Page 246 - Electromagnetics

initial intention of having only a single elemental function present. The sifting property
gives

e
e j ˇωt jξ(r) + e − j ˇωt − jξ(r)
e
ψ(r, t) = ψ 0 (r) = ψ 0 (r) cos[ ˇωt + ξ(r)]
2
as expected.
4.7.1 The time-harmonic EM fields and constitutive relations

The time-harmonic fields are described using the representation (4.121) for each field
component. The electric field is

3
ˆ E

E(r, t) = i i |E i (r)| cos[ ˇωt + ξ (r)]
i
i=1
for example. Here |E i | is the complex magnitude of the ith vector component, and ξ E is
i
E
the phase angle (−π< ξ ≤ π). Similar terminology is used for the remaining fields.
i
The frequency-domain constitutive relations (4.11)–(4.15) may be written for the time-
harmonic fields by employing (4.124). For instance, for an isotropic material where
˜
˜
˜
˜
D(r,ω) = ˜ (r,ω)E(r,ω), B(r,ω) = ˜µ(r,ω)H(r,ω),
with

µ
˜ (r,ω) =|˜ (r,ω)|e ξ (r,ω) , ˜ µ(r,ω) =| ˜µ(r,ω)|e ξ (r,ω) ,
we can write
3

D
ˆ
D(r, t) = i i |D i (r)| cos[ ˇωt + ξ (r)]
i
i=1
3
1 ∞
jξ (r) − jξ (r) jωt

E
E
= ˆ i i ˜ (r,ω)|E i (r)|π e i δ(ω − ˇω) + e i δ(ω + ˇω) e dω
2π
−∞ i=1
3
1
E E
i
i
= ˆ i i |E i (r)| ˜ (r, ˇω)e j( ˇωt+ jξ (r)) + ˜ (r, − ˇω)e − j( ˇωt+ jξ (r)) .
2
i=1
Since (4.25) shows that ˜ (r, − ˇω) = ˜ (r, ˇω), we have
∗
3
1
j( ˇωt+ jξ (r)+ jξ (r, ˇω)) − j( ˇωt+ jξ (r)+ jξ (r, ˇω))
E
E

D(r, t) = ˆ i i |E i (r)||˜ (r, ˇω)| e i + e i
2
i=1
3

E
ˆ
= i i |˜ (r, ˇω)||E i (r)| cos[ ˇωt + ξ (r) + ξ (r, ˇω)]. (4.125)
i
i=1
Similarly
3
ˆ

B
B(r, t) = i i |B i (r)| cos[ ˇωt + ξ (r)]
i
i=1
3
ˆ H µ

= i i | ˜µ(r, ˇω)||H i (r)| cos[ ˇωt + ξ (r) + ξ (r, ˇω)].
i
i=1
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