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The Lorentz Law of Force and Its Connections to Hidden Momentum

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The Lorentz Law of Force and Its Connections to Hidden Momentum The Lorentz force law and its connections to hidden momentum, the Einstein-Laub force, and the Aharonov-Casher effect Masud Mansuripur College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721 [Published in IEEE Transactions on Magnetics, Vol. 50, No. 4, 1300110, pp1-10 (2014)] Abstract. The Lorentz force of classical electrodynamics, when applied to magnetic materials, gives rise to hidden energy and hidden momentum. Removing the contributions of hidden entities from the Poynting vector, from the electromagnetic momentum density, and from the Lorentz force and torque densities simplifies the equations of the classical theory. In particular, the reduced expression of the electromagnetic force-density becomes very similar (but not identical) to the Einstein-Laub expression for the force exerted by electric and magnetic fields on a distribution of charge, current, polarization and magnetization. Examples reveal the similarities and differences among various equations that describe the force and torque exerted by electromagnetic fields on material media. An important example of the simplifications afforded by the Einstein-Laub formula is provided by a magnetic dipole moving in a static electric field and exhibiting the Aharonov-Casher effect. 1. Introduction. The classical theory of electrodynamics is based on Maxwell’s equations and the Lorentz force law [1-4]. In their microscopic version, Maxwell’s equations relate the electromagnetic (EM) fields, ( , ) and ( , ), to the spatio-temporal distribution of electric charge and current densities, ( , ) and ( , ). In any closed system consisting of an arbitrary distribution of charge and 푬current,풓 푡 Maxwell’s푩 풓 푡 equations uniquely determine the field distributions provided that the휌 sources,풓 푡 푱 and풓 푡 , are fully specified in advance. The EM force- density ( , ) and torque-density ( , ) that the and fields exert on material media, the seat of and , are then given by the Lorentz휌 law,푱 as follows: 푭1 풓 푡 푻1 풓 푡 푬 푩 =ρ +× 휌 푱 F1(,) rt (,) r tt Er (,) Jr (,) t Br (,), t (1) Tr11(,)tt= r × Fr (,). (2) (The subscript 1 here distinguishes the Lorentz force and torque from alternative expressions to be introduced later.) The force and torque densities in the above equations are purely electromagnetic. In general, there are other forces and torques that the various parts of the material media exert on each other. The spatio-temporal evolution of and is thus determined by the action of total force and total torque on charge carriers. However, since our analysis is based on the assumption that ( , ) and ( , ) are fully specified in휌 advance푱 , we need not be concerned with the internal interactions among the material parts. This is not to ignore the significance of such internal interactions휌 풓 푡 in푱 풓determining,푡 for instance, the constitutive relations, or the mechanical response of the host media to the EM fields; rather, by hiding away the internal processes and mechanisms that give rise to the specific forms of ( , ) and ( , ), we manage to focus our attention exclusively on the EM subsystem. The Lorentz force and torque expressions thus specify the rates of exchange of linear and angular moment휌 풓a푡 between푱 풓 the푡 EM subsystem and the host media. In other words, in the absence of information about interactions that take place within the media, Eq.(1) alone cannot tell the material subsystem how to react to the E and B fields; however, it does inform as to how much momentum per unit volume per unit time the EM fields give (or take away from) the material subsystem. Similarly, the Lorentz torque-density of Eq.(2) is a statement about the time-rate-of-exchange of angular momentum between the fields and the material media. Aside from Maxwell’s equations and the Lorentz force law, the classical theory contains a −1 few other ingredients, such as the Poynting vector Sr1o(,)t =µ E × B , and the linear and angular momentum densities pEM_1(,) rt =ε o EB × and LEM_1(,) rt = rp × EM_1 . (Here and are the permittivity and permeability of free space.) The Poynting theorem, derived from Maxwell’s 휀o 휇o equations in conjunction with the Poynting postulate that S1(r,t) represents the rate of flow of −1 EM energy per unit area per unit time, identifies E1(r,t)=½εo E·E + ½µo B·B as the EM energy-density, and E(r,t)·J(r,t) as the rate of exchange of energy with the material subsystem [1]. It is also important to know that (J, cρ) is a 4-vector, while E and B together form a 2nd rank tensor, both of which can be readily transformed between inertial frames in accordance with the Lorentz transformation rules [1-3]. The scalar and vector potentials ( , ) and ( , ), to which E and B are related through the identities ( , ) = / and ( , ) = × , also form a 4-vector, ( , ). 휓 풓 푡 푨 풓 푡 Suppose now that the EM sources include푬 풓polarization푡 −휵휓 −( 휕,푨) 휕푡and magnetization푩 풓 푡 휵 (푨, ), in addition to the aforementioned푐푨 휓 ( , ) and ( , ). Once again, we assume that the spatio- temporal distributions of , , and are fully specified푷 in풓 advance푡 , so that there 푴is no풓 푡need for any constitutive relations to enable휌 풓 푡 us to obtain푱 풓 푡 the source distributions from a knowledge of the EM fields. Similarly, we휌 푱do 푷not need푴 to know the internal forces and torques that the various parts of the media exert upon each other, which would be needed if one were to calculate the spatio-temporal evolution of the sources. Given the precise knowledge of the source distributions, the fields are now uniquely determined by solving Maxwell’s macroscopic equations [2,4], namely, ∇ ⋅=Dr(,)ttρfree (,), r (3) ∇ ×=Hr(,)t Jfree (,) r t +∂∂ Dr (,)/ tt , (4) ∇ ×=−Er(,)t∂∂ Br (,)/ tt , (5) ∇ ⋅=Br( ,t ) 0. (6) Here and are explicitly labeled as the densities of free charge and free current, is the electric field, is the magnetic field, is the displacement, and is the magnetic 휌free 푱free 푬 induction. By definition, Dr(,)t =εo E + P and Br(,)t =µo H + M . The speed of light in vacuum 푯 푫 푩 is related to the EM properties of free space via = 1/ . Once again, and form a 2nd rank tensor that can be Lorentz transformed between inertial frames. A second such tensor is o o formed by and , and a third one by and 푐 . If we� 휇now휀 define bound-푬charge푩 and bound- −1 current densities ρbound(,)rPt =−⋅∇ and Jrbound(,)tt= (∂∂ P / ) +× µo ∇ M , we may use them to 푷 푴 푫 푯 eliminate D and H from Eqs.(3-6). Maxwell’s macroscopic equations then reduce to his microscopic equations, where E and B are directly related to the total charge- and current- densities = + and = + . Once again, ( , ) acts as a 4-vector, and the continuity equation, + / = 0, is automatically satisfied. It total free bound total free bound total total is thus clear휌 that, as휌 far as 휌Maxwell’s equations푱 푱 are concerned,푱 ( , ) and 푱( , )푐 휌can be treated 휵 ∙ 푱total 휕휌total 휕푡 푷 풓 푡 푴 풓 푡 2 as distributions of charge and current, and that the knowledge of ( , ) and ( , ) is all that is needed to determine the corresponding EM fields throughout the entire space-time. total total The presence of magnetization in Maxwell’s equations, however,휌 풓 creates푡 difficulties푱 풓 푡 with regard to energy and momentum conservation [5-9], and also causes trouble with the Lorentz force and torque equations that leads to apparent violations of special relativity under certain circumstances [10]. One way to avoid these difficulties is to resort to the four-vector formulation of the Lorentz force, or to assume the existence of a hidden energy flux ( , ) × ( , ) and also a hidden momentum-density ( , ) × ( , ) wherever and whenever−1 the field is found o to interact with magnetic dipoles [5-9,11-17]. An alternative way is to automatically휇 푴 풓 푡 remove푬 풓 푡 the o hidden entities from the relevant휀 equations푴 풓 푡 푬and풓 푡to proceed by using these reduced푬 equations without ever invoking the hidden entities [13,18,19]. Removing the hidden energy flux from the aforementioned Poynting vector yields the reduced Poynting vector ( , ) = × which is recognized as the standard form used in many textbooks [2-4]. The energy density of the EM 2 field then becomes ( , ) = ½ · + ½ · , and the energy 푺exchange풓 푡 푬 rate푯 with the material subsystem becomes · + · / + · / . Also, eliminating the hidden ℰ2 풓 푡 휀o푬 푬 휇o푯 푯 contribution to EM momentum yields the reduced momentum density _ ( , ) = × / , free 푬 푱 푬 휕푷 휕푡 푯 휕푴 휕푡 2 which is readily recognized as the Abraham momentum-density of the EM 퐸푀field2 [20-28]. Finally, removing from the Lorentz force expression the contribution of hidden풑 momentum풓 푡 yields푬 푯 the푐 following formula for the EM force-density under all circumstances: −1 F2 (,)()(/) rtt=ρfree −⋅∇∇PE +[] Jfree +∂ P ∂ + µo ×MBME ×−∂ε(o × )/ ∂ t . (7) Equation (7) can be shown [29] to be very similar (but not identical) to another equation first proposed by Einstein and Laub in 1908 [30,31]. The Einstein-Laub force-density equation is Fr3 (,)t=ρfree E + J free × µ o H +⋅()(/)()(/). P∇∇ E +∂∂ P tt × µo H + M ⋅ H −∂ M ∂ × εo E (8) When Eqs.(7) and (8) are integrated over the volume of an isolated object (i.e., an object surrounded by free space), the total force experienced by the object in the two formulations turns out to be the same [29]. In other words, Eqs.(7) and (8) may predict different force distributions within a given object, but the total forces predicted by them are identical. As for the torque density, in the case of Eq.(7), one will have Tr22(,)tt= r × Fr (,), whereas with Eq.(8) it becomes necessary to use the alternative expression Tr33(,)tt=× r Fr (,) +×+ P E M × H.
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