Page 319 - Electromagnetics

Example of uniform cylindrical waves: fields of a line source. The simplest
example of a uniform cylindrical wave is that produced byan electric or magnetic line
˜
source. Consider first an infinite electric line current of amplitude I(ω) on the z-axis,
immersed within a medium of permittivity ˜ (ω), permeability ˜µ(ω), and conductivity
˜ σ(ω). We assume that the current does not varyin the z-direction, and thus the problem
is two-dimensional. We can decompose the field produced bythe line source into TE and
TM cases according to § 4.11.2. It turns out that an electric line source onlyexcites TM
˜
fields, as we shall show in § 5.4, and thus we need only E z to completelydescribe the
fields.
Bysymmetrythe fields are φ-independent and thus the wave produced bythe line
source is a uniform cylindrical wave. Since the wave propagates outward from the line
source we have the electric field from (4.330),

j (2)
˜
˜
E z (ρ, ω) =− E z0 (ω)H 0 (kρ), (4.343)
4
and the magnetic field from (4.332),
˜
k E z0 (ω) (2)
˜
H φ (ρ, ω) = H 1 (kρ).
ω ˜µ 4
˜
We can find E z0 byusing Ampere’s law:
"
˜ ˜ ˜
H · dl = J · dS + jω D · dS.
S S
˜
˜
˜
Since J is the sum of the impressed current I and the secondaryconduction current ˜σE,
we can also write
"
˜ ˜ ˜ ˜ c ˜
H · dl = I + ( ˜σ + jω˜ )E · dS = I + jω˜ E · dS.
S S
Choosing our path of integration as a circle of radius a in the z = 0 plane and substituting
˜
˜
for E z and H φ , we find that
˜ ˜ a
k E z0 (2) c − j E z0 (2)
˜
H 1 (ka)2πa = I + jω˜ 2π lim H 0 (kρ)ρ dρ. (4.344)
ω ˜µ 4 4 δ→0 δ
The limit operation is required because H (2) (kρ) diverges as ρ → 0. By(E.104) the
0
integral is
a a 1
lim H 0 (2) (kρ)ρ dρ = H 1 (2) (ka) − lim δH 1 (2) (kδ).
δ→0 δ k k δ→0
(2)
The limit maybe found byusing H (x) = J 1 (x) − jN 1 (x) and the small argument
1
approximations (E.50) and (E.53):

(2) kδ 1 2 2
lim δH 1 (δ) = lim δ − j − = j .
δ→0 δ→0 2 π kδ πk
Substituting these expressions into (4.344) we obtain
˜ ˜
k E z0 (2) c − j E z0 a (2) 2
˜
H 1 (ka)2πa = I + jω˜ 2π H 1 (ka) − j .
ω ˜µ 4 4 k πk 2

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