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

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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-and-forth 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.

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Notes

  1. The Helmholtz force-density associated with the action of the E-field on a dielectric material of mass density ρ and relative permittivity ε is often written as follows [41]:

    $$ \varvec{F}_{\text{H}} (\varvec{r},t) = - {\mathbf{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}\varepsilon_{\text{o}} ({\mathbf{\nabla }}\varepsilon )(\varvec{E} \cdot \varvec{E}) + {\mathbf{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}\varepsilon_{\text{o}} {\mathbf{\nabla }}\left( {\rho \frac{\partial \varepsilon }{\partial \rho }\varvec{E} \cdot \varvec{E}} \right). $$

    The first term on the right-hand side of the above equation arises naturally from the stress tensors of Abraham and Minkowski, as discussed in Sect. 8. The second term, which is associated with electrostriction, is derived phenomenologically, using arguments from the theories of elasticity and thermodynamics [40, 75].

  2. In the literature [41, 45, 47, 55], Abraham’s stress tensor is usually written as a symmetrized version of Minkowski’s tensor, that is,

    $$ {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\mathcal{T}} }}_{A} (\varvec{r},t) = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left[ {(\varvec{D} \cdot \varvec{E} + \varvec{B} \cdot \varvec{H}){\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {I} }} - (\varvec{DE} + \varvec{ED}) - (\varvec{BH} + \varvec{HB})} \right]. $$

    Abraham’s concerns, as well of those of his followers, were primarily with linear, isotropic media, namely, media for which \( \varvec{D} = \varepsilon_{\text{o}} \varepsilon \varvec{E} \) and \( \varvec{B} = \mu_{\text{o}} \mu \varvec{H} \). In such cases, since the stress tensor of Minkowski, given by Eq. (28), is already symmetric, the above act of symmetrization does not modify the tensor. In Abraham’s own paper [43], the stress tensor is written explicitly only twice, in Eqs. (5a) and (56), and in both instances it is identical to Minkowski’s (asymmetric) tensor. At several points in his papers [43, 44], Abraham mentions the symmetry of his tensor, but it appears that he has the special case of linear, isotropic media in mind. The special symmetry that Abraham introduced into Minkowski’s theory is, of course, that between the energy flow-rate, \( \varvec{E} \times \varvec{H} \), and the electromagnetic momentum density, \( \varvec{E} \times \varvec{H}/c^{2} \), which reside, respectively, in the fourth column and the fourth row of the stress-energy tensor. Be it as it may, in Eq. (28) we have chosen the asymmetric version of Abraham’s (3 × 3) stress tensor, as it simplifies the subsequent discussion. In any event, this does not affect the main results and the conclusions reached in Sect. 8, since the media chosen for analysis in that section are linear and isotropic.

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  1. College of Optical Sciences, The University of Arizona, Tucson, AZ, 85721, USA

    Masud Mansuripur

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Mansuripur, M. Force, torque, linear momentum, and angular momentum in classical electrodynamics . Appl. Phys. A 123, 653 (2017). https://doi.org/10.1007/s00339-017-1253-2

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