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cos
2
2
ϕ
2 ∑
ρ
n′ { ( ) + 1 n
( )
+ 1 iρ
n′
[ ] ( ) + 1 n
( ) + 1 iρ
[ ]
2
cos ϕ
= ∑ EE − −i −i e aτ π −τ − −i −i e bπ π −τ
2
2ρ
* 2 2
n n′ n n n′ n′ n n n′ n′
n′ () + 1 n
( ) + 1 n′
[ ] () + 1 n
( ) + 1
i i an n n′ n′ i i bn n[ n′ n′
]}
− − τ π +τ − − π π +τ =
n′ + 1 n+ 1 iρ
n′
+ 1 n+
1
{ −( − ) ( − ) [ π −τ ][ τ + π ] −() ( − ) [ π +τ ][ τ + π ]}
EE i i e a b i i a b
* 2
n n′ n′ n′ n n n n n′ n′
n n n n
n iρ
cosϕ
{ ( ) [ ]} ∑ ( )
n′
−iρ
{ ⎡ ⎤}
* ⎡cosϕ
+ 1 ⎤ ⎡ * + 1 * * ⎤
Es θEsθ
= ⎢ ∑En − −i e anτ n
+ bnπ n ⎥× ⎢ En′ − −i e ⎣an′ τ
n′ + bn′ π
n′
⎦ ⎥=
⎣ ρ
⎦ ⎣ ρ
⎦
2
cos ϕ
* n+ 1 n′
+ 1
* *
EE ( ) ( )
2 n n′ i i [ an n
bn n]
⎡an′ n′ bn′ n′
= ∑ − τ + π ⎣ τ + π
ρ
⎤⎦
n+ 1
{() −ρ i
n+
1 iρ
[ ] ( ) [ ]}
⎡ senϕ
E E = ⎢ ∑ E i e π −τ − −i e π +τ
⎣ 2ρ
*
iϕ
sϕ
n n n n n
n′+ 1 −ρ i
{()
⎡ ⎤}
⎡ senϕ
* * * ⎤
× ⎢ ∑ En′ i e ⎣an′ π
n′ + bn′ τ
n′
⎦ ⎥ =
⎣ ρ
⎦
2
sen ϕ
* 2 * * * *
= ∑ EE ′ π −τ ⎣ ′ π ′ − ′ τ ′ ⎦− − π +τ ⎣ ′ π ′ + ′ τ
2 n n
i i e
n n
an n
bn n
i i
n n
an n
bn n′
2ρ
⎦
n
{() + 1 n
() ′ + 1 − iρ
n
[ ]
( ) + 1 n
⎡ ⎤ ( ) ′ + 1
[ ] ⎡ ⎤}
⎤
⎥
⎦
sen
2ρ
2
2
ϕ
∑
⎡ senϕ
⎢
⎣ 2ρ
n+ 1 iρ
{( ) [ ]}
* ⎡ senϕ
⎤
Es ϕEiϕ
= ⎢ ∑ En −i e anπ n
+ bnτ n ⎥×
⎣ ρ
⎦
∑
n′ {( ) + 1 iρ
n′
[ ] () + 1 −iρ[ ]}
*
⎤
En′ −i e πn′ −τn′ − i e π
n′ +τ
n′
⎥ =
⎦
n
{( ) + 1 n
( ) ′ + 1 n
[ ][ ] ( ) + 1 n
− π + τ π −τ − − ( ) ′ + 1
[ π + τ ][ π −τ ]}
EE i i a b i i a b
*
n n′ n n n n n′ n′ n n n n n′ n′
+ 1 ′ + 1
{( ) ( ) [ ] ⎡ ⎤}
sen ϕ
n n
E E ∑ E E i i a b ⎣a b
ρ
2
* * * * * *
sϕ
sϕ
=
2
n n′ −
nπ n
+
nτn n′ π
n′ +
n′ τn′
⎦
Agora já podemos fazer as integrais em ϕ :
Para Fz
→ e ˆ = cosθ→
rz
2π
2π
2 2
∫ cos ϕ dϕ= ∫ sen ϕdϕ=π, e para
0 0
232