Рудакова Л.И., Соколова Е.Ю. Практический курс физики. Волновая оптика
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121
1.54.
θ
α
θ
λ
22
sin
1
2
cos
−
⋅=∆
n
x
.
1.55. x
3
= 0.31 ; N = 9 .
1.56. x =
2
22
102.5
sin2
−
⋅=
−
θα
λ
ntg
.
1.57. =
''5.8
2
=
nl
λ
1.58. r
9
= 0,39 .
1.59. l
2
= 3.56 .
1.60. k = 5; = 0.5 .
1.61. ) h = 0.15 ) h =
=
−
n
k
4
)12(
λ
0.47 .
1.62. R =
=
−
λ
16
2
1
2
2
dd
0.88 .
1.63.
;08,0
1
2
2
=
+
≈
R
R
V
.
1.64.
8,3,
2
)12(
6
2
0
=+−=
ρ
λ
ρ
rRm
m
.
1.65.
5,2
2
0
=−=
λρ
Rmr
m
.
1.66.
5,12
2
1
=−= hRrr
.
1.67.
hRRm 2
2
)12(
max
−−=
λ
ρ
(
h
ρ ),
λ
h
N
2
=
.
1.68.
7,41
)12(
2
=
−
=
λ
m
d
R
.
1.69.
1
2
2
2
−
=
m
m
ρ
λ
21
21
RR
RR +
⋅
.
1.70.
1.71.
73,1,
3
12
21
=
−
=
ρλρ
RR
RR
m
m
.
1.72.
33,1
1
=
+
=
m
m
n
.
122
1.73.
24,1;
5
2
==
ρ
λ
ρ
n
Rm
.
1.74. 15 ; 4 .
1.75.
( )
11,0;
2
12
4
1
4
=−=
ρ
α
λρ
tgm
m
.
1.76.
30;172;
5
2
0
5
==
−
=
+
m
dd
tg
mm
α
λ
α
.
1.77. = 2,26
.10
2−
⋅
1.78.
491
)(2
12
1
=
−
=
λλ
λ
m
1.79.
2
104,1
4
−
⋅=≤
α
λ
d
.
1.80.
6,1
)1(
2
min
≈
−∆
=
n
d
λ
λ
1.81.
0
2
36,0
2
A
dn
==∆
λ
λ
.
1.82.
max
d
∼ 0,3 .
1.83. ) ;
) ;
) ,
.
1.84.
b
∼ 0,2 .
1.85.
8,19=l
;
15
1
=∆
λ
;
2
N
=30.
1.86.
64,0
sin)(2
22
=
−∆
=
i
l
αλλ
λ
.
1.87.
3,1=
∆
=
λ
λρ
R
.
1.88.
4,62
2
=−
∆
= DR
λ
λ
ρ
.
123
II. .
2.1. )
0
16
9
II =
; )
0
16
1
II =
; )
0
4
1
II =
; )
2
0
)
2
1(
π
ϕ
−= II
.
2.2.
4
2
==
b
r
k
λ
, .
2.3.
580
b
1
b
1
r
21
2
=−=
λ
.
2.4.
17,322
21
21
=
−
=
bb
bb
D
λ
;
21
21
bb
bb
b
m
−
=
.
2.5.
82,1
2
2
1
2
2
=
−
=
λ
rr
b
;
1
2
2
1
2
2
=
−
=
rr
r
m
.
2.6. )
2
, )
3
2.7. r =
.1
2
l
=
λ
2.8.
67,1
2
==
λ
D
l
.
2.9.
(
)
(
)
b
a
barr
6,0
2
2
1
2
2
=
⋅⋅
+−
=
λ
.
2.10.
2;3
1
== bk
;
1
2
=
b
; .
2.11.
2,2
=
+
=
Fbx
2.12.
1
)(
min
+
=∆
k
λ
λ
.
2.13.
3
min
=k
.
2.14. )
0
2II =
; )
0
II =
; )
0
4
9
II =
.
2.15.
0
2
2
0
2
2
sin4 I
b
r
II ≈=
λ
π
.
2.16. )
0
16
25
II = ; )
0
16
49
II = ; )
2
0
)
2
1(
π
ϕ
+= II
; )
0
16
9
II =
;
)
0
16
1
II = ; )
2
0
)
2
1(
π
ϕ
−= II
.
2.17. )
0
=
I
; )
0
4
1
II =
; )
2
0
)1(
π
ϕ
−= II
.
124
2.18. ) h=0,6(2m+1) ; )h=0,3(2m+1) .
2.19. )
0
8,...,2,1,0),
4
1
(
)1(
IIkk
n
h ==+
−
=
λ
. )
,...2,1,0);
4
3
(
1
=+
−
= kk
n
h
λ
2.20. )
0
9,...,2,1,0),
2
1
(
1
IIkk
n
h ==+
−
=
λ
. )
0
,...,2,1,
1
IIkk
n
h ==
−
=
λ
.
2.21. )
)
8
3
(2,1)
8
3
(
1
kk
n
h +=+
−
=
λ
;
,...1,0),223(
0
=+= kII
)
)
8
7
(2,1 kh +=
;
).223(
0
−= II
)
kh 2,1
=
)
4
3
(2,1 += kh
.
2.22.
k
fb
bfk
r 9,0=
−
=
λ
; k=1,2,3,…
2.23. )
,...3,2,1,4,2
0max0
=≈= kIIkh
λ
)
0),12(
min0
≈
+
=
Ikh
λ
.
2.24.
.143
2
5
==
ϕλ
b
2.25.
2;43
2
=∆⋅=∆
′
≈=∆
ϕ
λ
ϕ
Fx
b
.
2.26.
6,0
2
1
1
2
=
∆
+
=
x
F
k
b
λ
.
2.27.
11,
2
)12( =≤+= N
b
kN
λ
(k – ).
2.28. , ,
,
(400 ÷ 760 ), a ≈ 0,7 .
2.29.
40
2
=
∆
=
x
F
bb
x
, 40 .
2.30.
5
2
==
b
l
A
λ
.
2.31.
5,0
12
==
l
ba
λ
.
125
2.32.
0
2≈∆
ϕ
.
2.33. )
0
0
9,7)sinarcsin( =−=∆
ϕϕϕ
n
;
)
λϕϕ
±=− )sin(sin
1
nb
0
11
3,7=−=∆
−+
ϕϕϕ
.
2.34.
045,0
0
=
II
.
2.35.
36,0
12
=II
.
2.36. 4 , ,
2 ,
2 – 1 – , 4 – - 2 –
. .; 2
; 2 .
2.37. 4 ; ,
, ;
2 ;
2 .
2.38. ):
2.39.
;15)12(
max
=+= mM
11=
p
M
.
2.40.
15,1
=
b
.
2.41.
580
=
λ
.
2.42.
6,0
2
=
λ
.
2.43.
0
2
7,54=
ϕ
.
2.44.
706
)(
2
1
sin
)(
2
1
sin
12
12
=
−
−
=
kk
k
ϕϕ
ϕϕ
λλ
.
2.45.
3
max
=m
;
462
0
max
′
=
ϕ
.
2.46.
5
sin
5
1
==
ϕ
λ
d
.
2.47.
8,2
sin
3
==
ϕ
λ
d
.
126
2.48. d=3,92 .
2.49. =3; =71,7 .
2.50. =6
2.51. ; =72,5 ;
1
=636 ; =477 .
2.52.
4,19
=
∆
λ
.
2.53. )
2,0
λ
k
d =
, k = 2 , d = 6 ;
)
5,1
4
==
d
b
.
2.54.
53,0
)cos(45
)sin(
=
∆−
∆
=
ϕ
ϕ
λ
d
.
2.55.
0
85,4=∆
ϕ
.
2.56.
30
=
≈
∆
λ
Lnx
.
2.57.
( )
0,8
1
d
1
1n
R2
x
2
=
−
−
=
λ
∆
.
2.58.
λ
λ
λλ
,1)(
23
2
12
−−=∆
d
k
d
k
Fx
k
∼
2
12
λλ
+
;
72,0
1
=∆
;
32,2
2
=∆
.
2.59.
65,72
′′
=≈∆
l
λ
ϕ
.
2.60.
111
2
1
2
′′
≈−=∆
λ
ϕ
d
N
.
2.61.
706,0
cos
2
3
==∆
ϕ
λ
l
F
x
.
2.62.
639,0)1(
2
2
=−
∆
∆
=
d
d
x
F
λ
λ
;
03,0
1
12
22
=
−
∆
∆
=∆
d
xd
l
y
λ
λ
λ
.
2.63. 5.63. )
6,
))(1(
1
2
=∆
−
∆
=∆ x
nk
nk
Fx
λ
λ
,
12
2
=
∆
x
.
) – , – .
2.64.
d
k
λ
θ
arccos=
.
127
2.65.
(
)
λ
θ
θ
nd
=
+
21
coscos
.
2.66.
6,0
2
2
0
2
=
−
≈
k
αα
λ
.
2.68.
290
min
=
∆
==
λ
λ
NR
.
2.69.
3,
min
=
∆
≥ k
l
d
k
λ
λ
.
2.70.
10
min
=
∆
=
λ
λ
k
d
l
.
2.71.
03,0≈
∆
≥
λ
λ
kl
d
.
2.72.
0
2
46arcsin =
∆
=
λ
λ
ϕ
l
.
2.73.
0
2
14,0 A
l
≈≥∆
λ
λ
,.
2.74.
2
)(1 dmd
m
D
λ
−
=
; )
3
1009,1
−
⋅
/ ;
)
3
1023,1
−
⋅
/ ; )
3
1054,1
−
⋅
/ ;
2.75.
[ ]
23
2
)(1 dmd
fm
D
λ
−
=
; ) 1,30 / ;
) 1,85 / ; ) 3,64 / .
2.76.
lDR
⋅
=
,
l
- .
2.77.
3
10=n
/ .
2.78.
1,21
sin
cos
3
==
ϕ
ϕ
λ
Df
.
2.79.
l
D 8.547
cos)(
21
=
−
=
ϕλλ
λ
2.80. =31 .
2.81. )
5,61
2
=−=
d
k
dkD
λ
. / , k = 2;
)
13sin1
2
0
=−−=
ϕ
λ
d
k
dkD
. / , k = 4.
2.82.
281
=
d
.
128
2.83.
580
=
λ
.
2.84.
281
2
cos2
2sin2
21
2
2
2
1
=−+=
θ
θ
α
kkkkd
,
2
1
=k
,
3
2
=k
,
0
302 =
θ
.
2.85. d=0,23 .
III. .
3.1. ) ; ) ;
) , y = - x.
3.2.
0
45=
ϕ
.
3.3. 1.2
1
2
1
0
≈
−
=
kJ
J
3.4.
2362
0
′
=
ϕ
.
3.5. ) 12 ; ) 65 .
3.6.
0
45=
ϕ
.
3.7. 2 .
3.8. 3,3 .
3.9. 3/2.
3.10.
60
cos
2
43
0
==
ϕτ
II
.
3.11.
0
1
2
30
2
arccos ==
η
η
ϕ
.
3.12.
12,0)(cos
2
1
)1(2
==
−N
ϕη
.
3.13.
1,0cos)
100
1(
2
1
22
21
=−=
α
P
II
.
3.14. 6.14. )
0
I
; ) 2
0
I
.
3.15. 6.15.
)1(2
)12(
−
+
=
n
m
d
λ
, m = 1,2,3,…,
0max
5II =
.
3.16.
0
3II =
.
3.17.
0
5II =
.
129
3.18. ) 25 ;
) 9 .
3.19.
nn
d
e
27
)(4
0
=
−
=
λ
3.20. , 2
3.21.
1
2
1
=
−
=
m
Je
J
k
3.22.
⋅⋅−−=
2
sin2sin2sin)(cos
222
δ
βαβα
aJ
1)
2
sin2sin
222
δ
α
⋅= aJ
2)
)
2
sin2sin1(
222
δ
α
⋅−= aJ
3.23.
3
min
=II
.
3.24. 1,23 .
3.25. = 0,348.
3.26.
1=
n
e
J
J
2
1
=p
3.27. p = 0.33
3.28. =
1II;
2
1
p
=
.
3.29.
5,0
1
2
=
I
I
.
3.30.
8,0)2cos1)(1(
=
−
−
=
ϕ
η
η
P
.
3.31.
3,0
1
=
−
=
P
P
I
I
.
3.32. )
905,0
1
1
P =
+
−
=
η
η
.
)
995,0
1
1
2
=−=
η
P
.
3.33. 6.33. )
89,01
42
2
=−++−=
η
ηη
P
;
)
2
1
2
P
P
P
+
=
, ).
130
3.34.
355
0
=
λ
;
395=
λ
.
3.35.
5
0
10516,0)(
−
⋅=−=∆
e
nndx
.
3.36.
715
0
′
=
ϕ
.
3.37.
21
12
))((
λλ
λ
λ
−
−
=
oe
nnd
k
= 10.
3.38.
10
)(
max
≈
−
=
λ
oe
nnd
m
.
3.39. 1)
2
cos
2
;
2
4
0
α
α
β
J
I ==
2)
2
sin
2
;
2
4
0
α
π
α
β
J
I =
−
=
3.40.
0
22
0
19,0
2
2sin
2
1
IsiinII ==
δ
α
,
π
λ
π
δ
4
3
)(
2
≈−=
oe
nnd
.
3.41. 6.41. )
=
′
2
cos
2
δ
II
; )
=
′
⊥
2
sin
2
δ
II
.
3.42. N = 4.
3.43.
6
1
1069,0
−
⋅=
λ
;
6
2
1043,0
−
⋅=
λ
.
3.44.
25,0)
2
1
(
1
=
∆
−=
n
kd
λ
, k = 4.
3.45.
06,5
min
=d
.
3.46. d = 0,07 .
3.47.
0
6000 A=
λ
.
3.48.
0
21=
∆
=
θ
π
α
xtg
/ ; I(x) ∼
x
x
∆
π
2
cos
, x –
.
3.49.
32arcsin
1
min
==
η
α
d
.
3.50.
7,8
min
=d
.
3.51. ) 0,490 ; ) 0,475 .
3.52. d=0,014