Page 206 - Electromagnetics

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
in § 4.5 and § 4.8.3.
We can also write the constitutive equations in amplitude–phase form. Letting

˜ ij =|˜ ij |e jξ ij , ˜ µ ij =| ˜µ ij |e jξ ij µ , ˜ σ ij =| ˜σ ij |e jξ σ ij ,
and using the field notation (4.6), we can write (4.13)–(4.15) as

3

E
˜
˜

˜
D i =|D i |e jξ i D = |˜ ij ||E j |e j[ξ +ξ ] , (4.18)
j
ij
j=1
3
µ
H

˜
˜
˜
B i =|B i |e jξ i B = | ˜µ ij ||H j |e j[ξ +ξ ] , (4.19)
ij
j
j=1
3
E
σ
˜
˜ J i =|J i |e jξ i J =
| ˜σ ij ||E j |e j[ξ +ξ ] . (4.20)
˜
ij
j
j=1
Here we remember that the amplitudes and phases may be functions of both r and ω.
For isotropic materials these reduce to

E
˜
˜
˜
D i =|D i |e jξ i D =|˜ ||E i |e j(ξ +ξ ) , (4.21)
i
H
˜
˜
˜
µ
B i =|B i |e jξ i B =| ˜µ||H i |e j(ξ +ξ ) , (4.22)
i
E
σ
˜
˜
˜ J i =|J i |e jξ i J =| ˜σ||E i |e j(ξ +ξ ) . (4.23)
i
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
˜ i
˜ c
˜
the current as a sum of impressed and secondary conduction terms (J = J + J ) and
substituting (4.13) and (4.15) into Ampere’s law, we find
˜ i
˜
˜
˜
∇× H = J + ˜ ¯σ · E + jω ˜ ¯ · E.
Defining the complex permittivity
˜ ¯ σ(r,ω)
c
˜ ¯ (r,ω) = + ˜ ¯ (r,ω), (4.24)
jω
we have
c
˜ i
˜
˜
∇× H = J + jω ˜ ¯ · 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
c∗
˜ ¯ (r, −ω) = ˜ ¯ (r,ω) (4.25)
© 2001 by CRC Press LLC