Page 428 - Electromagnetics

˜
˜
Using this we can obtain expressions for E and H in the far zone of the sources. The
approximation (6.17) leads directly to
˜ i
∇ · J k
˜ i
˜ i
i

˜ ρ ∇ G ≈ j (ˆ r jkG) =− ˆ r ∇ · (GJ ) − J ·∇ G .
ω ω
Substituting this into (6.13), again using (6.17) and also using the divergence theorem,
we have
k

˜ ˜ i ˜ i ˜ i

E(r,ω) ≈− jω ˜µ J − ˆ r(ˆ r · J ) GdV + ˆ r c (ˆ n · J )GdS ,
V ω˜ S
where the surface S surrounds the volume V that contains the impressed sources. If
we let this volume slightly exceed that needed to contain the sources, then we do not
change the value of the volume integral above; however, the surface integral vanishes
˜ i
˜ i
˜ i
˜ i
since ˆ n · J = 0 everywhere on the surface. Using ˆ r × (ˆ r × J ) = ˆ r(ˆ r · J ) − J we then
obtain the far-zone expression

˜ ˜ i
E(r,ω) ≈ jωˆ r × ˆ r × ˜ µ(ω)J (r ,ω)G(r|r ; ω) dV
V
˜

= jωˆ r × ˆ r × A e (r,ω) ,
˜
where A e is the electric vector potential. The far-zone electric field has no r-component,
and it is often convenient to write
˜
˜
E(r,ω) ≈− jωA eT (r,ω) (6.18)
˜
˜
where A eT is the vector component of A e transverse to the r-direction:
˜
˜
˜
˜
ˆ ˜
ˆ ˜

A eT =−ˆ r × ˆ r × A e = A e − ˆ r(ˆ r · A e ) = θA eθ + φA eφ .
˜ i
We can approximate the magnetic field in a similar fashion. Noting that J ×∇ G =
˜ i
J × ( jkˆ rG) we have
k
˜ i
˜

H(r,ω) ≈− j ˆ r × ˜ µ(ω)J (r ,ω)G(r|r ,ω) dV
˜ µ(ω) V
1
˜
≈− jωˆ r × A e (r,ω).
η
With this we have
˜
ˆ r × E(r,ω)
˜
˜
˜
E(r,ω) =−ηˆ r × H(r,ω), H(r,ω) = ,
η
in the far zone.
To simplify the computations involved, we often choose to approximate the vector
potential in the far zone. Noting that

2 2
R = (r − r ) · (r − r ) = r + r − 2(r · r )
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