Page 427 - Electromagnetics

where
i

˜ ρ (r ,ω)
˜
φ e (r,ω) = c G(r|r ; ω) dV ,
V ˜ (ω)

˜ ˜ i
A e (r,ω) = ˜ µ(ω)J (r ,ω)G(r|r ; ω) dV , (6.15)
V
˜ i
are the electric scalar and vector potential functions introduced in § 5.2. Using J ×∇ G =
˜ i
˜ i
−J ×∇G =∇ × (J G) we have
1
˜ ˜ i
H(r,ω) = ∇× ˜ µ(ω)J (r ,ω)G(r|r ; ω) dV
˜ µ(ω) V
1
˜
= ∇× A e (r,ω). (6.16)
˜ µ(ω)
These expressions for the fields are identical to those of (5.56) and (5.57), and thus the
integral formula for the electromagnetic fields produces a result identical to that obtained
˜ i
i
using potential relations. Similarly, with ˜ρ = 0, J = 0 we have

˜ ˜ i
E =− J ×∇ GdV ,
m
V

˜ ρ i
˜ m c ˜ i
H = ∇ G − jω˜ J G dV ,
m
V ˜ µ
or
1
˜ ˜
E(r,ω) =− ∇× A h (r,ω),
c
˜ (ω)
˜
˜
˜
H(r,ω) =−∇φ h (r,ω) − jωA h (r,ω),
where
˜ ρ (r ,ω)
i
˜ m
φ h (r,ω) = G(r|r ; ω) dV ,
V ˜ µ(ω)

˜ c ˜ i
A h (r,ω) = ˜ (ω)J (r ,ω)G(r|r ; ω) dV ,
m
V
are the magnetic scalar and vector potentials introduced in § 5.2.
6.2.1 The far-zone fields produced by sources in unbounded space
Many antennas may be analyzed in terms of electric currents and charges radiating in
unbounded space. Since antennas are used to transmit information over great distances,
the fields far from the sources are often of most interest. Assume that the sources are
contained within a sphere of radius r s centered at the origin. We define the far zone of
the sources to consist of all observation points satisfying both r r s (and thus r r )

ˆ
and kr 1. For points in the far zone we may approximate the unit vector R directed
from the sources to the observation point by the unit vector ˆ r directed from the origin
to the observation point. We may also approximate
d
e − jkR
1 + jkR e − jkR e − jkR
ˆ
∇ G = ∇ R = R ≈ ˆ r jk = ˆ r jkG. (6.17)
dR 4π R R 4π R 4π R

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