Graphics Programs Reference
In-Depth Information
H
v
()'
H
v
()
Multiplying Eq. (11.83) by and Eq. (11.84) by , and by subtraction
and using the Wronskian of the Bessel and Hankel functions, we get
πωȺI
()
(
)
2
b
=−
−−
0
e
H
kρ
(11.87)
n
ν
0
2
παβ
Substituting
b
n
in Eqs. (11.81) and (11.82) and solving for
c
n
yield
()() ()()
()
()()
()
()()
′
′
JkaJka kJkaJka
−
πωȺI
()
(
)
2
v
v
1
1
v
v
1
c
=
0
e
H
kρ
(11.88)
n
ν
0
2
παβ
−−
′
2
2
kH
ka J
k a
−
k H
ka J
′
k a
ν
v
1
1
ν
v
1
From Eqs. (11.86) through (11.88),
d
n
may be given by
()() ()()
()
()()
()
()()
′
′
πωȺI
kJ
ka J
k a
−
k J
ka J
k a
()
(
)
(
)
2
v
v
1
1
v
v
1
(11.89)
d
=
0
e
H
k
ℑ
−
J
kρ
n
v
0
v
0
2
παβ
−−
′
2
2
kH
ka J
k a
−
k H
ka J
′
k a
v
v
1
1
v
v
1
which can be written as
(
)
(
)
()
(
)
()
()()
′
′
2
2
kJ
k a
J
ka H
kρ
−
H aJ ρ
+
K
v
1
v
ν
0
ν
v
0
(11.90)
(
)
()
()() ()
()
(
)
′
2
2
kJ
ka
H
ka J
kρ
−
JkaH ρ
πωȺI
1
v
1
ν
v
0
v
ν
0
d
=
0
e
n
2
παβ
−−
()
()()
()
()()
′
2
2
′
kH
ka J
k a
−
k H
ka J
k a
ν
v
1
1
ν
v
1
Substituting for the Hankel function in terms of Bessel and Neumann func-
tions, Eq. (11.90) reduces to
(
)
( ) ( ) ( ) ( )
()()() ()()
()
()()
()
()()
′
′
kJ
k a
J
ka Y
kρ
−
YkaJ kρ
+
K
v
1
v
ν
0
ν
v
0
′
kJ
ka
Y ka J
kρ
−
JkaY ρ
πωȺI
1
v
1
ν
v
0
v
ν
0
(11.91)
d
=−
j
παβ
0
e
n
2
−−
2
′
2
kH
ka J
k a
−
k H
ka J
′
k a
ν
v
1
1
ν
v
1
With these closed form expressions for the expansion coeffiecients , ,
and , the field components and can be determined from Eq.
(11.69) and Eq. (11.72), respectively. Alternatively, the magnetic field compo-
nent
a
n
b
n
c
n
d
n
E
z
H
ϕ
H
ρ
can be computed from
11
z
∂
E
H
=−
(11.92)
ρ
j
ωµρ φ
∂
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