Microscopic derivation of electromagnetic force density in magnetic dielectric media

Macroscopic force density imposed on a linear isotropic magnetic dielectric medium by an arbitrary electromagnetic field is derived by spatially averaging the microscopic Lorentz force density. The obtained expression differs from the commonly used expressions, but the energy-momentum tensor derived from it corresponds to a so-called Helmholtz tensor written for a medium that obeys the Clausius–Mossotti law. Thus, our microscopic derivation unambiguously proves the correctness of the Helmholtz tensor for such media. Also, the expression for the momentum density of the field obtained in our theory is different from the expressions obtained by Minkowski, Abraham, Einstein and Laub, and others. We apply the theory to particular examples of static electric, magnetic and stationary electromagnetic phenomena, and show its agreement with experimental observations. We emphasize that in contrast to a widespread belief the Abraham–Minkowski controversy cannot be resolved experimentally because of incompleteness of the theories introduced by Abraham and Minkowski.

3 law, the own fields of the free and bound charges do not contribute to the force density exerted on them by the total field. Our derivation shows that the knowledge of these fundamental laws of nature is sufficient for also obtaining the forces due to electrostriction and magnetostriction. Our second discovery is a new equation for the momentum density of the field. We would like to stress the observation that this new equation is asymmetrical with respect to and µ, whereas both the Minkowski and Abraham momenta densities are symmetrical with respect to these quantities. While usually the contribution of the field momentum density to the time-averaged force density is negligibly small, it can play a significant role in the interaction of a strong low-frequency or pulsed electromagnetic field with a dielectric material.
Let us write the microscopic Maxwell's equations and the Lorentz force density as follows: Here e and b are microscopic electric and magnetic fields in the medium, f mic is the microscopic Lorentz force density, and the microscopic electric charge and current densities ξ and j, respectively, are given by The electric charges, defined by q i and having coordinates r i , can be divided into free and bound charges,q j and q k , respectively. The bound charges form localized groups that belong to individual molecules in the medium. For these groups of charges, we use a dipole approximation that is obtained by expanding the charge and current densities of each molecule into Taylor series around the center of mass of the molecule and truncating the series after the terms containing the molecular electric and magnetic dipole moments d l and m l , respectively [30]. This is often used when deriving the macroscopic Maxwell's equations from the microscopic ones. Within this approximation, the charge and current densities are and the molecules are treated as point particles with electric and magnetic dipole moments. We note, however, that physically the third term in equation (9) represents current loops originating from rotational motion of electric charges in the molecules [30]. 4 Using equations (8) and (9), we write equation (5) as where the force densities f (f) mic and f (b) mic experienced by free and bound charges, respectively, are given by The macroscopic force density f (f) ≡ f (f) mic , where the angle brackets denote spatial averaging over a representative elementary volume δV chosen to have a spherical shape, is found with the aid of integration of f (f) mic over this volume: The spherical averaging we use is a common one, since it is simple and most general (in a sense that it does not discriminate between different directions in space). From now on we consider the free charges to be identical and drop the subindex j fromq j . The integration property of the delta function yields After the integration, only those indices j that cite the free charges in δV survive in the sum.
Since δV is small, we can assume that the free charges are distributed uniformly in δV and move at the same speed ∂r j /∂t = v (this speed can be equated to the average speed of free charges in δV ). In this case, equation (14) can be written in terms of the fields e(r) and b(r) averaged over the coordinates of the charges, i.e.Ñ −1 δV j in δV e(r j ) andÑ −1 δV j in δV b(r j ), whereÑ δV is the number of charges in δV . Since the charges are assumed to be distributed uniformly, this averaging is equivalent to the volume averaging of e(r) and b(r) over δV . We note that the own electric and magnetic fields of the free charges exhibit strong inhomogeneities around each r = r j , but exactly at r = r j they are zero 4 , so that the fields e(r j ) and b(r j ) are the same as without the charge j. In other words, the fields e(r j ) and b(r j ) are independent of the charge j. The spatial averages of the own fields of the free charges are zero as well, as long as these charges are considered to be distributed uniformly in δV (see section 2.13 in [31] or the text above equation (27) in the present paper). For the volume averages of the fields, we have e(r) = E and b(r) = B, with E and B being macroscopic electric and magnetic fields, respectively. Thus, the force density f (f) is where ρ =qÑ δV /δV and J =qvÑ δV /δV are the macroscopic charge and current densities, respectively. The obtained quantities ρ, J, E and B are considered to have a coordinate at the center of δV . Equation (15) is a standard equation for macroscopic force density imposed by electric and magnetic fields on free charges and currents. The macroscopic force density due to bound charges, f (b) ≡ f (b) mic , is calculated by us as a sum of three terms, In obtaining the result in equation (18), we have used the fact that ∇ · b = 0. Using equation (3), we can rewrite equation (17) as from which it follows that The overall force density f (b) is For molecules within δV and an arbitrary coordinate r in δV , we can write e(r) = e ext (r) + e own (r) and b(r) = b ext (r) + b own (r), where e own (r) and b own (r) are the fields created by the molecules themselves and the fields e ext (r) and b ext (r) are external with respect to the molecules within δV . The external fields are independent of the coordinates of the molecules in δV , while the own fields are strongly inhomogeneous around each r l . Substituting these expansions into equation (21), we obtain where is equal to zero due to Newton's third law. Indeed, besides the factor of δV −1 , f (b) own is the total force imposed on the molecules in δV by the fields produced by the molecules themselves. 6 Removing f (b) own from equation (22) and assuming that in δV the molecules have the same dipole moments d and m, owing to the smallness of δV , we obtain where the quantities with subindex k are the Cartesian vector components of the corresponding vector quantities. In the small δV , the molecules can to first order be considered to be distributed uniformly. Taking into account the fact that e ext (r) and b ext (r) are the same as without the molecules, we can substitute the averaging of the fields over the coordinates of the molecules with volume averaging, which results in where E ext = e ext (r) and B ext = b ext (r) , and the electric polarization P and magnetization M are given by P = N δV d/δV and M = N δV m/δV , respectively, with N δV denoting the number of molecules in δV . Equation (25) is one of our key results. It obviously differs from the other commonly used expressions for the macroscopic force density on the bound charges [5]- [15], [32]. On the other hand, besides the term k M k ∇ B ext,k , equation (25) coincides in its form with equation (18) in [33], where the fields E and B are used instead of the external fields. The physical principles that have led to equation (25) have in fact been discussed by Brevik [5] (see 148), but neither Brevik nor Hakim, to whose paper [34] Brevik refers, have introduced this equation. The total macroscopic force density is thus The averaged external electric field can be found as E ext = E − e own . For an arbitrary charge distribution in the spherical volume δV , the field e own is calculated in a straightforward manner to obtain e own = −D δV /(3 0 δV ), where D δV = PδV is the total dipole moment of the medium within δV (section 2.13 in [31]). Therefore, the external field is where the medium is assumed to be linear, so that P = 0 ( − 1)E. The obtained field E ext is equal to the traditional local field with the Lorentz correction, which is explained by the fact that both the Lorentz sphere and our δV have spherical shapes. Similarly, the external magnetic field is calculated as B ext = B − b own . Note that while e own is directed oppositely to E, the field b own being created by electric current loops in the molecules is co-directed with B. This field is b own = 2µ 0 M/3 (see equations (9)- (22) in [31]), which leads to the following equation: where the expressions B = µ 0 µH and M = (µ − 1)H have been used. It has been shown that for some dense materials the Lorentz correction to the local field is insufficient and, consequently, the Clausius-Mossotti and Lorentz-Lorenz equations are not exact (see e.g. [27]- [35]). In principle, due to similar reasons, the fields E ext and B ext can also deviate from those given by equations (27) and (28). In such cases, one should make corrections to these equations.

7
Substituting equations (27) and (28) into equation (25) and expressing P and M through E and H as above, we obtain If the medium is non-magnetic, i.e. µ = 1, the third term is equal to Abraham's term ∂ ∂t { 0 µ 0 ( µ − 1)E × H} that represents the difference between the Abraham and Minkowski force densities [5]. This term has been proven to exist by the experiments of Walker and Walker [23], in which the material had µ = 1.
The fields E and H satisfy the macroscopic Maxwell's equations where D = 0 E + P = 0 E is the electric displacement. Using these equations and combining equations (15) and (29), we find that the total force density f = f (f) + f (b) is described by the following equation: where the energy-momentum tensorT and the momentum density G of the field are given bŷ In equations (35) and (36), EE and HH are the outer products of the field vectors,Î denotes the unit tensor and c is the speed of light in vacuum. It can be seen that the tensorT is equal to the Helmholtz tensor, when the latter is written for a Clausius-Mossotti medium [5]- [7]. Note that the field momentum density in equation (36) does not appear in the Helmholtz, Minkowski, Einstein-Laub or Abraham pictures. At high frequencies, the permeability µ is equal to 1, and the momentum density G becomes equal to which is in agreement with Planck's principle of inertia of energy. Let us describe some particular examples of application of the obtained equations. As a first example, we consider the well-known experiment on raising a dielectric liquid within a parallelplate capacitor. The capacitor is partially immersed in the liquid, and the liquid rises when a horizontal static electric field E is applied between the plates. According to equation (29), the force density due to the field should have two terms, The first term is the force density applied to the surface of the liquid and pushing it down, while the second term is the force density due to the inhomogeneous (fringing) electric field near the edges of the capacitor in the liquid. This second term leads to the elevation of the liquid.
Assuming that the size of the capacitor plates is large compared to their separation, the height h to which the liquid will rise is calculated from the following equation: where z is the coordinate along a vertical axis z drawn in the middle of the capacitor and the integration of f 1 and f 2 is performed over regions of inhomogeneous medium at the surface of the liquid and inhomogeneous electric field at the bottom edge of the capacitor, respectively; ρ l and g are the mass density of the liquid and gravitation acceleration, respectively. We point out that not only f 2 but also f 1 is of dipole (gradient) nature. The molecules of the liquid see the field E ext rather than E. While E is continuous across the surface, E ext is not. The gradient of E ext results in a dipole force acting on the molecules at the surface and pulling them down. This physical explanation of the surface force is usually missing from the description of the phenomenon. Note that in the Mikowski, Abraham and Einstein-Laub pictures the calculated height h is the same as in equation (40), but the reasons for rising of the liquid are different. In both the Minkowski and Abraham pictures, the liquid rises due to an upward directed surface force and the volume force due to the fringing electric field in the liquid is zero [5,36], whereas in the Einstein-Laub picture it is the volume force that raises the liquid and the surface force is zero [5]. The physical explanation given by us comes from equation (25) that is based on the microscopic interaction picture. It is straightforward to apply the theory to the case of elevation of a magnetic liquid by a horizontal static magnetic field. Owing to the symmetry of tensorT, we can immediately write Both equations (40) and (41) have been verified experimentally [22]. Moreover, these equations can be derived directly from energy principles and are equally correct also for solid materials (see e.g. [37]). Equations (34) and (35) describe correctly many other experiments. For example, the excess pressure of a dielectric liquid due to a static electric field, E, has been measured by Hakim and Higham and it has been shown that it satisfies the equation [20] which cannot be obtained within the Abraham, Minkowski and Einstein-Laub pictures but is readily obtained from equations (34) and (35) or from equation (29). Now let us turn to stationary optical phenomena. Suppose that a plane optical wave is reflected from a perfect mirror surrounded by a transparent dielectric medium. To calculate the radiation pressure on the mirror, we enclose the reflecting surface of the mirror by two auxiliary surfaces, of which one, A 1 , is chosen inside the mirror where the fields E and H are zero, and the other, A 2 , is chosen inside the dielectric but immediately on the surface of the mirror. The time-averaged pressure is calculated by using equations (34) and (35) and applying Gauss's integration law to obtainp =T · n 2 , where n 2 is the unit vector normal to the surface A 2 and pointing inward the mirror and the bar denotes time averaging; the time-averaged momentum density G is equal to zero. Since at A 2 the electric field is zero, we obtain Writing H 2 in terms of the incident field intensity I as and using the equation I =hωφ, wherehω and φ are the photon energy and photon flux density, respectively, we obtain In this equation, the vector k 0 is the wavevector in vacuum chosen to point along the propagation direction of the incident wave. If we assume that the mirror receives an average momentum of 2p ph per reflected photon, then the calculated pressurep must be equal to 2p ph φ, from which we find where n is the index of refraction of the medium. The last equality in equation (46) is obtained after setting µ to 1. Obviously, in the considered case the average photon momentum p ph in the medium is equal to the photon momentum in the Minkowski picture. It is worth mentioning that this momentum has been obtained in experimental studies of radiation pressure not only on a mirror in a dielectric medium [16,17] but also on atoms in a Bose-Einstein condensate [24] and charge carriers in a semiconductor [18]. The fact that each recoiled atom in [24] and charge carrier in [18] experiences the same photon momentum as a mirror in a dielectric is what in our opinion could be expected. We proceed to the calculation of a radiation pressure imposed by the reflected wave in the above example on the medium. The interference of the incident and reflected waves forms a standing optical wave, and we want to know the pressure on the medium between the mirror and the first interference maximum of the electric field. This maximum occurs on a surface A 3 located at a distance of λ/4 from the mirror. At this distance the magnetic field of the wave is equal to zero. The pressure is calculated as where the tensors are evaluated on surfaces A 2 and A 3 and n 3 is directed outward the mirror. Setting µ to 1, we obtain where E λ/4 is the electric field strength at a distance of λ/4 from the mirror. This result is in agreement with equation (42). Equation (48) shows that the medium is compressed toward the interference maxima of the electric field of the wave. It is similarly straightforward to evaluate the force density and radiation pressure in a medium that interacts with a laser beam instead of a plane wave. For example, an ordinary Gaussian laser beam propagating in a linear dielectric medium will compress the medium toward the beam axis. By applying equations (34) and (35), or equation (29), and the fact that µ = 1, it can be shown that the time-averaged compressive force density is given bȳ independently of beam polarization. If, on the other hand, the beam is normally incident from vacuum onto a flat surface of a dielectric liquid, then the resulting force density has two components. The first component acts on the surface, pushing it down with a pressure of The overall pressure that elevates the surface of the liquid is then This result is in agreement with the experiments of Ashkin and Dziedzic [21]. Note that equation (52) can also be obtained by assuming that the average momentum per photon is given by equation (46) and applying the momentum conservation law. However, the question of correctness of existing different expressions for the real photon momentum in a medium is not a topic we want to consider here. Furthermore, the classical macroscopic picture that we have used in the above examples is, in our opinion, not quite appropriate for making conclusions about the real photon momentum in a medium. In fact, such conclusions, being made on the basis of different experimental observations, continuously appear in the literature to contradict each other. Even a quantum mechanical description can yield different photon momenta in the same dielectric material under different experimental conditions, if it uses the macroscopic and, therefore, already averaged quantities and operators [38]. Equations (34)-(36) can be applied not only to stationary and static electromagnetic fields but also to such dynamical phenomena, in which the time derivative of the field momentum density plays a significant role. This, however, can be the case only within a short time interval so that the measured, time-averaged force density due to the field momentum is usually close or equal to zero. We nevertheless believe that it is possible to experimentally verify equation (36) by using a low-frequency or pulsed electromagnetic field and a medium with µ = 1. In particular, we expect that the difference between equation (36) and other expressions for the field momentum can become evident when dealing with the interaction of strong laser pulses with optical-frequency magnetic materials, such as recently developed metamaterials.
In summary, we have obtained a general expression, equation (26), for the macroscopic force density imposed by an electromagnetic field on a linear isotropic magnetic dielectric medium by spatially averaging the microscopic Lorentz force density. This equation is an important fundamental result exhibiting the true nature of the macroscopic force density in a medium. We have evaluated the volume-averaged fields produced by sources that are external with respect to the averaging volume and transformed equation (26) into equation (34) written in terms of the energy-momentum tensor and the momentum density of the field. The obtained tensor has been found to be equal to the so-called Helmholtz tensor written for a Clausius-Mossotti medium. By this, our microscopic derivation unambiguously proves the correctness of this tensor and insufficiency of the Abraham and Minkowski tensors. Moreover, different energy-momentum tensors are obtained if media are considered not to satisfy the classical Clausius-Mosotti law. It is important that in our derivation we have used only the microscopic Lorentz force density, Maxwell's equations and Newton's third law. Thus we have shown that the knowledge of these fundamental laws is sufficient to also obtain the electroand magneto-strictive forces. The expression derived by us for the momentum density of the field, equation (36), does not coincide with any other existing expression for this quantity. In particular, we would like to stress that our expression is asymmetrical with respect to and µ. We anticipate this new expression to attract the attention of experimentalists and find useful applications in the future.