Continuum Electrodynamics of a Piecewise-Homogeneous Linear Medium
CContinuum Electrodynamics of a Piecewise-Homogeneous Linear Medium
Michael E. Crenshaw
US Army Aviation and Missile Research, Development,and Engineering Center, Redstone Arsenal, AL 35898, USA (Dated: November 5, 2018)The energy–momentum tensor and the tensor continuity equation serve as the conservation lawsof energy, linear momentum, and angular momentum for a continuous flow. Previously, we de-rived equations of motion for macroscopic electromagnetic fields in a homogeneous linear dielectricmedium that is draped with a gradient-index antireflection coating (J. Math Phys.
I. INTRODUCTION
The energy–momentum tensor is an innate and com-pelling aspect of energy and momentum conservation ina continuous flow [1]. Recently [2–5], we used globalconservation principles to construct the total energy–momentum tensor for a thermodynamically closed sys-tem consisting of a quasimonochromatic optical pulse anda homogeneous simple linear medium that is draped witha gradient-index antireflection coating. Regarding the to-tal energy–momentum tensor and the tensor continuityequation as fundamental, we derived equations of motionfor the macroscopic fields. The formulation of contin-uum electrodynamics that was derived in our previouswork [2–7] was limited to homogeneous materials with agradient-index antireflection coating. In this article, wedevelop the theory of continuum electrodynamics for themore usual situation of a piecewise-homogeneous lineardielectric medium. One of the major differences with theMaxwell theory is that the Fresnel relations can no longerbe derived from the application of Stoke’s theorem to theFaraday and Maxwell–Amp`ere Laws. Instead, we derivethe Fresnel relations from the electromagnetic wave equa-tion and conservation of energy. We obtain the field andmaterial components of the total momentum, derive theradiation pressure, and derive the electromagnetic con-tinuity equations for a piecewise homogeneous medium.We provide an interpretation of the Jones and Richards[8] measurement of the optical force on a mirror immersedin a dielectric fluid.
II. EQUATIONS OF MOTION
We take as a given that propagation of the electromag-netic field is characterized by the wave equation ∇ × ( ∇ × A ) + n c ∂ A ∂t = 0 (2.1)in a limit in which absorption can be neglected. Dis-persion is treated parametrically for the arbitrarily longquasimonochromatic fields that are considered here. Theelectromagnetic wave equation, Eq. (2.1), is a mixedsecond-order differential equation that can be written interms of first-order differential equations. To that end,we define the macroscopic magnetic field B = ∇ × A (2.2)and a second macroscopic field Π = nc ∂ A ∂t . (2.3)A Maxwell–Amp`ere-like law ∇ × B + nc ∂ Π ∂t = 0 (2.4)results from the substitution of the definitions of themacroscopic fields, Eqs. (2.2) and (2.3), into the waveequation, Eq. (2.1). Two more equations of motion arederived from the definitions of the fields. Thompson’sLaw ∇ · B = 0 (2.5)is obtained from the divergence of Eq. (2.2) with the vec-tor identity that the divergence of the curl of any vector a r X i v : . [ phy s i c s . c l a ss - ph ] M a r is zero. The curl of Eq. (2.3) ∇ × Π − nc ∂ B ∂t = ∇ nn × Π (2.6)is our variant of the Faraday Law. Taking the divergenceof Eq. (2.4) and integrating with respect to time, we ob-tain the Gauss-like law ∇ · Π = − ∇ nn · Π . (2.7)A constant of integration has been suppressed in the ab-sence of charges. Note that each field equation is alge-braically equivalent to its counterpart in the macroscopicMaxwell equations ∇ × B − n c ∂ E ∂t = 0 (2.8) ∇ · B = 0 (2.9) ∇ × E + 1 c ∂ B ∂t = 0 (2.10) ∇ · ( n E ) = 0 (2.11)if we define Π = − n E , which we have every right to dounder the auspices of Maxwellian continuum electrody-namics. However, the different sets of motional equa-tions for macroscopic fields have different tensorial andrelativistic properties. This means that Maxwellian con-tinuum electrodynamics, which is fundamentally a vec-tor theory, admits improper tensor transformations ofcoordinates for the coupled equations [7]. The deriva-tion of the motional equations Eqs. (2.4)–(2.7) from La-grangian field theory appears elsewhere [7]. Now, it canbe argued that Eqs. (2.4)–(2.7) violate relativity becausethe Lorentz factor is γ d = 1 / (cid:112) − n v /c instead of γ v = 1 / (cid:112) − v /c . However, that argument presup-poses transformations between a coordinate system inthe dielectric and a coordinate system in a vacuum lab-oratory (lab) frame. If we consider, instead, transforma-tions between two coordinate systems in an arbitrarilylarge region of space in which the speed of light is c/n [9], as we should, then γ d is the correct Lorentz fator andEqs. (2.4)–(2.7) are, in fact, compatible with relativity[7].Returning to the equations of motion for the macro-scopic fields, Eqs. (2.4)–(2.7), we can identify two lim-iting cases of particular interest. The first is when thegradient of the index of refraction ∇ n is sufficiently smallthat reflections and Helmholtz forces can be neglected[2]. It is in this limit of an unimpeded continuous flow ofelectromagnetic radiation that the energy and momen-tum densities can be expressed through continuity equa-tions and a total energy–momentum tensor for a ther-modynamically closed system. This case was treated in Refs. [2–7]. Here, we consider a quasimonochromaticoptical pulse that passes from one simple linear mediuminto a second such medium through a planar interface atnormal incidence. In this limit of piecewise-homogeneousmedia, we need only retain the homogeneous parts ofEqs. (2.4)–(2.7) to obtain ∇ × B + nc ∂ Π ∂t = 0 (2.12) ∇ · B = 0 (2.13) ∇ × Π − nc ∂ B ∂t = 0 (2.14) ∇ · Π = 0 , (2.15)where the fields are connected by boundary conditions atthe interface between different homogeneous linear ma-terials. While the derivation of the equations of motionfor macroscopic fields in a homogeneous medium fromEqs. (2.4)–(2.7) is obvious, the usage of Eqs. (2.12)–(2.15) for piecewise-homogeneous matter remains to beinvestigated. III. THE FRESNEL RELATIONS
First, we demonstrate how our results apply to themost obvious issue relating to piecewise-homogeneousmatter, the Fresnel relations. Because the equations ofmotion for macroscopic fields in a piecewise-homogeneouslinear medium have changed, there is a question thatarises as to how the Fresnel relations survive. Apply-ing Stokes’s theorem to the macroscopic Maxwell curlequations, Eqs. (2.8) and (2.10), it was long ago foundthat the tangential components of the electric field E andthe magnetic auxiliary field H = B are continuous at aplanar interface between linear homogeneous dielectrics.Similarly, the normal components of the displacementfield D = n E and the magnetic field B were shown tobe continuous by the divergence theorem applied to theMaxwell divergence equations, Eqs. (2.9) and (2.11). Thesimultaneous continuity of the transverse E and H fieldsand the normal parts of the D and B fields leads to theFresnel relations. Because Eqs. (2.6) and (2.7) are inho-mogeneous, the transverse and normal components of themacroscopic field Π are not continuous at the materialinterface, even though the appearance of Eqs. (2.14) and(2.15) might seem to suggest otherwise.We treat the propagation of a quasimonochromatic op-tical pulse through a piecewise-homogeneous simple lin-ear medium. The material is initially stationary in thelaboratory frame of reference and is rigidly attached toa support. We take as a given that propagation of theelectromagnetic field is characterized by the wave equa-tion ∇ × ( ∇ × A ) + n ( r ) c ∂ A ∂t = 0 . (3.1) n - ( no un i t s ) Fig.1
FIG. 1: Amplitude of the incident field envelope (arb. units)
The pulse is sufficiently monochromatic that dispersioncan be neglected in accordance with the characterizationof a simple linear medium. Further, the simple linearmaterial with its support is assumed to be arbitrarilymassive so that n ( r ) can be treated as a time-independentfunction of space in the laboratory frame of reference.In order to not overly complicate matters, we assumenormal incidence and adopt the plane-wave limit for thefield. The vector potential, in the plane-wave limit, A ( z, t ) = 12 (cid:16) ˜A ( z, t ) e − i ( ω d t − k d z ) + ˜A ∗ ( z, t ) e i ( ω d t − k d z ) (cid:17) can be written in terms of an envelope function ˜ A ( z, t )and a carrier wave with center frequency ω d . Here, k d isthe amplitude of the wave vector k d = ( nω d /c ) ˆe z thatis associated with the center frequency of the field, and ˆe z is a unit vector in the direction of propagation alongthe z axis. The vector potential amplitude ˜A is notslowly varying if there is a backward propagating fieldcomponent. Figure 1 shows a one-dimensional represen-tation of the amplitude of the incident field ˜ A ( z ) = (cid:16) ˜A ( z, t ) · ˜A ∗ ( z, t ) (cid:17) / about to enter a simple linearmedium with n = 1 .
40 at normal incidence from thevacuum n = 1. Figure 2 presents a time-domain nu-merical solution of the wave equation at a later time t depicted by ˜ A ( z ) = (cid:16) ˜A ( z, t ) · ˜A ∗ ( z, t ) (cid:17) / . The re-flected field and the refracted field have separated andthe refracted field is entirely inside the medium. The re-fracted pulse has not propagated as far as it would havepropagated in the vacuum due to the reduced speed oflight c/n in the material. In addition, the spatial extentof the refracted pulse in the medium is w t = n w i /n interms of the width w i of the incident pulse. As shown inFig. 2, the amplitudes of the reflected and refracted fieldsare different from the amplitude of the incident field.We would like a formula for determining how muchof the incident pulse goes into the reflected field andhow much is refracted. Although the Fresnel relationsare the necessary formulas, their use is problematic at Fig.2 n - ( no un i t s ) FIG. 2: Refracted field entirely within the linear medium andseparated from the reflected field. this point because their provenance from the macroscopicMaxwell equations, Eqs. (2.8)–(2.11), is suspect. Becausethe macroscopic field Π is not continuous at a step indexboundary, we present a derivation of the Fresnel relationsthat is based on the wave equation and conservation ofenergy. Applying Stokes’s theorem to the wave equation,Eq. (3.1), we have (cid:73) C ( ∇ × A ) · d l = (cid:90) S ( ∇ × ( ∇ × A )) · ˆn da . (3.2)Consider a thin right rectangular box or “Gaussian pill-box” that straddles the interface between the two medi-ums with the large surfaces parallel to the interface.Then S is the surface of the pillbox, da is an elementof area on the surface, and ˆn is an outwardly directedunit vector normal to da . There is no contribution tothe surface integral from the large surfaces for our nor-mally incident field because ∇ × ( ∇ × A ) is orthogonalto ˆn . The contributions from the smaller surfaces can beneglected as the box becomes arbitrarily thin. Then (cid:73) C ( ∇ × A ) · d l = 0 . (3.3)We choose the closed contour C in the form of a rectan-gular Stokesian loop with sides that bisect the two largesurfaces and two of the small surfaces on opposite sidesof the pillbox. Here, d l is a directed line element thatlies on the contour, C . Then C , like S , straddles the ma-terial interface. For normal incidence in the plane-wavelimit, the field ∇ × A can be oriented along the longsides of the contour C . Performing the contour integra-tion in Eq. (3.3), the contribution from the short sides ofthe contour are neglected as the loop is made vanishinglythin and we obtain( ∇ × A ) · ∆ l + ( ∇ × A ) · ∆ l = 0 (3.4)from the long sides, 1 and 2, of the contour.For linearly polarized radiation, we can write the vec-tor potential of the incident, reflected, refracted, andtransmitted waves as A i = ˆe x ˜ A i e − i ( ω d t − k z ) (3.5) A r = ˆe x ˜ A r e − i ( ω d t + k z ) (3.6) A t = ˆe x ˜ A t e − i ( ω d t − k z ) (3.7) A T = ˆe x ˜ A T e − i ( ω d t − k z ) . (3.8)where k = n ω d /c , k = n ω d /c , and ˆe x is a unit polar-ization vector. It is understood that we use the real partof complex fields and neglect double frequency terms infield products. For convenience, the scalar amplitudesare taken to be real. Using the fact that the line ele-ments ∆ l and ∆ l in Eq. (3.4) are equal and opposite,we obtain a relation n ( ˜ A i − ˜ A r ) = n ˜ A t (3.9)between the amplitudes of the incident, reflected, andrefracted fields. In order to derive boundary conditions,we need another such relation.For a stationary simple linear material, the electromag-netic energy U = (cid:90) σ (cid:32) n c (cid:18) ∂ A ∂t (cid:19) + ( ∇ × A ) (cid:33) dv (3.10)is conserved. Here, the volume of integration, which in-cludes all fields present, has been extended to all-space σ . The total energy is invariant in time by virtue of be-ing conserved. The total energy at time t , the incidentenergy U ( t ) = U i , is equal to the total energy at a latertime t , U ( t ) = U r + U t , which is the sum of the re-flected energy U r and the refracted energy U t when therefracted field is entirely within the medium.In terms of the incident, reflected, and transmitted en-ergy, the energy balance U ( t ) = U ( t ) is U i = U r + U t . (3.11)Substituting Eqs. (3.5)–(3.7) into the formula for theenergy, Eq. (3.10), and expressing the energy balance,Eq. (3.11), in terms of the amplitudes of the incident,reflected, and transmitted vector potential results in (cid:90) σ n ˜ A i dv = (cid:90) σ n ˜ A r dv + (cid:90) σ n ˜ A t dv . (3.12)In order to facilitate the integration of Eq. (3.12), wechoose the incident pulse to be rectangular with a nom-inal width of w i . The pulse has a finite rise time and afinite fall time to reduce ringing, but the short transitionregion can be neglected compared to the arbitrarily large width of the pulse. The refracted pulse has a width of n w i /n due to the change in the velocity of light be-tween the two media. Then, evaluating the integrals ofEq. (3.12) results in n ˜ A i = n ˜ A r + n n ˜ A t . (3.13)Grouping terms of like refractive index, the previousequation n (cid:16) ˜ A i − ˜ A r (cid:17) = n ˜ A t (3.14)becomes more suggestive as n (cid:16) ˜ A i − ˜ A r (cid:17) (cid:16) ˜ A i + ˜ A r (cid:17) = n ˜ A t (3.15)by factoring the binomial. The second-order equationcan be written as two first-order equations. SubstitutingEq. (3.9) into Eq. (3.15), we have the unique decomposi-tion n (cid:16) ˜ A i − ˜ A r (cid:17) = n ˜ A t (3.16) (cid:16) ˜ A i + ˜ A r (cid:17) = ˜ A t (3.17)of Eq. (3.13). We eliminate ˜ A t from Eq. (3.16) usingEq. (3.17) to obtain ˜ A r ˜ A i = n − n n + n . (3.18)Subsequently, we eliminate ˜ A r to get˜ A t ˜ A i = 2 n n + n . (3.19)We see that the usual Fresnel relations can be derivedwithout invoking Maxwell’s equations. Conservation ofenergy, by itself, is sufficient to derive Eq. (3.13). How-ever, there are several ways that Eq. (3.13) can be decom-posed into two first-order equations. The application ofStoke’s theorem to the wave equation guarantees unique-ness of the decomposition represented by Eqs. (3.16) and(3.17) and the Fresnel relations, Eqs. (3.18) and (3.19). IV. MOMENTUM CONSERVATION INPIECEWISE HOMOGENEOUS MEDIA
The correct form for the momentum of the electromag-netic field in a dielectric is the subject of the century-oldAbraham–Minkowski controversy [10–16]. The currentlyaccepted resolution of the controversy, due to Møller [17],Penfield and Haus [18], Pfeifer et al. [10], and others, isthat the issue is undecidable because neither the Abra-ham momentum nor the Minkowski momentum is thetotal momentum. If we adopt this viewpoint, then the
Fig.3 n - ( no un i t s ) FIG. 3: Transmitted field has left the medium.
Abraham momentum and the Minkowski momentum areirrelevant. Here we use conservation of total energy andconservation of total momentum to show that each com-ponent of the total linear momentum, reflected, refractedor transmitted, and kinematic, has a definite expressionin terms of the macroscopic fields.The energy and momentum of an electromagnetic pulsein the vacuum U v = (cid:90) σ (cid:0) Π v + B v (cid:1) dv (4.1) G v = (cid:90) σ B v × Π v c dv (4.2)are well-defined and settled. Here, v denotes a quan-tity that is based in the vacuum, n = 1. The totalenergy and the total momentum of our system in theinitial configuration at t as shown in Fig. 1 are con-sidered to be given quantities. Figure 3 shows the re-sult of continuing the numerical solution of the waveequation, Eq. (3.1), until the pulse has propagated com-pletely through the medium. The incident, reflected, andtransmitted fields are in vacuum with well-defined ener-gies so that we can write an energy balance equation U i = U r + U T + U kinematic . Here, U kinematic is the kine-matic energy of a solid block of dielectric material. Writ-ing the components of the energy balance equation interms of the corresponding vector potential amplitudes,we have (cid:90) σ ω d c ˜ A i dv = (cid:90) σ ω d c ˜ A r dv + (cid:90) σ ω d c ˜ A T dv + U kinematic . (4.3)Conservation of linear momentum cause more prob-lems than conservation of energy because linear momen-tum is a directed quantity that changes sign upon reflec-tion. Surface reflection takes momentum from the fieldand transfers the momentum to the material throughradiation pressure. Once the field has passed entirely through the surface, as in Fig. 2, there are no more sur-face forces and the block of material moves with con-stant velocity carrying a momentum G kinematic . Notethat G kinematic is not the kinetic momentum describedby Barnett [19]. Here, in Fig. 3, all the fields have leftthe material and have well-defined momentums in thevacuum. Then, we can write G i = G r + G T + G kinematic (4.4)or (cid:90) σ α ˜ A i ˆe z dv = − (cid:90) σ α ˜ A r ˆe z dv + (cid:90) σ α ˜ A T ˆe z dv + G kinematic (4.5)by conservation of linear momentum. Here, α = ω d / (2 c )is a useful combination of coefficients. SubstitutingEq. (4.3) into Eq. (4.5) we find that the kinematic mo-mentum of the material is G kinematic = (cid:90) σ α ˜ A r ˆe z dv + U kinematic /c . (4.6)Taking U kinematic /c to be negligible, we find that thekinematic momentum is twice the momentum of the re-flected field, but in the forward direction as determinedby the direction of the incident field such that G kinematic = − G r = − (cid:90) σ B r × Π r c dv . (4.7)The sign is a bit awkward, but necessary, because B r × Π r is in the backward (negative) direction while the kine-matic momentum is in the forward direction.Now we can return to the situation of Fig. 2 with therefracted pulse entirely within the medium. There is alinear momentum that is associated with the propagatingfield in the medium [20] although its composition in termsof what portion is field momentum and what portion ismaterial momentum remains disputed [10]. Whateverthe composition, the momentum that travels with thefield through the material, G t , must be equal to the well-defined momentum of the field G t ( t ) = G T ( t ) = (cid:90) σ B T × Π T c dv (4.8)that has exited the material through the antireflectioncoating. Applying conservation of energy, Eq. (3.10), wefind that Π t = √ n Π T , B t = √ n B T , and G t ( t ) = (cid:90) σ B t × Π t c dv (4.9)is the total momentum that travels with the field insidethe medium. When comparing Eqs. (4.8) and (4.9), re-call that the refracted field is spatially narrower thanthe transmitted field. Substituting Eqs (4.7)–(4.9) intoEq. (4.4), we find that the total momentum G tot is thesum of well-defined quantities for the refracted momen-tum, the reflected momentum, and the kinematic mo-mentum G tot = (cid:90) σ B t × Π t c dv + (cid:90) σ B r × Π r c dv − (cid:90) σ B r × Π r c dv. (4.10)The identification of G tot with G i is proven by demon-strating conservation of the total momentum. Substi-tuting the definitions of the fields, Eqs. (2.3) and (2.4),and the Fresnel relations, Eqs. (3.18) and (3.19), intoEq. (4.10) proves that the total momentum is conserved.The total momentum has a definite electromagnetic com-ponent G em = (cid:90) σ B t × Π t c dv + (cid:90) σ B r × Π r c dv (4.11)that is associated with the propagating field. The mate-rial component G matl = − (cid:90) σ B r × Π r c dv (4.12)is the momentum of the block of dielectric. In the nextsection, we will relate the kinematic movement of theblock to the Fresnel surface force. Now we consider thegeneral case of a dielectric block with index n in aninviscid dielectric fluid of index n . The momentum ofthe field in the dielectric block is known to be G em = (cid:90) σ B × Π c dv (4.13)by Eq. (4.9) and by prior work [2–5]. Then repeat-ing the above analysis, the total, electromagnetic, andmaterial momentums are still given by the formulas,Eqs. (4.10)–(4.12), although the transmitted and re-flected fields are different in accordance with the Fresnelrelations, Eqs. (3.18) and (3.19). V. ELECTROMAGNETIC CONTINUITYEQUATIONS
The field imparts a surface force to the material dueto the change of sign of the electromagnetic momentumupon reflection. By Newton’s third law, the material ac-celerates, increasing in momentum. By Newton’s secondlaw, the material imposes an equal force on the electro-magnetic field and momentum is extracted from the field.Clearly, the subsystems are open systems as momentumis removed from the field and transferred to the materialby the surface force, but the total system is thermody-namically closed and the total linear momentum, as wellas the total energy, is conserved.Consider a quasimonochromatic pulse incident on anarbitrarily large homogeneous medium. The mediumis draped with a gradient-index antireflection coatingand the index changes sufficiently slowly that Helmholtz forces are negligible. Then a stationary medium remainsstationary, the system is thermodynamically closed, andenergy and momentum are conserved. The tensor conti-nuity equation is ¯ ∂ β T αβ = 0 , (5.1)where ¯ ∂ β = (cid:18) nc ∂∂t , ∂∂x , ∂∂y , ∂∂z (cid:19) (5.2)is the material four-divergence operator [2–5, 7, 21], T = ( Π + B ) / B × Π ) ( B × Π ) ( B × Π ) ( B × Π ) W W W ( B × Π ) W W W ( B × Π ) W W W (5.3)is the total energy–momentum tensor [2–5], and W ij = − Π i Π j − B i B j + 12 ( Π + B ) δ ij (5.4)is the stress tensor [2–5].The imposition of a step-index interface on the incidentsurface of the solid material is accompanied by a surfaceforce F due to Fresnel reflection. For a field of cross-sectional area A with square temporal dependence,∆ G matl = − A B r × Π r c c ∆ tn (5.5)is found by integration of Eq. (4.7). Here, as in the pre-ceding section, n is the refractive index of the regionfrom which the field originates, that is, the index of thedielectric fluid (or vacuum n = 1) in which the dielectricblock is immersed. Then F = n c ∆ G matl ∆ t = − A B r × Π r c . (5.6)The force must represent a source or sink of electromag-netic momentum and it must therefore have the sametime dependence as the momentum continuity equation.For a field in the plane-wave limit that is normally inci-dent on the block of material, the radiation pressure F A = 1 A n c c ∆ G matl ∆ t = − B r × Π r (5.7)acts on the incident surface at z = 0. Then the radiationpressure can be represented in terms of a force density as f = ( − B r × Π r ) δ ( z ) . (5.8)There is no source or sink of electromagnetic energy sowe can write a four-force density f α = (0 , ( − B r × Π r ) δ ( z )) . (5.9)Then the tensor continuity equation for the unimpededflow of the electromagnetic field, Eq. (5.1), becomes¯ ∂ β T αβ = f α (5.10)for a piecewise homogeneous medium. Because the forcedensity is a sink of the electromagnetic momentum den-sity, there is an equal and opposite force that acts as asource for the kinematic momentum of the material. Us-ing the results of Ref. [7] we can derive Newton’s secondlaw F = M d v d ( t/n )for a material body of mass M immersed in a dielectricfluid of index n . Then the momentum conservation lawfor the solid block of material is F = M n c cd v dt = (cid:90) σ − B r × Π r δ ( z ) dv . (5.11)Note that the tensor continuity equation for a flow ofnon-interacting material particles that is based on a dusttensor that is used in Refs. [10] and [22], for example,does not apply here because we have posited a solid di-electric. The components of the tensor continuity equa-tion, Eq. (5.10), are the energy continuity equation, nc ∂∂t (cid:18) (cid:0) Π + B (cid:1)(cid:19) + ∇ · B × Π c = 0 , (5.12)and the momentum continuity equation, nc ∂∂t ( B × Π ) + ∇ · W = ( − B r × Π r ) δ ( z ) . (5.13)The momentum continuity equation, Eq. (5.13), explic-itly displays the relation between the change in momen-tum on the left-hand side and the effect of the force,acting as a momentum sink, on the right-hand side. Themomentum continuity equation is not an exact compo-sition of Eqs. (2.12)–(2.15) because boundary conditionsimpose additional constraints.One of the enduring questions of the Abraham–Minkowski controversy is why the Minkowski momentumis so often measured experimentally while the Abrahamform of momentum seems to be so favored in theoreticalwork. We now have the tools to answer that question.The Minkowski momentum is not measured directly, butinferred from a measured index dependence of the opticalforce on a mirror placed in a dielectric fluid [8, 10, 19].Because the field is completely reflected at the mirror,the force on the mirror is F = nc ddt (2 c G ) = nc ddt (cid:90) V B × Π δ ( z ) dv . (5.14) The measured force on the mirror is directly proportionalto the refractive index n = n of the fluid [8, 10]. On theother hand, if we were to assume F = d G /dt , then wecan write Eq. (5.14) as F = 1 c ddt (cid:90) V D × B δ ( z ) dv (5.15)using D = − n Π . Then one might infer that the momen-tum of the field in the dielectric fluid is the Minkowskimomentum. Instead, we see that the electromagnetic mo-mentum that is obtained from an experiment that mea-sures the optical force on a mirror depends on the theorythat is used to interpret the results. However, based onthe changes to continuum electrodynamics that are ne-cessitated by conservation of energy and momentum bythe propagation of light in a continuous medium, we findthat Eq. (5.14) is the correct relation between the forceon the mirror and the momentum of the field in a dielec-tric. VI. CONCLUSION
The extraordinary persistence of theoretical and ex-perimental inconsistencies surrounding the Abraham–Minkoswski controversy [10–16] regarding the energy–momentum tensor for light in a linear medium suggestedthe need to re-examine the role of conservation of energyand conservation of momentum in classical continuumelectrodynamics. We found that it is necessary to giveup the classical macroscopic Maxwell equations in orderto preserve the tensor form of the energy and momen-tum conservation laws [2–6]. Then we must re-examinethe body of work that has been built upon the historicalforms of the macroscopic Maxwell equations. In this arti-cle, we derived equations of motion, boundary conditions,continuity equations and radiation forces for the limitingcase of macroscopic fields B and Π propagating througha piecewise-homogeneous linear dielectric medium. [1] See for example: R. W. Fox and A. T. McDonald, Intro-duction to Fluid Dynamics , 2nd. ed. (Wiley, 1978).[2] M. Crenshaw and T. B. Bahder, Opt. Commun.
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