Page 326 - Electromagnetics
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Figure 4.26: Geometryof a perfectlyconducting wedge illuminated bya line source.
This in turn gives us the surface current induced on the cylinder. From the boundary
˜
˜
˜
˜
ˆ ˜
condition J s = ˆ n × H| ρ=a = ˆρ × [ˆρH ρ + φH φ ]| ρ=a = ˆ zH φ | ρ=a we have
∞
j # n j −n (2)
˜
J s (φ, ω) =− ˆ zE 0 (2) J (k 0 a)H n (k 0 a) − J n (k 0 a)H n (2) (k 0 a) cos nφ,
n
η 0 H n (k 0 a)
n=0
and an application of (E.93) gives us
˜ ∞ −n
2E 0 # n j
J s (φ, ω) = ˆ z cos nφ. (4.365)
(2)
η 0 k 0 πa H n (k 0 a)
n=0
Computation of the scattered field for a magnetically-polarized impressed field pro-
ceeds in the same manner. The impressed electric and magnetic fields are assumed to be
ˆ
˜
˜ i
˜
E (r,ω) = ˆ yE 0 (ω)e − jk 0 x = (ˆρ sin φ + φ cos φ)E 0 (ω)e − jk 0 ρ cos φ ,
˜
˜
E 0 (ω) − jk 0 x E 0 (ω) − jk 0 ρ cos φ
˜ i
H (r,ω) = ˆ z e = ˆ z e .
η 0 η 0
For a perfectlyconducting cylinder, the total magnetic field is
∞ n j −n
˜ #
˜ E 0 (2) (2)
H z = J n (k 0 ρ)H n (k 0 a) − J (k 0 a)H n (k 0 ρ) cos nφ. (4.366)
n
(2)
η 0 H n (k 0 a)
n=0
The details are left as an exercise.
Boundary value problems in cylindrical coordinates: scattering by a perfectly
conducting wedge. As a second example, consider a perfectlyconducting wedge im-
mersed in free space and illuminated by a line source (Figure 4.26) carrying current
˜ I(ω) and located at (ρ 0 ,φ 0 ). The current, which is assumed to be z-invariant, induces
a secondarycurrent on the surface of the wedge which in turn produces a secondary
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