Page 266 - Electromagnetics

˜
This is the general frequency-domain wave equation for E. Using ˜ ¯ −1 we can write (4.200)
as
−1
−1
˜
˜ ¯
˜
¯
˜
jωE = ˜ ¯ · ∇− jωξ · H − ˜ ¯ · J.
Substituting this into (4.199) we get
˜ ¯ −1 ˜ ¯ 2 ˜ ¯ −1
˜
˜
¯
˜
¯
¯
∇+ jωζ · ˜ ¯ · ∇− jωξ − ω ˜ ¯µ · H = ∇+ jωζ · ˜ ¯ · J − jωJ m . (4.202)
˜
This is the general frequency-domain wave equation for H.
Wave equation for a homogeneous, lossy, isotropic medium. We may specialize
(4.201) and (4.202) to the case of a homogeneous, lossy, isotropic medium by setting
˜
¯
¯
˜ ¯
˜ ¯
˜ c
˜ i
ζ = ξ = 0, ˜ ¯µ = ˜µI, ˜ ¯ = ˜ I, and J = J + J :
˜
˜ i
˜ c
˜
˜
2
∇× (∇× E) − ω ˜µ˜ E =−∇ × J m − jω ˜µ(J + J ), (4.203)
˜ i
˜
˜ c
2
˜
˜
∇× (∇× H) − ω ˜µ˜ H =∇ × (J + J ) − jω˜ J m . (4.204)
˜
˜ c
Using (B.47) and using Ohm’s law J = ˜σE to describe the secondary current, we get
from (4.203)
˜
˜
˜
˜
2 ˜
˜ i
2
∇(∇· E) −∇ E − ω ˜µ˜ E =−∇ × J m − jω ˜µJ − jω ˜µ ˜σE
˜
which, using ∇· E = ˜ρ/˜ , can be simplified to
1
2
˜ i
2 ˜
˜
(∇ + k )E =∇ × J m + jω ˜µJ + ∇ ˜ρ. (4.205)
˜
˜
This is the vector Helmholtz equation for E. Here k is the complex wavenumber defined
through

˜ σ
2 2 2 2 c
k = ω ˜µ˜ − jω ˜µ ˜σ = ω ˜µ ˜ + = ω ˜µ˜ (4.206)
jω
c
where ˜ is the complex permittivity (4.26).
By (4.204) we have
˜ c
˜
2
2 ˜
˜
˜
˜ i
∇(∇· H) −∇ H − ω ˜µ˜ H =∇ × J +∇ × J − jω˜ J m .
Using
˜
˜
˜
˜
˜ c
∇× J =∇ × ( ˜σE) = ˜σ∇× E = ˜σ(− jωB − J m )
˜
and ∇· H = ˜ρ m / ˜µ we then get
1
2 2 ˜ ˜ i c ˜
(∇ + k )H =−∇ × J + jω˜ J m + ∇ ˜ρ m , (4.207)
˜ µ
˜
which is the vector Helmholtz equation for H.
4.11.2 Field relationships and the wave equation for two-dimensional
fields
Many important canonical problems are two-dimensional in nature, with the sources
and fields invariant along one direction. Two-dimensional fields have a simple structure
© 2001 by CRC Press LLC