Page 360 - Electromagnetics

Show that when the cylinder radius is small compared to a wavelength the radar cross
section may be approximated as

π 2 1
σ 2−D (ω, φ) = a
2
k 0 a ln (0.89k 0 a)
and is thus independent of the observation angle φ.
4.20 A TE-polarized plane wave is incident on a material cylinder with complex per-
c
mittivity ˜ (ω) and permeability ˜µ(ω), aligned along the z-axis in free space. Apply the
boundary conditions on the surface of the cylinder and determine the total field both
internal and external to the cylinder. Show that as ˜σ →∞ the magnetic field external
to the cylinder reduces to (4.366).

4.21 A TM-polarized plane wave is incident on a PEC cylinder of radius a aligned
along the z-axis in free space. The cylinder is coated with a material layer of radius b
c
with complex permittivity ˜ (ω) and permeability ˜µ(ω). Apply the boundary conditions
on the surface of the cylinder and across the interface between the material and free
space and determine the total field both internal and external to the material layer.

4.22 A PEC cylinder of radius a, aligned along the z-axis in free space, is illuminated
˜
by a z-directed electric line source I(ω) located at (ρ 0 ,φ 0 ). Expand the fields in the
regions a <ρ <ρ 0 and ρ> ρ 0 in terms of nonuniform cylindrical waves, and apply the
boundary conditions at ρ = a and ρ = ρ 0 to determine the fields everywhere.

4.23 Repeat Problem 4.22 for the case of a cylinder illuminated by a magnetic line
source.

4.24 Assuming
k
f (ξ, ω) = A(k x ,ω) sin ξ,
2π
use the relations

cos z = cos(u + jv) = cos u cosh v − j sin u sinh v,
sin z = sin(u + jv) = sin u cosh v + j cos u sinh v,
to show that the contour in Figure 4.29 provides identical values of the integrand in

˜ − jkρ cos(φ±ξ)
ψ(x, y,ω) = f (ξ, ω)e dξ
C
as does the contour [−∞ + j , ∞+ j ] in

∞+ j
1
˜ − jk x x ∓ jk y y
ψ(x, y,ω) = A(k x ,ω)e e dk x . (4.433)
2π
−∞+ j
4.25 Verify (4.409) by writing the TE fields in terms of Fourier transforms and apply-
ing boundary conditions.

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