Page 386 - Electromagnetics

˜ NZ
˜ NZ
◦
We note that H and E are 90 out-of-phase. Also, the electric field has the same
spatial dependence as the field of a static electric dipole. The terms that dominate far
from the source are called the far-zone or radiation fields:
˜
jkIl e − jkr
ˆ
˜ FZ
H (r,ω) = φ sin θ, (5.92)
4π r
˜
jkIl e − jkr
ˆ
˜ FZ
E (r,ω) = θη sin θ. (5.93)
4π r
The far-zone fields are in-phase and in fact form a TEM spherical wave with
˜ FZ
ˆ r × E
˜ FZ
H = . (5.94)
η
We speak of the time-average power radiated by a time-harmonic source as the integral
of the time-average power density over a very large sphere. Thus radiated power is the
power delivered by the sources to infinity. If the dipole is situated within a lossy medium,
all of the time-average power delivered by the sources is dissipated by the medium. If
the medium is lossless then all the time-average power is delivered to infinity. Let us
compute the power radiated by a time-harmonic Hertzian dipole immersed in a lossless
medium. Writing (5.90) and (5.91) in terms of phasors we have the complex Poynting
vector
c
ˇ
ˇ ∗
S (r) = E(r) × H (r)
2 2
ˇ
ˇ
|I|l 2 2 2 |I|l k 2 1 2
ˆ
= θη j k r + 1 cos θ sin θ + ˆ rη 1 − j sin θ.
3 5
4π kr 5 4π r 2 k r
c
We notice that the θ-component of S is purely imaginary and gives rise to no time-
3
average power flux. This component falls off as 1/r for large r and produces no net
flux through a sphere with radius r →∞. Additionally, the angular variation sin θ cos θ
integrates to zero over a sphere. In contrast, the r-component has a real part that varies
2
2
as 1/r and as sin θ. Hence we find that the total time-average power passing through
a sphere expanding to infinity is nonzero:
 
2

ˇ
2π π 1  |I|l k 2 
2
2
P av = lim Re rη 2 sin θ · ˆ rr sin θ dθ dφ
ˆ
r→∞ 0 2 4π r
0  
2
π l

ˇ 2
= η |I| (5.95)
3 λ
where λ = 2π/k is the wavelength in the lossless medium. This is the power radiated by
ˇ 2
the Hertzian dipole. The power is proportional to |I| as it is in a circuit, and thus we
may define a radiation resistance
2

2P av 2π l
R r = = η
ˇ 2
|I| 3 λ
that represents the resistance of a lumped element that would absorb the same power as
radiated by the Hertzian dipole when presented with the same current. We also note that
the power radiated by a Hertzian dipole (and, in fact, by any source of finite extent) may
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