Force, torque, linear momentum, and angular momentum in classical electrodynamics

Abstract

Abstract. The classical theory of electrodynamics is built upon Maxwell's equations and the concepts of electromagnetic (EM) field, force, energy, and momentum, which are intimately tied together by Poynting's theorem and by the Lorentz force law. Whereas Maxwell's equations relate the fields to their material sources, Poynting's theorem governs the flow of EM energy and its exchange between fields and material media, while the Lorentz law regulates the back-andforth transfer of momentum between the media and the fields. An alternative force law, first proposed by Einstein and Laub, exists that is consistent with Maxwell's equations and complies with the conservation laws as well as with the requirements of special relativity. While the Lorentz law requires the introduction of hidden energy and hidden momentum in situations where an electric field acts on a magnetized medium, the Einstein-Laub (E-L) formulation of EM force and torque does not invoke hidden entities under such circumstances. Moreover, total force/torque exerted by EM fields on any given object turns out to be independent of whether the density of force/torque is evaluated using the law of Lorentz or that of Einstein and Laub. Hidden entities aside, the two formulations differ only in their predicted force and torque distributions inside matter. Such differences in distribution are occasionally measurable, and could serve as a guide in deciding which formulation, if either, corresponds to physical reality.

A heuristic resolution of the Abraham–Minkowski controversy

“…It is not known whether Einstein in 1916 agreed with such an expansion, but it is certain that Einstein in 1908 must have held a negative attitude towards it, because he and Laub were the first physicists to criticize Minkowski momentum, suggesting the so called Einstein-Laub momentum density [82]. However, in 1918, Einstein changed his view and supported Minkowski momentum density, criticized Abraham momentum density, and declared that the momentum density given by himself and Laub was not tenable [83].…”

Torque Analysis for Rotational Devices with Nonmagnetic Rotor Driven by Magnetic Fluid Filled in Air Gap

“…However, since the magnetic force density is intertwined with numerous physics, it is very difficult to determine the correct solution from it [21,22]. We include the discussions on this issue in the references [23][24][25][26][27].…”

“…Solving the Naiver-Stokes equation with magnetic body force density distribution provides the best information: pressure and velocity distribution in all space. However, there have been controversies about magnetic body force density within the magnetic materials, and there is no generally accepted conclusion yet [15][16][17][18][19][20][21][22][23][24][25][26][27]. On the other hand, the virtual work principle, with appropriate approach, gives up distributive information.…”

A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction

“…For instance, the vectorial equivalent of Eq. ( 25) yields (31) It is easy to see that the first two terms of the second integral on the right-hand side of Eq. (31) vanish since both 𝐺𝐺𝐸𝐸 𝑥𝑥 and 𝐺𝐺𝐸𝐸 𝑦𝑦 go to zero in the far out regions of the 𝑥𝑥𝑥𝑥-plane.…”