Abstract
Maxwell's macroscopic equations combined with a generalized form of the Lorentz law of force are a complete and consistent set of equations. Not only are these five equations fully compatible with special relativity, they also conform with conservation laws of energy, momentum, and angular momentum. We demonstrate consistency with the conservation laws by showing that, when a beam of light enters a magnetic dielectric, a fraction of the incident linear (or angular) momentum pours into the medium at a rate determined by the Abraham momentum density, E x H/c(2), and the group velocity V(g) of the electromagnetic field. The balance of the incident, reflected, and transmitted momenta is subsequently transferred to the medium as force (or torque) at the leading edge of the beam, which propagates through the medium with velocity V(g). Our analysis does not require "hidden" momenta to comply with the conservation laws, nor does it dissolve into ambiguities with regard to the nature of electromagnetic momentum in ponderable media. The linear and angular momenta of the electromagnetic field are clearly associated with the Abraham momentum, and the phase and group refractive indices (n(p) and n(g)) play distinct yet definitive roles in the expressions of force, torque, and momentum densities.
Highlights
Standard textbooks on electromagnetism tend to treat the macroscopic equations of Maxwell as somehow inferior to their microscopic counterparts [1,2]
This is due to the fact that, for real materials, polarization and magnetization densities P and M are defined as averages over small volumes that must contain a large number of atomic dipoles
This is an unfortunate state of affairs, considering that the macroscopic equations of Maxwell are a complete and self-consistent set, provided that the fields are treated as precisely-defined mathematical entities, i.e., without attempting to associate P and M with the properties of real materials
Summary
Standard textbooks on electromagnetism tend to treat the macroscopic equations of Maxwell as somehow inferior to their microscopic counterparts [1,2]. If we proceed to assume that the energy contained in a given volume of the material is N ħω, with ħ being the reduced Planck constant, ω the angular frequency of the light, and N the number of photons, the Abraham momentum per photon turns out to be ħω/(ngc) This is the general formula for the photon’s electromagnetic momentum in a transparent medium having group refractive index ng. Upon entering a homogeneous, linear, isotropic, and transparent medium, the electromagnetic angular momentum per photon shrinks by a factor of npng, while, according to Eq(14), the balance of the incident, reflected and transmitted angular momenta is transferred to the medium as a torque (via the leading edge of the beam). This result applies to negativeindex media as well, where the sign of the electromagnetic angular momentum is reversed relative to that of the incident beam (because np< 0)
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