In most instances, the presence of an imaginary part in the constitutive parameters implies that the material is either dissipative(lossy ), transforming some of the electro- magnetic energy in the fields into thermal energy, or active, transforming the chemical or mechanical energy of the material into energy in the fields. We investigate this further We can also write the constitutive equations in amplitude-phase form. Letting 石=同;e周, l周,=同Gle, and using the field notation(4.6), we can write(4.13)-(4. 15)as D=D=∑Ee+, (4.18) B1=1B=∑la1e 方=1=∑GEA Here we remember that the amplitudes and phases may be functions of both r and a For isotropic materials these reduce te D1=|D|e"=|ele+), (4.21) Bi= biles=lall hilejt (4.22) 1=1 4.4.1 The complex permittivity As mentioned above, dissipative effects may be associated with complex entries in the permittivity matrix. Since conduction effects can also lead to dissipation, the permittivit and conductivity matrices are often combined to form a compler permittivity. Writing the current as a sum of impressed and secondary conduction terms(J=Ji+Jc)and substituting(4.13) and(4.15)into Ampere's law, we find V×H=J+aE+jo是.E. Defining the complex permittivity e(r.o-o(r, o (4.24) V×H=+joeE. Using the complex permittivity we can include the effects of conduction current by merely replacing the total current with the impressed current. Since Faraday s law is unaffected any equation(such as the wave equation) derived previously using total current retains its form with the same substitutio By(4.16)and(4.17) the complex permittivity obeys E(r,-a)=e*(r,o) (4.25) 2001 by CRC Press LLCIn most instances, the presence of an imaginary part in the constitutive parameters implies that the material is either dissipative (lossy), transforming some of the electro￾magnetic energy in the fields into thermal energy, or active, transforming the chemical or mechanical energy of the material into energy in the fields. We investigate this further in § 4.5 and § 4.8.3. We can also write the constitutive equations in amplitude–phase form. Letting ˜i j = |˜i j|e jξ  i j, µ˜ i j = |µ˜ i j|e jξµ i j, σ˜i j = |σ˜i j|e jξ σ i j, and using the field notation (4.6), we can write (4.13)–(4.15) as D˜ i = |D˜ i|e jξ D i = 3 j=1 |˜i j||E˜ j|e j[ξ E j +ξ  i j] , (4.18) B˜i = |B˜i|e jξ B i = 3 j=1 |µ˜ i j||H˜ j|e j[ξ H j +ξµ i j] , (4.19) J˜ i = |J˜ i|e jξ J i = 3 j=1 |σ˜i j||E˜ j|e j[ξ E j +ξ σ i j] . (4.20) Here we remember that the amplitudes and phases may be functions of both r and ω. For isotropic materials these reduce to D˜ i = |D˜ i|e jξ D i = |˜||E˜i|e j(ξ E i +ξ  ) , (4.21) B˜i = |B˜i|e jξ B i = |µ˜ ||H˜i|e j(ξ H i +ξµ) , (4.22) J˜ i = |J˜ i|e jξ J i = |σ˜||E˜i|e j(ξ E i +ξ σ ) . (4.23) 4.4.1 The complex permittivity As mentioned above, dissipative effects may be associated with complex entries in the permittivity matrix. Since conduction effects can also lead to dissipation, the permittivity and conductivity matrices are often combined to form a complex permittivity. Writing the current as a sum of impressed and secondary conduction terms (J˜ = J˜i + J˜ c) and substituting (4.13) and (4.15) into Ampere’s law, we find ∇ × H˜ = J˜i + σ˜¯ · E˜ + jω˜¯ · E˜ . Defining the complex permittivity ˜¯ c (r,ω) = σ˜¯ (r,ω) jω + ˜¯(r, ω), (4.24) we have ∇ × H˜ = J˜i + jω˜¯ c · E˜ . Using the complex permittivity we can include the effects of conduction current by merely replacing the total current with the impressed current. Since Faraday’s law is unaffected, any equation (such as the wave equation) derived previously using total current retains its form with the same substitution. By (4.16) and (4.17) the complex permittivity obeys ˜¯ c (r, −ω) = ˜¯ c∗ (r,ω) (4.25)