Page 385 - Electromagnetics

where ˜ p is the dipole moment defined by
˜
Il
˜
˜ p = Ql ˆ p = ˆ p.
jω
That is, we consider a Hertzian dipole to be a “point source” of electromagnetic radiation.
With this notation we have

˜

A e = ˜µ jω˜ pδ(r − r p ) G(r|r ; ω) dV = jω ˜µ˜ pG(r|r p ; ω),

V
which is identical to (5.87). The electromagnetic fields are then
˜
H(r,ω) = jω∇× [˜ pG(r|r p ; ω)], (5.88)
1
˜
E(r,ω) = c ∇× ∇× [˜ pG(r|r p ; ω)]. (5.89)
˜
˜
˜
Here we have obtained E from H outside the source region by applying Ampere’s law.
By duality we may obtain the fields produced by a magnetic Hertzian dipole of moment
˜ I m l
˜ p m = ˆ p
jω
located at r = r p as
˜
E(r,ω) =− jω∇× [˜ p m G(r|r p ; ω)],
1
˜
H(r,ω) = ∇× ∇× [˜ p m G(r|r p ; ω)].
˜ µ
We can learn much about the fields produced by localized sources by considering the
simple case of a Hertzian dipole aligned along the z-axis and centered at the origin. Using
ˆ p = ˆ z and r p = 0 in (5.88) we find that
˜ I e 1 1 k
− jkr
˜
˜
ˆ
H(r,ω) = jω∇× ˆ z l = φ Il + j sin θe − jkr . (5.90)
jω 4πr 4π r 2 r
By Ampere’s law
1
˜ ˜
E(r,ω) = ∇× H(r,ω)
jω˜ c
η 2 2 − jkr η k 1 1 − jkr
˜
= ˆ r Il − j cos θe + θ ˆ ˜ Il j + − j sin θe .
4π r 2 kr 3 4π r r 2 kr 3
(5.91)
3
The fields involve various inverse powers of r, with the 1/r and 1/r terms 90 out-of-
◦
2
phase from the 1/r term. Some terms dominate the field close to the source, while others
1
dominate far away. The terms that dominate near the source are called the near-zone
or induction-zone fields:
˜ Il e − jkr
ˆ
˜ NZ
H (r,ω) = φ sin θ,
4π r 2
˜ Il e − jkr
˜ NZ
ˆ
E (r,ω) =− jη 2ˆ r cos θ + θ sin θ .
4π kr 3
1 Note that we still require r l.
© 2001 by CRC Press LLC