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Macroscopic Electromagnetics

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Electromagnetic and Optical Pulse Propagation

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 224))

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Abstract

In the classical Maxwell–Lorentz theory, matter is regarded as being composed of point charges (e.g., point electrons and point protons and nuclei) that produce microscopic electric and magnetic fields. The microscopic equations of electromagnetics given in Eqs. (2.33)–(2.36) together with the Lorentz force relation given in Eq. (2.21) describe the detailed classical behavior of the charged particles and fields, as presented in Chaps. 2 and 3. The macroscopic equations of electromagnetics, in turn, describe the average behavior of the charged particles and fields. It is then expected that, through a suitable averaging procedure, the macroscopic electromagnetic field equations may be derived from the microscopic equations, a viewpoint that was initially developed by H. A. Lorentz in 1906 and has since been extended by J. H. van Vleck, R. Russakoff, and F. N. H. Robinson.

“A curious case of coincidence.” H. A. Lorentz on the Lorentz–Lorenz relation.

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Notes

  1. 1.

    Notice that the quantity ∇⋅P(r, t) is sometimes referred to as the bound charge density. In order to avoid confusion with the bound charge density described in Eq. (4.34), the quantity ∇⋅P(r, t) is referred to here by its proper name, the polarization charge density.

  2. 2.

    Notable exceptions are bi-anisotropic and bi-isotropic materials that exhibit chirality [14, 15]. A brief description of their general formulation is included at the end of this section.

  3. 3.

    A simple polarizable dielectric is defined here as one for which the quadrupole and all higher-order moments of the molecular charge distribution identically vanish.

  4. 4.

    This equation is taken here as the definition of a simple magnetizable medium.

  5. 5.

    Lorentz [3] attributed Eq. (4.176) with j = 1 to the earlier work by R. Clausius (1879) and O. F. Mossotti (1850); see p. 50 of [43].

  6. 6.

    See Eq. (1.20) with complex phase velocity given by v(ω) = c∕n(ω), where n(ω) ≡ (𝜖(ω)∕𝜖 0)1∕2 is the complex index of refraction with μ∕μ 0 = 1. This notation for the absorption coefficient should not be confused with that for the molecular polarizability, as each is determined by the context it is being used in.

  7. 7.

    This maximum frequency is below the frequency f P ≡ 1∕t P ≃ 1.855 × 1043 s−1 associated with the Planck time \(t_P \equiv \sqrt {\hbar G/c^5} \simeq 5.39\times 10^{-44}\,\mbox{s}\), a unique combination of the gravitational constant G, the speed of light constant c in special relativity, and the rationalized Planck constant ħ ≡ h∕2π from quantum theory. The Planck time supposedly provides an estimate of the quantized time scale at which quantum gravitational effects may appear. Smaller time intervals would then be meaningless, implying that frequencies above f P could not possibly exist. A more realistic bound is obtained from the realization that macroscopic quantities such as the index of refraction become meaningless when the wavelength decreases below intermolecular distances, so that f max ≈ 1 × 1020 s−1 for water. .

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  1. College of Engineering and Mathematical Sciences, University of Vermont, Burlington, VT, USA

    Kurt E. Oughstun

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Oughstun, K.E. (2019). Macroscopic Electromagnetics. In: Electromagnetic and Optical Pulse Propagation . Springer Series in Optical Sciences, vol 224. Springer, Cham. https://doi.org/10.1007/978-3-030-20835-6_4

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