 kinetic− energy is what the -field delivers to (or takes away from) the moving dipole . Unless the changes of with position along the -axis are independently specified, the Einstein-Laub formalism cannot guarantee that 1 1 the net𝐻𝐻 work done by the -field on the permanent dipole would𝑚𝑚 be𝒛𝒛� zero. 𝑚𝑚 In contrast to 𝑧𝑧the Einstein-Laub formalism, magnetic dipoles in the Lorentz formulation are treated 1 as Amperian current loops,𝐻𝐻 1-3 with the current-density𝑚𝑚 associated𝒛𝒛� with an arbitrary spatio-temporal distribution of magnetization ( , ) being = × .1,2,5 These current loops must then act as

ordinary current loops in the presence of externalbound EM0−1 fields. Given that, in their internal structure, magnetic dipoles (and also electric𝑴𝑴 𝒓𝒓 𝑡𝑡 dipoles)𝑱𝑱 do not 𝜇𝜇abide𝜵𝜵 by𝑴𝑴 the rules of classical physics, it is by no means obvious if realistic (i.e., physical) dipoles are structurally capable of responding “properly” to classical EM fields. In other words, in the interaction between a magnetic field and magnetic dipoles, it is not a priori clear if the quantized spin and orbital angular momenta of realistic dipoles would allow the necessary adjustment of the magnetic dipole moment in order to ensure that no net mechanical work is performed by the applied magnetic field. 4. Energy considerations for a stationary dipole. With reference to Fig.2, let a small, uniformly- magnetized spherical particle of radius have a magnetization ( , ) = ( ) ( ) ( ) ( ) , where the temporal variations of ( ) are assumed to be fairly slow. The volume occupied by the spherical dipole is = 4 3. Here we are using the -function𝑅𝑅 notation to represent𝑴𝑴 𝒓𝒓 a𝑡𝑡 small𝑚𝑚 spherical𝑡𝑡 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑦𝑦volume𝛿𝛿 𝑧𝑧 𝒛𝒛� centered at 𝑚𝑚 𝑡𝑡 the0 origin 3of coordinates. In what follows, depending on the mathematical needs of the situation, we shall 𝑉𝑉go back𝜋𝜋 𝑅𝑅and⁄ forth between this -function𝛿𝛿 notation and the small-sphere representation of ( , ).

𝛿𝛿 𝑴𝑴 𝒓𝒓 𝑡𝑡 3

In the Lorentz formalism and in the absence of external fields, the time-rate-of-change of the stored EM energy inside the dipole, where ( ) = ( ), is given by

= ½ 𝑩𝑩 𝑡𝑡 ⅔(𝑴𝑴) 𝑡𝑡 = ( ) ( ) dr stands for volume . element dxdydz. (9) 𝑑𝑑ℰstored 𝑑𝑑 −1 2 4 −1 −1 ′ 0 sphere 0 0 The time-varying𝑑𝑑𝑑𝑑 magnetization,𝑑𝑑𝑑𝑑 � 𝜇𝜇 ∫ being𝑩𝑩 equivalent𝑡𝑡 𝑑𝑑𝒓𝒓� to9 𝜇𝜇 a time𝑉𝑉 -𝑚𝑚varying𝑡𝑡 𝑚𝑚 𝑡𝑡-field inside the sphere, gives rise to an -field circulating around the -axis in accordance with Maxwell’s equation × = . At the surface of the sphere, around a circle of radius sin , we have 𝐵𝐵 = ( ) sin . This ( ) 𝑡𝑡 -field performs𝐸𝐸 mechanical work on the𝑧𝑧 magnet’s bound electric current-density ′ 𝜵𝜵 =𝑬𝑬2 −2𝜕𝜕× 𝑩𝑩 ( ) 𝑅𝑅 𝜃𝜃 ∮ 𝑬𝑬 ∙ 𝑑𝑑𝓵𝓵 −⅔𝑀𝑀𝑒𝑒 𝑡𝑡 𝜋𝜋𝑅𝑅 𝜃𝜃 and exchanges energy with the dipole at the rate of ( ) . Given that the productbound of loop0− current1 and 𝐸𝐸loop area integrated over equals ( ), we see that the time-rate-of-change𝑱𝑱 of the 𝜇𝜇dipole’s𝜵𝜵 𝑴𝑴total𝑡𝑡 ∮ 𝑬𝑬 ∙ 𝑱𝑱 𝑑𝑑ℓ internal energy is 0−1 𝜃𝜃 𝜇𝜇 𝑚𝑚 𝑡𝑡 = ( ) ( ) ( ) ( ) = ( ) ( ). (10) 𝑑𝑑ℰtotal 4 2 2 0−1 0−1 ′ 0−1 0−1 ′ 0−1 0−1 ′ It is thus𝑑𝑑𝑑𝑑 seen 9that,𝜇𝜇 𝑉𝑉when𝑚𝑚 𝑡𝑡 (𝑚𝑚) is𝑡𝑡 on− 3the𝜇𝜇 rise,𝑉𝑉 𝑚𝑚electromagnetic𝑡𝑡 𝑚𝑚 𝑡𝑡 − 9(EM)𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡z energy leaves the interior of the dipole and enters the surrounding space in the form of EM fields. Conversely,𝒎𝒎 𝑡𝑡 when ( ) is in decline, the external field energy leaves the surrounding space and enters the dipole’s spherical volume. 𝒎𝒎 𝑡𝑡 𝑣𝑣𝒛𝒛� R Fig.2. A uniformly magnetized sphere of radius , volume , and magnetic dipole y moment ( ) , moves along the -axis at constant velocity . The internal - 0 𝑚𝑚𝒛𝒛� field of the dipole is ( ) , while its𝑅𝑅 internal 𝑉𝑉-field is ( ) . ( ) 𝑚𝑚 𝑡𝑡 𝒛𝒛� −1 −1 𝑧𝑧 ( ) 𝑣𝑣𝒛𝒛� −1 𝐻𝐻 ( ) The externally applied magnetic0 0 field ( ) does not vary with time.0 x −⅓𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝒛𝒛� ext 𝐵𝐵 ⅔𝑉𝑉 𝑚𝑚 𝑡𝑡 𝒛𝒛� ext Let us now switch to the Einstein𝑯𝑯 -Laub𝒓𝒓 formalism and confirm that the overall rate of𝑯𝑯 flow𝒓𝒓 of energy in or out of the sphere is the same in the two formulations. Here ( ) = ( ) inside the sphere, and the time-rate-of-change of the energy stored within the dipole is given by 0−1 𝑯𝑯 𝑡𝑡 −⅓𝜇𝜇 𝑴𝑴 𝑡𝑡 = ½ ( ) = ( ) ( ). (11) 𝑑𝑑ℰstored 𝑑𝑑 2 1 −1 −1 ′ 0 sphere 0 0 In this case, the 𝑑𝑑𝑑𝑑-field plays𝑑𝑑𝑑𝑑 � no𝜇𝜇 role,∫ but𝑯𝑯 the𝑡𝑡 exchange𝑑𝑑𝒓𝒓� 9 𝜇𝜇 rate𝑉𝑉 of 𝑚𝑚energy𝑡𝑡 𝑚𝑚-density𝑡𝑡 between the field and the dipole is given by ( ) ( , ) = ( , ) ( , ), which must then be integrated over 𝐸𝐸 the volume of the sphere. Thus, the time-rate-of-0−change1 of total (internal) energy is 𝑯𝑯 𝒓𝒓 ∙ 𝜕𝜕𝑡𝑡𝑴𝑴 𝒓𝒓 𝑡𝑡 −⅓𝜇𝜇 𝑴𝑴 𝒓𝒓 𝑡𝑡 ∙ 𝜕𝜕𝑡𝑡𝑴𝑴 𝒓𝒓 𝑡𝑡 = ( ) ( ) ( ) ( ) = ( ) ( ). (12) 𝑑𝑑ℰtotal 1 1 2 0−1 0−1 ′ 0−1 0−1 ′ 0−1 0−1 ′ Note that𝑑𝑑𝑑𝑑 Eqs.(109 𝜇𝜇) and𝑉𝑉 (12𝑚𝑚) predict𝑡𝑡 𝑚𝑚 𝑡𝑡 identical− 3 𝜇𝜇 𝑉𝑉rates𝑚𝑚 for𝑡𝑡 𝑚𝑚the 𝑡𝑡overall− 9flow𝜇𝜇 𝑉𝑉 of energy𝑚𝑚 𝑡𝑡 𝑚𝑚 into𝑡𝑡 or out of the dipole, despite the differing physical mechanisms behind the two formulations. If the dipole happens to be immersed in an externally applied static magnetic field ( )( ), the results will be essentially the same. In this case, the Lorentz formulation predicts that ext 𝑯𝑯 𝒓𝒓 = ½ ( )( ) + ( , )

𝑑𝑑ℰstored 𝑑𝑑 −1 ext 2 0 ( sphere) 0 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 � (𝜇𝜇) ∫ (0)�𝜇𝜇+𝑯𝑯 𝒓𝒓 (⅔)𝑴𝑴 (𝒓𝒓 )𝑡𝑡. � 𝑑𝑑𝒓𝒓� (13) 2 ext 4 ′ 0−1 0−1 ′ The external -field will have3 𝑚𝑚 𝑡𝑡no𝐻𝐻 effect𝑧𝑧 on the9 𝜇𝜇induced𝑉𝑉 𝑚𝑚 -𝑡𝑡field𝑚𝑚 and,𝑡𝑡 therefore, the sole contribution of ( ) to the overall time-rate-of-change of energy will be the addition of the first term on the right- 𝐻𝐻 ( ) 𝐸𝐸 hand sideext of Eq.(13), namely, ( ) (0), to found in Eq.(10). 𝑯𝑯 ext ′ total ⅔𝑚𝑚 𝑡𝑡 𝐻𝐻𝑧𝑧 𝑑𝑑ℰ ⁄𝑑𝑑𝑑𝑑 4

In contrast, in the Einstein-Laub formalism, when ( ) ( ) is present in the system, the time-rate- of-change of the energy stored inside the sphere will be ext 𝑯𝑯 𝒓𝒓 = ½ ( )( ) ( , ) 𝑑𝑑ℰstored 𝑑𝑑 2 0 ext 0−1 sphere( ) 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 � 𝜇𝜇( ∫) �𝑯𝑯(0) + 𝒓𝒓 − ⅓𝜇𝜇 (𝑴𝑴) 𝒓𝒓 𝑡𝑡( �). 𝑑𝑑𝒓𝒓� (14) 1 ext 1 ′ 0−1 0−1 ′ This time, however, the rate of− energy3 𝑚𝑚 𝑡𝑡 exchange𝐻𝐻𝑧𝑧 between9 𝜇𝜇 the𝑉𝑉 𝑚𝑚-field𝑡𝑡 𝑚𝑚 and𝑡𝑡 the dipole becomes ( , ) ( , ) = ( )( ) 𝐻𝐻 ( , ) ( , ) ext −1 sphere sphere ( ) 0 ∫ 𝑯𝑯 𝒓𝒓 𝑡𝑡 ∙ 𝜕𝜕𝑡𝑡𝑴𝑴 𝒓𝒓 𝑡𝑡 𝑑𝑑𝒓𝒓 =∫ ( )�𝑯𝑯 (0𝒓𝒓) − ⅓𝜇𝜇 𝑴𝑴 𝒓𝒓 (𝑡𝑡 )� ∙ 𝜕𝜕𝑡𝑡(𝑴𝑴). 𝒓𝒓 𝑡𝑡 𝑑𝑑𝒓𝒓 (15) ext ′ 0−1 0−1 ′ Upon adding Eq.(14) to Eq.(15), we find 𝑚𝑚 𝑡𝑡 𝐻𝐻𝑧𝑧 − ⅓𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡 ( ) = ( ) (0) ( ) ( ). (16) 𝑑𝑑ℰtotal 2 ext 2 ′ 0−1 0−1 ′ It is seen, once again,𝑑𝑑𝑑𝑑 that the3 𝑚𝑚 sole𝑡𝑡 contribution𝐻𝐻𝑧𝑧 − 9of𝜇𝜇 the𝑉𝑉 external𝑚𝑚 𝑡𝑡 𝑚𝑚 field𝑡𝑡 to is the addition of ( ) the term ( ) (0) to the result obtained previously in Eq.(12). We concludetotal that the overall rate 𝑑𝑑ℰ ⁄𝑑𝑑𝑑𝑑 of EM energy′ flow extinto or out of a stationary dipole in a static magnetic field is independent of whether one works⅔ in𝑚𝑚 the𝑡𝑡 formalism𝐻𝐻𝑧𝑧 of Lorentz or that of Einstein and Laub. 5. Energy considerations in the case of a moving dipole. A small spherical dipole of radius and volume , immersed in the static external field ( )( ) and moving slowly with velocity along the 𝑅𝑅 -axis, can0 be represented by its magnetization (ext, ) = ( ) ( ) ( ) ( ) ; see Fig.2. In what follows, 𝑉𝑉we will evaluate the integrals over a slightly𝑯𝑯 expanded𝒓𝒓 sphere of radius (denoted𝑣𝑣 by𝒛𝒛� sphere+) 𝑧𝑧in order to accommodate slight displacements of 𝑴𝑴the𝒓𝒓 dipole𝑡𝑡 𝑚𝑚during𝑡𝑡 𝛿𝛿 an𝑥𝑥 infinitesimal𝛿𝛿 𝑦𝑦 𝛿𝛿 𝑧𝑧 − 𝑣𝑣 𝑣𝑣time+ 𝒛𝒛� interval . In the Lorentz formalism, the time-rate-of-change of the EM energy stored𝑅𝑅 inside the (slightly enlarged) sphere is written ∆𝑡𝑡

= ½ [ ( )( ) + ( , )] 𝑑𝑑ℰstored 𝑑𝑑 0−1 0 ext 2 sphere+ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝜇𝜇 �( ) 𝜇𝜇 𝑯𝑯 𝒓𝒓 ⅔𝑴𝑴 𝒓𝒓 𝑡𝑡 𝑑𝑑𝒓𝒓� = ( )[ ( ) ( ) ( ) ( )] ( ) ( ) + ( ) ( ) 2 ext 4 ′ ′ 0−1 0−1 ′ 3 + 𝑧𝑧 9 �sphere 𝐻𝐻( ) 𝒓𝒓 𝑚𝑚 𝑡𝑡 𝛿𝛿 𝑧𝑧 − (𝑣𝑣𝑣𝑣 )− 𝑣𝑣𝑣𝑣 𝑡𝑡 𝛿𝛿 𝑧𝑧 − 𝑣𝑣𝑣𝑣 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑦𝑦 𝑑𝑑𝒓𝒓 𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡 = ( ) ( ) + ( ) ( ) + ( ) ( ). (17) 2 ext 2 ext 4 ′ ′ 0−1 0−1 ′ As for 3the𝑚𝑚 contribution𝑡𝑡 𝐻𝐻𝑧𝑧 𝑣𝑣𝑣𝑣 of the3 𝑣𝑣 𝑣𝑣-field𝑡𝑡 𝐻𝐻𝑧𝑧 induced𝑣𝑣𝑣𝑣 by9 a𝜇𝜇 time𝑉𝑉 -varying𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡( ), the exchanged energy will continue to have a time-rate given (to a good approximation) by ( ) ( ); see the 𝐸𝐸 𝑚𝑚 𝑡𝑡 paragraph immediately after Eq.(9). The time-rate-of-change of total energy within0−1 0− the1 dipole′ will thus be −⅔𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡 ( ) ( ) = ( ) ( ) + ( ) ( ) ( ) ( ). (18) 𝑑𝑑ℰtotal 2 ext 2 ext 2 ′ ′ 0−1 0−1 ′ In contrast, in the𝑑𝑑𝑑𝑑 Einstein3 𝑚𝑚-Laub𝑡𝑡 𝐻𝐻 formulation,𝑧𝑧 𝑣𝑣𝑣𝑣 3we𝑣𝑣𝑣𝑣 will𝑡𝑡 𝐻𝐻have𝑧𝑧 𝑣𝑣𝑣𝑣 − 9 𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡

= ½ [ ( )( ) ( , )] 𝑑𝑑ℰstored 𝑑𝑑 ext −1 2 0 + 0 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝜇𝜇 �sphere( ) 𝑯𝑯 𝒓𝒓 − ⅓𝜇𝜇 𝑴𝑴 𝒓𝒓 𝑡𝑡 𝑑𝑑𝒓𝒓� = ( )[ ( ) ( ) ( ) ( )] ( ) ( ) + ( ) ( ) 1 ext ′ ′ 1 −1 −1 ′ 𝑧𝑧 0 0 − 3 �sphere+𝐻𝐻( ) 𝒓𝒓 𝑚𝑚 𝑡𝑡 𝛿𝛿 𝑧𝑧 − (𝑣𝑣𝑣𝑣 )− 𝑣𝑣𝑣𝑣 𝑡𝑡 𝛿𝛿 𝑧𝑧 − 𝑣𝑣𝑣𝑣 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑦𝑦 𝑑𝑑𝒓𝒓 9 𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡 = ( ) ( ) ( ) ( ) + ( ) ( ). (19) 1 ext 1 ext 1 ′ ′ 0−1 0−1 ′ − 3 𝑚𝑚 𝑡𝑡 𝐻𝐻𝑧𝑧 𝑣𝑣𝑣𝑣 − 3 𝑣𝑣𝑣𝑣 𝑡𝑡 𝐻𝐻𝑧𝑧 𝑣𝑣𝑣𝑣 9 𝜇𝜇 𝑉𝑉 𝑚𝑚 𝑡𝑡 𝑚𝑚 𝑡𝑡 5

As for the exchange rate of EM energy between the field and the dipole, we write

( , ) ( , )

+ sphere ( 𝑡𝑡 ) ∫ = 𝑯𝑯 𝒓𝒓 𝑡𝑡[ ∙ 𝜕𝜕 𝑴𝑴(𝒓𝒓)𝑡𝑡 𝑑𝑑𝒓𝒓 ( , )][ ( ) ( ) ( ) ( )] ( ) ( ) ext −1 ′ ′ 𝑧𝑧 0 �sphere+ (𝐻𝐻 ) 𝒓𝒓 − ⅓𝜇𝜇 𝑀𝑀( 𝒓𝒓 )𝑡𝑡 𝑚𝑚 𝑡𝑡 𝛿𝛿 𝑧𝑧 − 𝑣𝑣𝑣𝑣 − 𝑣𝑣𝑣𝑣 𝑡𝑡 𝛿𝛿 𝑧𝑧 − 𝑣𝑣𝑣𝑣 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑦𝑦 𝑑𝑑𝒓𝒓 = ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( , ). (20) ext ext ′ ′ 0−1 0−1 ′ 0−1 Thus𝑚𝑚 the𝑡𝑡 sum𝐻𝐻𝑧𝑧 of Eqs.(1𝑣𝑣𝑣𝑣 𝑣𝑣9)𝑣𝑣 and𝑡𝑡 𝐻𝐻(2𝑧𝑧0) agrees𝑣𝑣𝑣𝑣 −with⅓ 𝜇𝜇the 𝑉𝑉result𝑚𝑚 obtained𝑡𝑡 𝑚𝑚 𝑡𝑡 −in ⅓the𝜇𝜇 Lorentz𝑣𝑣𝑣𝑣 𝑡𝑡 formalism,𝜕𝜕𝑧𝑧𝑀𝑀 𝒓𝒓 𝑡𝑡 Eq.(18), except for the last term on the right-hand side of Eq.(20), which appears to have no Lorentz counterpart. In fact, this last term is a mathematical artifact, as it represents the self-action of a fixed (i.e., time- invariant) dipole moving at constant velocity in free space, which must vanish. This last term also needs to be corrected, because the -field of the dipole immediately outside its north and south poles needs to be included in the equation. The discontinuity of the spherical dipole’s -field at the sphere’s surface requires that the self -field 𝐻𝐻at the surface be replaced by its average value across the discontinuity. The contributions of the last term in Eq.(20) to the integral, which are localized𝐻𝐻 at the upper and lower hemispherical surfaces,𝐻𝐻 thus cancel out. As was the case in Sec.4 with regard to a stationary dipole, we conclude that the overall rate of EM energy flow into or out of a moving dipole in the presence of a static magnetic field does not depend on whether one works in the formalism of Lorentz or that of Einstein and Laub. 6. Concluding remarks. This paper started by asking the question: Why do appear to perform mechanical work by pushing or pulling on other magnets (and also by attracting iron filings or pieces of scrap iron) when the Lorentz force law explicitly forbids such work? We argued that the answer depends on the model that one uses to represent magnetic dipoles. In the case of the Amperian current-loop model, the diamagnetic susceptibility of the dipole in the presence of an external magnetic field ensures that internal work (performed to speed up or slow down the current of the loop itself) cancels out the external work that is performed when the dipole is pushed or pulled by the external field. In the case of dipole as a north-south pair of adjacent magnetic monopoles (i.e., the Einstein-Laub model), the external work by the applied field is the same as that performed on an equivalent Amperian current loop. However, without additional information about the changes (if any) induced in the paired monopole type of dipole upon exposure to the external field, it is not possible to decide if the external work performed on the dipole is indeed cancelled by any internal work. Both the Amperian current-loop model (in the context of what we called the Lorentz electro- dynamics) and the model in which an adjacent pair of north-south monopoles comprise a magnetic dipole (i.e., in the Einstein-Laub formulation) can handle static as well as dynamic magnetic dipoles without violating Maxwell’s equations or compromising the energy-momentum conservation laws; see Secs.4 and 5. The question remains as to which model can more accurately predict all the experimental observations.

Acknowledgment. The author is grateful to Ewan Wright and David Griffiths for commenting on an early version of the manuscript. References 1. J. D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York (1999). 2. D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Prentice-Hall, New Jersey (1999). 3. J. H. Van Vleck, Theory of Electric and Magnetic Susceptibilities, Clarendon Press, Oxford (1932). 4. A. Einstein and J. Laub, “Über die im elektromagnetischen Felde auf ruhende Körper ausgeübten pondero-motorischen Kräfte,” Ann. Phys. 26, 541-550 (1908); reprinted in translation: “On the ponderomotive exerted on bodies at rest in the ,” The Collected Papers of Albert Einstein, vol.2, Princeton University Press, Princeton, NJ (1989). 5. M. Mansuripur, “The Force Law of Classical Electrodynamics: Lorentz versus Einstein and Laub,” Optical Trapping and Optical Micro-manipulation X, edited by K. Dholakia and G.C. Spalding, Proceedings of SPIE 8810, 88100K~1:18 (2013). 6. M. Mansuripur, “On the Foundational Equations of the Classical Theory of Electrodynamics,” Resonance 18, 130-155 (2013).

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