Page 332 - Electromagnetics

˜
Since we require H r = 0 we must have
∂
˜
[sin θ E φ (r,θ,ω)] = 0.
∂θ
˜
˜
˜
This implies that either E φ ∼ 1/ sin θ or E φ = 0. We choose E φ = 0 and investigate
whether the resulting fields satisfythe remaining Maxwell equations.
˜
˜
In a source-free, homogeneous region of space we have ∇·D = 0 and thus also ∇·E = 0.
Since we have onlya θ-component of the electric field, this requires
1 ∂ cot θ
˜
˜
E θ (r,θ,ω) + E θ (r,θ,ω) = 0.
r ∂θ r
˜
˜
From this we see that when E φ = 0, the field E θ must obey
˜ f E (r,ω)
˜
E θ (r,θ,ω) = .
sin θ
By(4.378) there is onlya φ-component of magnetic field which obeys
˜ f H (r,ω)
˜
H φ (r,θ,ω) =
sin θ
where
1 ∂
˜ ˜
− jω ˜µ f H (r,ω) = [r f E (r,ω)]. (4.379)
r ∂r
So the spherical wave is TEM to the r-direction.
˜
We can obtain a wave equation for f E bytaking the curl of (4.378) and substituting
from Ampere’s law:
1 ∂ 2
˜
˜
˜
˜
˜
∇× (∇× E) =−θ ˆ (r E θ ) =∇ × − jω ˜µH =− jω ˜µ ˜σE + jω˜ E ,
r ∂r 2
hence
d 2 2
˜
˜
[r f E (r,ω)] + k [r f E (r,ω)] = 0. (4.380)
dr 2
c
c 1/2
Here k = ω( ˜µ˜ ) is the complex wavenumber and ˜ = ˜ + ˜σ/jω is the complex
˜
permittivity. The equation for f H is identical.
The wave equation (4.380) is merelythe second-order harmonic differential equation,
with two independent solutions chosen from the list
sin kr, cos kr, e − jkr , e jkr .
We find sin kr and cos kr useful for describing standing waves between boundaries, and
e jkr and e − jkr useful for describing waves propagating in the r-direction. Of these, e jkr
represents waves traveling inward while e − jkr represents waves traveling outward. At this
˜
point we choose r f E = e − jkr and thus
e − jkr
˜
ˆ ˜
E(r,θ,ω) = θE 0 (ω) . (4.381)
r sin θ
By(4.379) we have
˜
E 0 (ω) e − jkr
˜
ˆ
H(r,θ,ω) = φ (4.382)
Z TE M r sin θ
© 2001 by CRC Press LLC