Page 412 - Electromagnetics

where a n is given by (5.187).
The total transverse electric field is found by superposing the TE and TM transverse
fields found from the total potentials. We have already computed the transverse incident
fields and may easily generalize these results to the total potentials. By (5.183) and
(5.191) we have

˜ ∞
˜ t j E 0 " ˆ ˆ (2) 1
E (a) = sin θ cos φ a n J (ka) + b n H (ka) P (cos θ) −
θ n n n
ka
n=1
˜ ∞
E 0 " 1
ˆ (2)
ˆ
− cos φ a n J n (ka) + c n H n (ka) P (cos θ) = 0,
n
ka sin θ
n=1
where
(2)
ˆ (2)
H (x) = xh (x).
n n
By (5.184) and (5.192) we have
˜ ∞
˜ t j E 0 " ˆ ˆ (2) 1
E (a) = sin φ a n J (ka) + b n H (ka) P (cos θ) −
φ n n n
ka sin θ
n=1
˜ ∞
E 0 " 1
ˆ (2)
ˆ
− sin θ sin φ a n J n (ka) + c n H n (ka) P (cos θ) = 0.
n
ka
n=1
These two sets of equations are satisfied by the conditions
ˆ
ˆ
J (ka) J n (ka)
n
b n =− a n , c n =− (2) a n .
ˆ
ˆ
(2)
H n (ka) H n (ka)
We can nowwrite the scattered electric fields as
∞
"
˜ s
1
ˆ (2)
˜

ˆ (2)
E =− j E 0 cos φ b n H (kr) + H (kr) P (cos θ),
r n n n
n=1
˜ " 1
∞
˜ s E 0 ˆ (2) 1 ˆ (2) 1
E = cos φ jb n sin θ H (kr)P (cos θ) − c n H (kr)P (cos θ) ,
θ n n n n
kr sin θ
n=1
˜ " 1
∞
˜ s E 0 ˆ (2) 1 ˆ (2) 1
E = sin φ jb n H (kr)P (cos θ) − c n sin θ H (kr)P (cos θ) .
φ n n n n
kr sin θ
n=1
Let us approximate the scattered field for observation points far from the sphere. We
may approximate the spherical Hankel functions using (E.68) as
ˆ (2)
n − jz
(2)
ˆ (2)
ˆ (2)
e
e
H (z) = zh (z) ≈ j n+1 − jz , H (z) ≈ j e , H (z) ≈− j n+1 − jz .
n n n n
˜
Substituting these we find that E r → 0 as expected for the far-zone field, while
∞
e − jkr " 1
˜ s ˜ n+1 1 1
E ≈ E 0 cos φ j b n sin θ P (cos θ) − c n P (cos θ) ,
n
n
θ
kr sin θ
n=1
∞
e − jkr " 1
˜ s ˜ n+1 1 1
E ≈ E 0 sin φ j b n P (cos θ) − c n sin θ P (cos θ) .
n
n
φ
kr sin θ
n=1
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