e(,-0)=e(r,m),(r,-a)=-(r,o) For an isotropic material it takes the particularly simple form +60+∈0 d we have (r,-①)=e(r,o),e"(r,-)=-e"(r,o) (4.27) 4.4.2 High and low frequency behavior of constitutive parameters At low frequencies the permittivity reduces to the electrostatic permittivity. Since 2 is even in o and e" is odd. we have for small o If the material has some dc conductivity oo, then for low frequencies the complex per mittivity behaves as If E or H changes very rapidly, there may be no polarization or magnetization effect at all. This occurs at frequencies so high that the atomic structure of the material cannot respond to the rapidly oscillating applied field. Above some frequency then, we can assume xe =0 and im =0 so that P=0.M=0, =μ In our simple models of dielectric materials($ 4.6) we find that as o becomes large Our assumption of a macroscopic model of matter provides a fairly strict upper frequency limit to the range of validity of the constitutive parameters. We must assume that the wavelength of the electromagnetic field is large compared to the size of the atomic struc ture. This limit suggests that permittivity and permeability might remain meaningful even at optical frequencies, and for dielectrics this is indeed the case since the values of P remain significant. However, M becomes insignificant at much lower frequencies, and B=10H[107] 4.4.3 The Kronig-Kramers relations The principle of causality is clearly implicit in(2. 29)-(2.31). We shall demonstrate that causality leads to explicit relationships between the real and imaginary parts of the frequency-domain constitutive parameters. For simplicity we concentrate on the isotropic case and merely note that the present analysis may be applied to all the dyadic com- ponents of an anisotropic constitutive parameter. We also concentrate on the complex permittivity and extend the results to permeability by induction. 2001 by CRC Press LLCor ˜ c i j(r, −ω) = ˜ c i j(r, ω), ˜ c i j (r, −ω) = −˜ c i j (r, ω). For an isotropic material it takes the particularly simple form ˜ c = σ˜ jω + ˜ = σ˜ jω + 0 + 0χ˜e, (4.26) and we have ˜ c (r, −ω) = ˜ c (r, ω), ˜ c(r, −ω) = −˜ c(r, ω). (4.27) 4.4.2 High and low frequency behavior of constitutive parameters At low frequencies the permittivity reduces to the electrostatic permittivity. Since ˜ is even in ω and ˜ is odd, we have for small ω ˜  ∼ 0r, ˜  ∼ ω. If the material has some dc conductivity σ0, then for low frequencies the complex per￾mittivity behaves as ˜ c ∼ 0r, ˜ c ∼ σ0/ω. (4.28) If E or H changes very rapidly, there may be no polarization or magnetization effect at all. This occurs at frequencies so high that the atomic structure of the material cannot respond to the rapidly oscillating applied field. Above some frequency then, we can assume χ˜¯ e = 0 and χ˜¯ m = 0 so that P˜ = 0, M˜ = 0, and D˜ = 0E˜ , B˜ = µ0H˜ . In our simple models of dielectric materials (§ 4.6) we find that as ω becomes large ˜  − 0 ∼ 1/ω2 , ˜  ∼ 1/ω3 . (4.29) Our assumption of a macroscopic model of matter provides a fairly strict upper frequency limit to the range of validity of the constitutive parameters. We must assume that the wavelength of the electromagnetic field is large compared to the size of the atomic struc￾ture. This limit suggests that permittivity and permeability might remain meaningful even at optical frequencies, and for dielectrics this is indeed the case since the values of P˜ remain significant. However, M˜ becomes insignificant at much lower frequencies, and at optical frequencies we may use B˜ = µ0H˜ [107]. 4.4.3 The Kronig–Kramers relations The principle of causality is clearly implicit in (2.29)–(2.31). We shall demonstrate that causality leads to explicit relationships between the real and imaginary parts of the frequency-domain constitutive parameters. For simplicity we concentrate on the isotropic case and merely note that the present analysis may be applied to all the dyadic com￾ponents of an anisotropic constitutive parameter. We also concentrate on the complex permittivity and extend the results to permeability by induction