Page 248 - Electromagnetics

then forming
ˇ
j ˇωt
D i (r, t) = Re D i (r)e
we reproduce (4.125). Thus we may write
ˇ
ˇ
D(r) = ˜ (r, ˇω)E(r).
Note that we never write ˇ or refer to a “phasor permittivity” since the permittivity does
not vary sinusoidally in the time domain.
An obvious benefit of the phasor method is that we can manipulate field quantities
without involving the sinusoidal time dependence. When our manipulations are complete,
we return to the time domain using (4.126).
The phasor Maxwell equations (4.128)–(4.131) are identical in form to the temporal
frequency-domain Maxwell equations (4.7)–(4.10), except that ω = ˇω in the phasor
equations. This is sensible, since the phasor fields represent a single component of the
complete frequency-domain spectrum of the arbitrary time-varying fields. Thus, if the
phasor fields are calculated for some ˇω, we can make the replacements

ˇ
ˇ
˜
˜
ˇ ω → ω, E(r) → E(r,ω), H(r) → H(r,ω), ...,
and obtain the general time-domain expressions by performing the inversion (4.2). Simi-
˜
larly, if we evaluate the frequency-domain field E(r,ω) at ω = ˇω, we produce the phasor
˜
ˇ
field E(r) = E(r, ˇω) for this frequency. That is
3
˜
j ˇωt
E
Re E(r, ˇω)e = ˆ ˜
i i |E i (r, ˇω)| cos ˇωt + ξ (r, ˇω) .
i=1
4.7.3 Boundary conditions on the phasor fields
The boundary conditions developed in § 4.3 for the frequency-domain fields may be
adapted for use with the phasor fields by selecting ω = ˇω. Let us include the effects of
fictitious magnetic sources and write
ˇ
ˇ
ˇ
ˆ n 12 × (H 1 − H 2 ) = J s , (4.133)
ˇ
ˇ
ˇ
ˆ n 12 × (E 1 − E 2 ) =−J ms , (4.134)
ˇ
ˇ
ˆ n 12 · (D 1 − D 2 ) = ˇρ s , (4.135)
ˇ
ˇ
ˆ n 12 · (B 1 − B 2 ) = ˇρ ms , (4.136)
and
ˇ
ˇ
ˇ
ˆ n 12 · (J 1 − J 2 ) =−∇ s · J s − j ˇω ˇρ s , (4.137)
ˇ
ˇ
ˇ
ˆ n 12 · (J m1 − J m2 ) =−∇ s · J ms − j ˇω ˇρ ms , (4.138)
where ˆ n 12 points into region 1 from region 2.
4.8 Poynting’s theorem for time-harmonic fields
We can specialize Poynting’s theorem to time-harmonic form by substituting the time-
harmonic field representations. The result depends on whether we use the general form

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