б юс а ь н г жб сы ь 0

 1
/0 &1"' ,
$ 234 . ,5
$ !6&7
X1-010 (2002/07/14)
mamwad@mailaps.org
' R „ ) = 8 ( *
4& d
# A X e w 2 * $ 7,PT * 7,* _q
0
8
U A , 8 _ (x, y) u
, $
* e w 2 (u, v)
X e w 2 A * x = a cosh u cos v,
DP ( *
0 ≤ u < ∞ u = const
6
*
X
v = const
6 *
v
(x = ±a, y = 0)
6
X 6 y = a sinh u sin v,
(u, v)
+ )! D* q( # !
u = const
6 1
0≤v≤π
*
C
a -
$
−∞ < u < ∞ 2π
4* q ( q q/ % &' ,w2 6
-$ q(*
v = const
qq/6
RS 7 ds2 := dx2 + dy 2 ,
= D2 (u, v)(du2 + dv 2 ),
2
- $
D2 (u, v) := a2 (cosh2 u − cos2 v),
= a2 (sinh2 u + sin2 v).
( '
Y09
$ *" #+ 2 2 ( #+ 0 # / % *#+ 0 3
# ( * 1
1" Z0[= #+
0
http://staff.alzahra.ac.ir/mamwad/x1/x1-010.pdf
)Rutqu 0 ∇2 :=
=
∂2
∂2
+
,
2
∂x
∂y 2
1
D2 (u, v)
1
7 8_ A z‹ Œ N6
∂2
∂2
+
2
∂u
∂v 2
(x, y) ,
.
4
u ew2 .)
$
(u, v)
ew2
u
V A* , $
ew2 7 8_ k ew2 k)(
u ew2 )Ru tqu 8_ k 6
, $
(u, v)
u u
ew2 )R tq 8_ + 6 M & $
ew2 h
U A,
(ρ, φ, z) I*
$ U A, X ew2 I * A ew2 * (u, φ, v)
ew2 D* I ,+ U A, =
ρ = a cosh u cos v,
*
*
−∞ < u < ∞
−(π/2) ≤ v ≤ (π/2)
G C * - ! X u = const
* z = a sinh u sin v.
u = const
*
0 ≤ u < ∞
* v = const
(u, v)
D* q ( # !
4* q ( = C7 - ! q q/ 6 * =C7 -!qq/
5
+
) !
0 ≤ v ≤ (π/2)
v = const
* * * GC * -!X6
# - * $ q q/ X z d D (
(ρ = a, z = 0)
- * , w 2 + -f -$
7 ds2 := dρ2 + ρ2 dφ2 + dz 2 ,
= D2 (u, v)(du2 + dv 2 ) + a2 cosh2 u cos2 v d2 φ,
6
)Rutqu *
∇2 :=
=
∂2
∂2
∂2
+
+ 2,
2
2
∂x
∂y
∂z
1
D2 (u, v)
+
∂
1 ∂
∂
1
∂
cosh u
+
cos v
cosh u ∂u
∂u
cos v ∂v
∂v
∂2
1
.
a2 cosh u cos2 v ∂φ2
2
7
U
A, =
(ρ, φ, z)
e w 2 * (s, φ, t)
e w 2 4* I ρ = a sinh s cos t,
- ! X s = const
* z = a cosh s sin t.
−(π/2) ≤ t ≤ π/2
, w 2 * = C* - ! q q/ 6 (ρ = 0, z = ±a) -$ *
8
0 ≤ s < ∞ (s, t) t = const
* *
qq/ * X + )!
+ f $ *
z dD(
#- *
7 ds2 := dρ2 + ρ2 dφ2 + dz 2 ,
= F 2 (s, t)(ds2 + dt2 ) + a2 sinh2 s cos2 t dφ2 ,
9
$
F 2 (s, t) := a2 (cosh2 s − sin2 t),
= a2 (sinh2 s + cos2 t).
10
)Rutqu
∇2 :=
=
∂2
∂2
∂2
+
+
,
∂x2
∂y 2
∂z 2
1
F 2 (u, v)
+
∂
∂
1 ∂
∂
1
sinh s
+
cos t
sinh s ∂s
∂s
cos t ∂t
∂t
1
a2
2
sinh s
cos2
∂2
.
2
∂φ
t
11
u u
qA
8( R5 : 0 6 $ U A, $ ew2 I ‹ Œ NRtq a
+ ew2 7,PT * 7,* _q R„) –A
1
40 $ 9' #: $ '# ,
1 &1"' $ 9'
$ '# ,
$
$
'# ,
$ &:' . 9 ; L
- c y 6 R„) * :H A* q' - ', R„) D* R„) &' -: _ _q - #- * 4* R„) &'
‹nŒ :5 d3a )PT
π/2
+ ,- _ _q 8)C - 6
λ
qP
qK=
2a 1.1
BC 6$ ")(
6 ! z
d d *
y = 0, −a ≤ x ≤ a
‹ Œ NRutqu qA %&' +C -* $ a
y
*
+ C % & ' x %, _
_q 8)C e a
c *
∇2 Φ = 0,
* v
* 6 K 3 u
, (y = 0 ∨ |x| > a).
u=0
R „ ) ∂2
∂2
+
2
∂u
∂v 2
12
y 8)C NC Φ = 0,
u > 0,
x M + C
2π
DP (
13
*
Φ(u = 0, v) = Φ0
2π
c + * + *
v
14
.) 8)C +*]5 *
Φ(u, v + 2π) = Φ(u, v).
4M 6
u → ∞ 15
8)C # ) #$ 8)C - * y Q R 7 . • J ρ $ $ ")( 0 NC
Φ≈−
ρ
λ
ln ,
2π ρ0
ρ → ∞.
16
u → ∞ A ρ → ∞ 6
6 *
ρ≈
a eu
,
2
0 /! 7 ρ → ∞.
17
V - $
Φ≈−
)
v
.) Φ
λ
u + const,
2π
#$
Φ
u → ∞.
18
A, 15
* 18
14
13
&
b
c ‹ Œ #
Φ=
∞
αk (u)ei k v ,
19
k=−∞
V - $
d2
− k2
du2
αk (u) = 0.
20
qA "0
αk (u) = βk e|k| u + γk e−|k| u ,
k = 0,
α0 (u) = γ0 + β0 u
21
18
V y k = 0,
βk = 0,
22
*
β0 = −
λ
.
2π
23
8)C Φ=−
λ
γk e−|k| u ei k v .
u + γ0 +
2π
24
k=0
14 V k = 0.
γk = 0,
y 25
0 Φ=−
λ
u + γ0 .
2π
26
14
* 0 , y 0 , $ - & - * C " 0
! 6 K u
% , ' # D* 8 ) C
13
,+ + 7 q A *
18
y ‹ Œ NRu tqu qA "0 ,7 XS
"0 -: ‹ Œ R f c
a
3
‹ Œ 6 u
6 _ J # : *
26
: A , 8w# Q
F P ' c a2
1
y
*
x
1
&
k)(
x2
y2
+ 2
= 1.
2
cosh u
a sinh2 u
27
c
‹ Œ R f y R„) 7 8_ M R„) X ew2 +3 e a - d
u
$ 6 , $
e w 2 R „ ) ‹ Œ -O -* d3a
c
‹ Œ Rf B * * c yR2 R„)
c !+ 6 _ _q -
E=
λ
∇u.
2π
d& qK= * 28
3a
Ex
u=0
* R:0 σ(x) = sgn(y) Ey (x, y)|y→0 ,
= sgn(y)
=
λ
- =
6 $ λ
2π
√
λ
,
2π a sin v
λ
.
a2 − x2
29
% & ' * e . 5 D K + !+ qP qK=
26
# x NC B B 8)CF
]
-f qA
3a DP (*
./ 0123 3R„
* $ v # $ S*
K
Q I
- = $ q ( 7 0 8 ( *
2a
1.2
D P 3 a ) P T - : 5
8.S R„) _ _q - * )PT - e a u
u
c‹nŒ B =
=
µ = 1 6 #
0
I
ẑ × E,
λ
I
ẑ × ∇u,
2π
! * 28
30
c d& - 0 qK=
Js (x) = ẑ
qK= z F P
- 0
2π
√
E
& _ _q -
29 - $
f
I
.
a2 − x2
31
+/! - 0qK=
- 0qK= $ 0,
' * 8$
- &'. 9 ' '# < #: ,
$ =>
_ _q R„) + - 2a
-* F P d 3 a
n̂
: 5
BC $ K
$
E0 n̂
2
Q d3a
* _ _ q - $ $ - 6 ) P T R „ ) e* 3 d 3 a F P* B 0
R„)* d t̂
$ B0 t̂ × n̂
* )PT -
‹nŒ 8., 6 !+ 6
d3a 8 $ K
Q 6)  + "2 y =
- 7 d 3 a F P
* F _ * _ _ q - * F _ 3 R „ 7 d 3 a F P
F _ * * d 3 a _ J _ NC :5 3R„ '# * _ _q - -=
- 7 d3a
* d3a
_ q
3a - :
:5 F_ * d3a F P* 7 -
:5 3R„ '# $ - #H _ d3a F P* $ !-
c + D* - y
R „ ) " 0 $ J = 4* R „ ) - - : K F P S F P 7 *
* 7 0 : _ _ q - 0
d3a F_ * )PT - $ v # 6 )PT R„) d3a :5 3R„ 4RA M DM f DM 7
- 7 F_ * - NC -7 d3a F P*
7 -
- $ $ JV, o& - 7 #H d3a F P* -7
L :5 R„) )PT - A*]
K F P S d3a F P 7 F_ * L* Jq u 0: :5 F_ d
4,"! 5 ( 6 47 48
9 ,- * 6
L:H
2.1
P $ d 3 a e a 6 ! 8 . S 1 2 8 _ - : K v=π
, *
v=0
D A X e w 2 + _ ,
$ y 0≤v≤π
*
−∞ < u < ∞
v = 0 ∨ v = π.
Φ = 0,
32
*
Φ ≈ −E0 |y|,
(x2 + y 2 ) → ∞.
33
$ 0 y n̂ = sgn(y) ŷ.
b
6 ‹ Œ # ) v
. ) Φ
32
* 6 $ + 3 34
y c6
Φ=
∞
αk (u) sin k v.
35
k=1
0- * *
20 αk
V
*
‹ Œ NRutqu qA a
Φ
$ αk (u) = βk ek u + γk e−k u .
36
V βk = γk = 0,
k = 1,
33
y 37
*
β1 = γ1 = −
a E0
.
2
38
NC
Φ = −a E0 cosh u sin v,
a2
sin v
39
* 27 cosh u
y2
x2
− 2 2 = 1.
2
cos v
a sin v
- $
40
6 _ _q -
E = a E0 ∇(cosh u sin v).
41
c d& qK=
29 f
σ(x) = sgn(y) Ey (x, y)|y→0 ,
= sgn(y) E0
= E0 √
7 qK= 7 D K *
cosh u
,
sinh u
|x|
.
− a2
42
x2
|x| < R , qK= !
D K F]
+ !+
3a R → ∞ |x| < R
, 4,"! 5 ( 6 47 48
9 0123 R„)
x=0
2.2
3a )PT - :5 3R„ * $ v # 6 0
c‹nŒ 8., 8.S R„)
B = a B0 ẑ × ∇(cosh u sin v).
c 41
qK= . 6 d& - 0 Js (x) = ẑ B0 √
|x|
.
x2 − a2
43
t̂ = ẑ $ 6 0,
44
|x| < R ,
7 qK= 7 D K *
qK= D K F]
+ 6 0
!
3a R → ∞ |x| < R ,
3
#: $ ? @ ,
$ 1 &1"' $ 9'
" ) ( - _ Q
_ q - 6 a
I A ? S 7 z=0
d3a ? S 6$
8 (
ew2 ? S 6 ! ew2 Y . - J$ *
$ R„) Φ = Φ0 ,
u=0
X
u = 0,
45
*
Φ → 0,
q A 8 ) C
# !
u > 0
φ 8') ? S -*
(ρ2 + z 2 ) → ∞.
Q
46
? S 8 $ $ - = 8)C - ': -= *
Φ0
L : H
‹ Œ NRu tqu
a
e a 1
∂
∂
1 ∂
∂
Φ = 0.
cosh u
+
cos v
cosh u ∂u
∂u
cos v ∂v
∂v
X ew2 k)(
46
y $ 0
v
*
u
47
ew2 .) 8)C
Φ → 0,
c6 ! 8)C u → ∞.
v 8') +B 7 NC
)
v
48
y $ +
%, d
d
1
cosh u
Φ = 0.
cosh u du
du
49
c 8')&
"0 * qA P̃0 (u) := 1,
Q̃0 (u) := tan−1 (sinh u).
*
Φ = Φ0 1 −
50
48 * 45 y
2
tan−1 (sinh u) ,
π
$ "0
=
2Φ0
cot−1 (sinh u)
π
51
c $ +3 ? S * Ra# 8)C # Φ ≈
≈
Φ0
- * 2Φ0
,
π sinh u
4Φ0
.
π eu
52
Φ ≈
≈
4π
Q
ρ2 + z 2
Q?
S -= ,
Q
.
2π a eu
53
0 Φ0 =
Q
,
8a
54
*
Φ=
+ 8.,
Q
cot−1 (sinh u).
4π a
(ρ, z) (x, y) - 55
$ 27
f & u
_ _q -
E=
Q
∇u,
4π a cosh u
56
d& qK= - * *
Q
σ=
4π a
a2 − ρ 2
.
57
54
c ? S # x C = 8a.
4V 58
4
< #: ,
=> ,
1. # . 1 &1"' $ 9'
$
$
- &'. 9 ' ? @
* _ _q - + - a
IA ? S $ K
0, d3a -* F P * d3a
3
12 f ew2 "2 Q d3a 7
n̂
:5 _ 2
$ E0 n̂
? S 12 †d . y :
y 8)C +
Φ = 0,
v = 0,
59
*
Φ ≈ −E0 |z|,
8 ) C L : H 6 # !
φ 8') (ρ2 + z 2 ) → ∞
0 ≤ v ≤ (π/2)
*
60
−∞ < u < ∞
(u, v)
8)C - ': -= * *
e w 2 k ) (
60
y * 47
+ ) ! *
6 NRutqu qA
8 ) C e a # !
X
Φ ≈ −a E0 | sinh u| sin v,
.) K *
u .) 7 8) 3 * -6 +b* k)(
|u| → ∞.
61
47
!8:5 * I:V ^= F P !8:5
8)C NC 00 6 $ v
c )
Φ=
Uµ (u)Vµ (v),
62
µ
$
d
d
1
cosh u
U = µU
cosh u du
du
d
1 d
cos v
V = −µ V.
cos v dv
dv
63
ξ := sin v
T T, 4* qA
64
c 8., ‹ Œ bqO qA e
(1 − ξ 2 )
µ = l(l + 1)
! 3a 1
3a *
$ ) 8_ d2
d
− 2ξ
V = −µ V.
dξ 2
dξ
62 NC
Φ=
±1
#p
4
‹ Œ ∞
3
65
#p
"0 qA 8w# 3 * oda 5
ζl (u)Pl (sin v),
l
$
66
l=1
+3 ‹ Œ bqO R:0=
e
P1 (ξ) = ξ,
66
$
67
‹ Œ bqO R:0= &
D]' J *
61
e
F P V c
3a s
l=1
R:0 '#
Φ = ζ1 (u) sin v,
68
Pl
ζ1
$
1
d
d
ζ1 = 2ζ1
cosh u
cosh u du
du
69
c 8')&
"0 * M qA *
P̃1 (u) := sinh u,
Q̃1 (u) := 1 + sinh u tan−1 (sinh u).
NC 6 K
Φ=−
Q1
" X ζ1
70
61 V y 2a E0
[1 + sinh u tan−1 (sinh u)] sin v.
π
71
_ _q - 0 E=
2a E0
∇{[1 + sinh u tan−1 (sinh u)] sin v},
π
z → 0 E · ẑ sgn(z) A
σ=
E0
2E0
π
a
ρ 2 − a2
+ tan−1
ρ 2 − a2
a
72
d& qK= *
.
73
qK= D K qK= D K F]
-f !+
7 !
3a R → ∞ R
ρ=0
IA , 5
< #: ,
=> ,
1. # . 40 $ 9'
$
$
- &'. 9 ' ? @
? S * )PT - $ v # *
d3a -* ( c# !
K
Q 8.S 12 R„) -:
3a - o& :5 3R„ o& * * B0 sgn(z)x̂
q_ 8)C 7 if )PT - 0- NC 6+
B = −∇ΦM ,
v = 0.
74
y * ‹ Œ NRutqu qA 8)C a
ΦM ≈ −a B0 cosh u cos v cos φ sgn(z),
|u| → ∞,
75
*
∂ΦM ∂v b
76
v=0
y e a ‹ Œ # )
c
φ .) 8)C *
cos φ k 8)C ΦM = Ψ(u, v) cos φ.
0 *
75
V 77
‹ Œ NRutqu qA a
1
∂
∂
1 ∂
∂
cosh2 u − cos2 v
cosh u
+
cos v
−
cosh u ∂u
∂u
cos v ∂v
∂v
cosh2 u cos2 v
Φ
Ψ = 0.
78
∂
∂
1
∂
1
1 ∂
1
cosh u
+
cos v
−
+
Ψ = 0.
cosh u ∂u
∂u
cos v ∂v
∂v
cos2 v
cosh2 u
NC 00 6 $ 8) 3 c )
!8:5 * I:V ^= F P !8:5 6 0
!8:5* -6 q(+b* k)(
Ψ=
Xµ (u)Yµ (v),
79
Ψ
80
µ
$
d
1
d
1
X = µX
cosh u
+
cosh u du
du
cosh2 u
1 d
d
1
Y = −µ Y.
cos v
−
cos v dv
dv
cos2 v
81
c 8., ‹ Œ bqO 7., )* qA 4* qA
e
(1 − ξ 2 )
! ξ
.) Z* *
80
= 64
ξ=1
T T, 1
d2
d
−
− 2ξ
Y = −µ Y.
dξ 2
dξ
1 − ξ2
#p
"0 4
e a ‹ Œ 3
ξ = ±1 82
#p
"0 qA 8 w # 3 * o d a
l
$
µ = l(l + 1)
Ψ=
∞
ηl (u)P1l (sin v),
83
l=1
+3 P11 (ξ) =
75 V 1
., ‹ Œ bqO )* %,
e
1 − ξ2,
m = 1 R:0 m .,
c
$
84
‹ Œ bqO%, &
D]' J *
3a s
e
l=1
R:0 '# 83
F P
Ψ = η1 (u) cos v,
85
P1l
η1
$
d
1
d
1
η1 = 2η1
cosh u
+
cosh u du
du
cosh2 u
86
c 8')&
"0 * M qA *
P̃11 (u) := cosh u,
Q̃11 (u) := tanh u + cosh u tan−1 (sinh u).
NC 6 K
Ψ=−
Q11
" X η1
87
75 V 2a B0
[tanh u + cosh u tan−1 (sinh u)] cos v,
π
y 88
0- *
ΦM = −
2a B0
[tanh u + cosh u tan−1 (sinh u)] cos v cos φ.
π
89
)PT - k, , B=
2a B0
∇{[tanh u + cosh u tan−1 (sinh u)] cos v cos φ}.
π
90
6 d& - 0 qK=
Js = ẑ × B sgn(z)|z→0 ,
91
73
$ + f - $
AB; 6
[1] James Ward Brown & Ruel V. Churchill; “Complex variables and applications”,
6th edition (Mc Graw-Hill, 1996) chapter 9
“
7 J# RV z - - ', * H 5 A* )PT* _q -
”
z :d ‹nŒ
n>> , n>n lm~{ > _e
[3] John David Jackson; “Classical electrodynamics”, 3rd edition (John Wiley & Sons,
1998)
? ,
$ AC:'
[a] Laplace
[b] Fourier
[c] Dirichlet
[d] Neumann
[e] Legendre
7