Page 411 - Electromagnetics

˜ "
∞
˜ E 0 ˆ 1
E θ =− cos φ a n J n (kr)P (cos θ), (5.191)
n
kr sin θ
n=1
˜ "
∞
˜ E 0 ˆ 1
E φ =− sin θ sin φ a n J n (kr)P (cos θ). (5.192)
n
kr
n=1
The total field is then the sum of the TE and TM components.
Example of spherical TE–TM decomposition: scattering by a conducting
sphere. Consider a PEC sphere of radius a centered at the origin and imbedded in a
c
homogeneous, isotropic material having parameters ˜µ and ˜ . The sphere is illuminated
by a plane wave incident along the z-axis with the fields
˜
˜
˜
E(r,ω) = ˆ xE 0 (ω)e − jkz = ˆ xE 0 (ω)e − jkr cos θ ,
˜
˜
E 0 (ω) − jkz E 0 (ω) − jkr cos θ
˜
H(r,ω) = ˆ y e = ˆ x e .
η η
We wish to find the field scattered by the sphere.
The boundary condition that determines the scattered field is that the total (incident
plus scattered) electric field tangential to the sphere must be zero. We sawin the previous
example that the incident electric field may be written as the sum of a field TE to the
r-direction and a field TM to the r-direction. Since the region external to the sphere
is source-free, we may also represent the scattered field as a sum of TE and TM fields.
˜ s
˜ s
These may be found from the functions A and A , which obey the Helmholtz equations
e h
(5.165) and (5.174). The general solution to the Helmholtz equation may be found using
the separation of variables technique in spherical coordinates, as shown in § A.4, and is
given by
˜ s

∞ n
A /r = " " C nm Y nm (θ, φ)h (kr).
(2)
e
˜ s
A /r n
h n=0 m=−n
Here Y nm is the spherical harmonic and we have chosen the spherical Hankel function h (2)
n
as the radial dependence since it represents the expected outward-going wave behavior
of the scattered field. Since the incident field generated by the potentials (5.180) and
˜ s
˜ s
(5.181) exactly cancels the field generated by A and A on the surface of the sphere, by
e h
orthogonality the scattered potential must have φ and θ dependencies that match those
of the incident field. Thus
˜
˜ s
∞
A E 0 k "
e (2) 1
= cos φ b n h (kr)P (cos θ),
n
n
r ω
n=1
˜
˜ s
∞
A E 0 k "
h (2) 1
= sin φ c n h (kr)P (cos θ),
n
n
r ηω
n=1
where b n and c n are constants to be determined by the boundary conditions. By super-
position the total field may be computed from the total potentials, which are the sum of
the incident and scattered potentials. These are given by
˜
˜ t
∞
A e E 0 k " (2) 1
= cos φ a n j n (kr) + b n h (kr) P (cos θ),
n
n
r ω
n=1
˜ t
˜
A E 0 k ∞
h " (2) 1
= sin φ a n j n (kr) + c n h (kr) P (cos θ),
n
n
r ηω
n=1
© 2001 by CRC Press LLC