Electromagnetic force density in dissipative isotropic media

Abstract derive an expression for the macroscopic force density that a narrowband electromagnetic field imposes on a dissipative isotropic medium. The result is obtained by averaging the microscopic form for Lorentz force density. The derived expression allows us to calculate realistic electromagnetic forces in a wide range of materials that are described by complex-valued electric permittivity and magnetic permeability. The three-dimensional energy-momentum tensor in our expression reduces for lossless media to the so-called Helmholtz tensor that has not been contradicted in any experiment so far. The momentum density of the field does not coincide with any well-known expression, but for non-magnetic materials it matches the Abraham expression. Electromagnetic force density in dissipative isotropic media 2 After the first well-known theoretical model for static electromagnetic forces in ponderable media, introduced by H. von Helmholtz [ 1 ], several other descriptions have been proposed to account for the action of both static and oscillating electromagnetic fields on a medium. The two most famous of them were proposed by H. Minkowski [ 2 ] and M. Abraham [ 3 ]. Their models give different predictions for the value of the force density in certain particular cases, which has created a century-long scientific debate on the topic [ 4]?[15 ]. It is well known that both these models ignore the existing electroand magnetostrictive forces [ 7, 16 ]. A model by Einstein and Laub [ 17 ] was intended to include these forces, but it turned out to be in disagreement with experiments (see, e.g., Ref. [ 7 ] and references therein). A reliable way to also include electro- and magnetostrictive forces can be based on the so-called Helmholtz energy-momentum tensor. Previously we have shown that the Helmholtz tensor can be derived in a relatively simple way starting with the microscopic field-matter interaction picture [ 18 ]. However, this tensor and the other models discussed so far are not applicable to dissipative media and, therefore, they do not allow calculating electromagnetic forces in many realistic materials. The aim of this work is to derive a general expression for the macroscopic force density imposed on a dissipative and electrically conducting inhomogeneous isotropic medium by a narrow-band electromagnetic field. We do this by spatially averaging the microscopic Lorentz force density (that assumes a unique and commonly accepted form, in contrast to the macroscopic electromagnetic force, energy, and momentum densities). The result is written in terms of the three-dimensional electromagnetic energy-momentum tensor and the momentum density of the field, which explicitly depend on the complex relative permittivity and permeability of the material. In the limit of lossless medium, the obtained energy-momentum tensor becomes the Helmholtz tensor. We note that the force density expressed in terms of complex-valued quantities has been introduced previously (see, e.g., [ 19 ] and [ 20]), but not in connection with the Helmholtz tensor. The expression obtained by us for the field momentum density does not coincide with any of the well-known expressions, but for lossless dielectrics it converges to an expression obtained by us previously [ 18 ], and if the material is not magnetic, it matches the Abraham expression. The derivation of the force density in dissipative media is similar to that introduced by us in Ref. [ 18 ] for lossless materials. Since essential new details appear along the derivation, we present it here in its entirety. Within classical electrodynamics, the interaction of an electromagnetic field with matter is unambiguously described by the microscopic Maxwell's equations,