Page 204 - Electromagnetics

after we differentiate under the integral signs and combine terms. So
˜
˜
∇× H = jωD + J ˜ (4.7)
by the Fourier integral theorem. This version of Ampere’s law involves only the frequency-
domain fields. By similar reasoning we have
˜
˜
∇× E =− jωB, (4.8)
˜
∇· D = ˜ρ, (4.9)
˜
∇· B(r,ω) = 0, (4.10)
and

˜
∇· J + jω ˜ρ = 0.
Equations (4.7)–(4.10) govern the temporal spectra of the electromagnetic fields. We may
manipulate them to obtain wave equations, and apply the boundary conditions from the
following section. After finding the frequency-domain fields we may find the temporal
fields by Fourier inversion. The frequency-domain equations involve one fewer derivative
(the time derivative has been replaced by multiplication by jω), hence may be easier to
solve. However, the inverse transform may be difficult to compute.

4.3 Boundary conditions on the frequency-domain fields

Several boundary conditions on the source and mediating fields were derived in § 2.8.2.
For example, we found that the tangential electric field must obey

ˆ n 12 × E 1 (r, t) − ˆ n 12 × E 2 (r, t) =−J ms (r, t).

The technique of the previous section gives us
˜
˜
˜
ˆ n 12 × [E 1 (r,ω) − E 2 (r,ω)] =−J ms (r,ω)
as the condition satisfied by the frequency-domain electric field. The remaining boundary
conditions are treated similarly. Let us summarize the results, including the effects of
fictitious magnetic sources:
˜
˜
˜
ˆ n 12 × (H 1 − H 2 ) = J s ,
˜
˜
˜
ˆ n 12 × (E 1 − E 2 ) =−J ms ,
˜
˜
ˆ n 12 · (D 1 − D 2 ) = ˜ρ s ,
˜
˜
ˆ n 12 · (B 1 − B 2 ) = ˜ρ ms ,
and
˜
˜
˜
ˆ n 12 · (J 1 − J 2 ) =−∇ s · J s − jω ˜ρ s ,
˜
˜
˜
ˆ n 12 · (J m1 − J m2 ) =−∇ s · J ms − jω ˜ρ ms .
Here ˆ n 12 points into region 1 from region 2.
© 2001 by CRC Press LLC