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45
1
4
3
8
6
32
4sin
4
2sin
8
3
sin
22
2
0
2
2
0
42
=
π
=
π
=
ϕ
+
ϕ
−
ϕ
=ϕϕ
π
π
∫
AAAdA
,
откуда
π
=
3
4
2
A
.
( )
=ϕϕ
ϕ∂
∂
ϕ−=ϕϕϕ=
∫∫
ππ 2
0
2
2
2
222
2
0
2222
sinsinsin
€
sin
€
dAdALAL
zz
h
( )
=ϕϕ−ϕϕ=
=ϕ−−=ϕ−=ϕ
ϕ∂
∂
=ϕ
ϕ∂
∂
=
∫∫
ππ 2
0
422
2
0
222
22
2
2
sin4sin2
sin2122cos22sinsin
dAdA hh
=
ϕ
+
ϕ
−
ϕ
−=
ϕ
−
ϕ
=
ππ 2
0
22
2
0
22
32
4sin
4
2sin
8
3
4
4
2sin
2
2 hh AA
2
22
222222
3
2
24
3
2
8
23
4
2
2
2 h
h
hhh =
π
=
π−π=
π⋅
−
π
=
A
AAA
.
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45
2π 2π
2 3ϕ
sin 2ϕ sin 4ϕ 2 6π 3π 2
∫ A sin ϕdϕ = A 8 − 4 + 32 = A 8 = 4 A = 1,
2 4
0 0
4
откуда A =
2
.
3π
2π 2π
L€2z = 2 2
(2
)
∫ A sin ϕL€z A sin ϕ dϕ = − A h ∫ sin ϕ
2 2 2 ∂2
∂ϕ 2
sin 2 ϕdϕ =
0 0
=
∂2
∂ϕ 2
sin 2 ϕ =
∂
∂ϕ
(
sin 2ϕ = −2 cos 2ϕ = −2 1 − 2 sin 2 ϕ = )
2π 2π
= 2 A 2 h 2 ∫ sin 2 ϕdϕ − 4 A2 h 2 ∫ sin 4 ϕdϕ =
0 0
2π 2π
ϕ sin 2ϕ
2 2 2 3ϕ sin 2ϕ sin 4ϕ
= 2A h −2
= −4 A h − + =
2 4 0 8 4 32 0
2π 3 ⋅ 2π 3 A2 h 2 π 2 2
= 2 A2 h 2 − 4 A2 h 2 = 2 A2 h 2 π − π = = h .
2 8 4 2 3
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