Abstract
The present chapter is of central importance to the whole book. Its principal aim is the formulation of the basic balance laws of mechanical and energetic nature which govern the Galilean-invariant electrodynamics of continua, irrespective of the exact mechanical and electromagnetic responses of bodies. To do this, it is necessary to evaluate the electromagnetic contributions of these balance laws. In order to avoid an arbitrary choice of these contributions, and to keep in touch with the microscopic model developed in Chapter 2, we deduce these contributions from microscopic equations by performing statistical averaging as introduced in Section 3.2. Maxwell’s equations for the macroscopic fields are then obtained, in Section 3.3, from the corresponding equations of the microscopic fields derived in Section 2.5. In a natural way, this leads to the notion of macroscopic concepts such as polarization, magnetization, charge, and current, and their expressions in terms of microscopic notions. Galilean invariance of Maxwell’s equations is shown in Section 3.4, which leads to the introduction of fields expressed in a moving frame of reference.
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References
See Penfield and Haus [1967], Pao [1978], and Hutter and Ven [1978].
A similar method was used by Masson and Weaver [1929] for the electrostatic case. Independently, Lax and Nelson [1971] repeated the analysis of Dixon and Eringen.
Such approaches are favored by other authors, e.g., Fano et al. [1960], Penfield and Haus [1967], Pao and Hutter [1975], and Grot [1976]. Basic equations of electromagnetic elastic solids may be derived by means of the global energy balance law which is postulated to be invariant under Euclidean transformations (Alblas [1974], Parkus [1972], Van de Ven [1975]). For an excellent comparative discussion of different models, see Hutter and Van de Ven [1978].
See Jackson [1975, p. 552].
See Maugin and Collet [1974]. Such identities were also obtained by Tiersten and Tsai [1972], for dielectric insulators, by Livens [1962], and by others. See also Eringen [1980, Sects. 10.6 and 10.7].
The controversy about the forms of the electromagnetic stress tensor and the electromagnetic momentum has been discussed, among others, by Møller [1952, Sect. 72], and extensively, by Penfield and Hauss [1967] Hutter and Ven [1978], and Maugin [1980b]. Penfield and Hauss remark (p. 297) that the wrong choice in the expressions of the electromagnetic momentum, once the electromagnetic stress tensor is chosen, “leads to errors in force that are small, that have not time average value, and that are not easily measured.” This certainly is true. However, formulations (3.6.1) and (3.6.8) are not subject to this criticism, since they are pure identities, after the ponderomotive force and couple have been evaluated from such a sound physical theory as that of Lorentz.
E.S. Şuhubi, private communication, 1985. This result was obtained by Kafadar [1971] in connection with the surface polarization.
While equivalent equations, in connection with the electrodynamics of continua and continuum mechanics, were given by Maugin and Eringen [1977] and Eringen [1967], [1980], these elegant formal results are due to E.S. Şuhubi (private communication).
We follow the work of Maugin and Eringen [1977].
In polar theories of continua, the mechanical surface body couple densities, and angular momentum must be taken into account. In this case, (3.10.6) is more complicated and the jump condition survives. For the polar theories, see Eringen and Kafadar [1976].
More generally, the entropy flux vector should be taken as τk = (qk/θ) + sk, where sk is the excess over the classical value qk/θ (Eringen [1966a]). For mixture and nonsimple materials, sk may not vanish (see Eringen and Ingram [1966]). See Müller [1968] for a discussion of the form to be assumed a priori or a posteriori by the entropy flux. Application of Müller’s idea to the electromagnetic fluids was made by Liu and Müller [1972].
Tiersten ad Tsai [1972] considered global balance laws of the form (3.10.13)–(3.10.15) in the absence of discontinuity surfaces.
See Liu and Müller [1972], Benach [1974], and Benach and Müller [1974].
This fact is especially emphasized by Møller [1952], Maugin [1971a], [1980b], de Groot and Suttorp [1972], and others.
General theory was given by Germain [1973].
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© 1990 Springer-Verlag New York Inc.
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Eringen, A.C., Maugin, G.A. (1990). Macroscopic Electromagnetic Theory. In: Electrodynamics of Continua I. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3226-1_3
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DOI: https://doi.org/10.1007/978-1-4612-3226-1_3
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