(-第二版)-习题解答-张三慧-清华大学出版社

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( ) 158
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( , 100084)
ht tp: / / www. t up. tsinghua. edu. cn
8501168 1/ 32: 15625: 391
2000 7 1 2001 12 4
ISBN 7302038104/ O228
30014000
30. 00
( )

1 ( 1)
1 ( 1)
2 ( 19)
3 ( 42)
4 ( 61)
5 ( 89)
6 ( 111)
2 ( 129)
1 ( 129)
2 ( 138)
3 ( 158)
4 ( 181)
3 ( 195)
1 ( 195)
2 ( 215)
3 ( 218)
4 ( 240)
5 ( 250)
6 ( 269)
7 ( 277)
8 ( 294)
9 ( 321)
10 ( 329)

11 ( 345)
4 ( 355)
1 ( 355)
2 ( 383)
3 ( 402)
4 ( 417)
5 ( 430)
5 ( 439)
1 ( 439)
2 ( 453)
3 ( 459)
4 ( 477)
5 ( 483)

1
11( 118)
1920 021 / a ( [ ] , a )
, 180
, ? (
1054
)
1054
1920- 180/ 021= 1060
12:
27410 9 l. y.
( l. y. , 1l. y. = 94610 1 5 m ) , 39310 7 m/ s
,
? ( ,
)
94610 1 5 27410 9
39310 7 = 65910 17 s= 20910 1 0 a
20910 1 0
1

13 1
200km, ?
1 180m / s 2 , 1
v = 2gh = 218020010 3 = 849 m/ s
14, 180s
1600km/ h ,
25 g ? 180s ?
a =
v t
=
25 g
160010 3
3600180 = 247102 m/ s 2 25 g
180s
s = v 2 t = 1600103
23600 180 =
400 m
15, 90k m/ h
, 80m
65k m/ h ,
070s( ) ,
75m/ s 2 , ?
, ?
v1 0 = 90km/ h= 25m/ s, v 20 = 65k m/ h = 18m / s, s0 =
80m , t = 070s, a = 75m/ s 2
,
s0 = s0 - ( v10 + v2 0) t = 80 - ( 25 + 18)070
= 50 m
2
s1 = v2 1 0
2a = 25 2
275 =
417 m

s2 = v2 2 0
2a = 18 2
275 =
217 m
s1 + s2 > s0 ,
t ,
s0 = v 10t -
,
1
2 at 2 + v2 0t -
50 = ( 25 + 18) t - 75t 2
t =
1
2 a t 2 = ( v1 0 + v2 0) t - a t 2
162 s, 411 s( )
v 1 = v 10 - a t = 25 - 75162 = 129m / s = 46 km/ h
16 1200m ,
, 200km/ h
50m ,
18km / h
g / 2 , ?
h 0 = 1200m, v1 = 200k m/ h = 556m/ s, v2 =
18k m/ h = 5m / s, h2 = 50m
t 1 =
v 1
h 1 =
v1
g / 2 =
v 2 1
2g / 2 = v 2 1
g =
t 1 = h 0 - h 1 - h 2
v1
=
2556
98
= 113 s
5562
98 = 315 m
1200 - 315 - 50
556
= 150 s
3

v 2
t 2 = h 2
v 2
= 50
5 = 10 s
t = t1 + t 1 + t2 = 113 + 150 + 10 = 363 s
17q= 280L/ min,
v0= 26m/ s,
? ?
h = v 0 2
2g = 26 2
298 =
t = 2v 0
g = 226
98
345 m
= 531 s
t
V = qt = 280531/ 60 = 247 L
18v0
= 150m/ s ,
( 1) t1 = 10s , h =
110m , ;
( 2) t2 = 13s , h =
110m ,
( 1) t ,
,
4
h = v 0t -
11 = 15t -
1
2 g t2
1
2 98t 2

t 1 = 184 s, t 1 = 122 s
t u = v 10/ g = 15/ 98 = 153 s
t 1 > t u, ;
t 1 < t u,
v 20 v20t 1 t1
,
h = v 20( t 1 - t 1) -
1
2 g ( t 1 - t 1) 2
v2 0=
,
h + 1 2 g ( t 1 - t 1) 2
= 172 m/ s
t1 - t 1
=
11 + 1 2
98( 184 - 1)
2
184 - 1
h +
v2 0 =
= 511 m/ s
1
2 g ( t 1 - t 1) 2
t1 - t 1
=
11 + 1 2
98( 122 - 1)
2
122 - 1
( 2) t 2 = 13s > t 1 ,
t 1 ,
h + 1 2 g ( t 1 - t 2) 2
v2 0 =
t1 - t 2
= 230 m/ s
=
11 + 1 2
98( 184 - 13)
2
184 - 13
19 ,
1
, x = ut + u - t ln ( 1 - bt) ,
b
u , , b
5

( 1) ;
( 2) ;
( 3) u = 3010 3 m/ s, b = 7510 - 3 / s, 120s
, t = 0s t = 120s ;
( 4) t = 0s t = 120s
( 1) v =
dx = - uln ( 1 - bt)
dt
( 2) a = dv
dt = ub
1 - bt
( 3) t = 0s , v = 0;
t = 120s , v = - 310 3 ln( 1 - 7510 - 3 120)
= 69110 3 m / s
( 4) t = 0s , a = 3103 7510 - 3
t = 120s , a = 3103 7510 - 3
1 - 7510 - 3 120
1
= 225 m/ s 2
= 225 m/ s
2
110 x y ,
x = 2t, y = 4t 2 - 8( SI)
;
( 1)
( 2) t1 = 1s t 2 = 2s ,
( 1) t,
y = x 2 - 8
11
( 2) r = 2ti + ( 4t 2 - 8) j
v =
dr
dt =
2i + 8tj
11 110
6

a =
du
dt = 8j
t = 1s ,
r 1 = 2i - 4j , v 1 = 2i + 8j , a1 = 8j ;
t = 2s
r 2 = 4i + 8j , v 2 = 2i + 16j , a 2 = 8j
111 12 , h ,
, v0, s
12 111
l , v 0 = -
dl/ dt12
s = l 2 - h 2
v =
ds
dt =
a =
dv
dt = -
l dl
l 2 - h 2 dt = - s 2 + h 2
s
d
dl
l
l 2 - h 2 v 0
v 0
dl
dt = - h 2 v0
a ,
s 3
2
7

112 v 0 =
110km/ h,
=
45 ( 13)
( 1)
L ?(
)
( 2) ,
L = 165m,
?
13 112
( 1) 13 ,
x =
Lcos45 = v 0t, y = Lsin45 = 1 2 g t 2
,
t =
2v0
g
2
v 0t
L =
cos45
= 2 v0
g cos45
= 22 11010 3 2
= 269 m
98 2 3600
( 2) L ,
113
v0 = 25m/ s,
L =
50m , h = 13m
?
?
14 ,
8
14 113

y = x tan -
cos 2 = ( 1 + tan 2 ) - 1
gx 2
2 tan 2 - x tan+
2 v0
1
2
g x 2
v 0 2 cos 2
g x 2
2 v 0 2 + y = 0
,
2
tan = v0 g L 1 1 -
,
1 -
2g
v 0
2 h + g L 2
2g
2 v0 2
v0 2 h + g L 2
2 = - 0022 < 0
2 v 0
, ,
1 -
, ,
L =
h v 0 2
2g - g L 2
2 v 0 2 =
123m
2g
h + g L 2
0
v0 2 2 v0 2
25 2
298 - 9850 2
225 2 = 123 m
50m ,
= arct an v 0 2
g L = arct an 25 2
9850 =
519
114 , 1997 6 1
( 121) 265m,
, 150km / h, 5,
57m,
?
( 1) ,
9

( 2) ?
( 3) 100m,
?
( 4) ?
15 114
15 , s = 265m, v0 = 150km/ h, = 5 ,
L = 57m, h 1 = 10m
( 1)
a = v 2
0
2s = 1
2265
t1 = 2s
v 0
( 2)
t 2 =
L
v 0cos =
15010 3
3600
( 3)
2
= 328 m / s 2
= 22653600
15010 3 = 127 s
573600
15010 3 cos5
= 137 s
2
H = h 1 + v0 sin 2
= 10 + ( 15010 3 ) 2 sin 2 5
2g
3600 2 298 = 1067 m
( 4)
h 2 - h 1= v 0sin t 2 -
10
1
2 gt 2 2
= 15010 3
sin5137 -
3600
1
2 981372

= - 422 m
422m
115
610 4 r/ min , 10cm
?
2 r
g = 4
2 2 r
g
= 4
2 ( 610 4 ) 2 01
= 410 5
60 2 98
116 399
v =
a =
2 RE
T
cos = 2
2
2
R Ecos=
T
= 25910 - 2 m/ s 2
6378103
86400
cos399 = 356m / s
2
2
637810 3 cos399
86400
117 ,
?
a =
v= 2
T R S E = 2
31610 7 1510 11
= 29810 4 m/ s = 29. 8 k m/ s
2
2
R S E =
T
= 59310 - 3 m/ s 2
2
2
1510 11
31610 7
118( )
2910 4
l. y. ?
11

a =
v= 2
T R G S =
229104
94610 15
2510 8 31510 7
= 2210 5 m / s = 220 km / s
2
2
T
RG S =
= 1810 - 1 0 m/ s 2
( 2 ) 2 2910 4 94610 15
( 2510 8 31510 7 ) 2
119, ,
2210 6 m/ s, 053
10 - 1 0 m
=
a =
v
2 r = 2210 6
= 6610 1 5 H z
205310 - 10
v2
r
= ( 22106 ) 2
05310 - 10 = 9110 22 m/ s 2
120240m,
?
v 2
Rg =
121 R =
( 3108 ) 2 2
= 2410 14
24098
400m
, v = 10m/ s, a t =
02m/ s 2 ?
16 ,
12
a n = v2
R = 10 2
400
= 025 m/ s
2

16 121
a = a 2 n + a 2 t = 025 2 + 02 2 = 032 m/ s 2
180 - = 180 - arctan a n
a t
= 180 - arctan 025
020 =
122 R = 10m ,
, A ( 17)
, A , t =
20s h = 04m
t = 3s ,
A
a =
2h = 204 = 02 m / s 2
t 2 2 2
t = 3s
,
a t = 02 m/ s 2
t = 3s
128 40
17
122
13

an = v
2
R =
( a t ) 2
R = ( 023) 2
10
= 036 m/ s 2
123( CD) R 1 = 22cm,
R 2 = 56cm (
18) , N = 650
/ mm CD ,
,
,
v = 13m/ s
( 1) ?
18123
( 2) r = 50cm ,
?
( 1) r ,
dr 2 r N dr
dt = 2 r N dr/ v
T = dt =
R 2
R 1
( 2)
=
d
dt = -
= -
2 rN dr
v
= N v ( R2 - R 2 1)
= 65010 3 ( 0056 2 - 0022 2 )
13
= 41610 3 s = 694 min
v
r 2
=
dr
dt = -
v r
v
r 2
= 13
005 =
26 rad/ s
v
2 rN = - v 2
2 N r 3
13 2
= - 33110 - 3 rad/ s 2
265010 3 005 3
124 12m/ s 2
14
,

05s 15m
a = 12m/ s 2 , t 0 = 05s, h 0 = 15m
19 , , ,
v 0 = at 0 = 1205 = 06 m/ s
t ,
,
h = v0t + 1 2 g t 2
h = v0t + 1 2 a t 2
19
124
h 0 = h - h = 1 2 ( g - a ) t2
t =
2h 0
g - a = 215
98 - 12 = 059 s
h= v 0t + 1 2 g t2
= 06059 + 1 2 980592
= 206 m
125 30m / s , ,
?
344m/ s
v1 = 30 m/ s, v 2 = 344 m/ s
15

v E = v1 + v2 = 30 + 344 = 374 m / s
vW = v2 - v1 = 344 - 30 = 314 m/ s
2
2
vN = v 2 - v 1
= 344 2 - 30 2 = 343 m / s
126 18km/ h ,
36k m/ h ,
120 ,
v m 1 = 18km/ h, v m 2 = 36km/ h, = 90 , = 120v rm 1
v rm 2 , v r
110
110 126
vrm 1 + v m1 = vr = vrm 2 + v m 2
16
vr = vm 2 = 36 km / h
= 90 - 60 = 30

30
127 A v A = 1000km/ h
( ) ,
B v B = 800km/ h
( ) 30
A B B
A
111 = 60
A B
v A B = v 2 A + v 2 B - 2v AvBcos
111 127
= 1000 2 + 800 2 - 21000800cos60
= 917 m/ s
=
arccos
40 56
vB cos30
= arccos 8000866
vA B
917
B A v BA = - v A B ,
vB A =
40 56
917 m / s
= 40 56
128 150k m/ h ,
, 750km/ h
950km/ h
vW = 150km/ h , v 1W = 750km/ h , v 21 = 950km / h, = 45
v 1 v2 ,
112
17

112 128
v 1 = v 2 W + v 2 1W = 150 2 + 750 2 = 765 km/ h
1 = arctan vW
v1W
= arct an 150
750 = 113
2 = 45 - 1 = 45 - 113 = 337
v2= v 2 1 + v 2 2 1 - 2v 1v2 1cos 2
= arct an
= 765 2 + 950 2 - 2765950cos337
= 527 km / h
v 1W - v2 1sin
= arctan
v 21cos - v W
750 - 950sin 45
950cos45 - 150 = 8 32
527km/ h 8 32
18

2
21F M ( 21)
F ,
k s
( 1) , F ? , F
?
( 2) , F
?
( 1) ,
, ,
x : F mi ncos - f m a x= 0
y : N - F m insin - Mg = 0
f m a x= sN
2121
F
F mi n=
sMg
cos - ssin
, F
F =
kMg
cos - ksin
( 2) F m in , cos - ssin0, F m in
, ,
cos - ssin 0
19

= arct an 1 s
22m= 050kg = 30
( 22)
( 1) a = 20m/ s 2 ,
?
( 2) , ?
2222
( 1) ,
x : T cos - N sin = ma
y : T sin + N cos - mg = 0
,
T = m( a cos + g sin ) = 05( 2cos30 + 98sin 30 ) = 332 N
N = m( g cos - a sin ) = 05( 98cos30 - 2sin 30 ) = 375 N
,
N = N = 375 N
( 2) N = 0,
T cos = ma , T sin = mg
a = g / t an = 98/ t an30 = 170 m/ s 2
2323 A , B , 3
: m 1= 200g, m 2= 100g, m 3= 50g
( 1)
20

( 2) T 1 T 2
( 1) ,
a1 a2 a3,
23 a m 2
B ,
a2= a - a1, a3= a + a1
m1, m 2 m 3
:
m 1g - T 1= m 1a 1
m 2g - T 2= m 2a 2= m 2( a - a1)
m 3g - T 2= - m 3a 3= - m 3( a + a1)
B ,
T 1- 2T 2= 0
4 ,
23
23
a 1 =
m1m 2+ m1m 3- 4m 2m 3
g
m1m 2+ m1m 3+ 4m 2m 3
0201+ 02005- 401005
=
0201+ 02005+ 401005 98
= 196 m/ s 2
2m1m 2- 2m 1m 3
a =
m 1m 2+ m1m 3+ 4m 2m g
3
20201- 202005
=
0201+ 02005+ 401005 98
= 392 m / s 2
a 1, a 2, a 3
a 2= a - a 1= 392- 196= 196 m/ s 2
a 3= a + a 1= 392+ 196= 588 m/ s 2
, 23
21

( 2) T 1= m1( g - a 1) = 02( 98- 196) = 157 N
T 2= m2( g - a 2) = 01( 98- 196) = 0784 N
24 5000kg 1500kg
060m/ s 2
( 1) ?
( 2) ?
( 1) 24
,
F - ( M+ m) g = ( M+ m) a
F = ( M+ m) ( g + a )
= ( 5000+ 1500)( 98+ 06)
24
= 67610 4 N
24 ( 2) ,
T - mg = ma
T = m( g + a ) = 1500( 98+ 06)
= 15610 4 N
2525 ,
( 1) ,
?
( 2) 10% ,
? 75kg
,
2T - Mg = Ma
T = M( g + a ) / 2
( 1) , a = 0,
T = Mg / 2= 7598/ 2= 368 N
22
25
25

( 2) T = 11T = 11Mg / 2,
2T
a = - Mg ( 11- 1) Mg
= = 01g = 098 m/ s 2
M
M
26 m
( 26) ,
( 1) ;
( 2)
( )
2626
26 , ,
x : T 1cos = T 2
y : T 1sin = mg / 2
,
( 1)
( 2)
T 1= mg / ( 2sin )
T 2= mg / ( 2tan )
27 A B m A = 150kg, mB = 285kg,
, = 30( 27) A B
kA = 015, kB = 021
( 1) A B
( 2)
( 3) A B , ( 1) ( 2) ?
23

A ,
2727
, x ,
B,
m Ag sin - kA mgcos = mA aA
mB gsin - k Bmgcos = mBa B
a A = g ( sin - k A cos ) = 98( sin30 - 015cos30 ) = 363 m/ s 2
a B = g ( sin - k Bcos ) = 98( sin 30 - 021cos30 ) = 312 m/ s 2
( 1) 27 , A , B a A > a B
, , aA = aB = a T T A
B ( T = T ) , , x
A :
B:
m A gsin - k A mA gcos - T = mA a
m Bg sin - kB mBg cos + T = m Ba
k Am A + k BmB
a = g sin -
g cos
m A+ m B
01515+ 021285
= 98sin30 -
98cos30
15+ 285
= 329 m/ s 2
24

( 2) 27
T =
mA gsin - kA m Ag cos - mA a
= 1598sin30 - 0151598cos30
- 15329 = 051 N
( 3) , A , B , a A > a B ,
, T = 0, A B
a A = a A = 363 m / s 2 , aB = aB = 312 m/ s 2
28 M = 150k g , m =
245k g,
= 025, ?
2828
28 , f A B = f BA = mg , f B =
( M+ m) g m, , x
a m= g
mg = ma m
M, , x
F - ( M+ m) g - mg= MaM
F = MaM + ( M+ 2m) g
, aM a m
F
F Mam+ ( M+ 2m) g = 2 ( M+ m) g
F m in= 2 ( M+ m) g = 2025( 150+ 245)98
25

= 194 N
29, v= 90km/ h
s= 35m ,
110 ,
?
v= 90km/ h = 25m/ s,
, f = mg,
s= v 2 / 2a = v 2 / 2 g
2929
, 29 , ,
x : mgcos - mg sin = ma
a = g cos - g sin
= v2
2s cos - g sin
s = v 2
2a
= v 2
25 2
=
= 393 m
v 2
s cos 25 2
- 2g sin 1- 2981
35 10
210k 1 k2
210210
( 1) ( 210( a ) ) , k
k= k 1k2
k1+ k 2
( 2) ( 210( b) ) , k
26

k= k1+ k 2
( 1) , F , x
= x 1+ x 2F = k 1 x 1= k2 x 2
k= F x =
F
= 1
x 1+ x 2
x 1
F + x 2
F
= 1
1
k1
+ 1 k2
= k 1k 2
k1+ k2
( 2) , x , F =
F 1+ F 2, F 1= k1 x , F 2= k 2 x
k= F x = F 1
x + F 2
x = k 1+ k 2
211211 , m1 L 1
,
m 2, L 2
, m1, m2
211 , m 2,
T 21= m 2
2 ( L 1+ L 2)
m 1,
T 1- T 12= m 1
2 L 1
T 12 =
T 21, ,
T 1= m 1
2 L 1+ m2
2 ( L 1+ L 2)
m 1
,
T 2 1= T 12= m2
2 ( L 1+ L 2) 211211
212212 ,
, , ,
s, r
, ( r, , s ) ?
f m
, m ,
, 212 ,
27

x : N sin - f mcos = m
2 m inr
y : N cos + f msin - mg = 0
f m= sN
,
m in=
( sin - scos ) g
( cos + s sin ) r
m in= 1
( sin - scos ) g
2( cos + ssin ) r
, ,
, f m
212
ma x= 1 ( sin + scos ) g
2( cos - ssin ) r
, ,
m ax min
212
212
213213 , m= 1200kg ,
, v= 25m/ sR= 400m, ,
= 6
213213
( 1)
( 2) s= 09,
, ?
28

( 1) 213 , ,
x : N sin + f cos = m v 2
R
y : N cos - f sin - mg = 0
f = m v2
R cos - mg sin
= 1200 252
400 cos6 - 120098sin 6 = 635 N
f n = m v 2
2
R = 120025 400 = 188103 N
( 2) v m , f m a x= sN vm
f m ax v f ,
v m=
Rg ( sin + scos )
cos - ssin
=
40098( sin 6 + 09cos6 )
cos6 - 09sin 6
= 660 m/ s
214 5988kg,
1829m,
30577k m/ h g ?
,
, 1 , ,
?
a n= v 2
R = ( 30577103 ) 2
3600 2 1829 = 3945 m/ s 2 = 402 g
29

N 1 , , ,
,
N - N 1- mg = 0
N = N 1+ mg
N = F = ma n
N
N 1=
man
- mg = m an - g
= 5988 3945
1
- 98 = 17710 4 N
215 73 000km
136 000km 10 -
6 m 10m
142h,
?
m , M
, ,
G Mm
R 2
= m 4
2
R
T 2
M= 4
2 R 3
4 ( 2 13610 8 ) 3
=
GT 2 66710 - 11 ( 1423600) 2
= 5710 2 6 kg
21616 , 4
1610 ( 244) 4
, 10710 6 k m,
716d
M m ,
,
30

M= 4
2 R 3
=
GT 2
G Mm
R 2
= m 4
2
R
T 2
4 ( 2 10710 9 ) 3
= 18910 27 kg
66710 - 11 ( 71686400) 2
217 :
p =
2 3 G
2 R 2
, R
, 13
10 3 k g/ m 3 , 7010 7 m ,
( atm ) ( 1atm= 101310 5 P a)
214 , r dr, dS
dSdr, r
214217
dF = G 4 3 r 3 dm/ r 2 = 4 3 G
2 rdrdS
31

, dm dS
dp = dF
dS =
4 3 G
2 rdr
p = dp =
R
, ,
0
4
3 G
2 r dr = 2 3 G
2 R 2
p =
2 3
11 66710- ( 1310 3 ) 2
( 7010 7 ) 2 / 101310 5
= 1110 7 atm
218
( 1) T min = 3 / ( G ) ,
( 2) 3010 3 kg/ m 3 ,
( 3) 16ms, 10km ,
( ) ?
( 1) M m
, ,
m R
G Mm = m 2
2
R 2 T
R
m in
T m in=
M= 4 3 R3
3 / ( G )
( 2) = 3010 3 kg/ m 3 T m in,
32

T m in=
3
G
= 3
= 6910 3
66710 - 11 3010 3 s
( 3) T min
M= 4 3 R3 =
=
4
2 R 3
GT 2
= 3
GT 2
4
2 ( 10 4 ) 3
= 23110 2 9 k g
66710 - 11 ( 1610 - 3 ) 2
= 012( )
219
,
? ( )
?
18910 27 kg, 71410 4 km,
10h
r ( m)
,
r =
=
G Mm = m 2
2
r
r 2 T
219
GMT 2 1 / 3
4
2
66710 - 11 18910 27 ( 103600) 2
= 16110 8 m
4
2 1 / 3
215
33

h= r - R = 16110 8 - 71410 7
= 89610 7 m
215
S= 2 R 2 ( 1 -
= 2 R 2 1 -
= 2 R 2 h/ r
cos )
R
r
= 2( 71410 4 ) 2 89610 7 / ( 16110 8 )
= 17710 10 km 2
220: M
, R,
216 ,
,
GM 2
3
F = 2
( 3 R) 2 2
GM 2
=
3 R 2
GM 2
3 R 2= M 2
2
R
T
216220
T =
4
48 R 3 / 2
( GM) 1/ 2
221 m
M ,
, r
34

, ,
,
,
G Mm
r 2 +
m 2
= m 4
2
r
( 2r ) 2 T 2
T = 4 r 3/ 2 G( 4M+ m)
222 ,
a , - 02v
t 1 / 2 =dt = - v 0 / 2
a = dv
dt = - 02 v
dt= - dv/ 02 v
v 0
dv
02v =
223
, R
( 217) ,
k
A
v 0, t
A
217 ,
N = m v2
R
ln2
02 =
347 s
217223
35

v v 0
f = kN
- f = m dv
dt
dv
dt = - v 2
k
R
dv
v 2
v=
= -
t
t
t
s= vdt =
0
= R ln 1 +
k
0
k
R dt
v 0R
R+ v 0 kt
v 0R
t
0
v 0 kt
R
dt
R + v0 kt
224510 4 r/ min,
200cm, 100cm
( 218)
( 1)
g
( 2) 120g
,
? ?
218224
( 3) 10 5
?
( 1)
510 4
a n1/ g = 4
2 2 r 1/ g = 4
2 60
36
2
002/ 98= 05610 5

a n 2/ g = 4
2 2 r 2/ g = 4
2 5104
60
2
010/ 98= 28010 5
( 2) 218 , r
dm= Sdr
, S (
)
F + dF - F = dF
dF = dm a= S
2 r dr
F r b
2
F b = dF =
0
S
2 rdr = S
2
r 2 2 - r 2 1
2
r 1
= S ( r 2 - r 1)
2 ( r 2 + r 1)
2
= m
2
2 ( r 2 + r 1)
= 1210- 3
2
= 19710 4 N
510 4
60
2
4
2 ( 002 + 010)
, 19710 4 N ,
201t
( 3)
F i= m
2 r = 16710 - 2 7 10 5 4
2 5104
60
= 4610 - 16 N
2
010
225597m
, 400r/ min ,
?
37

219 , r
dm= Sdr
, S
(
)
F - ( F + dF ) = - dF
- dF = dma= S
2 r dr
L
F 0 = F 0
0
dF = - L S
2 r dr = -
0
219225
1
2 S
2 L 2 = -
1
2 m
2 L
r , ,
©¦F 0©¦
mg =
2 L
2g = 4
2 400597
= 534
29860 2
226a = 70m/ s 2
, ,
l= 2m
? k= 050
220 , ,
f = kmg F 0= ma , , ,
220
226
F 0- f = ma
a = F 0- f
m
= a - kg
l
38

v = 2a l =
2( a - kg ) l =
= 29 m/ s
2( 7- 0598)2
227
( ) , 18km
g ,
? ?
,
=
g
r =
m
2 r = mg
98
1810 3 = 7410 -
2
rad/ s
228 R
( 221)
(
) ,
?
221 ,
m, :
mg, N F 0
: N cos - mg = 0
:
N sin - F 0= 0
F 0= m
2 Rsin
msin (
2 Rcos - g ) = 0
221
228
= 0,
; = , ; = arccos
g
2 R 39

,
F t = F 0cos - mg sin = m
2 sin cos - mgsin
, F t= 0,
=
dF t
d
= m
2 R 2cos 2 - 1-
dF t
d =
dF t d,
= 0
:
m
2 R 1 +
dF t
d = m
2 R 1-
g
2 R cos
g
2 R > 0
g
2 R
<
g
R , dF t
d, ;
g R , dF t
= arccos
:
d, dF t
= 0,
d
g
2 R
dF t
d = m
2 R g 2
- 1
4 R 2
>
g
R , dF t
d, ;
g R , = 0, dF t
= 0,
d
229
20km40k m ,
40

,
50m,
0062kg/ m 3 , 40m / s,
0016,
10mN / W, ? (
)
,
F = f =
1 2 C Av 2
= 1 2 00160062252
40 2 =
1560 N
15610 3 / 1010 - 3 = 15610 5 W= 156 KW
41

3
F = -
t =
31 ( 31) ,
kx , x = Acos t, , k, A, t = 0
/ ( 2 )
/ ( 2 )
I = F dt = -
0
3131
k / ( 2 )
Acos tdt = -
0
x
kA
32 m = 50g, v = 20m/ s
, 1/ 4 ?
3232
32 , v1 = v2 = v,
I = mv2 - mv1 = 2 mv
42

= 2 00520 = 141 N ¡¤s
33 m = 014kg v 1 = 50m/ s
, , = 45,
v2 = 80m/ s
t =
?
002s,
3333
33 , x
I = mv2 - mv 1
= m v 2 1 + v 2 2 + 2v1v 2cos
= 014 50 2 + 80 2 + 25080cos45
= 169 N ¡¤s
= 180 - arct an
= 180 - arct an
F =
I t =
mv2sin
mv1 + mv 2cos
80sin45
50 + 80cos45
= 152
169
002 =
845 N
43

F / ( mg ) = 845/ ( 01498)
= 616
34120 ,
m = 790g , 735m/ s( )
,
F =
N mv
t
= 12000079735
60
= 116 N
, ,
, 116N
35 90 3
10 3 k g/ s, 10m / s
34 , t
AA v BB
p = p B B-
p A A
= p BA + p A B - p A B + p B A
= p A B - p A B = mv A - mvA
m AB A B,
v A = v A = v
F =
,
p = m ( v A - vA )
t t
3435
F = m t
2 v
= Q 2 v = 310 3 210 = 42410 4 N
90
44

, 424
10 4 N , 90
36
,
v0 : l
, F = llg + lv 2 0,
l
35 , dt
dl = v 0dt ,
dt
3536
v 0 l ( l + dl) - v0 ll = v0 ldl
, ,
( F - llg ) dt =
F =
l lg +
2
v0 l dl = v 0 ldt
v 2 0 l
37 , ,
,
3
3637
36 , t
, l
l,
m = l l, l
v =
2g l dt , dm
= l dl = l vdt ,
v 0m + dm ,
45

[ ( m + dm) g - N ] dt = dm( 0 - v)
dmdt ,
= - lv 2 dt = -
= - 2mg dt
N = 3mg
N = N = 3mg
3
l 2g ldt
3834
mv 1( 1 - cos
f = )
t
t m
37 , m
t
p = p 2 - p 1 = m( v 2 - v1)
V
p V = - m( v 2 - v 1cos )
= - mv1( 1 - cos )
m
f =
p V
=
dt
- mv1( 1 - cos t
, f
f = - f = mv 1( 1 - cos )
t
3738
39,
p 1= 92210 -
46
21 kgm/ s ,

p 2= 53310 - 2 1 k gm / s
p 3
,
p 1 + p 2 + p 3 = 0
3839
38 , p 3
p 3 = p 2 1 + p 2 2
= 10 - 21 922 2 + 533 2
= 10710 - 20 k g ¡¤m / s 2
p 3 p 1 p 2 , p 1
= 90 + arctan p 1
p 2
= 90 + arct an 922
533 =
149 58
310M ,
,
m, u
,
, , ,
2m( u - v1) - Mv1 = 0
,
47

v 1 =
2mu
M + 2m
, , - v 2
,
m( u - v2 ) - ( M + m) v 2 = 0
, - v 2 ,
v 2 =
m( u - v 2) - Mv 2 = - ( M + m) v2
mu + ( M + m) v 2
= mu
M + m
v 1 v 2 , v1< v 2
1
M + m + 1
M +
311v 0= 7600m/ s
m 1 = 2900kg m 2 =
1500kg ,
u= 9100m/ s
,
, v0 , v 1 v2
( m 1 + m 2) v 0 = m 1v 1 + m 2v2
, u= v 2- v1, v 2= u+ v 1
v 1, v 2
v1 = v0 -
( m1 + m2) v 0 = m 1v1 + m2( u + v 1)
m2u
m 1 + m 2
= 7600 -
150910
150 + 290
v2 = u + v 1 = 910 + 7290 = 8200 m/ s
2m
= 7290 m/ s
, , v 0
48

312v 0 ,
,
45 , 45
,
?
39 ,
v0cos , ,
v 1 v2
,
39312
:
mv 0cos= m 2 v 1cos45 + m 2 v 2cos45
:
,
0 = m 2
v 1sin 45 - m
v2sin 45
2
v1 = v 2 = 2 v 0cos
313,
310313
v =
2 kg s =
s= 25m
k = 080
( 14m/ s) ,
?
- kmg / m
= - kg ,
2089825 = 198 m/ s
, v l
( 310)
,
49

v = 1 2
( mv1) 2 + ( mv2) 2 = ( 2mv ) 2
v 2 1 + v 2 2 =
2
2 vl = 2
14 = 99 m/ s
2
v> v , ,
314 m 1= 20g, m 2= 50g,
, v 1= 10icm/ s, v 2= ( 30i+ 50j ) cm/ s
,
v
m1 v1 + m2 v2 = ( m 1 + m 2) v
2010i + 50( 30i + 50j ) = ( 20 + 50) ( v xi + vyj )
vx =
2010 + 5030
20 + 50
vy = 5050
20 + 50
= 357 cm/ s
= 50 cm / s
v = v 2 x + v 2 y = 5 2 + 357 2 = 614 cm/ s
x
= arct an vy
vx
= arct an 357
50 = 35 32
315 6090kg, 105
m / s253m/ s
800kg , ?
( ( 311) )
v f =
50
vi + uln M i
M f
= 105 + 253ln 6090
6010
= 108 m/ s

316, ,
, ?
v f
u
, vf- u vf= v i+
uln( M i/ M f)
v f = 0 + uln Mi
M f
M i/ M f= e
Mi - M f
M i
= 1 -
M f
M i
= 1 -
= u
1
e =
0632
31727210 6 kg ,
12910 3 kg/ s
( 1)
?
( 2) 55010 4 m/ s ,
155s?
( 3) ( 2) , , ?
Mi = 27210 6 k g, =
10 4 m / s, T = 155s, g = 98m/ s 2
dM
= 12910 3 kg/ s, u = 550
dt
( 1) ,
u 1
dM
dt
= Mig
u 1
= M ig
dM
dt
= 272106 98
12910 3
= 20710 4 m/ s
( 2) , ,
udM + Mdv = - Mgdt
51

dv
dt = -
t 0 T
v =
v
0 dv = - M f
= uln M i
( 3)
Mf
M i
u
M
dM
dt
u
M dM - T 0 g dt
- g T = uln
= 5510 4 ln
Mi
Mi -
- g
T -
gT
27210 6
27210 6 - 12910 3 155
- 98155 = 26810 3 m/ s
dv = -
u
M dM -
gdt
0 t v
v = uln
M i
M i -
t -
0 T , ,
g t
h = T 0 vdt = T 0
uln
M i
M i -
t -
gt dt
= uT lnM i + u ( M i - T ) ln( Mi - T ) - 1
- Mi lnM i - 1 -
h =
1
2 g T 2
172 k m
318 210m/ s ,
75kg , 30k g
490m/ s
52

v= 210m / s, dm 1 dM
= 75kg/ s, = - 30k g/ s, u= 490m/ s
dt dt
, dt
dm 10 + Mv = ( dm 1 - dM) ( v - u ) + ( M + dM) ( v + dv)
F = M dv
dt = -
Mdv = - udM + ( u - v) dm1
u dM
dt
+ ( u - v) dm1
dt
= - 490( - 30) + ( 490 - 210)75
= 22510 4 N
319311
00958nm,
105
, B ,
AB= 00958cos 105 = 00583 nm
2
C
311319
ACm O = BC2mH = ( AB - A C)2mH
AC =
AB2m H
m O + 2mH
= 0058321
16 + 21
320
= 000648 nm
312 , R, m, A =
2m/ ( R 2 ) x ,
53

x C
x dm
=
m = m 1
R x
0 A 2 R 2 - x 2 dx
= 4R
3
312320
313321
321, a
a / 4 ( 336)
313
,
O
, A
AO , B
,
m1BO = m 2AO
m 2= a 2
a , m 1= a 3 - m2= a 3 1-
4
16
( ) m1 m 2
54

BO =
m2
m1
A O =
/ 16
1 - / 16 AO = a
4( 16 -
0061a
) = 0061a
322 ,
8m , ,
? 3500kg,
140kg
,
, , x ,
x3500 = ( 8 - x ) 140
x =
8140
3500 + 140 = 031 m
323 h= 25m, m= 75kg
, ?
l E l p
,
mEl E = mpl p
, l p = h ,
l E = m p
mE
l p =
m E l E = mp l p
7525
59810 24 = 31410 - 2 2 m
31410 -
22 m
324 m= 20g , 1s
m= 50g t( > 1) s ,
55

v C =
a C =
m 1v 1 + m 2v 2
m 1 + m 2
m 1a 1 + m 2a2
m 1 + m 2
=
=
20g t + 50g ( t - 1)
20 + 50
20g + 50g
20 + 50 = g
= t -
5
7 g
325 m= 2200kg v= 60k m/ h
d = 50m
? ?
P 1
314325
314 , d = 50m
L 1
P 2
L 2
= mvr 1sin 1 = mvd
= 2200 6010 3
3600 50
= 18310 6 kg ¡¤m 2 / s
= mvr 2sin 2 = mvr 2sin0 = 0
326
r 1 = 87510 10 m, v 1 = 54610 4
m / s v2= 90810 2 m / s,
r 2 ?
,
, ,
56

,
r 2 =
r 1v1
v 2
mr 1v1 = mr 2v 2
= 875101 0 54610 4
90810 2 = 52610 12 m
327
, 273d
L M = mM R M v M = 2 mM R 2 M / T M
dS
dt =
L M
2mM
= 273510
22 ( 38210 8 ) 2
27386400
= 28610 3 4 kg ¡¤m 2 / s
=
28610 34
273510 22 = 19410 1 1 m 2 / s
3281988 12
,
2055km, 358357k m
102km / s
( 1) ;
( 2) ( :
)
315 , h P = 2055km, h A = 358357k m, vP =
102k m/ s, R E= 6378km
( 1)
vA =
vP
r P
r A
= vP
= 159 km/ s
R E
+ h P
RE + h A
6378 + 358357
= 102
6378 + 2055
57

315328
( 2) S =
2 ( r A + r P ) r A r P
dS
dt = 1 2 vP r P
T =
S
dS / dt =
( r A + r P ) r A r P
vP
r P
= ( r A + r P )
v P
r A
r P
= ( 422137 + 65835)
102
422137
65835 = 38057s =
106 h
329 (
316) , r 0, v0,
r , v
316329
58

,
, m ,
mr 0v0 = mrv
v = v 0
330 m , a
, ,
m
,
?
r 0
r
( 1) ?
( 2) ,
? , ?
( 3) , ?
( 1) 317 , C ,
,
317330
2ml 1 = m( a - l 1)
l 1 = a 3
C a / 3
, ,
59

, , C
vC =
mv + m( - v) + m ¡¤0
m + m + m
= 0
( 2) , ( )
v= a/ 2C 0,
a / 2, C
L 1=
m a 3 a / 2 + m 2a
3 a / 2 + m a 3 0
= 1 2 ma 2
, ,
, C , , ,
L 2=
1 2 ma 2
( 3)
C ,
L 2 = 2m a 3
a
3 + m 2a
3
= 3 4
2a
3 = 1 2 ma 2
60

4
41
( 41) ( ) M=
1 200k g, m= 1 000k g
, a =
15m/ s 2
?
( 1)
( 2) t= 10s,
? ( )
( 3) t = 10s ,
h= 10m ,
?
( 1) 41 , ,
M m
T 1- Mg = Ma
mg - T 2= ma
41
T 1= M( g + a ) = 1200( 98+ 15) = 13610 4 N
T 2= m( g - a ) = 1000( 98- 15) = 08310 4 N
41
( 2 ) t = 10s , h =
1
2 a t 2 , ,
61

A= ( T 1- T 2) h = ( 136- 083) 1 2 151 2 = 39510 3 J
( 3) , T 1 =
Mg , T 2 = mg h
A = ( T 1 - T 2 ) h = ( M - m) g h
= ( 1200 - 1000)9810 = 19610 4 J
42
R,
,
m,
k45 ,
?
?
42 , F
, ,
: F - mg sin - f = 0
: N - mg cos = 0
f = kN
F = kmg cos + mg sin
A F = ( kmg cos+ mg sin ) Rd
0
= R[ kmg sin - mg ( cos - 1) ]
= Rmg [ ksin45 - cos45 + 1]
4242
= mg R 1 -
2
2
+
2
2 k
62

A g= - mg sinRd = mg R( cos - 1)
0
= mg R
2
2
- 1
A f = - mg kcosRd= -
0
mg ksin R
= -
2
2
mg kR
43, S = 50m 2 , h 1= 15m
h 2
= 5m
, ?
80% , P =
35k W,
?
43 , dh
Sg hdh ,
4343
A=
h 1
+ h 2
h 2
Sg hdh =
h 1Sg h 1
2 + h 2
= 110 3 155098 15
2 + 5
= 4210 6 J
T , A = P T
T =
A
P
= 42310 6
351008 = 151 s
3
63

44 ,
q= 50kg/ s, v= 15m/ s
, ?
?
4444
44 , dt dm
, F 0 v
,
F dt= dm( v- 0)
F = dm
dt v = qv
, F ,
, F
P = F v= qv 2 = 5015 2 = 113 W
dE k
dt = 1 2 dm v 2 / dt= 1 2 qv 2 = 1 2 P
dm 0
v , , s = vdt/ 2
64

, F
A = -
F s
dt = - qv 2
2
45 45 , A B
mA = m B, B
k = 020,
A h=
10m
45 ,
- f h+ m Bgh = 1 2 ( m A + m B) v 2 4545
f = kmA g h,
v= ( 1- k) gh = ( 1- 02)981= 28 m/ s
4646 , M
m v s
( 1) ?
?
( 2)
s
46 , ,
, V ,
mv= ( m+ M) V
V=
m
m+ M v 65

4646
( 1) s1 ,
s= s1+ s, f
,
- f ( s1+ s ) = 1 2 mV 2 -
1
= 1 2 mv2 2 mv m
2
m+ M
2
- 1
, f
f s1= 1 2 MV 2 = 1 2 M m
m+ M
( 2) f = f , ,
- f s =
1
2 mV 2 + 1 2 MV2 -
2
v 2
1
2 mv2
s
47S S u
: S ,
( 46) , S ,
, A A B =
1 2 m ( v B) 2 -
)
1
2 ( mv A ) 2 ( :
dr F S dt
, dr S , dr 0= udt dt
S S S , A B
F
66

A A B = B F ¡¤dr B = F ¡¤( dr - dr 0) = B F ¡¤dr B - F ¡¤udt
A A A A
= A A B -
B
A
F dt ¡¤u
( )
B F dt = m( vB - vA )
A
A A B = A A B- m( vB - vA ) u
S , A B
1
2 mv B 2 -
1
2 mv A 2
= 1 2 m( v B - u) 2 -
= 1 2 mv 2 B -
1
2 m( v A - u) 2
1
2 mv 2 A - m( vB - v A ) ¡¤u
, , S , A A B =
1
2 mv 2 B -
1
2 mv 2 A ,
A A
B =
1 2 mv 1 2
B-
2 mv
2 A
S ( A A B A A B,
1
2 mv
2 A 1 2 mv 2 A ,
1
2 mv
2 B 1 2 mv 2 B)
4847 , AB, k,
A , m ,
a F
F
47 ,
67

, f = ka,
F = ka + mg cos
4748
, F
A = F dr =
F ad=
0
ka 2 d
0
+
0
mg acos d
= 1 2 ka 2
2 + mga sin
49 ( k)
,
y ,
: y ,
1 2 ky 2
48
y 0,
mg = ky 0
y , y 0
48
49
,
68

E p 1= 1 2 k ( y + y 0) 2 -
1
2 ky 2 0= 1 2 ky 2 + ky 0y
,
E p2= - mgy= - ky 0y
E p= E p1+ E p 2= 1 2 ky 2 + ky 0y - ky 0y = 1 2 ky 2
410 l 0, k, ,
m , ,
, ?
?
v y ,
- mgy + 1 2 ky 2 + 1 2 mv 2 = 0
, v= 0, y = y m ax ,
y m ax = 2mg / k
f m a x= ky m a x= 2mg
, mg = ky 0
- mgy 0+ 1 2 ky 2 0+ 1 2 mv 2 0= -
v 0= y 0( 2mg - ky 0) / m = y 0g =
m 2 g 2
k + 1 2 k m2 g 2
k 2 + 1 2 mv2 0= 0
m
k g
41149 , A( m= 05kg)
B C s= 3m
k= 400N / m = 45A
, ?
69

49411
m 1, m2
x ,
mg ( s+ x ) sin = 1 2 kx 2
0598( 3+ x ) sin 45 = 1 2 400x 2
,
m 1 m2
m 1
x = 023 m
412410 ,
,
,
? k= 89
N / m , m1= 500g, m 2= 300g
m1 m2 , m 1+ m 2
,
( m 1+ m2) g = ky 0
410
412
( m 1+ m 2) g ( 05+ 03)98
y 0= = = 088 m
k
89
, m 1 y 1
m 1
y 1= m 1g
k = 0598 = 055
89
m
,
v m
1
2 m 1v2 m+ 1 2 ky 2 1- m1gy 1= 1 2 ky 2 0- m1gy 0
vm=
1
m1
[ k( y 2 0 - y 2 1) - 2m 1g ( y 0 - y 1) ]
70

=
1
05 [ 89( 0882 - 055 2 ) - 20598( 088 - 055) ]
= 140 m/ s
413m 1 m2 A BC ( 411)
m 1 ( m 1 ) , , m1
B m 2
x 0
k,
?
411413
( 1) m1 < m2, m1 x
( 2) m1= m 2, x ?
( 3) m1< m 2, m 2 h C ,
x 0
?
( 1) , m 1
v 1=
k
x 0
m 1
m 1< m 2 , m 1
m 2- m1
v 1 = v 1=
m 1+ m2
m 2- m1
m 1+ m2
k
x 0
m 1
71

x =
m 1
k v m 2- m1
1 =
m 1+ x 0
m2
( 2) m 1= m 2 , m 1 , ,
( 3) m2< m 1 ,
2m 1
v 2 =
m1+ m v 1=
2
2m 1
( m 1+ m2)
, m 2 ,
m 2gh=
1 2 m 2v 2
2 = 1 2 m 2
x 0= ( m 1+ m 2)
k
m1 x 0
4m 2 1kx 2 0
( m1+ m 2) 2 m 1
g h
2m 1k
414 412 , k,
M A B,
m v 0
,
412414
A ,
V 0 , , B A
, A ,
mv0= MV 0+ mV 0
72
V 0=
mv 0
m+ M

B , B A
, x m V A B
,
mv 0= ( M+ m+ M) V
V=
A V 0
mv0
2M+ m
,
A B V
1
2 ( m+ M) V 2 0= 1 2 ( 2 M+ m) V 2 + 1 2 kx 2 m
x m= mv0
M
k( m+ M) ( m+ 2M)
415m , M
,
R, / 2( 413)
,
( 1) ,
?
( 2) A B
, A
( 3) B
1 / 2
413415
( 1) 413 , ,
, N N ,
v V
,
mg R= 1 + 1 2 mv2 2 MV 2 73

, ,
v V ,
v=
mv- MV= 0
2Mg R
M+ m , V= m
2g R
M( M+ m)
( 2) , N ,
,
,
A N =
1 2 MV 2 = m2 g R
M+ m
( 3) B , ,
, ,
v = v+ V=
2MgR
M+ m + m 2g R
M( M+ m)
N - mg = m v
2
R
N = mg + m v
2
R = 3+ 2m M mg
N = N = 3+ 2m M mg
416 :
, (
423 424)
74

v 10 , v1 v 2
m 1= m2
,
v1 0= v 1+ v2
v 2 1 0= v 2 1+ 2 v 1v 2+ v 2 2
,
v 2 1 0= v 2 1+ v 2 2
v 1v 2= v1v 2cos = 0
,
v 1 v2 , 90
417 :
, ,
v 10- v2 0= - ( v1- v 2)
v1= m1- m 2
v 10+
m1+ m 2
2m2
m 1+ m2
v 20
v 2= m 2- m 1
v 0+
m 1+ m 2
2m1
v 0
m 1+ m2
v 2 - v 1=
m2 - m1
m1 + m2
-
2m2
m1 + m 2
v 20
+
2m1
m 1 + m 2
-
m 1 - m 2
m 1 + m 2
v 10
= - v 20 + v 10
v 10- v2 0= - ( v1- v 2)
418 414 , ,
75

, m k,
l 0m 1
v 0
l ,
v
v 0
414418
, ,
m 1v 0= ( m 1+ m) v0
v 0 = m 1v0
m 1+ m
O
, O ,
( m+ m 1) l 0v0 = ( m+ m 1) lvsin
( ) ,
,
1
2 ( m+ m 1) v 0
2 = 1 2 k ( l- l 0) 2 + 1 2 ( m+ m 1) v 2
,
k( l- l 0) 2
v= v0
2 -
m+ m 1
= arcsin v0 l 0
=
vl
arcsin
1/ 2
=
419: m, E k
76
m 2 1v 2 0
( m+ m1) 2 -
k( l - l 0) 2
m+ m 1
1 / 2
m 1v 0l 0
[ m 2 1v 2 0- k( m+ m 1) ( l - l 0) 2 ] 1/ 2 l
M

, M
M+ m E k
m M ,
V=
mv
m+ M
v m
E av= 1 2 mv 2 -
1
2 ( m+ M) V 2 = 1 2 mv 2 1-
m
m+ M = M
M+ m E k
420m ,
2 ( 2R) m, R, G,
M :
( 1) ;
( 2) ;
( 3)
( 1)
GmM
= mv2
( 3R) 2 3R
v
( 2)
E k=
1 2 mv 2 = GmM
6R
E p= -
( 3)
GmM
3R
E = E k+ E p= GmM
6R - GmM
3R
= - GmM
6R
421:
E = -
GMm
r 1+ r 2
M, m , r 1, r 2
77

v1 v 2
,
,
mr 1v1= mr 2v2
, ,
1
2 mv 2 1-
,
GMm
= 1 r 1 2 mv 2 2-
1
2 mv2 2=
E = 1 2 mv 2 2-
GMmr 1
( r 1+ r 2) r 2
GMm
= -
r 2
GMm
r 2
GMm
r 1+ r 2
422 173kg,
439km, 2 348km ,
421
E = -
= -
GMm
r 1 + r 2
= -
GMm
h 1 + h 2 + 2R
66710 - 11 59810 2 4 173
439 + 2384 + 63702
= - 44310 9 J
423 ( 415)
500kg
1 400k m ,
36 000km,
( 1) ?
( 2) ?
78

415423
( 1 ) ( ) E s =
GMm
- GMm/ ( 2r 1) , ( ) E h= -
r 1+ r 2
E 1= E h - E s = GMm
r 2 - r 1
2r 1( r 1 + r 2)
= 66710 - 11 59810 24 500
42300 - 7800
27800( 7800 + 42300)10 3
= 88010 9 J
E g= -
GMm
2r 2
E 2= E g - E h = GMm
r 2 - r 1
2r 2( r 1 + r 2)
79

= 66710 - 11 59810 24 500
42300 - 7800
242300( 7800 + 42300)10 3
= 16210 9 J
( 2)
v s=
GM
r 1
=
66710 - 1 59810 1 24
= 715 km/ s
780010 3
v hP ,
1
2 mv 2 hP -
GMm
= -
r 1
GMm
r 1+ r 2
vh P =
2GMr 2
r 1( r 1 + r 2)
=
266710 - 11 59810 24 42310 7
780010 3 ( 7800 + 42300)10 3
= 929 km/ s
v hA ,
vh A= v hP
r 1
= 929 7800
r 2 42300 = 171 km / s
vg=
GM
r 2
=
66710 - 11 59810 24
= 307 km/ s
4230010 3
v 1= v hP - vs = 929- 715= 214 k m/ s
v2= v g- v hA = 307- 171= 136 km/ s
4243 500kg
, 100km
80

( 1) 100km
( 2) ,
? ? 39810 5
J/ kg, 10510 7 J/ kg
( 1) 100km
E 1 = -
GMm
= -
2r
= - 10810 11 J
66710 -
11 59810 24 3500
2( 6370+ 100)10 3
E 2 = -
GMm
R = - 66710 - 11 59810 24 3500
637010 3
= - 21910 11 J
E = E 1- E 2= - 10810 11 - ( - 21910 1 1 ) = 11110 11
( 2)
Q 1= 39810 5 3500= 13910 9 J
Q 2= 10510 7 3500= 36810 10 J
E> Q 1, E > Q 1+ Q 2,
425 10 30 kg, 20k m,
10 10 m,
( 1) , ?
( 2) , ?
( 1) , r
,
J
81

-
Gmm
= -
r
Gmm
r / 2 + 21 2 mv 2 1
v 1=
Gm
=
r
66710 - 1 1 10 3 0
10 10 = 8210 3 m/ s
( 2) , ,
-
Gmm
= -
r
v2= Gm 1
2R - 1
r
Gmm
2R + 21 2 mv 2 2
= 66710 - 11 10 30 1
2210 4 -
= 4110 7 m/ s
1
10 10
426( )
6 500
10km, 6010 3 kg/ m 3 ( ) ,
?(
42) ?
E = -
GM EmA
r A
- -
GM EmA
r A R E,
RE
E=
GMEm A
R E
= GME 4 3 r 3 A
RE
= 66710 - 11 59810 24 4( 10 4 ) 3 6010 3
640010 3 3
82

= 1610 2 4 J
( 10 18 J) 10 6
427 10 18 kg
? 210 30 kg, 10km
E = 0- -
Gm1m 2
R
= Gm 1m 2
R
= 66710- 11 210 30 10 18
1010 3
= 1310 3 4 J
428
5010 1 0 m , 7510 4 m / s
,
416428
( 416) , ,
0+ 1 2 mv2 0= -
GMm
+ 1 r m in 2 mv2
83

v0= v 2 -
2GM
r m in
= ( 7510 4 ) 2 -
= 1810 4 m / s
266710 - 11 19910 30
5010 1 1
,
v0b= vr m in
b= r m in
v
v 0
= 510 10 51010
1810 4 = 2110 11
429 ,
,
, , ( ve =
m
2GM/ R )
, ( R= 7
10 8 m ) ? ?
c ,
Rb = 2GM
c 2 = 266710- 11 19910 30
( 3010 8 ) 2 = 29510 3 m
b= 3M =
4 R 3 b
319910 3 0
= 18510 19
4( 29510 3 ) 3 k g/ m 3
430
,
10 -
6 K ,
10 67
84

3 ,
200 ,
?
T M 3 ,
M mi = M s
T m i
T s
1/ 3
= 510 11 kg
( c=
T m i
=
T s
Mm i
M s
= 19910 3 0
3
210
10
10 67 1/ 3
2GM/ R)
Rm i= 2GMm i
c 2 = 266710- 11 510 11
( 310 8 ) 2 = 7410 - 16 m
431 235 ,
,
,
( 2 ) ( 12
)
, 1MeV
, 003eV ?
( m n) ( m )
v 10
E 0
v 1=
E 1= 1 2 m nv 2 1= 1 2 m nv 2 10
m- mn
v1 0
m+ mn
m- m n
m+ m n
2
= E 10
m- mn
m+ mn
2
85

E = E 10
m- mn
m+ mn
2
= 1 2
= 1 2
, 8
= 1 2
, 52
lgE 10- lgE
lg( m+ mn) - lg( m- mn )
lg10 6 - lg003
lg( 2+ 1) - lg( 2- 1) = 788
lg10 6 - lg003
lg( 12+ 1) - lg( 12- 1) = 518
432 238 U 427MeV ,
238 U
238 U 2 34 T h
E =
mT hv Th = m v
1
2 m T hv 2 T hm Th = 1 2 m v 2 m
E
1+ m / mT h
E T hm Th = E m
E Th + E = E
=
427
= 420 MeV
1+ 4/ 234
433 1 4 N
, 1 7 O
119MeV
( 14 N )
86

E a vE k
E av=
M N
E k
m+ M N
E k, = m + M N
MN
E av=
4+ 14
119= 153 MeV
14
434
E p( x ) = A x 1 2 -
A , B , x
B
x 6
F = 0
F = -
dE p( x )
dx
= 12A
x 13 -
6B
x 7
x =
6
2A
B
435: ,
E k, r=
1 2 v2 2 1= 1 2
m1m 2
( v2- v 1) 2
m 1+ m 2
E k , int =
1 2
m 1v 2
1 + 1 2 m 2
2v 2 = 1 2 m1( v1- vC) 2 + 1 2
m2( v 2- vC)
2
v C=
m 1v1+ m 2v2
m1+ m 2
v C
87

E k, int = 1 m 2 2( v 2 - v 1) 2
2 m1 + 1 m 2 1( v 2 - v 1) 2
( m 1 + m2) 2 2 m2 ( m 1 + m 2) 2
= 1 2
m 1m 2
m1 + m2
( v 2 - v 1) 2
E k , r= E k , in t
436 M m
, k
m , v 0
M ,
1
2 v2 mM = 1 2 kx 2 ma x
= mM
m+ M , vmM = v 0,
x ma x=
mM
( m+ M) k v 0
88

5
51 125 ,
v = 25m/ sR = 10m
:
( 1) ;
( 2) ;
( 3) ( )
( 1)
= v/ R = 25/ 10 =
( 2)
=
2
2 = 25 2
22125 =
( 3)
t =
2
=
22125
25
25 rad / s
398 rad / s2
= 0628 s
52 70s 200r/ min
3 000r/ min
( 1) ;
( 2) ;
( 3) r = 02m ,
70s
( 1)
0 = 2200/ 60 =
209 rad/ s
89

=
=
- 0
=
t
( 2)
=
0 + t =
2
( 3)
a t =
23000/ 60 =
314 - 209
70
314 rad/ s
= 419 rad/ s 2
209 + 314
7 = 11710 3 rad = 186( )
2
R = 41902 = 838 m/ s 2
an =
2 R = 314 2 02 = 19710 4 m / s 2
a = a 2 t + a 2 n = 837 2 + ( 19710 4 ) 2 = 19710 4 m/ s 2
= arct an a n
a t
= arct an 19710 4
837
= 89 59
53 1987 365
1900 114s1900 1987 ,
-
= 0
=
t
-
3652
-
T 0 + t
3652
t
= -
87T 3 0
= - 9610 - 22 rad/ s 2
3652
87T 0
T 0
3652114
87( 31510 7 ) 3
54 P ,
( 51)
51 ,
90

v =
a =
5154
2
Rcos =
86400 6370103 cos
2 Rcos =
= 463cosm/ s
2
2
637010 3 cos
86400
= 33710 - 2 cosm / s 2
55 , h(
52) ,
( 1) :
h,
( 2) : ,
,
h
2h/ g
52 55
91

h =
30m , (
,
, ( 2h / 3) 2h / g )
( 1) vt = ( R + h) , vb
= Rvt - v b = h
( 2) R ,
t =
2h
g
h = 30 m, =
s = ( vt - v b) t = h
2h/ g
2
86400 = 72710- 5 rad / s
s = 72710 - 5 30 230/ 98 = 5410 - 3 m
56 53
AA
J A A = 19310 -
BB J B B =
4 7 kg ¡¤m 2 ,
114
10 - 4 7 kg ¡¤m 2
d
J A A = 2m Hd 2 sin 2
2
53 56
92

J B B = 2mH d 2 cos 2
2
,
d =
J A A +
,
2mH
J A A + J B B = 2mH d 2
J BB
=
( 193 + 114)10 - 47
216710 - 27
= 95910 - 11 m
J A A / J B B = t an 2
2
= 2arctan
J A A
= 2arct an
J B B
193
114 =
104 54
57C60( F ullerene, )
60
32 60
( 54)
71n m
( 1) ,
?
( 2) C60
41410 - 21 J
54 57
( 1) J = 2 3 mR 2
= 2 3 1216710- 27 60
7110 - 9
2
2
= 10110 - 39 kg ¡¤m 2 93

( 2) E = 1 2 J
2 = 1 2 J ( 2 ) 2
= 1 2E
2 J
=
1 241410 - 21
= 45610 8 Hz
2 10110 - 39
58 m, R
l ( 55) ,
?
5558
AA
BB
J A A = 2 2 5 mR 2 + m l 2 + R 2
= m 14
5 R2 + 2lR + l 2
2
J B B = 2 2 5 mR 2 = 4 5 mR 2
59 R
R ,
R/ 2 ( 56) ,
m
,
J
94
56 59

J 1
J 2 , a ,
J 1= 1 2 m 1 R
2
= 3 2 R 2
2
+ m 1
2
a R 2
2
R
2
2
=
3
32 a R 4
R - R 2
J 2= 1 2 m 2R 2 = 1 2 a R 4
J = J 2 - J 1 = 13
32 a R 4
2
a = m ,
a R 2 = 4 3 m
J ,
J = 13
24 mR2
510 m, R
, ?
57 ,
dm = mdS
dS = Rd cos2Rcos
2 R2 = 4 mcos 2 d
AA ,
dJ =
R 2 sin 2 dm =
4 mR2 sin 2 cos 2 d
57
510
95

AA
/ 2
J = dJ =
0
4
mR 2 sin 2 cos 2 d=
511 58 , m1
m, r , m 2
1 4 mR2
m2,
k , m 1 ?
,
m 1
m 2,
,
58511
,
m 1g - T 1 = m 1a
T 2 - km2g = m 2a
( T 1 - T 2) r = 1 2 mr 2
,
,
a =
= a / r
m1 - km 2
m 1 + m2 + m/ 2 g
T 1 = ( 1 + k ) m2 + m/ 2
( 1 +
m1g , T 2 = k ) m 1 + km/ 2 m 2g
m 1 + m 2 + m/ 2 m 1 + m 2 + m/ 2
96

512 59 R 1 R2,
M 1 M 2
,
,
m1 m 2
, m 2
,
m 1: T 1 - m 1g = m 1a 1
m 2: m 2g - T 2 = m 2a 2
,
T 2R 2 - T 1R 1 =
59512
1
2 M 1R 2 1 + 1 2 M 2R2 2)
,
= a1/ R1 = a 2/ R 2
=
( m2R 2 - m 1R 1) g
( M1/ 2 + m1) R 2 1 + ( M 2/ 2 + m 2) R 2 2
513 , 60cm ,
,
?
510 , m,
I = 1 3 m 1l 2 1 + 1 3 m 2l 2 2
= 1 3 2 5 m042 + 1 3 3 5 m06 2
= 0093m
97

510513
, ,
=
mgOC
I
mgOC = I
= m9801
0093m
,
= 105 rad/ s 2
mgOC =
1 2 I
2
=
2mgOC
I
=
2m9801
0093m
= 458 rad/ s
514 E k = 1 2 mv2 ,
E k = 1 2 J
2
v =
r m ,
r ,
E k = 1 2 mv2 =
1 2 mr 2
2
98

= 1 2 mr 2
2 = 1 2 J
2
515 ,
( 511)
R , m,
k ,
? ?
, , ?
?
511515
511 , dS =
dm = mrd dr / ( R 2 ) ,
dM =
rdf = r kdmg =
mg kr 2 d dr / ( R 2 )
,
M =dM =
R 2 R dr
2 dr =
2
0 0
kmg
2 3 kmg R
rd dr ,
, 0
99

t =
=
M/
= 3R
1
2 mR 4 kg
2
A = M ¡¤ = M ¡¤ t =
1 2 mR 2
2
E k = 1 2 J
2 = 1 2
1
2 mR 2 = 1 4 mR 2
2
516 k =
20N / m
, m1 = 80g
( 512) , r =
005m , m = 100gm1,
, m 1 h = 05m ?
,
512516
- m 1gh + 1 2 kh 2 + 1 2 m 1v 2 + 1 2 J
2 = 0
J =
1 2 mr 2 , = v/ r ,
100

v =
2m 1g h - kh 2
m 1 + m/ 2 = 20089805 - 205 2
008 + 005
= 148 m / s
517 ( 515)
, J 1 = 2k g ¡¤m 2
, 1 = 15r/ min ,
J 2 = 080kg ¡¤m 2 , 2 ?
, ? ??
, ,
2 =
1
J 1 ¡¤21 = J 2 ¡¤22
J 1
J 2
= 15 2
08 =
375 r / min
, ( ) ,
A =
= 1 2
1 2 J 2
2 2 -
= 370 J
1
2 J 1
2 1
08 2 375 60
2
- 2 2
15
60
518 513 , L M,
m,
v 0
d
( 1) ?
( 2) ( t) ,
?
2
101

( 3) , ?
( 1) ,
O ,
,
mv0d =
=
1
3 ML2 + md 2
3mv 0d
ML 2 + 3md 2
( 2) ,
mv - mv 0
f =
t
f = -
= m d - mv 0
t
x : F x + f = Ma Ct = M L
2 t
F x = - f + M L =
2 t
mv0 - m
f = d
t
ML
2 + md
t - mv 0
t
y : F y - Mg - ( m
2 d + mg ) = Ma Cn = M
2 L/ 2
F y =
ML
2
+ md
2 + ( M + m) g
( 3) F x = 0,
ML
2 + md
t -
( 1) ,
d = 2L / 3
mv0
= 0
t
513
518
519 514 L = 040m, M = 10kg,
102

m = 80g
v = 200m/ s ,
d =
3L/ 4
( 1) ;
( 2)
( 1) O
mv 3 4 L= 1
3 ML2 + m 3L 4
3mv
=
4 1 3 ML + 9
16 mL
=
2
30008200
4 1 3 104 + 9
16 000804
= 889 rad/ s
( 2) ,
1
2
514
519
1
3 ML 2 + 9
16 mL 2
2 = Mg L 2 + mg 3 4 L ( 1 - cos )
= arccos 1 -
1
3 M + 9
16 m L
2
M +
3 2 m g
= arccos 1 -
1
3 1 + 9 0008 048892
16
1 + 3 0008 98
2
= 94 18
520 , t =
10s, J = 1200kg ¡¤m 2 M = 80k g
103

, , ,
r = 2m ?
,
,
2 =
J
J +
Mr 2 1 =
J 1 = ( J + Mr 2 ) 2
1200
2 = 0496 rad / s
1200 + 802 2 10
521 1200kg
72k m/ h ,
05m, , ,
2500kgm 2 , :
( 1) ;
( 2)
( 1)
=
2mvr
I
( 2)
2 1 2 mv2 -
2mvr = I
= 2120020025
2500
1
2 I
2 = 120020 2 -
= 4510 5 J
= 48 rad / s
1
2 250048 2
522
( 1) ,
? 25m,
( 2) , ,
104

? 70kg,
310 5 kgm 2
( 3) 30 , ?
( 1) v 2 / r = g
v = gr = 9825 = 495 m / s
( 2)
=
3mvR
J
3mvR - J = 0
= 37049525
310 5 = 86710 - 3 rad/ s
( 3) 30t = / ( 6 ) ,
+ v/ R, t
= ( + v/ R) t/ ( 2 ) = 1 12 1 + v
R
= 1
12 1 + 495
86710 - 3 25
= 19( )
523 ,
52
?
25d ,
J S =
2 5 mR2
= 2 2
0
5 199103 ( 69610 8 ) 2
2586400
= 1110 4 2 kg ¡¤m 2 / s
01110 43
( 011 + 32)10 43 = 33%
524( 118) , PSR
105

0531+ 21( 0033s)
, ,
, 126
10 - 5 s/ a
( 1)
( 2) 1510 30 kg( ) ,
10km( J/ s ) (
)
( 3) ,
( 1)
= d
dt =
= -
d dt
2
T = - 2 dT
T 2
dt
2
( 0033) 2 12610 5
31510 7 = - 2310 - 9 rad/ s 2
( 2)
dE
dt =
d dt
1
2 J
2 = 1 2 J2d = 2 dt 5 mR 2 d
dt
= 2 5 15103 0 ( 10 4 ) 2
2
0033 ( - 2310- 9 )
= - 2610 31 J/ s
( 3)
106
t=
=
E
dE
dt
= 1 2 J dE
2
dt
21510 30 ( 10 4 ) 2 ( 2 ) 2
25( 0033) 2 2610 31

= 41810 10 s = 1300 a
525 033MR 2 , M
, R
, ,
16s ,
? ? ( : dE k/ E k =
2d / = - 2dT / T )
E =
1 2 J
2 = 1 2 033MR 2 2
2
T
= 1 2 03359810 24 ( 6410 6 ) 2
2
2
86410 4
= 21410 29 J
dE
dt =
d dt
2
1
2 J 2
= - J ( 2 ) 2
dT
T
T 3
dt
= - 03359810 24 ( 6410 6 ) 2
4
2
16
( 86410 4 ) 3 10 6 31510 7
= 2610 12 J/ s = - 2610 9 kW
2610 9 k W,
1999
2610 12 31510 7
410 18
M= J = J d
dt = -
20
J2 dT
T 2
dt
107

= -
03359810 24 ( 6410 6 ) 2 216
( 86410 4 ) 2 10 6 31510 7
= - 3510 16 N ¡¤m
526 ,
,
?,
26d
T 2 = T 1
J 2
J 1
= T 1
J 1
2
= J 2
T 1
2
,
T 2
R 2 2
= 262460 ( 64106 ) 2
R 2 1
( 69610 8 ) 2
= 31 min
527
235 ( 515) ,
, 26 000a
J = 80510 37 kg ¡¤m 2
, ©¦ dL/ dt©¦ ,
?
dL
dt
= Lsin d
dt
= J sin d
dt
= 80510 37 2
86400 sin235
2
2600031510 7
= 17910 22 kg ¡¤m/ s 2
108

M =
515527
dL
dt
= 17910 2 2 N ¡¤m
528 500t ,
200m 800r / min
( 1) 74610 4 W ,
?
( 2) 100 / s
, ?
( 1) P t =
1 2 J
2
t= J
2
2P = mR2 ( 2) 2
22P
= 5010 3 2 2 4
2 800 2
= 784 min
2274610 4
109

( 2) M = L = J = 1 2 mR 2 2
= 5010 3 2 2
800
60 180
= 14610 5 N ¡¤m
529 034kg m 2 ,
036m 50m/ s , ,
60kg 004m
M =
L ,
= M L = dmg
J v/ R =
0046098036
03450
= 498 rad/ s
530 h/ ( 4 )
= 5310 - 3 5 J ¡¤s
, = 2410 7 rad/ s,
, 516
M =
L = 5310 - 35 2410 7
= 12710 - 2 7 N ¡¤m
516
530
110

6
61 S , l , x
, S x
x = x 1 -
y =
S
u 2
= lcos1 -
c 2
y = lsin
l = ( x ) 2 + ( y ) 2 = l 1 - cos 2 u
2
lx
= arct an
u 2
c 2
lsin
lcos1 - u 2 / c 2 = arctan tan 1 -
c 2 1/ 2
u 2 - 1/ 2
c 2
62 a , u
S, S
?
S
a = a 1 - u 2 / c 2 , a
V =
a a 2 = a 3 1 - u 2 / c 2
63S x ,
Sx
, S , S
S S
111

S S
( x 1, t 1) ( x 2, t 2) ( x 1, t 1) ( x 2, t 2) ( 61) , ,
6163
x
2 - x
1 =
( x 2 - x 1) - u( t 2 - t 1)
1 - u 2 / c 2
t 2 = t 1, x 2 - x 1 = 1m, S
x
2 - x = 1/ 1 - u 2 / c 2 > 1m
S , ,
-
x , x 2 , , x 1
, S
1m
64 ,
( v = 0998c) 22
10 - 6 s, 8000m
t=
t/ 1 - v 2 / c 2 ,
l = v t =
v t
1 - v 2 / c = 0998310 8 2210 - 6
2
1 - 0998 2
= 10410 4 > 8000 m
112

65 S ,
2sS 3s
S
S , x 1 = x 2,
t = t/ 1 - u 2 / c 2
u = c 1 - ( t/ t ) 2
S
x = x - u t
1 - u 2 / c = - u t = - c 1 - ( t/ t ) 2 t
2
= - 310 8 1 - ( 2/ 3) 2 3 = - 67110 8 m
66 S x ,
1mS2mS
S t =
x =
0, x =
x
1 - u 2 / c 2
u = c 1 - ( x / x ) 2
1m,
S
t=
=
67 l =
u
t - x
c 2 = u x = x 1 - ( x / x ) 2
c 2 c
1 - u 2 / c 2
2
310 8 1 - ( 1/ 2) 2 = 57710 - 9 s
[ ( x ) 2 +
( y) 2 + ( z) 2 ] 1 / 2 ( x 1, y 1,
z 1) ( x 2, y 2, z2) , :
,
113

( x ) 2 + ( y ) 2 + ( z ) 2 = ( x ) 2 +
s = [ ( c t) 2 - ( x ) 2 -
( y ) 2 -
( y ) 2
+ ( z ) 2
( z) 2 ] 1/ 2 ( x 1,
y 1, z 1, t 1) ( x 2, y 2, z 2, t 2) :
, ( c t ) 2 - ( x ) 2 - ( y ) 2 - ( z ) 2 = ( c t) 2 -
( x ) 2 - ( y) 2 - ( z ) 2
, ( y ) 2 = ( y ) 2 , ( z ) 2 = ( z ) 2 ,
( x ) 2 = ( x ) 2 - ( u t) 2
, t =
0, ( x ) 2 = ( x ) 2 ,
( x ) 2 + ( y ) 2 + ( z ) 2 = ( x ) 2 +
,
( c t ) 2 - ( x ) 2 =
c
2
u
t - x
c 2
1 - u 2 / c 2
= ( c t) 2 - ( x ) 2
-
( y ) 2
+ ( z ) 2
2
x - u t
1 - u 2 / c 2
( ) ( y ) 2 = ( y ) 2 ,
( z ) 2 = ( z ) 2 ,
( c t ) 2 - ( x ) 2 - ( y ) 2 - ( z ) 2
= ( c t) 2 - ( x ) 2 - ( y) 2 - ( z ) 2
68 , 4 5 c
, , 60s
?
?
( 1) ,
( 2) ,
( 1) , ,
, , 30s
114

,
6268
c30 =
910 9 m
( 2) , ,
l = c30 -
4
c30 = 6c
5
, ,
l = l / 1 - u 2 / c 2 = 6c/ 1 - ( 4/ 5) 2 = 10c = 310 9 m
, t =
60s, ,
t =
t / 1 - u 2 / c 2 = 60/ 1 - ( 4/ 5) 2 = 100 s
, 10c ,
l 1
= u t =
4c 100 = 80c
5
, , ,
l + l 1 = 10c + 80c = 90c = 2710 1 0 m
69 x ( S , ) ,
u =
080c , ( S ,
) , t = - 6010 8 s, x = 18010 17 m, y =
12010 17 m, z = 0 ,
, ?
115

:
x =
x + ut
1 - u 2 / c
2
= 18101 7 + 08310 8 ( - 6. 010 8 )
1 - 08 2
= 6010 16 m
y = y = 1210 17 m, z = z = 0
t =
=
t + u x
c 2
1 - u 2 / c 2
- 6010 8 +
= - 2010 8 s
08
310 8 1810 17
1 - 08 2
610 ,
, ,
( 1) ( S ) ?
( 2) ,
( S ) ?
( 3) ( S ) ?
63 , 1,
2, 3,
4
116
t 1 = - 6010 8 s, x1 = 1810 17 m ,
y 1 = 1210 17 m, z1 = 0
t 1 = - 2010 8 s, x 1 = 6010 16 m,
y 1 = 1210 17 m, z 1 = 0

63610
( 1) S
t 2 = t 1 + t 12 = t 1 +
x 1
2 + y 1
2
c
= - 6010 8 + 101 7 18 2 + 12 2
310 8 = 1210 8 s
( 2)
t 3 = t 2 + t2 3 =
=
=
t2 + u x 2
c 2
1 - u 2 / c + x 2
2 c
t 2
1 - u 2 / c + ut 2
2
c 1 - u 2 / c 2
t 2
1 - u 2 / c 2 1 + u c
= 12108 1 + 083108
1 - 08 2 310 8
= 3610 8 s
( 3)
t 4= t 1 + t 14 = t1 +
x 2 1 + y 2 1
c
117

= - 2010 8 + 1017 06 2 + 12 2
310 8 = 2510 8 s
611 120km 9
9 0 00003
u = 08c
,
?
1, 2,
u
t2 - t1 - ( x
c 2 2 - x 1)
t 2 - t 1 =
1 - u 2 / c 2
=
00003 -
= - 3310 - 5 s
08
310 8 1210 5
1 - 08 2
t 2 < t 1 ,
612 060c
5s 080c
( 1)
( 2) ,
64 612
( 1) ,
v2 - v 1
v 2 =
=
v2v 1
1 -
c 2
118
- 08c - 06c
1 - ( - 0806) = - 095c

095c
( 2)
( 5s ) 1,
2,
,
t = t 1 - v 2 1/ c 2 = 5 1 - 06 2 = 4 s
613 S O ,
x y x , S
, S c
S v = c
S
v x =
vx + u
vx
1 + u = v cos
+ u
1 + v cos u = ccos + u
1 + ucos
c 2 c 2 c
v y = v y1 - u 2 / c 2
1 + v x u = v sin1 - u 2 / c 2
1 + v cos u
c 2 c 2
= csin1 - u 2 / c 2
1 + ucos
c
S x
= arct an vy
vx
= arct an
sin 1 - u 2 / c 2
cos + u/ c
v= v 2 x + v 2 y
=
=
c
c + ucos ( u + ccos ) 2 + c 2 sin 2 1 -
c +
c
ucos
[ u
2 cos 2 + 2uccos + c 2 ] 1/ 2 = c
u 2
c 2 1/ 2
119

614 69 S S ,
S x v = v ( t) , a : S
a
a = ( 1 - u 2 / c 2 ) 3/ 2
a
( 1 - uv/ c 2 ) 3
, u, v c ,
?
,
a = dv
dt
= dv dt
dt dt = d dt
=
v - u d
1 - uv/ c 2 dt
( 1 - uv/ c 2 ) + u c 2 ( v - u )
( 1 - uv/ c 2 ) 2 1 - uv/ c 2
= ( 1 - u 2 / c 2 ) 3/ 2
( 1 - uv/ c 2 ) 3 a
1 - u 2 / c 2
t - ux / c 2
1 - u 2 / c 2
S S x x ,
a a
u, v c , a = a,
615 m 0 F = F i
, t, v x ?( t
m 0c/ F ) ( t m 0c/ F ) , v x
?
F , , F =
d ( mv) / dt t
120
v =
t
v =
0
F dt/ m = F t 1 - v 2 / c 2
m0
F t/ ( m0c)
[ 1 + ( F t/ ( m 0c) ) 2 ] 1 / 2 c

t
t
x = vdt =
0
0
F t/ m 0
[ 1 + ( F t/ ( m 0c) ) 2 ] 1 / 2 dt
= [ 1 + ( F t/ ( m0c) ) 2 ] 1/ 2 - 1 m 0c2
F
( t m 0c/ F ) ,
v F t/ m 0
x [ 1 + 1 2
= a t
F t
m0c
2
- 1 m0c2
F = F t 2
2m 0
= 1 2 a t 2
a = F / m 0 ,
( t m 0c/ F ) ,
vc
x F t
m 0c
m0c 2
F = ct
c
616 ?
,
m 0v
= 2m 0v
1 - v 2 2
/ c
v = 3 c/ 2 = 0866c
mc 2 - m 0c 2 = 2 1 2 m 0v 2
1 = 1 + v 2
c 2 2
1 -
v 2
c 2
121

v =
0786c
617 ,
E k = 2810 9 eV
( 1) m / s?
( 2) ?
( 3) 240m ,
??
( 1) E k =
c 2 - v 2 =
m0
1 - v 2 / c 2 - m 0 c 2
m0c 3
E k + m0c 2 2
c v, c 2 - v 2 = ( c + v) ( c - v) 2c( c - v) ,
c - v = 1 2c
m 0c 3
E k + m 0c 2 2
E k = 2810 9 eV m0c 2 = 051110 6 eV ,
c - v = m 2 0c 5
2E 2 k
( 2)
E k
=
p =
m 0c 2 ,
( 091110 - 30 ) 2 ( 310 8 ) 5
2( 2810 9 1610 - 1 9 ) 2 = 502 m/ s
E 2 - m 2 0c 4
c 2 =
E k( E k + 2m 0c 2 )
c
p E k
c = 28109 1610 - 19
310 8 = 14910 - 18 kg ¡¤m/ s
( 3)
122
F = mv 2
R
mc2 R = E R E k
R
= 2810 9 1610 - 19
240/ ( 2 )
= 1210 - 11 N

B = F ev F ec = 1210 - 11
1610 - 19 310 8 = 025 T
618 50J ,
, m/ s?
E = m0c 2 / 1 - v 2 / c 2
v c, v +
c - v = m2 0c 5
c 2 - v 2 = m 2 0c 6
c 2c,
E 2
2c( c - v) = m 2 0c 6
E 2
= ( 16710 - 2 7 ) 2 ( 310 8 ) 5
2E 2 250 2
= 13610 - 1 5 m/ s
619 m p = 1672 6510 -
27 kg,
m n = 1674 9510 - 27 kg,
m D = 3343 6510 - 2 7 kg
MeV ? ,
?
, 1358eV ,
?
m H = 1673 2310 - 27 kg
( E ) D = ( m) D c 2 = ( 1672 65 + 1674 95 - 3343 65)
10 - 27 ( 310 8 ) 2
= 35510 - 1 3 J = 222 MeV
123

D =
( E ) D
=
m Dc 2
35510 - 13
3343 6510 - 27 ( 310 8 ) 2 = 012%
H = ( E ) H
=
m H c 2
= 14510 - 6 %
13581610 - 1 9
1673 2310 - 27 ( 310 8 ) 2
620 ,
:
1 1H + 1 1H + 1 1H + 1 1H 2 4He +
2 0 1e
( 1 1H ) mp = 1672 610 -
27 kg,
( 2 4He) mH e = 6642 510 - 27 kg( 0 1e)
m e = 0000 910 - 27 kg
( 1) ?
( 2) ?
( 3) 1kg ?
( 4) P = 3910 2 6 W,
?
( 5) m = 1510 30 kg
( 4) , ?
( 1)
E = mc 2 = ( 41672 6 - 6642 5 - 20000 9)10 - 27
( 310 8 ) 2
= 41510 - 12 J
( 2)
=
E
=
4mpc 2
41510 - 12
41672 610 - 2 7 ( 310 8 ) 2 = 069%
( 3) 1kg
41510 - 12 ( 41672 610 - 2 7 ) = 62010 14 J/ k g
124

( 4)
3910 26 ( 62010 14 ) = 62910 1 1 k g/ s
( 5)
1510 30 ( 62910 11 ) = 23810 18 s = 75610 1 0 a
6211996
( 618) 8m 2 ,
120V 10A
1010 3 W / m 2
=
10120
1010 3 8 = 15%
62220 60
( 1) 1kg ?
( )
?
( 2) 1kg
? ( 10km)
?620
?
( 1)
( E ) E = 0 -
- GM Em
RE
= 66710- 1 1 59810 24 1
63710 6
= 6310 7 J
125

E = ( E ) E
m = 6310 7
= 710 - 8 %
1( 310 8 ) 2
( 2)
( E) N = 0 -
- GMN m
R N
= 66710- 11 19910 3 0 1
1010 3 = 1310 1 6 J
=
( E ) N
m = 1310 1 6
= 14%
1( 310 8 ) 2
( 069% ) 20
623 = 05 , :
( 1) ;
( 2)
( 1) p 1 = m 1v1 = m 0 c 1 -
2 = 05m 0c/ 1 - 05 2
= 058m 0c
E 1 = m 1c 2 = m 0c 2 1 -
2 = m0c 2 1 - 05 2 = 115m 0c 2
( 2)
2 =
+
=
1 +
2
1
125
p 2 = m 2v2 = m0 2c 1 -
2 2
= m0c
125
1 -
1
125
2
= 133m 0c
E 2 = m 2c 2 = m 0c 2 1 -
2 2
126

= m 0c 2 1 -
1
125
2
= 167m 0c 2
624 05310 -
110 -
110 -
34 J ¡¤s
1 8
m,
18
m, ,
??
,
,
L = J = 2 vm ax
5 mR2 ¡¤
R = 2 mRv ma x
5
vm ax = 5L
2mR = 505310 - 3 4
2091110 - 30 10 - 18
= 14610 14 m / s
,
?
625 22GeV ,
m =
00005GeV/ c 2 , M
1GeV/ c 2 , E k = 22GeV
E av = 2Mc 2 E k + [ ( m + M) c 2 ] 2
= 2122 + ( 0000 5 + 1) 2 = 67 GeV
626 28GeV
?
,
?
, 282 =
56GeV
127

,
E a v = 2Mc 2 E k + [ ( m + M) c 2 ] 2
E k = E a v 2 - [ ( m + M) c 2 ] 2
2Mc 2
M = m = 0000 51GeV / c 2 E a v = 56GeV ,
E k = 562 - [ 20000 51] 2
20000 51
= 3110 4 GeV
627 E 2 -
c 2 p 2
( S S : E 2 - c 2 p 2 = E
2 - c 2 p
2 )
E
2 - c 2 p
2 =
2 ( E - cp x ) 2 - c 2
2 p x -
2
+ p y 2 + p 2 z
c
=
2 [ ( E - cp x) 2 - ( cp x - E ) 2 ] - c 2 ( p 2 y + p 2 z )
=
2 [ ( E - cp x + cp x - E ) ( E -
+ E ] - c 2 ( p 2 y + p 2 z )
=
2 ( 1 - ) ( E + cp x ) ( 1 + ) ( E - cp x )
- c 2 ( p 2 y + p 2 z )
= E 2 - c 2 ( p 2 x + p 2 y + p 2 z ) = E 2 - cp 2
cp x - cp x
128

2
1
11
, 66510 3 Pa
?
( 1) 37315K , ?
( 2) 22010 3 P a , K?
( 1) , ,
,
( 2)
T 1/ T 3 = p 1/ p 3
p 1 = p 3 T 1/ T 3 = 6. 6510 3 373. 15/ 273. 16
= 9. 0810 3 P a
T 2 = T 3 p 2/ p 3 = 273. 162. 2010 3 / ( 6. 6510 3 )
= 90. 4 K
= - 182. 8
12 l = 0. 7m ,
,
h = 02m ,
, , ,
129

, p 0= 99810 4 Pa,
?
11 , ,
,
p ( l -
h) = p ( l - h )
mmH g ( 1mmH g 133Pa )
,
( p 0 + h) ( l - h) = ( p 0 - h ) ( l - h )
( p 0= 0. 7510 3 mmHg)
h
2 - 1. 45h + 0. 05 = 0
h = 0. 035 m
( h= 1. 4m > l , )
1112
1328 7112cm ( 28 ) ,
3cm- 3
30cm, 15cm20 , ,
? 7
T 1 = 273 -
3 = 270K , V 1 = 201. 5 2 30cm 3 , p 1 =
1atm , T 2= 273+ 7= 280K , V 2= 71. 121. 5 2 cm 3
p 2 =
,
p 1V 1T 2
T 1V 2
p 1V 1
T 1
= p 2V 2
T 2
= 1201.
5 2 30280
= 2. 8 at m
27071. 121. 5 2
14,
290g/ mol
130

km / 10 2 Pa / ( kg/ m 3 )
20 5. 510 8. 810 - 2
32 87 1210 - 2
53 5810 - 1 7110 - 4
90 1810 - 3 3210 - 6
p V= m M RT
T =
p VM
mR = p M R
( ) ,
218 K, 253 K , 285 K , 196 K
15 2110 4 m 3 ,
4510 3 kg, 20, ,
?
0 , 0= 1. 29kg/ m 3 1
2 ,
( 1at m)
1 = 0T 0/ T 1, 2 = 0T 0/ T 2
( 1 - 2) Vg = mg
0T 0
1
T 1
-
1
T 2
T 2 =
V = m
V 0T 0T 1
V 0T 0 - mT 1
131

=
2. 110 4 1. 29273293
2. 110 4 1. 29273 - 4. 510 3 293
= 357 K = 84
16 ,
47, 86110 4 Pa ,
1/ 17, 42510 6 Pa,
( K )
,
T 2 = p 2V 2T 1
p 1V 1
p 1V 1
T 1
= p 2V 2
T 2
= 4. 25106 V 1320
8. 6110 4 V 117
= 929 K = 656
1720, 12atm, 15m
, 10, 14m,
11atm
m= m 1 - m 2 = M R
p 1V 1
T 1
-
p 2V 2
T 2
= 210 - 3
8. 31 1. 241.
53 / 3
293
1. 0110 5 = 0. 32 kg
-
1. 141. 4 3 / 3
283
18,
p 1, Vm 1T 1,
, , p V m
C ,
p =
CV m
( 1) C( p 1, T 1 R )
( 2) T 1= 200K , 2V m1 ,
132

?
( 1)
p 1 = CVm 1
V m 1
p 1V m1 = RT 1
C = p 1 2
RT 1
( 2) p = CV m, V m 2= 2V m1,
1, 2
p 2 = p 1V m2/ V m 1 = 2p 1
p 1Vm 1
T 1
= p 2V m 2
T 2
T 2 =
p 2Vm 2T 1
p 1V m 1
= 4T 1 = 4200 = 800 K
1910010 -
1cm 3
=
18 atm
? 20
p
kT = 1. 0010- 18 1. 0110 5
1. 3810 - 23 293
= 2. 510 7 m - 3 = 25 cm - 3
1101997 7 26
: 671mbar ( 1bar = 10 5 Pa ) ,
- 133, 1cm 3 ?
=
p
kT = 6. 7110 2
1. 3810 - 23 ( 273. 15- 13. 3)
= 1. 8710 23 m - 3 = 1. 8710 1 7 cm - 3
11110 1 0 / m 3 ,
10 4 K
133

p = kT = 10 10 1. 3810 - 23 10 4 = 1. 410 - 9 P a
112( 1) 20%
70% 30% 9010 6 K 36
10 4 kg/ m 3 at m? (
)
( 2) , ,
35% 65% 1510 7 K ,
1510 5 kg/ m 3 , atm ?
p V =
m RT
M
p ( 1) H =
p =
H RT
M H
RT
M
= 3. 6104 0. 78. 319. 010 6
110 - 3
= 1. 910 15 Pa
p H e =
H eRT
MH e
=
= 3. 6104 0. 38. 319. 010 6
410 - 3
2. 010 14 Pa
p = p H + p H e = 1. 910 15 + 0. 210 1 5
= 2. 110 15 Pa = 2. 110 10 at m
p ( 2) H = H RT
M H
= 1. 5105 0. 358. 311. 510 7
110 - 3
= 6. 510 1 5 Pa
p H e =
H eRT
M H e
= 1. 5105 0. 658. 311. 510 7
410 - 3
= 3. 010 15 Pa
p = p H + p H e = ( 6. 5 + 30)10 15 = 9. 510 15 Pa
= 9. 410 1 0 atm
113, 50
134

750mmH g , 590mmH g
290g/ mol
p = p 0exp -
h =
RT
Mg ln p 0
p =
Mg h
RT
8. 31278 750
ln
2910 - 3 9. 8 590 = 1. 95103 m
114,
T T s
( T - T s) ,
-
,
dT
dt
= A( T - T s)
A , T s
T 1
, t,
T = T s + ( T 1 - T s ) e - A t
,
25 , - 10
40min 20,
15?
-
dT
dt
= A ( T - T s )
Adt = -
- t 0 Adt = T T 1
dT
( T - T s)
dT
T - T s
- At = ln T - T s
T 1 - T s
135

T = T s + ( T 1 - T s ) e - A t
t 1= 40 min ,
20 = - 10 + ( 25 + 10) e - A t 1
t 1 = 1 A
t 2
ln
35
30
2520,
15 = - 10 + ( 25 + 10) e - A t 2
t 2 = 1 ln 35
35
ln
A 25 = t 25
1
ln 35
30
2015
t2 - t 1 = t 1
= 40
ln ( 35/ 25)
ln ( 35/ 30) - 1
ln ( 7/ 5)
ln ( 7/ 6) -
1 = 47 min
115T y
:
T 0
T = T 0 - y
,
( 1) ,
p = p 0exp Mg
R ln 1 -
y
T 0
( 2) 0 , ( 116)
( 3) = 0. 6/ 100m,
M= 29. 0g/ mol, T 0 = 273K , p 0 = 1. 00atm
( 1) p y ( 115) ,
T = T 0- y ,
136

p p 0
( 2) 0 , ln 1-
( 116)
dp
p = - Mgdy
R( T 0 - y )
dp
p = - y 0
ln ( p / p 0) =
Mg dy
R( T 0 - y)
Mg T 0 -
ln y
R T 0
p = p 0exp Mg
R ln 1 -
y
-
T 0
y
T 0
y
,
T 0
p = p 0ex p -
Mg y
RT 0
( 3) , y = 8848m, ,
p = p 0exp
= 1. 00exp
= 0. 29 atm
T = T 0 - y = 273 - 0. 610 - 2 8848
= 220 K = - 53
Mg
R ln 1 -
y
T 0
2910 - 3 9. 8
0. 610 - 2 8. 31 ln 1 -
0. 610 - 2 8848
273
137

2
21 , 85g, 17,
65km / h :
( 1) ( )
;
( 2) ;
( 3)
( 1)
t
t =
2 kT = 3 2 13810- 23 290
= 60010 - 2 1 J
( 2) E = 2 RT =
r = r 2 kT = 2 2 13810- 23 290
= 40010 - 2 1 J
k = 2 kT = 5 23 2 13810- 290
= 100010 - 21 J
5
2 85 28 831290 = 183103 J
( 3) E k = 1 2 mv2 = 1 3 2 8510- 65000
2
= 139 J
3600
22 27 , 1mol ?
1g ?
E m, H e = 2 RT = 3 2 831300 = 374103 J/ mol
138
E m, H 2
= 2 RT = 5 2 831300 = 623103 J/ mol

1g
E m, O 2
= 2 RT = 5 2 831300 = 623103 J/ mol
E H e
= E m, H e = 1 4 374103 = 093510 3 J
E H 2
= E m, H 2
= 1 2 62310 3 = 31210 3 J
E O 2
= E m, O 2
= 1
32 62310 3 = 019510 3 J
23 10 12 eV ,
001mol,
, ?
E =
2 RT E =
2 R T ,
T =
2 E
R = 2101 2 1610 - 19
= 12810 - 6 K
3001831
24 10 8 K ,
, , : ( 1)
?( 2) ?
( 1)
k = 3 2 kT = 3 23 2 13810- 10 8
= 20710 - 15 J
=
12910 4 eV
3RT
( 2) v 2 =
M = 383110 8
= 15810 6 m/ s
110 - 3
25 , ,
250K; , 310K
, H e p 1V 1 = 1RT 1, O 2 p 2V2 =
2RT 2p 1V1 = p 2V2,
139

1T 1 = 2T 2
E 0 = E 1 + E 2 = 3 2 1RT 1 + 5 2 2RT 2 =
8 2 1RT 1
, T ,
E =
3 2 1RT +
, , E 0
T =
5 2 2RT = 3
2 + 5T 1
2T 2
8
2 1RT 1 = 3
2 + 5T 1
2T 2
8T 1
3 + 5T 1/ T 2
=
= E ,
1RT
1RT
8250
3 + 5250/ 310 = 284 K
26N ,
f ( v) =
( 1) ;
( 2) v0 C;
dN
N dv =
C( v 0 v 0)
f ( v) = 0 ( v > v 0)
( 3)
( 1) 21
( 2)
0
Cdv =
v 0
0
Cdv = Cv 0 = 1
C= 1/ v 0
2126
( 3) v
v 0
= vf ( v) dv =
0 vCdv =
1 v 2 0
0 v0 2 = v0
2
27 N ,
140

f ( v) = a v/ v 0( 0 v v0)
f ( v) = a ( v 0 v 2v0)
f ( v) = 0 ( v > 2v 0)
( 1)
a ;
v 0
( 2) v0
;
( 3)
( 1)
22
0
f ( v) dv =
v 0
0
a v
v0
dv +
2v 0
2227
v 0
a dv
= a v 0
2 + a v 0 = 3 2 a v0 = 1
a =
( 2) v > v0
v < v 0
( N ) 1 =
v 0
dN =
( 3) v
= vf ( v) dv
=
v 0
0
0
2
3v 0
v 0
N f ( v) dv = 2 v 0
( N ) 2 = N - ( N ) 1 = 1 3 N
v av
2v 0
dv +
v 0 va dv = 11v 0
v 0
9
v 0
N a dv = 2 3 N
28 N
,
141

v 2
v
[ : ( v -
( v - v ) 2 0,
v ) ]
( v - v ) 2 = v 2 + v 2 - 2vv = v 2 + v 2 - 2v v = v 2 - v 2
v 2 - v 2 0
v 2
v
29 300K , ( 1) v p
; ( 2) 10v p ,
( v = 1m/ s)
? , 10 5 mol ?
29g/ mol
T =
vp =
N
N
300K ,
v = 1m / s,
= 4m 2 kT
=
4
v 2
vp 3 e -
3/ 2
v 2 e - mv2 / 2kT
v
v2 / v 2 p
v
2kT
m = 2RT
M = 2831300
2910 - 3 = 415 m / s
( 1) v = vp
N
=
N 1
4
v 2 p
e -
v 3 p
( 2) v = 10 v p
142
=
v2 p / v2 p
v
4 1
415 e- 1 1 = 0002 = 02%

N
N 2
=
=
4
( 10 vp) 2
3 e - ( 10 v p ) 2 / v 2 p v
vp
4 100
415 e- 100 1 = 2010 - 4 4 = 2010 - 42 %
10 5 mol , vp
, v= 1m / s
N N N 1= 60210 23 10 5 02%
10 v p
, v =
= 1210 2 6 ( )
1m/ s
N N N 2
= 60210 23 10 5 210 - 4 2 %
= 1210 - 15 = 0( )
210 t
F ( t) d t =
2
( kT ) -
3/ 2
1 / 2
t e - t / kT d t
, 1 2 mv 2 p
t =
1 2 mv 2
v =
2 t
m , dv =
f ( v) dv = 4 m
2 kT
3 / 2
v 2 e -
( 2m t ) - 1
2
d t
m v2 / 2kT dv
F ( t ) d t = 2 ( kT ) - 3 / 2 1
t
2 e
- / kT t
d t
t , F (
dF ( t )
d t
=
t, p
t ) t ,
1 ( kT ) 1
- 3 / 2
e - t / kT 2
- t , p 1/ 2
+
1 / 2
t , p -
1
kT
= 0
143

t , p = kT 2
t , p 2
t ( vp ) = 1 2 mv2 p = 1 2 m 2kT
m
= kT
211 210 6 K ,
27K , ,
1994 Na
, 2410 -
1 1
K , N a
v rm s, e =
3kT
me
=
313810 - 23 210 6
9110 - 3 1
= 09510 7 m / s
v rm s, H =
3RT
MH
=
= 2610 2 m / s
383127
110 - 3
Na
vrm s , N a =
3RT
M N a
=
= 1610 - 4 m/ s
38312410 - 11
2310 - 3
212 0108 ,
0531 , ? 240K ,
144

CO 2 H 2 ?CO 2
H 2 ( , 96% CO 2 )
318 , 112 ,
? 130K , H 2
? H 2 ( 78%
H 2, H e, ,
)
ve , M =
2GM M
RM
=
2G0108ME
0531R E
=
vr ms , CO 2
=
v rms , H 2
=
266710 - 11 010859810 24
053163710 6
= 5010 3 m/ s
3RT
M CO 2
=
= 36810 2 m/ s
3RT
MH 2
=
= 17310 3 m/ s
3831240
4410 - 3
3831240
210 - 3
CO 2 , H 2 , CO 2 H 2
v e, J =
2GMJ
R J
=
2G318M E
112R E
=
266710 - 1 1 31859810 2 4
11263710 6
= 6010 4 m/ s
145

v rms , H 2
=
3RT
MH 2
=
3831130
210 - 3
H 2
= 12710 3 m/ s
, H 2
213
,
1610 -
16 kg, 300K
300K ,
?
vrm s =
3kT
m = 313810 - 23 300
1610 - 16
= 8810 - 3 m/ s
,
214 6210 -
14 g 27 ,
14cm/ s R
N A
= 3RT
v 2 m
=
v 2 =
3kT
m = 3RT
N A m
3831300
( 1410 - 2 ) 2 6210 - 14 10 - 3
= 61510 2 3 mol - 1
215 89g/ mol 50
10 4 g/ mol 37
?
146

vr ms , a m =
3RT
M am
v rm s, pr =
=
3RT
M pr
=
3831310
8910 - 3 = 2910 2 m/ s
3831310
510
= 12 m/ s
216 2 35 U 23 8 U :
235 U F 6 23 8 U F 6 23 5 U F 6
, ( v ) ,
23 5 U F 6
993% 238 U 07% 235 U , 235 U
99% , 2 35 U F 6 2 38 U F 6
349 352
1 2 238 U 23 5 U ,
1/ 2 = 993235/ ( 07238)
1 , 1v 1/ 2v 2,
, ( 1v 1) v 1/ [ ( 2v 2) v 2] =
1v 1 2 / ( 2v 2 2 ) , , 1v 1 / 2v 2 ,
1v 1
238/ ( 2v
2 235) , 235 U
2v
A =
2 235
1v 1
238 + 2v
2 235
( v
=
2/ v 1)
993/ 07 + ( v 2/ v 1)
v 1 =
8kT
, v2 =
m 1
8kT
m2
147

A =
A =
099,
= 2ln 99993
07
= 2ln 99993
07
( m 1/ m 2) / 2
993/ 07 + ( m 1/ m 2) / 2
ln m 1
m 2
ln 352
349 = 223103
217
RT
ln 0
N A =
4 3 a 3 ( -
) g h
0 h ,
, , a
( 1) ;
( 2) a = 021210 - 6 m ,
- = 0206 7g/ cm 3 , t = 20, 3010 - 6 m
7 160, 3 360, 1 620, 860
,
,
( 1)
ln 0 =
mg h
kT
mg h h ,
( m -
h
m W ) g , mg ,
( m - mW ) g h = 4 3 a 3 ( -
) g h
m W
ln 0 = 1
kT 4 3 a 3 ( -
148
) g h = N A
RT 4 3 a 3 ( -
) g h

RT
ln 0
N A =
4 3 a 3 ( -
( 2)
) g h
860/ 1620 = 053, 1620/ 3360 = 048, 3360/ 7160 = 047
,
N A =
=
3RT
4 a 3 ( -
1
ln
) gh 2
3831293ln ( 7160/ 3360)
4 ( 021210 - 6 ) 3 0206710 3 983010 - 6
= 7610 23 mol - 1
218 20L, 11k g CO 2 ,
13, ( a = 36410 5
P a ¡¤L 2 / mol 2 , b =
00427L/ mol) ,
CO 2 ?
p V =
=
mRT
MV - mb - m 2 a
M 2 V 2
11831286
4410 - 3 2010 - 3 - 110042710 - 3
-
11 2 36410 5 10 - 6
44 2 10 - 6 20 2 10 - 6
= 25710 6 Pa
p I =
p in t =
mRT
MV =
m 2 a
M 2 V 2 =
11831286
4410 - 3 2010 - 3 = 29710 6 Pa
11 2 36410 5 10 - 6
( 4410 - 3 ) 2 ( 2010 - 3 ) 2 = 56910 5 Pa
219 212 215, CO 2
149

K
( 1) ( ( 236) ) p = f ( v)
a = 3V 2 m , cp c , b = Vm , c
3
( 2) 212 K
CO 2
a b
( 1) ( ( 236) )
p =
p c =
RT
V m - b - a
V 2 m
RT c
V m , c - b - a
V 2 m , c
,
dp
dV m c = - RT c
( Vm , c - b) 2 +
2a
V 3 m , c
= 0
RT cV 3 m , c = 2a ( V m, c - b) 2
,
/ ,
d 2 p
dV 2 m c = 2RT c
( V m, c - b) 3 -
6a
V 4 m , c
= 0
RT cV 4 m , c = 3a ( V m, c - b) 3
V m, c = 3 2
( Vm , c - b)
150

=
RT c =
,
a =
V m, c
3
8a
9V m , c
3V 2 m, cp c
( 2) 212 V m, c = 95510 -
10 6 Paa , b ,
a = 3V 2 m , cp c
5 m 3 / mol, p c = 732
= 3( 95510 - 5 ) 2 73210 6 = 0200 J ¡¤m 3 / mol 2
b = Vm , c
3
= 95510- 5
3
= 31810 - 5 m 3 / mol
220 , 100,
1atm, 30110 4 cm 3 / mol
( 1) cm 3 ?
( 2) 1cm 2 ?
( 3) ,
1cm 2
?
( 4) ,
?
( 1) cm 3
= N A
V m
= 6021023
30110 4 = 20010 19 cm - 3
( 2) 1cm 2
= 1 4 v = 1 4 8RT
M
151

= 1 4 20010 8831373
19
10 6 1810 - 3
= 33110 2 7 m - 2 ¡¤s - 1 = 33110 23 cm - 2 ¡¤s - 1
( 3) , 1cm 2
1cm 2 , 331
10 2 3 cm - 2 ¡¤s - 1
( 4) ,
, 1at m
221 : , u
v :
u
=
[ : u 1 2 v1, v2
v1
2 v
, ]
v 2 ,
u 12 = v 1 - v 2
u 2 1 2 = v 2 1 + v 2 2 - 2v 1v2 cos
v1 v2 ,
u 2 1 2 = v 2 1 + v 2 2 - v 1 v2 cos
, 0180,
cos = 0
u 2 12 = u 2 , v 2 1 = v 2 2 = v 2 ,
u 2 = 2 v 2
u 2 = 2 v 2
T
,
152

u
=
2 v
222 3810 -
=
kT
2 d 2 p
13810 -
23 273
=
2 ( 3810 - 10 ) 2 10110 5
= 5810 - 8 m
10 m
=
v =
M
8RT
= 5810 - 8 2810 - 3
8831273 = 1310- 10 s
223 10 -
2 m, 133
10 - 3 Pa, 310 - 10 m, 27
=
p
kT = 13310 - 3
13810 - 23 300 = 7 32101 m - 3
1 =
2 d 2
=
( 10 -
1
= 78 m
2 ( 310 - 10 ) 2 17
3210
2
m) ,
,
,
= 10
- 2 m,
z = v
=
1
8RT
M
153

=
1 8831300
10 - 2 2910 - 3
= 4710 4 s - 1
224 160km , 1510 -
500K 3010 -
=
=
1 =
2 d 2 =
9 kg/ m 3 ,
1 0
m ,
M
2 d 2 A
2910 - 3
2 ( 310 - 10 ) 2 1510 - 9 60210 23
80 m
=
v =
M
= 80 2910 - 3
8RT 8831500 = 013 s
225 ,
,
d
: ( 1) ; ( 2)
( )
( 1) = d 2 + d e
2
d e d ,
v e
2
= d 2 / 4
( 2)
v e
=
z = u
, z
v e
, u = ve
154
1 =
= 4
d 2

226 60km/ h, ,
500m/ s
110 -
??
N =
1 0 s1cm ,
s1
vW =
0013600
6010 3 10 - 1 0 = 610 6
s = N
= N v
= 610
650010 - 10 = 03 m
227 ( He) =
10 - 5 Pa ¡¤s, M = 0004kg/ mol, v = 12010 3 m/ s, :
( 1) ;
( 2)
( 1) = 1 3 v ,
= 3
v = 318910 - 5
000412010 3 / ( 22410 - 3 )
= 26510 - 7 m
( 2)
= 1/ ( 2 d 2 ) =
d =
=
1/ 2
kT
2 p
kT
2 d 2 p
13810 - 23 1/ 2
273
2 10110 5 26510 - 7
= 17810 - 1 0 m
189
228 l = 04cm, t = 27
N 2, N 2 d = 3710 - 10 m,
, N 2 ?
155

k =
1 3 mv
cV
1 = ,
2 d 2
, 1/ , k p = kT , , k
= l , ,
l , k , p
:
p =
=
kT
2 d 2 l
=
kT
2 d 2 p =
13810 - 2 3 300
= 17 Pa
2 ( 3710 - 10 ) 2 - 2
0410
2291mm 20K ,
,
, ?
a ir = 23810 - 2 W / ( m ¡¤K) , gla ss = 072W / ( m ¡¤K )
Q 1
= Q 2,
Q 1 = - a ir
S t
x 1
Q 2 = - gl as s
l
T
S t
x 2
T T
a ir = gl as s
x 1 x 2
156

x 2 = x 1
gla s s
ai r
= 1 072 = 303 mm
23810 - 2
230 R ,
,
, ,
D , W :
( 1) W = 4 D( - ) R;
( 2) t = W R 2 / ( 3D( - ) )
( 1) , r
dM
dt
= - D d
dr 4 r 2 = - 4 D d
dr r 2
,
d=
d
dr r 2 = C
C = ( -
C dr
R
r 2
) R
W =
dM dt
= - 4 D C =
4 D ( -
) R
( 2)
t =
4 3 R dM
3
W
dt =
W R 2 / ( 3D ( - ) )
157

3
3131
,
BC p V
( 1) A
T A = 300K , B, C D
( 2) A D
?
( 3) VT
3131
,
V B
( 1) AB : T B = T A
V A
BC : T C = T B = 600 K
= 300 20
10 = 600 K
CD : T D
= T C
V D
VC
= 600 20
40 = 300 K
( 2) A = A A B + A B C + A C D
= p A ( V B - V A ) + p B VB ln VC
V B
+ p C( V D - V C)
= 2( 20 - 10) + 220ln 40
20
+ 1( 20 - 40) 10110
2
158
= 28110 3 J

( 3) VT 32
32 31 ( 3)
321mol V 1 V 2
?a = b = 0 ,
p =
A=
V m2
V m1
p dV m
RT
V m - b - a
V 2 m
=
V m2
V m1
= RT ln V m2 - b
V m1 - b - a 1
V m 1
, a = b = 0,
A =
RT
V m - b - a
V 2 m
RT ln V m2
V m1
-
1
V m2
dV m
33 33 a a cb
b , 560J , 356J
159

( 1) a d b b , 220J ,
?
( 2) b ba
a , 282J
, ?
, ?
E b - E a = Q a cb - A a c b =
560 - 356 = 204 J
( 1) Q a d b = E b - E a + A a d b =
204 + 220 = 424 J
( 2) Q ba = E a - E b + A ba =
- 204 + ( - 282) = - 486 J
486J
3333
3464g 050, ( 1) ;
( 2) ?
??
( Q= 1) CV , mT = 64
32 5 831( 50 - 0)
2
= 20810 3 J
= Q = 20810 3 J
A= 0
( 2) Q= Cp , m T = 64
32 5 + 2 831( 50 - 0)
2
= 29110 3 J
= 20810 3 J
A= Q - E = ( 291 - 208)10 3
= 08310 3 J
3510g 10 3 J ,
160

300K, ?
Q = C p , m T =
m M
2QM
T 2= T 1 +
( + 2) Rm
= 300 +
+ 2
R( T 2 - T 1)
2
210 3 4
( 3 + 2)83110 = 319 K
36 40010 5 Pa
, 00 500 , 6010 4 J
( 1) ?
( 2) ?
( 3) ?
( 4) ,
?
( 1) Q = C p , m T = + 2 R T
2
2Q
=
( + 2) R = 26010 4
= 413 mol
( 5 + 2)83150
( 2) E =
C V , m T =
2 RT
= 413 5 2 83150 = 429104 J
( 3) A = Q - E = ( 60 - 429)10 4 = 17110 4 J
( 4) Q = = 42910 4 J
37 , 1at m, 10L,
300K ( 1) ; ( 2) 400K
, ? ?
( Q= 1) C V , m T = 2 R T =
p 1V 1
T
2 T 1
= 5 1010 - 3
2 1013105 100
300
161

= 084410 3 J
( 2) Q= Cp , mT = + 2 R T = + 2
2
2
p 1V 1
T
T 1
= 7 5 2 101310 1010 - 3
100
300
= 11810 3 J
38 p V
p V ( p , V) ,
dp
dV T
dp
dV S
=
=
dp
dV S
d
dV
d
dV
C
V
T
C
V
S
dp
dV T
= -
p
V ,
= - p V
= > 1
393mol 2atm 40L,
,
;
( 1) ;
( 2)
( 3) p V
( 1) :
p 2= p 1( V 1/ V 2) = 2( 40/ 20) 14 = 528 atm
T 2= p 2V 2
R = 5281013105 2010 - 3
3831
( 2) Q = 0 + RT 2ln V 1
V 2
162
= 429 K

= 3831429ln 40
20
A=
= 74110 3 J
=
1
V1
( p 1V 1 - p 2V 2) + RT 2ln
- 1 V2
1
( 240 - 52820)1013102
14 - 1
+ 3831429ln 40
20
= 09310 3 J
= Q - A = ( 741 - 093)10 3 = 64810 3 J
( 3) 34
3439( 3)
3102mol 300K 005m 3 ( 1)
; ( 2) ; ( 3) ,
025m 3
?p V
( 1) :
T 2=
T 1( V 1/ V 2) - 1
A S= - E = CV , m( T 1 - T 2)
163

35 310
= 2 5 2 831300[ 1 - ( 005/ 025) 1 4- 1 ]
= 59110 3 J
( 2) :
A T = RT 1ln V 2
V 1
( 3) :
= 80210 3 J
A p= p 1( V 2 - V1)
= RT 1
( V 2 - V1)
V 1
= 2831300ln 025
005
= 2831300 ( 025 - 005) = 19910 3 J
005
, ,
, p V ( 35) :
311 : p h
164

p = p 0 1 -
Mg h
Cp , mT 0
/ ( - 1)
p 0, T 0 h = 0 , Cp , m
, ,
,
( 115) ,
dp = -
p Mg
RT dh
, ( 325) ,
,
,
dp =
dT = -
p - 1 T - = C3
T = T 0 -
p
dT
( - 1) T
- 1 Mg
R dh
dp
p = - Mg dh
= -
- 1 Mg
R T 0 -
R h
,
ln p p 0
=
- 1 Mg
R h
- 1 ln 1 - - 1 Mg
h
RT 0
Mg dh
RT 0 1 -
- 1 Mg
h
RT 0
165

p = p 0 1 -
- 1
R
Mg
h
T 0
/ ( - 1 )
= p 0 1 -
Mg h
Cp , m T 0
/ ( - 1)
312 366m ,
0152m, 454kg,
098m, 311m/ s, 24310 8 Pa
, :
=
( 1) ?
12
( 2) ( )
l = 366m, D = 0152m , m = 454kg, l 1 =
098m , v 1 = 311m/ s, p 1 = 24310 8 P a, = 12
( 1) :
p 2= p 1
A=
V 1
V 2
= p 1
= 050010 8 Pa
=
l 1
l
1
( p 1V1 - p 2V 2)
- 1
= 24310 8 098
366
1
( 243098 - 0500366)
12 - 1
4 01522 10 8
= 49910 6 J
( 2) A= 1 2 mv 2 -
v =
2A
m + v 2 1 =
1
2 mv 2 1 ,
249910 6
454
12
+ 311 2 = 563 m/ s
313 : p 1,
V 1, ( 312) p 2,
V 2,
166

E 1 + p 1V1 = E 2 + p 2V2
E + p V H , ,
, ,
, p 1
, ,
p 2, ,
A = p 2V 2 - p 1V1
,
0 = E 2 - E 1 + A = E 2 + p 2V 2 - ( E 1 + p 1V 1)
E 1 + p 1V1 = E 2 + p 2V2
314:
, p = p 1 = p 2
Q= E 2 - E 1 + A = E 2 - E 1 + p ( V 2 - V 1)
= E 2 - E 1 + p 2V 2 - p 1V 1
= ( E 2 + p 2V 2) - ( E 1 + p 1V 1) = H 2 - H 1
315 1mol :
( 1) , ,
, CV , mT ( CV , m
)
( 2) , , ,
,
p in = a / V 2 m(
( 235) )
: , V m, 0 Vm
a / V m , 0 - a / V m
( 3)
E = CV , mT -
a
V m
+ E 0
167

E 0
( 1) E k = CV , mT
( 2) ,
,
E p0 - E p = -
V m
V m, 0
p indV =
a
Vm
-
a
V m, 0
( 3) ,
E = E k + E p = CV , mT -
E 0 E p0, a / V m , 0
a
V m
+ E 0
316 :
, T 0, p 0, V 0,
,
T 1 p 1 T 2
V 1,
= ( p 1 - p 0) V 0
( V 1 - V 0) p 0
T 0 = p 0V 0
R
T 1 = p 1V 1
R = p 1V 0
R
T 2 = p 2V 2
R = p 0V 1
R
,
,
168
CV , m ( T 1 - T 0) = Cp , m( T 2 - T 0)

=
Cp , m
CV , m
= T 1 - T 0
T 2 - T 0
=
( p 1 - p 0) V0
( V 1 - V 0) p 0
317
p V =
,
( 1) = 0, 1, ?
( 2) :
p 2V2 - p 1V 1
- 1
( 3) :
Cm
= CV , m
( 1) Cm
-
1 -
( 1) = 0, p V = p = , ;
= 1, p V = p V = , ;
= , p V = p V = , ;
= , p 1 V = p
1
V = V =
( 2)
A = - V 2
p dV =
V 1
- V 2
V 1
=
1
( p 2V 2 - p 1V 1)
- 1
C
V dV =
,
C
- 1 ( V 1-
2 - V 1-
1 )
( 3) 1mol T 1 T 2
Q= E + A = E -
= CV , m( T 2 - T 1) -
Cm=
Q
T 2 - T 1
= CV , m -
= CV , m -
CV , m - CV , m
- 1
A
R
- 1 ( T 2 - T 1)
R
- 1
= CV , m
-
1 -
169

= 0 , , Cp , m = CV , m;
= 1 , , CT , m = , ;
= , , CS , m = 0, ;
= , , CV , m = CV, m
318 10mol ,
, 209J, 1K
, ?
E =
CV , m T = 10 3 8311 = 1247 J
2
Q = + W = 1247 - 209 = - 843 J
Q
Cm =
T = - 843
101 = - 843J/ ( mol ¡¤K)
319 36 40L
, ,
, A , B
1mol ,
101310 3 Pa ,
B , A
, :
( 1) B ,
;
( 2) ;
( 3) B
( 1) A ,
p A V
A =
C = p A 1V
A 1
170
= 101310 5 002 1 4 = 4210 2
p A = p B, VA = V - VB = 004 - VB
36
319

p B( 004 - V B) = 4210 2
B , p B
= RT B
VB
,
( 2)
T B ( 004 - V B) = 51V B
T A 2= T A 1
T B 2=
= p A 1V A 1
R
= 322 K
V A 1
V A 2
- 1
V A 1
V A 2
- 1
51V B 2
( 004 - V B 2) 1 4 =
( 3) Q B = E B + A B
= 1013105 002
002
831
001
51003
( 004 - 003) 14 = 965 K
0 4
= 2 R( T B 2 - T B 1) + V B2
V B1
p BdV B
= 2 R T B 2 - p B 1V B 1
R
+ V B2
4210 2
dV
V B1
( 004 - V B) B
= 2 R T B 2 -
p B 1V B 1
R
+ 42102
1 -
( 004 - V B1) 1- - ( 004 - VB 2)
1-
= 5 2 831 965 - 101310 5 002
831
+ 4210 2
1 - 14 [ 002 - 0 4 - 001 - 04 ]
= 16610 4 J
320 37 T V
mol , CV , m
a
171

T 1, V 1, b V 2, ca :
( 1) c :
( 2)
( 1) ca ,
T c =
T a
Va
V c
- 1
= T 1
V 1
V 2
- 1
( 2) a b ,
37 320
Q 1 = RT 1ln V 2
V 1
bc ,
Q 2= CV , m ( T b - T c ) = CV , mT 1 1 -
= CV , mT 1 1 -
= 1 -
Q2
Q1
= 1 -
V 1
V 2
- 1
CV , m
R
321
( )
a b cd bc
d a ( 38) ,
1 -
V 1
V 2
ln V 2
V 1
T c
T 1
- 1
= 1 -
V 1
V 2
V
1
V2
- 1
- 1
V
1
- 1
V 2
bc
38321
Q 1 = Cp , m ( T c - T b)
172

d a
Q 2 = CV , m ( T d - T a )
= 1 -
Q 2
Q 1
= 1 -
cd ,
a b ,
bc ,
T d
T a
CV , m( T d - T a )
Cp , m ( T c - T b) = 1 - 1
T c
T d
T b
T a
=
=
T c
T b
V 1
- 1
V 1
V1
V2
- 1
= V 1
V 2
= T d
T b
T c
T c T a T b
,
= 1 -
V 2
- 1
V1
V2
=
V 1
- 1
V 1
V 2
V 1
- 1
V 2
T b
T a
T d
T a
T c
T b
- 1
- 1
322 1854 39
, ab, cd , ef T , T 2 T 1
, bc, d e, f a
cd ef
, T 1, T 2, T ,
a b
173

Q a b =
cd
Q cd
ef
Q ef
RT ln V b
Va
= RT 2ln Vd
V c
= RT 1ln V e
T V
b
- 1
Vf
= T 2V
c - 1
, T 2 = T 1
V e
Vd
39 322
- 1
, T = T 1
V f
V a
- 1
Q cd
= Q ef ,
V b
V a
= V e Vc
V f V d
ln ( V d/ V c)
ln ( V e/ V f ) = T 1
T 2
= 1 -
Q ef
Q a b +
Q cd
= 1 -
R T ln V b
V a
RT 1ln Ve
V f
+ T 2ln V d
V c
= 1 -
T
ln Ve
V f
T 1ln V e
V f
- ln V d
Vc
+ T 2ln Vd
V c
= 1 -
T 1T 2
T ( T 2 - T 1) + T 1T 2
323 ,
174

1 2
:
=
1 + ( 1 - 1) 2
T 1, T 2, T 3 , Q 1 Q2
, A 1 A 2
,
A 1 =
1Q1, A 2 = 2Q 2 = 2( Q 1 - A 1) = Q 1 2( 1 - 1)
1 = 1 -
= A 1 + A 2
Q1
T 2
, 2 = 1 -
T 1
= 1 + ( 1 - 1) 2
T 3
,
T 2
= 1 -
T 2
T 1
+ 1 - 1 -
T 2
T 1
1 -
T 3
T 2
= 1 -
324
25, 300m 5
( 1) ?
( 2)
1MW, ?
( 3) 25
5 , 25
?
T 2 278
( 1) = 1 - = 1 -
T 1 298 = 67%
( 2) = 1 - Q 2/ Q 1 = 1 - Q2/ ( A + Q 2)
T 3
T 1
175

Q 2 =
A( 1 - )
= 10 6 ( 1 - 0067)
0067
14MW
( 3) Q 1 = A + Q2 = CmT
= 1410 6 J
m = A + Q 2
C T = 1106 + 1410 6
41810 3 ( 25 - 5) = 1810 2 kg
1810 2 kg/ s =
6510 2 t / h
325 , -
10,
15 , 10 3 J
, ?
Q 2 =
AT 2
T 1 - T 2
w = Q 2
A = T 2
T 1 - T 2
= 10 3 263
288 - 263 = 105104 J
326 310 (
212 ) , d a bc
a , b, c, d
:
T / p / kPa V/ m 3 E / kJ
a 80 2 305 0068 2 1 969 0%
b 80 2 305 0009 46 1 171 100%
c 5 363 0220 2 1 015 54%
d 5 363 0451 3 1 641 5%
?
( 1) ,
176

?
310 326
( 2) ,
( 3) , ?
( 4) ,
?
( Q 1) cd = E d - E c + A cd
= E d - E c + p c( V d - V c)
= ( 1641 - 1015)10 6
+ 36310 3 ( 04513 - 02202)
= 71010 5 J
( 2) - Q a b= - A a b - ( E b - E a )
= p a ( V a - V b) - ( E b - E a )
= 230510 6 ( 00682 - 000946)
- ( 1171 - 1969)10 6
= 93310 5 J
( 3)
A = A d a + A a b -
A bc -
A cd
177

= E a - E d + p a ( V a - V b) - ( E b - E c) - p c( V d - V c)
= ( 1969 - 1641)10 6 + 230510 6
( 00682 - 000946) - ( 1171 - 1015)10 6
- 36310 3 ( 04513 - 02202)
= 22310 5 J
( 4) w = Q 2
A
= Q cd
A = 71010 5
= 318
22310 5
w C =
T 2
T 1 - T 2
=
278
80 - 5 = 371
327 32,
21 3810 4 kJ/ h ,
?
A = Q 2
w > Q 2
w C
=
Q 2( T 1 - T 2)
T 2
= 38104 ( 305 - 294)
294
= 1410 3 K J/ h
P m in = A m in
t
= 14106
3600
= 039 kW
328
311 ,
1 2 ( T 1) 3 4
( T 2) ,
,
(
4 1
2 3
178
311328

, )
12
34
Q 1 = RT 1ln V 1
V 2
Q 2 = RT 2ln V 1
V 2
41 23 ,
w = Q A = Q2
Q1 - Q2
=
T 2
T 1 - T 2
= w C
329 : ,
,
t 1 = 210,
t 2 = 15,
t 3 = 60
,
,
1k g ,
?
?
33410 7 J/ k g
312 329
312 , T 1 = 483K , T 2 = 333K , T 3 = 288K,
Q 1 = 33410 7 J
A = Q 1 = 1 -
T 2
T 1
Q 1
179

Q 2= Q 1 - A = T 2
Q 1
T 1
Q 2 = Q 1 + A = A( + 1) = A
T 3
T 2 - T 3
+ 1
Q 2
+ Q 2 =
= A
=
T 2
T 1
+
333
483
= 99810 7 J
T 2
T 2 - T 3
T 2( T 1 - T 2)
T 1( T 2 - T 3)
Q1
= ( T 1 - T 2) T 2
T 1( T 2 - T 3) Q 1
333( 483 - 333)
+
483( 333 - 288) 334107
99810 7
44310 7 = 299( )
180

4
4. 11mol ( ) 4. 1
a b c
4141
a bc ,
A= 1 2 ( p a + p b) ( Vb - V a ) + 1 2
( p b + p c) ( Vc - V b)
= 1 2 ( 8 + 6)10 5 ( 2 - 1)10 - 3
+ 1 2 ( 8 + 4)105 ( 3 - 2)10 - 3
= 1. 310 3 J
Q= E c - E a + A = CV , m( T c - T a ) + A
=
2 R( T c - T a ) + A =
2 ( p cVc - p a V a ) + A 181

= 5 2 ( 43 - 61) 105 10 - 3 + 1. 310 3
= 2. 810 3 J
S= CV ln T c
T a
= R 2 ln p c
p a
+ Rln V c
V a
=
+ + 2 ln V c
2 V a
p cVc
Rln
2 p a V a
= 8. 31 5 2 ln 4 6 + 5 + 2
2 ln 3 1
= 23. 5 J/ K
+ Rln V c
V a
4. 2 30g, - 40 100
c1 = 2. 1J/ ( g K ) , , c2 = 4. 2
J/ ( gK ) , 1. 01310 5 Pa = 334J/ g,
L= 2260J/ g
- 400
d Q
T = T 2 c1mdT
R T
S 1 = T 2
R T 1
0
T 1
S 2 = R
d Q
T
2
T 2
= Q 2
T 2
0100
d Q
T = T 3 c2mdT
R T 2
T
S 3 = T 3
R T 2
= c1mln T 2
T 1
= m
T 2
= c2mln T 3
T 2
100100
d
S 4 =
Q
= Q4
= Lm
R T 3 T 3 T 3
T 3
- 40100
182
S= S 1 + S 2 + S 3 + S 4

= m c1ln T 2
T 1
+
+ c2ln T 3
+ L T 3
T 2
T 2
= 30 2. 1ln 273
233 + 334
273
= 268 J/ K
373
+ 4. 2ln
273 + 2260
373
4. 3 1mol 1. 01310 5 Pa 300K
1200K Cp , m =
a + bT , a = 2. 310 4 J/ ( mol ¡¤K ) , b = 592 J/ ( mol ¡¤K 2 )
S =
T 2
R T 1
= a ln T 2
T 1
d Q
T = R T 2
T 1
Cp , mdT
T
+ b( T 2 - T 1)
= T 2
R T 1
( a + bT )
dT
T
= 12. 310 4 ln 1200 + 15. 92( 1200 - 300)
300
= 3. 7210 4 J/ K
4. 4 810 6 J ,
? , 273K
T 1 = 36= 309K, T 2 =
273K ,
S = S 1 + S 2 = - Q
T 1
= 3410 3 J/ K
+ Q T 2
= 810 6 - 1
309 + 1
273
45210 8 J/ h
20, - 20, ( J/ ( K
s) ) ?
S = S 1 + S 2 = - Q + Q - 1
= 210 8
T 1 T 2
293 + 1
253
= 10810 5 J/ K
183

S
t =
108105 / 3600 = 30 J/ ( K ¡¤s)
46 ,
20k W ( J/ ( K ¡¤s) ) ?
12
S
t =
Q
T t =
20103
2851 =
70 J/ ( K ¡¤s)
47 65m,
23m 3 / s20, ?
T 1
:
g h
c =
mg h = cm( T 2 - T 1)
T 2 = g h
c + T 1
9865
4210 3 = 015 T 1 = 293
, 1
S= T 2
R T 1
d Q
T = R T 2
T 1
= cmln 1 + gh
cT 1
cmdT
T
= 23109865
293
= cmln T 2
T 1
cm gh
cT 1
T 2 T 2
= cmln T 1 + g h/ c
T 1
= mg h
T 1
= 5010 4 J/ K
48 ( 1) 1kg, 0 100 ,
, ,
?
184

( 2) 1kg , 0, 50
, 100
( ) , ( 1)
( 1) S =
T 2
S w a t er + S r es er voir =
T 1
= cmln T 2
+
T 1
- cm( T 2 - T 1)
T 2
cmdT
T
+ Q T 2
= 41810 3 1
ln 373
273 + - 41810 3 1( 373 - 273)
373
= 184 J/ K > 0
( 2) S = S w 1 + S w 2 +
= S w + S r 1 +
S r1 + S r 2
S r2
= cmln T 2 - cm( T
+ - T 1)
T 1 T
= 41810 3 1ln 373
273
-
41810 3 1( 323 - 373)
323
-
41810 3 1( 373 - 323)
373
= 97 J/ K ,
- cm( T 2 - T
+ )
T 2
, ,
, ,
49 25kg 07kg , 0
( 1) 0
( 2) 100
185

( 1) , 09kg
0
S w = - Q
T = - 334103 09
= - 11010 3 J/ K
273
, ( 0)
S e n = Q T = 33410 3 09
273
S w +
( 2)
S w = -
S e n = -
S w =
S en = 0
Q
T
= - 33410 3 09
373
= 11010 3 J/ K
1. 1010 3 J/ K
= - 08110 3 J/ K
S w + S en = ( 110 - 081)10 3 = 02910 3 J/ K
410 1mol ( = 14)
42 , 13
, 14
S =
S 3- S 1:
( 1) 123;
( 2) 13;
( 3) 143
( 1) S = S 12 + S 23
42410
186

= T 2
T 1
Cp , mdT
T
+ T 3
T 2
CV , mdT
T
= Cp , mln T 2
+ CV , mln T 3
T 1
T 2
T 1 = T 3, Cp , m = CV , m + R,
( 2) S = Q T 1
( 3) S =
S =
=
S 1 4 +
1
T 1
= Cp , mln T 3
T 4
Rln T 2
T 1
RT 1ln V 2
V 1
= Rln V 2
V 1
T 3
S 43 = 0 +
T 4
= Cp , mln T 1
T 4
= Rln V 2
V 1
Cp , mdT
T
p 1V 1 = p 3V 2 T 1/ T 4 = ( p 4/ p 1) 1 -
1-
= ( p 3/ p 1)
=
( V 1/ V 2)
1-
,
S =
Cp , mln T 1
T 4
= Cp , m
S ,
S =
Rln V 2
V 1
1 -
ln V 1
V 2
= 18. 31ln 60
20
= Rln V2
V1
= 913 J/ K
411 , ,
, m, , T 1
T 2 >
T 2,
T 1
, cp
T 0 = ( T 1 + T 2) / 2
( m mol)
S= S 1 +
S 2 =
T 0
T 1
mcp
T dT + T 0
T 2
mcp
T dT
187

= mcpln T 2 0
T 1T 2
= mcp ln ( T 1 + T 2) 2
4T 1T 2
( T 1 + T 2) 2 > 4T 1T 2, S >
0
412 T 1 = 300K , p 1 = 3039
10 5 Pa , V 1 = 4m 3 16m 3 ,
,
14)
( 1) p V T S
( 2) ( =
( 1) 43
43412
( 2)
A T=
RT 1ln V2
V1
= p 1V 1ln V2
= 303910 5 4ln 16
4 = 16910 6 J
V1
188

S T = Q Rln
T 1= V2
V1
= p 1V 1
ln V 2
T 1 V 1
= 3039105 4
ln 16
300 4 = 563103 J/ K
A V = 0,
T 3
S V =
T 2
CV , mdT
T
= CV , mln T 3
T 2
T 3/ T 2 = T 3/ T 1 = ( V 1/ V 2) - 1 ,
S V =
CV, mln V 1
V 2
= p 1V 1
( - 1) CV , mln V1
RT 1
V2
- 1
= ( - 1) CV , mln V 1
V 2
= p 1V 1
ln V 1
T 1 V 2
= 3039105 4
ln 4 300 16 = - 563103 J/ K
A S=
1
p 1V 1
( p 2V 2 - p 1V 1) =
- 1 - 1
= 3039105 4
14 - 1
= - 13010 6 J
4
16
14- 1
S S = 0
A = A T + A V + A S
- 1
V1
V2
- 1
- 1
= 16910 6 + 0 + ( - 13010 6 ) = 03910 6 J
S =
S T + S V + S S
= 56310 3 + ( - 56310 3 ) + 0 = 0
413 mol
, p 1
p 2,
189

S > 0
S = Rln ( p 1 + p 2) 2
4p 1p 2
T T , V1 V 2
,
, ,
1
2 CV , mT +
1
2 CV , mT =
1
T = T
2 ( + ) CV , mT
, ,
, V = ( V 1 + V 2) / 2,
V 1
+ V 2
p =
2 RT
V 1 + V 2
=
2 RT
RT
+ RT =
p 1
p 2
2p 1p 2
p 1 + p 2
V ,
S= S 1 +
= Rln
S 2 = Rln V + Rln V
V1 + V2
2V 1
= Rln ( V 1 + V 2) 2
4V 1V 2
= Rln ( p 1 + p 2) 2
4p 1p 2
V 1
+ Rln V 1 + V 2
2V 2
, , ,
190
V2

( p 1 + p 2) 2 > 4p 1p 2, S >
0
414 ,
( m, cp , T ) ( m ,
cp , T ) ,
,
cpm( T - T ) = cp m ( T - T )
T T
S = cpmln T cp m ln
T + T
T
d( S) = cpm
T dT +
m dT
T = 0
cp
- cp mdT = cp m dT
cpm
T dT -
cp mdT
T = 0
T = T
415 ( Cp
)
T i( ) ,
T f:
A min = Cp
T i
2
T f
+ T f - 2T i
T f ,
191

A + Cp ( T i - T f) = Cp ( T f - T i)
T f = A Cp
+ 2T i - T f
( , 0)
S = Cpln T f
T i
S 0
+ Cp ln T f = Cpln
T i
T fT f
1
T i
2
T f
A
Cp
A min = Cp
+ 2T i - T f T 2 i
T f
T 2
i
T f
+ T f - 2T i
T fT f
T 2 i
416 ( ( 236) )
( 315) : 1mol (
CV , m
)
S = CV , mlnT + Rln( V -
1mol
( 416)
E p ,
192
E = CV , mT -
p =
a
V m
RT
V m - b - a
V 2 m
T dS = dE + d A =
+ E 0
b) + S 0
dE + p dV

T dS = CV , mdT +
= CV , mdT +
a
dV
V 2 m +
m
RT
Vm - b dV m
dS = CV , mdT / T + RdVm / ( V m - b)
dS =CV , mdT / T + R
V m - b dV m
RT
Vm - b dV m - a
V 2 dVm
m
S - S 0 = CV, mlnT + Rln( V m - b) + C
C , C + S 0 = S 0,
S = CV , mlnT + Rln( V m - b) + S 0
417
, 44 ,
( H m , S m )
:
?
?
?
( 1) 80K, 50MP a 20MPa ,
( 2) 70K, 20MPa 01MPa ,
( 3) 76K , 50MPa 01MPa ,
44
( 1) , T 1 = 95K ;
( 2) , T 2 = 65K ;
( 3) , T 3 = 20K,
,
193

194
44

3
1
11 a , q, 2q,
- 4q2q, ,
11 , 2q
q 0 , q 0 q
- 4q ,
F =
=
qq0
+
4 0r 2
5q
2 0a 2
4qq0
=
4 0r 2
12 -
5qq0
2 0a 2
q
r
1111
, Q( Q > 0)
, Q ?
12 , Q 0, Q
- q
2F 1cos30 - F 3 = 2 q 2
3
4 0r 2 2
-
qQ
4 0l 2
195

=
1212
q
( 3 q - 3Q)
4 0r 2
Q = 3 q/ 3
13 13 , ,
- q, Q, - q Q
, cot 3 = q 2 / Q 2
13 , -
q
q 2
= 2qQcos
4 0( 2r cos ) 2 4 0r 2
q/ ( 2Q) = 4cos 3
, Q
Q/ ( 2q) = 4sin 3
, ,
cot 3 = q 2 / Q 2
1313
14 u d u
196

2 3 e , d 1 3 e ,
( 1010 -
15 m)
F =
qdqu
=
4
2 0r 2
910 9 1 3 2 3 ( 1. 610- 19 ) 2
( 1010 - 15 ) 2 = 512 N
15,
10 - 21 e , 10 - 21 e
, (
8 8 8 ) ?
, ,
?
qm a x = ( 8 + 8 + 8)( 10 - 2 1 e) = 2410 - 21 e
f G = G m 2
f e/ f G =
r 2
©¦ f e ©¦=
q 2 ma x
4 0r 2
qm 2 a x
= ( 2410 - 2 1 1. 610 - 19 ) 2 910 9
4 0Gm 2 66710 - 11 ( 161. 6710 - 27 ) 2
= 2. 810 - 6 1
16 p = ql,
r ( r l ) E= 2p / ( 4 0r 3 )
+ q - q r
197

E = E + - E - =
=
q
4 0 r -
2p r
4 0 r 2 -
l
2
l 2 4
2 -
2
q
4 0 r + l 2
2
r l,
E =
2p
4 0r 3
17 p
, x
x , xy
E ( x , y ) r P ( x , y ) , p r
r , 11 16
E ( x , y ) =
p ( 2x 2 - y 2 )
i +
4 0( x 2 + y 2 ) 5 / 2
14 , p
p r
P
3p x y
j
4 0( x 2 + y 2 ) 5/ 2
= p cos , p = p sin
1417
198

E r
= E rcos i + E rsin j
=
2p r
cos i +
4 0r 3
= 2p cos 2
i +
4 0r 3
=
E = -
= -
=
2p x 2
i +
4 0( x 2 + y 2 ) 5/ 2
2p r
sin j
4 0r 3
2p cos
sin j
4 0r 3
E sin i + E cos j
p
sin i +
4 0r 3
- p y 2
i +
4 0( x 2 + y 2 ) 5/ 2
P
E = E r + E =
2p x y
j
4 0( x 2 + y 2 ) 5/ 2
p
cos j
4 0r 3
p ( 2x 2 - y 2 )
i +
4 0( x 2 + y 2 ) 5 / 2
p x y
j
4 0( x 2 + y 2 ) 5/ 2
3p x y
j
4 0( x 2 + y 2 ) 5/ 2
18 , 2a ,
+ - ,
E = / ( 4 0a ) ,
F =
E =
2
4 0a
,
19L,
L r( r > L/ 2)
15 , dq= dx P
1519
199

dE =
dx
4 0( r - x ) 2
P
L / 2
E = dE
dx
= =
- L/ 2 4 0( r - x ) 2
x
11016,
, R,
= 0sin
16 , dq
= Rd = 0sin Rd
dE =
0sin d
4 0R
L
4 0( r 2 - L 2 / 4)
dE x = dE cos = 0sin cos d
4 0R
dE y = - dE sin= -
0 2,
E x = dE x = 0
E y = dE y
=
2
E = E xi + E yj = -
0
16110
0sin 2 d
4 0R
- 0sin 2 d
= - 0
4 0R 4 0R
0
4 0R j
111, ,
05m, 2cm , 31210 -
,
200
9 C

,
,
E =
d
=
4 0r 2
qd
4 0r 2 ( 2 r - d )
= 9109 3. 1210 - 9 0. 02
05 2 ( 20. 5 - 0. 02)
= 072 V/ m
,
112 17 , 2a ,
O
, 17 , d
dq dqO
, dq= adO
dE =
dq
=
4 0a 2
a d
=
4 0a 2
d
4 0a
O dqdq = rd /
sinO
dq
dE = =
4 0r 2
O dq
r d
4 0r 2 sin =
17112
d
4 0r sin =
d
4 0a
dE = dE ,
, dO
, O
113 ,
:
201

,
,
,
,
,
114 ( 1) q a ,
?
( 2) ,
?
( 1) 6 ,
q/ ( 6 0)
( 2)
q/ ( 8 0) ,
1 3 q =
8
q/ ( 24 0)
0
115, 0,
120V / m, ,
?
, S ,
q =
0 E ¡¤dS =
0E S
= q S = 0E
202

=
e =
0E e = 88510- 12 120
1. 610 - 1 9
= 6. 6410 9 / m 2 = 6. 6410 5 / cm 2
116,
100m 150N / C, 300m
100N / C ,
100m 300m S
,
q =
0 E ¡¤dS =
0S ( E 1 - E 2)
= q
ehS =
0( E 1 - E 2)
eh
= 1. 3810 7 / m 3
q ,
= 88510 - 1 2 ( 150 - 100)
1. 610 - 19 200
117, a ,
,
r,
E r
, ,
l r
,
E2 r l = qi nt / 0
, r < a , q int = 0, E in t = 0
, r > a , q int = 2 a l , E = a
0r
E r 18
18117
203

118 R 1 R 2,
+ -
,
( ) :
, r < R 1: E = 0
, R 1< r < R2: E =
2 0r
, R< r: E = 0
119 ,
1= 410 - 11 C/ m 2 2= - 210 - 11 C/ m 2
E 1 =
E 2 =
1
=
2 0
2
=
2 0
410 - 11
28. 8510 - 12 = 2. 26 V / m
210 - 11
28. 8510 - 12 = 1. 13 V / m
1 : E= E 1- E 2= 1. 13V / m, 1 ;
: E = E 1+ E 2= 3. 39V / m , 2 ;
2 : E= E 1- E 2= 1. 13V / m, 2
120 , D , ,
E d d ,
,
, 19( a)
E ¡¤2S = qin t / 0
d < D / 2 , qint = 2d S , E = d
0
204

19120
d > D / 2 , qint = DS , E = D
2 0
E d 19( b)
121R , ,
0,
x
R (
) ,
E =
0
-
2 0
0
1 -
2 0
x
( R 2 + x 2 ) 1/ 2 =
122 ,
R, ,
r ( r < R) ,
,
,
(
0x
2 0( R 2 + x 2 ) 1/ 2
110) 110122
205

P ,
E + =
E = E + + E - =
= a
3 0
r +
3 0
E - = - r -
3 0
( r + - r - )
3 0
a O+ O -
,
123 :
+ e ( r ) = - Ce - 2 r/ a 0 , a 0
, a 0= 05310 - 10 m, C - e
a 0 ? a0
a 0
a 0
E =
e = ( r) 4 r 2 dr = 4 Ce -
0
C = e/ ( a 3 0)
q= e -
a 0
206
( r ) 4
0
r 2 dr = e -
a 0
= 5ee - 2 = 1. 0810 - 19 C
0
0
?
2 r
a 0 r 2 dr = C a 3 0
4e
ex p -
a 3 0
2r
a 0
r 2 dr
q
= 5ee- 2
= 9109 51610 - 19 e - 2
4 0a 2 0 4 0a 2 0
( 05310 - 1 0 ) 2

= 34610 11 V/ m
124,
, :
=
b , b= 0. 2310 -
r = 1. 010 -
e
e -
8 b 3
r / b
15
m r
15
m
,
E =
qint
=
04 r 2
r = 1. 010 -
=
1
4 r 0r 4 r 2 dr =
2
0
r 0 r 2 e -
e
8 0b 3
r / b dr
e
[ ( - r 2 - 2br - 2b 2 ) e - r / b + 2b 2 ]
8 0b 2 r 2
15
m ,
E = 1. 210 2 1 N / C
125 ,
2e/ 3,
05010 -
1 5 m ; - 2e/ 3,
05010 - 1 5 m 1010 - 1 5 m
1010 - 1 5 m, 07510 - 1 5 m, 05010 - 1 5 m 025
10 - 1 5 m
+ =
2e/ 3
=
4
3 R3 1
- =
- 2e/ 3
=
4
3 ( R3 2 - R 3 1)
1610 - 1 9
= 2. 0410 26 C/ m 3
2( 0. 510 - 15 ) 3
- 1610 - 19
2 [ ( 10 - 15 ) 3 - ( 0. 510 - 1 5 ) 3 ]
207

= - 2. 9110 25 C/ m 3
,
r 1= 1. 010 - 15 m, E 1= 0;
r 2= 0. 7510 -
;
1 5 m
E 2 = [ 2 3 e + 4
-
3 ( r 3 2 - R 3 1) ] / ( 4 0r 2 2)
= [ 2 3 1. 610- 19 - 2. 9110 25 4 3
( 0. 75 3 10 - 4 5 - 1. 2510 - 46 ) ]
910 9 / ( 0. 7510 - 15 ) 2
= 1. 1410 21 V / m
r 3= 0. 510 - 15 m
;
E 3 = 2e/ 3 = 9109 21. 610 - 19
4 0r 2 3 3( 0. 510 - 1 5 ) 2
r 4= 0. 2510 -
E 4 =
= 3. 8410 21 V / m
1 5
m
4
+
3 r 3 4
= + r 4
= 2. 0410 26 0. 2510 - 1 5
4 0r 2 4 3 0 38. 8510 - 12
= 1. 9210 21 V/ m
126
31710 -
27 kg, 3480 ,
, ,
2910 -
7410 -
15 m,
15
m , 92e
208

=
92e
=
4
3 r 3 0
E =
- 1 9
921. 210
= 8. 6710 24 C/ m 3
4
3 ( 7. 15 410- ) 3
r = 8. 671024 2. 910 - 1 5
= 9. 4710 20 V / m
3 0 38. 8510 - 1 2
,
v =
E er
m = 9. 4710 20 1. 610 - 19 2. 910 - 1 5
3. 1710 - 2 7
= 1. 210 7 m/ s
E k = 1 2 mv 2 = 1 2 3. 1710- 27 ( 1. 210 7 ) 2
= 2. 210 - 13 J
L = mvr = 3. 1710 - 27 1. 210 7 2. 910 - 15
= 1. 110 - 34 kg ¡¤m 2 / s
=
v
2 r = 1. 210 7
= 6. 510 20 H z
22. 910 - 1 5
127 , r 0= 0. 53
10 - 1 0 m , - e,
E = 310 6 V / m ( ) ,
( ) ?
?
=
- e
=
4
3 r 3 0
- 1. 610 - 19
= - 2. 5710 1 1 C/ m 3
4
3 ( 0. 510 - 1 0 ) 3
209

,
r =
r
e + E e = 0
3 0
3 0E
- = 38. 8510 - 1 2 310 6
= 3. 110 - 16 m
2. 5710 1 1
p = er = 1. 610 - 19 3. 110 - 1 6 = 5. 010 - 35 C ¡¤m
128 ,
005nm ,
2e, ,
r
e 2
=
4 0( 2r ) 2
r = R/ 2
1 2e 2
r
3 0 4
3 R 3
2r = R = 0. 05 nm
1. 29111
E x = bx 1 / 2 , E y = E z = 0,
b= 800Nm -
1/ 2 / C:
( 1) ;
( 2) ?
a = 10cm
( 1)
210
111129

e = - E x , a a 2 + E x , 2a a 2 = - ba 1/ 2 a 2 + b( 2a ) 1/ 2 a 2
= ( 2 - 1) ba 5 / 2
= ( 2 - 1)80001 5/ 2
= 1. 05 N ¡¤m 2 / C
( 2)
q = 0 e = 8. 8510 - 12 1. 05 = 9. 2910 - 12 C
130x = + a x = - a + q
( 1) O ( dE / dx ) 0= - q/ ( 0a 3 ) ; ( 2)
p = p i ,
p ( dE / dx ) 0= - p q/ ( 0a 3 )
x
E = -
q
+
4 0( a - x ) 2
dE
dx x = 0
= -
q
= -
4 0( a + x ) 2
q 4a 5 + 8a 3 x 2 - 12a x 4
= -
4 0 ( a 2 - x 2 ) 4 x = 0
q 4a x
4 0 ( a 2 - x 2 ) 2
q
0a 3
p = q0l ,
+ q 0( + l/ 2 ) - q0( - l/ 2 ) ,
F = q0E + l / 2 - q0E- l / 2 = q0
dE
dx x = 0
= p dE
dx x = 0
l
= -
p q
0a 3
131 : p E
,
, p Ecos = p E
M= p E sin
, d
- Md = - p Esin d / 2
211

A = - Md= / 2
- p Esin d= p E cos = p ¡¤E
/ 2
132 + 1. 010 - 6 C - 4. 0
10 - 6 C, 10cm
( 1) q0 , ?
( 2) , q0
? q0
( 3) q0 ?
( 1) 112 , q 0 q1 q2 , q0
q 1 q2 q 1 x q0
, q1 ,
112132
q1q0
=
4 0x 2
©¦ q2©¦ q 0
4 0( x + r ) 2
x = r , - r / 3x = r= 0. 1m
( 2) q0 x x ,
dF
dx
F =
x = r
q0
q1
-
4 0 x 2
= - q 0 q1
-
2 0 x 3
= - q 0 1
-
2 0 r 3
©¦ q2©¦
( x + r ) 2
©¦q2©¦
( x + r) 3 x = r
1
2r 3 10 - 6
q 0> 0 dF < 0, q0 ; q0 < 0 , dF
dx x = r
dx
> 0, q 0
212
x = r

( 3) q0 x = r y ,
x
F =
dF
=
dy y= 0
q 0q 1
y
-
4 0( y 2 + r 2 ) 2 ( y 2 + r 2 ) 1/ 2
y
( y 2 + 4r 2 ) 1 / 2
= q0y q1
-
4 0 ( y 2 + r 2 ) 5/ 2
=
q0
q1
-
4 0 ( y 2 + r 2 ) 5/ 2
q0 1
-
4 0 r 5
4
( 2r ) 5 10 - 6
©¦ q 2©¦
( y 2 + 4r 2 ) 5/ 2
©¦ q 2©¦
( y 2 + 4r 2 ) 5/ 2 y = 0
q 0©¦ q2©¦
4 0( y 2 + 4r 2 ) 2
q 0 > 0 , dF > 0, q 0 ; q0 < 0 ,
dy y= 0
dF
< 0, q0
dy y= 0
133: ,
( : P + q,
, P ?
)
P q,
P , P
P P P
P ,
, P P
q ,
134113
, 10 -
5 m( 10 5
213

, ) ,
, ,
, ,
1. 510 -
10 k g,
- 1410 - 13 C , 20m/ s ,
1. 6cm. 1. 610 6 N/ C,
(
, ) ?
113134
= 1 2
Eq
m
l
v
2
= 1 2 1610 6 1. 410 - 13
1. 510 - 10
1610 - 2
20
= 4. 810 - 4 m = 0. 48 mm
2
214

2
21 , 2cm,
20cm, 10cm,
+ 2510 - 9 C - 2510 - 9 C ,
? ?
08c ,
?? ? 06c
, ?
,
E =
N = q e = 2510- 9
1610 - 19 = 1610 10 ( )
=
0
q
0S = 2510 - 9
88510 - 12 201010 - 4
= 1410 4 V / m
08c ,
l x = l x 1 - u 2 / c 2 = 20 1 - 08 2 = 12 cm
l y = l y = 10 cm
N = N = 1610 10 ( )
E = q
=
0
0S
= 2510 - 9
88510 - 12 121010 - 4
= 2410 4 V / m
06 c ,
215

l x = l x = 20cm , l y = l y = 10 cm
N = N = 1610 10 ( )
E = q
2510 - 9
= =
0
0S 88510 - 1 2 201010 - 4
= 1410 4
V/ m
22 ,
,
x y , v = vi x
?
,
E = k
2 0
E = rE =
2 0 1 - v 2 / c k 2
,
23 099c
10 -
E =
q
( 1 -
4 0r 2
2 )
10 m ?
= 9109 1610 - 19
( 10 - 10 ) 2 ( 1 - 099 2 )
= 2910 9 V / m
E =
216
q
( 1 -
4 0r 2
2 ) - 1/ 2
= 9109 1610 - 19
( 10 - 10 ) 2 ( 1 - 099 2 ) - 1/ 2

= 1010 12 V / m
24 :
v a
,
E =
2 0a
2124
21 , x dx P
dE =
( 1 -
) dx
4 0r 2 ( 1 -
sin 2
) 3/ 2 er
, , x
, P
21
E = dE =
+
r 2 =
E =
( 1 -
2 )
4
-
( 1 -
) dx
4 0r 2 ( 1 -
sin 2 ) 3 / 2
a 2 / sin 2 , x = acot , dx = -
0a 0
sin d
( 1 -
2 sin 2 ) 3/ 2
= ( 1 -
2 )
cos
4 0a ( 1 -
2 ) ( 1 -
2 sin 2
=
2 0a
a d
sin 2
) 1/ 2 0
217

3
31 , 10cm 30cm ,
110 - 8 C, 1510 - 8 C
( 1) 20cm , ( 2) 50cm
q 1
( 1) =
4 0r + q 2
4 0R2
= 900 V
= 910 9 110 - 8
+ 1510- 8
2010 - 2 3010 - 2
q1 + q2
( 2) = = 910 9 ( 1 + 15)10 - 8
4 0r
5010 - 2
= 450 V
32 , R 1 R 2( R 1 <
R 2) , U 12,
q,
U 1 2 = R 2
R 1
Edr = R 2
E =
R 1
q
dr =
4 0r 2
q 1
4 0
q
= U 1 2 R 1R 2
4 0r 2 r 2 R 2 - R 1
33 , R 1 = 50cm, R 2
= 200cm, 1 = 60V , 2 =
- 30V
( 1) ;
218
R 1
-
1
R 2

,
( 2) ?
( 1) q 1 q2
1 =
2 =
1
4 0
q1
R 1
1 q 1 +
4 0
R2
+ q2
R 2
q2
= 60
= - 30
R 1 R 2
( 2)
q1 = 6710 - 1 0 C, q 2 = - 1310 - 9 C
r =
q 1
- q2
=
1
4 0 r
q1
+ q2
R 2
= 0
R 2 = 6710 - 1 0
20 = 10 cm
1310 - 9
34 , R 1, R 2( R 1 < R 2) ,
q1, q 2 ,
q1
, ,
; ,
1 =
q1
+
4 0R 1
q2
4 0R 2
1 R1
-
1
R2
2 =
q1
+
4 0R 2
U 12 = 1 - 2 =
q2
4 0R 2
q1 1
4 0
R 1
-
1
R 2
> 0, U 12 q1 q1 > 0
219

, U 1 2 > 0, q 1 < 0 , U 12 < 0,
, q1
35 z z = -
a z = a ,
, , x x > 0
, P ( x , 0, 0)
= a -
= q d
=
ln
4 0
a
dz
4 0( x 2 + z 2 ) 1 / 2
x 2 + a 2 + a
x 2 + a 2 - a
36 , l = 150cm, = 20
10 -
7 C/ m :
( 1) a = 50cm ;
( 2) b = 50cm
( 1) x , x ,
,
1= l 0
dx
4 0( l + a - x ) =
ln a + l
4 0 a
= 910 9 2010 - 7 ln
= 2510 3 V
( 2) 35 ,
2=
ln
4 0
b 2 + l 2 / 4 + l / 2
b 2 + l 2 / 4 - l / 2
= 910 9 2010 - 7 ln
= 4310 3 V
50 + 150
50
5 2 + 15 2 / 4 + 15/ 2
5 2 + 15 2 / 4 - 15/ 2
37118
220

U 12 = R 2
R 1
E dr = R 2
R 1
dr
2 0r =
ln R 2
2 0 R 1
38 20cm ,
12710 -
110 3 V, :
( 1) ;
( 2) ;
,
U 1 2 =
R 2
D / 2
Edr =
R 1
d / 2
5 m
dr
2 0r =
ln D 2 0 d
= 2 0U 12
ln D d
E =
( 1)
2
2 0d = 2U 1 2
= 21410 7 V / m
( 2) ,
E =
d ln( D / d ) = 210 3
12710 - 5 ln
2
2 0D = 2U 1 2
= 13610 4 V / m
D ln( D/ d ) = 210 3
002ln
002
12710 - 5
002
12710 - 5
39 , , a
( 1) ;
( 2) , ;
( 3) E r r
( 1) r , l
221

( )
E ¡¤2 rl =
qi nt
0
r a , qin t = r 2 l , E i nt =
r
2 0
r a , qint = a 2 l , E ou t = a 2
2 0r
( 2) r a
r a
0
0
int = E in tdr =
r
a
out = E r ou tdr 0 + E a int dr a = r
= a 2
2ln a 4 0 r
- 1
r
r dr = -
2 0
r 2
4 0
a 2
2 0r dr 0 + a
r dr
2 0
3139
( 3) E r U r 31
310 R ,
: ( 1) 34 ;
( 2) 1 16
222

( 1) r , dr
x
d=
2 r dr
4 0 r 2 + x 2
, x
= d= R 0
2 r dr
4 0 r 2 + x =
( x 2 + R 2 - x )
2 2 0
( 2) x
E =
1 -
2 0
x
=
=
x
E dx =
2 0 x
x
( R 2 + x 2 ) 1 / 2
1 -
( R 2 + x 2 - x )
2 0
x
( R 2 + x 2 ) 1/ 2 dx
311 , R, ,
R/ 2
, ?
310
= ( R 2 + x 2 ) 1 / 2 - x -
2 0
=
( R 2 + x 2 ) 1 / 2 -
2 0
, x = 0,
R 2
1 / 2
2 0 4 + x 2
R 2
4 + x 2 1/ 2
0 = R
4 0
, ,
- x
223

312( 1) 040mm, 16pC,
? ( 2)
, ?
( 1) 1 =
q1
= 9109 1610 - 1 2
= 36 V
4 0R 1 0410 - 3
( 2) , 3 2 R 1
2=
q2
=
4 0R 2
4 3
2q1
2 0R1
= 910 9 21610 - 12
3
2 0410 - 3 = 57 V
313 , 79 e,
7010 -
1 5 m, ?
1=
q
4 0R = 910 9 791610 - 19
= 1610 7 V
7010 - 15
R
2=
0
r
dr +
3 0
q
dr = 3
R 4 0r 2 2
= 2410 7 V
q
4 0R =
3 2 16107
314 32 , ,
R, + q -
q, l, l R
( 1) O , x
;
ql
( 2) : x R =
4 0x 2
( 1)
224
=
1 q
4 0 ( R 2 + r 2 ) 1/ 2

32 314
32 x
=
q
4 0
1
R 2 + x -
l
2
2 1/ 2 -
1
R 2 + x +
l
2
2 1/ 2
l R,
q
1
=
-
4 0 ( R 2 + x 2 - xl ) 1/ 2
1
( R 2 + x 2 + xl ) 1/ 2
=
q
4 0x
1 -
1
l
x + R 2 1/ 2 -
x 2
1 +
1
l
x +
1/ 2
R2
x 2
( 2) x R , ,
=
q
4 0x
1 -
1
l
x
1/ 2 -
1 +
1
l
x
1/ 2
x l ,
=
q
4 0x 1 + l
2x - 1 - l
2x
=
ql
4 0x 2
315 35 x x > 0
225

x ,
x
E y = -
=
E = E x = -
ln
4 0
x 2 + a 2 + a
x 2 + a 2 - a
y = 0, E z = -
z = 0
x =
a
2 0x x 2 + a 2
316 q1 q2 x = -
b
x = + b , = 0
, ?
33316
33 , x y , P ( x , y )
=
1
4 0
= 0
q 2
y 2 + ( b - x ) 2 1 / 2 +
y 2 + ( b - x ) 2
y 2 + ( b + x ) 2 =
y 2 + x + b k + 1
k - 1
226
2
q1
y 2 + ( b + x ) 2 1/ 2
q2
q1
+ b 2 1 -
2
= k
k + 1
k - 1
, k
+ 1
k + 1
b, 0 ,
k - 1 k - 1
2
= 0
2
- 1
1/ 2
b

q1 q2 x , x
x y x ,
, x x = -
k + 1
k - 1
2
- 1
1 / 2
b
k + 1
b ,
k - 1
©¦ q 1©¦= ©¦ q 2©¦ , k = 1x ,
, yz ( , q1
q2 , = 0 )
317 -
+ z , x x = -
:
a x = + a
( 1) x y ,
;
( 2) x y
( 1) 34 , P ( x , y)
= + + - = - ln r + +
2 0
= -
ln r +
2 0 r -
+ C
lnr - + C
2 0
= r + / r - = r + / r - = k, r + = kr -
( a - x ) 2 + y 2 = k 2 [ ( a + x ) 2 + y 2 ]
, 1 + k 2
x 2 + y 2 - 2a 1 + k2
1 - k 2 x + a 2 = 0
1 - k 2 a , 0 , 2ka
1 - k 2 , 34
227

E=
( 2) P ( x , y)
= -
2 0r +
er +
+
( a - x )
2 0r 2 +
34 317
-
er
2 0r -
-
+ - ( a + x )
2 0r 2 -
i +
y
+ - y
2 0r 2 + 2 0r 2 j
-
E , : y = y ( x )
,
dy
dx =
E y
E x
=
1/ y 2 ,
228
y
+ - y
2 0r 2 + 2 0r 2 -
- ( a - x )
+ - ( a + x )
2 0r 2 + 2 0r 2 -
x 2 dy - 2x ydx - y 2 dy - a 2 dy = 0
=
2x y
x 2 - y 2 - a 2

x 2 dy - 2x ydx
- dy -
y 2
d -
x 2
y + y - a 2
c
x 2 + y -
c
2
a 2
y 2 dy = 0
x 2
y - y + a2
y = 0
2
y = c
- a 2 + c2
4 = 0
, ( 0, c/ 2) , a 2
+ c 2 / 434
,
318 1010 9 V,
30C
( 1) ?
( 2) 20kW h
?
( 1)
W = qU = 301010 9 V = 3010 10 J
( 2)
t = W P = 3010 10
2010 3 3600 =
416( )
319 , 35 K ,
A ,
, ,
, , A - K = 25
10 4 V , :
( 1) ( ) ;
( 2)
229

35319
( 1) E k = e( A - K ) = 1610 - 1 9 2510 4
= 410 - 15 J = 2510 4 eV
2E k
( 2) v =
m = 2410 - 1 5
= 9410 7 m/ s
9110 - 31
-
v = 1 -
1
1 + E k / ( m 0c 2 )
1/ 2
c =
6510 7 m/ s
320 a , q, -
2q , + Q
, ?
0=
=
q
4
0r + - q
4 0r + - 2q
4 0r
3q
4 0 3 a ( 1 - 1 - 2) = - 3 q
2 0a
q
230
A = - A = - Q( -
0) = Q 0 = -
3 qQ
2 0a

321 36 , ,
1 = 1210 - 4 C/ m 2 , 2 = 20
10 - 5 C/ m 2 , 3 = 1110 - 4 C/ m 2 A
50cm, B 70cm
( 1) A, B ;
( 2) q0 = - 1010 - 8 C
A B , ?
( 1) 36 ,
E =
1
( 1 - 2 - 3)
2 0
E =
U A B= E l A + E l B
=
=
1
( 1 + 2 - 3)
2 0
36
321
1
( 1 - 2 - 3) l A + ( 1 + 2 - 3) l B
2 0
10 - 4
288510 - 12 ( 12 - 02 - 11)005
+ ( 12 + 02 - 11)007
= 9010 4 V
( A 2) A B = - A A B = - q0U A B
= - ( - 1010 - 8 )9010 4
= 910 - 4 J
322
, 37 ,
,
( ) (
, )
231

( 1)
, L 1 1/ 2 , L 1
, ( )
( 2) U 0, , L 1
( 3) , ?
37322
( 1) v 1 = 2qU 0/ me ,
L 1 = v 1T / 2
v2 = v 2 1 + 2qU 0/ me = 2 v1
L 2 = v 2T / 2 = 2 L 1
v3 = v 2 2 + 2qU 0/ me = 3 v1
,
L 3 = v 3T / 2 = 3 L 1
L = L 1
( 2) L 1 = v 1T / 2 = 1 2U 0e
2 m e
( 3) E k = 1 2 m e v 2
= me
2
v 1
2
= eU 0
232

323 ( 1) ,
?
( 2) , m 0
E k = m 0c 2 1
1 - v 2 / c 2 - 1
?
( 1)
eU =
U =
1 2 m0c2
m 0c2
2e = 9110- 31 910 1 6
21610 - 19 = 2610 5 V
( 2)
v
c = 1 - 1
2
eU
1 +
m 0c 2
eU = m 0c 2 1
1 - v 2 / c 2 - 1
1/ 2
= 1 -
1
1 + 1 2
2
1/ 2
= 75%
324 , ,
02010 -
?( )
W =
1 e( - 2e)
2 +
4 0 r 2
10 m
( - e) ( - e)
( 2r) 2
= 910 9 - 4( 1610 - 1 9 ) 2
+ ( 1610- 19 ) 2
02010 - 10 0410 - 10
= - 4010 - 17 J
233

325 ,
, 12 C
3 30
10 - 1 5 m, 3 ?
( )
W = 3 ( 2e) 2
9 4 0r = 3910 4( 1610 - 19 ) 2
310 - 15
= 9210 - 14 J = 58 MeV
326
, e, a
:
ln( 1 + x ) )
e 2
( 2) -
4 0a ln2
( 1)
1 = 2
( 1) e
2 0a ln2 ( :
e
4 0a - 1 + 1 2 - 1
3 + = - e
2 0a ln 2
( 2) N ( N 1) ,
( )
W =
1 N
2
= 1
W1 = W / N = 1 2 e 1 = -
e = 1 2 N e 1
e 2
4 0a ln2
N , ,
327 R, e
mec 2 ,
234

e m e R ?R
?( R ,
10 -
18 m )
,
,
R=
e 2
4 0R = m ec 2
R =
e 2
4 0mec 2
( 16010 - 19 ) 2
488510 - 1 2 91110 - 31 ( 30010 8 ) 2
= 28110 - 15 m
328 1010 -
15 m
, ?
?
W = 3 5
e 2
4 0R = 3( 1 9 1610- ) 2 910 9
5110 - 15
= 13810 - 1 3 J = 8610 5 eV
( 938MeV )
086
938 = 0092%
329 92e,
7410 -
15 m
, , 46e,
,
, ?
?
235

, 1kg
?( )
WU = 3 5
( 92e) 2
4 0RU
= 1610 - 10 J
WP d= 2 3 5
= 3( 921610- 19 ) 2 910 9
57410 - 15
2
46 e)
(
4 0RP d
= 23( 19 461610- ) 2 910 9
57410 - 15 3
/ 2
= 1010 - 1 0 J
W 1 = WU - WP d = ( 16 - 10)10 - 10 = 6010 - 1 1 J
1kg
W = W1 1000
235 60210 23 = 1510 1 4 J
330 40MeV ,
2e, 79e,
66810 -
10 - 2 5 kg, ?
r m in=
236
qqA u
4 0r min
= E k ,
27
kg, 329
q qA u
= 910 9 279( 1610 - 19 ) 2
4 0E k , 410 6 1610 - 1 9
= 5710 - 14 m

,
E p= -
Gm m A u
r 2 mi n
= 66710 - 1 1 66810 - 27 32910 - 25
5710 - 1 4
= 25710 - 48 J
= 1610 - 3 5 MeV
331 , 317
10 - 2 7 kg
,
??
92e , 7410 -
,
qqU
4 0RU
v1=
=
- 2qqU
4 0R U m
+ 1 2 mv 2 1 = 0
15 m
- 2( - 1610 - 1 9 )921610 - 19 910 9 1/ 2
7410 - 15 31710 - 2 7
= 4310 7 m/ s
3
2
qU
q + 1 4 0R U 2 m v 2 2 = 0
v2 =
- 2qqU
3 4 0R U m 2 = v1 3 2 = 5210 7 m/ s
332p 1
p 2,
r , ,
237

-
p 1p 2
2 0r 3
l 1, l 2 , p 1 r
p 2
1 =
q 1 1
4 0 r - l 1/ 2 - 1
r + l 1/ 2 = q1l 1
=
4 0r 2
r ,
W = -
= -
q2 1 + q2 1 + d 1
dr l 2 = q2l d 1
2
dr
p 1p 2
2 0r 3
p 1
4 0r 2
333100V / m
( 1) ?
( 2) 10km
, kW h ?
( 1) w e =
( 2) W=
0E 2
= 88510- 12 ( 100) 2
2
2
= 4410 - 8 J/ m 3
4 R 2 Ehw e
= 4(
64106 ) 2 1010 3 4410 - 8
3610 6
= 6310 4 kW ¡¤h
334 ,
, ,
h = 10510 - 3 4 J ¡¤s, 2 h, 3 h
r =
( 1) : h( = 1, 2, 3, ) ,
4 0
2 h 2 ;
me e 2
( 2) : , ( + )
238

W = -
mee 4
1
2( 4 0) 2 h 2 2
( 3) = 1 ( eV )
( 1)
,
e 2
=
4 0r 2 me
me rv = h
r =
( 2)
v 2
r
4 0
2 h 2
m ee 2
W = 1 2 mev 2 + - e2
4 0r = - e 2
8 0r
= -
( W1= 3) -
= -
mee 4
2( 4 0) 2 2 h 2
m ee 4
2( 4 0) 2 h 2
91110 - 3 1 ( 1610 - 19 ) 4 ( 910 9 ) 2
2( 10510 - 34 ) 2
= - 21810 - 19 J
= - 136 eV
239

4
41
, S , S
S ( )
( S )
,
/ 0( ( 42) )
41 ,
S
/ 2E 1E 2 , S
S
E , ,
S
E i nt = E + E 1 = 0
E = - E 1 = E 2
S ,
E ex t =
2E 2
4141
E e xt 2 / ( 2 0) = / 0,
42 R 1,
R 2 R 3 , 1,
Q
q1, -
q1 + Q,
q 1,
240

1 =
,
1
4 0
q1
R 1
-
q1
R 2
+ Q + q1
R 3
q1 = 4 0R1R 2R3 1 - R 1R 2Q
R 2R 3 - R 1R3 + R1R 2
r < R 1:
R 1
= 1, E = 0
q 1
1
< r < R 2: =
4 0 r
+ - q1
R2
+ Q + q 1
R 3
,
E=
q1
4 0r 2
R 2 < r < R 3: = Q + q1
, E =
4 0R 3
0
r > R 3: = Q + q1 Q + q1
, E =
4 0r 4 0r 2
43 R1 = 60cm A
BB R 2 = 80cm, R 3 =
100cmA QA = 310 - 8 C, B
Q B = 210 - 8 C
( 1) B A B
;
( 2) B , A A
B A B
( 1)
Q B, int = - Q A = - 310 - 8 C
Q B , e xt = Q B + QA = 510 - 8 C
241

, A =
1
4 0
QA
R1
+
QB , i nt
R 2
+ Q B , ex t
R 3
= 910 9 310- 8
= 5610 3 V
B =
1
4 0
QB , e xt
R 3
= 4510 3 V
006
+ - 310- 8
008
= 910 9 510- 8
010
+ 510- 8
010
( 2) B , Q B = Q B, int = - 3
10 - 8 CA , A = 0A qA ,
qA =
1
A =
4 0
qA
R1
+
- QB / R 3
1/ R1 - 1/ R 2 + 1/ R 3
= 2110 - 8 C
qB , in t = - qA = - 2110 - 8 C
qB , ex t = Q B + qA = - 0910 - 8 C
B =
QB
- qA
+ Q B + qA
= 0
R 2 R 3
=
310 - 8 / 010
1/ 006 - 1/ 008 + 1/ 010
+ qA ( - 3 + 21)10 - 8
= 910 9
4 0R 3
01
= 8110 2 V
44 , R,
q r ,
( ) , q
q/ 4 0r , q , q
q / 4 0R,
242
q
4 0r +
q = -
q
4 0R = 0
R
r q

4542 , ,
,
13
10 - 5 C/ m 2
?(
)
42 ,
1,
2 A, B C ,
1 - 2 - 3 - 4 - 5 - 6 = 0
1 + 2 + 3 - 4 - 5 - 6 = 0
1 + 2 + 3 + 4 + 5 - 6 = 0
A C ,
U A B
= U C B
( 1 + 2 - 3 - 4 - 5 - 6) d A B
= ( 5 + 6 - 1 - 2 - 3 - 4) d C B
: B
A C
3 + 4 = = 1310 - 5 C/ m 2
1 + 2 + 5 + 6 = 0
6 ,
1= 6510 - 6 C/ m 2
2= - 3 = - 4910 - 6 C/ m 2
4= - 5 = 8110 - 6 C/ m 2
6= 6510 - 6 C/ m 2
4245
46 A ,
243

, qb qc,
r qd ( 43) qb, qc q d
?
4346
qb qc ,
, E = 0, qb qc
, ( q b + qc)
qd
,
, qb
F d =
( qb + qc) qd
4 0r 2
47 f =
2
en, , en
2 0
,
41 , S
, / ( 2 0) ,
, S S
f = ( S ) ( / ( 2 0) ) =
2 S / ( 2 0) ,
244

f =
f
S =
2 / ( 2 0)
, f ,
en
S ,
f =
2
en
2 0
48 ( I
) ,
915MV ,
?
( 1) 100MV / m,
( 2) , ,
320 C/ s ?
( 3) 485cm, 330m/ s
( 1) 0 =
q
4 0R , E =
q
E b,
4 0R 2
R 0
= 915106 = 91510 - 2 m
E b 10010 6
( 2)P mi n= Uq/ t = 0q/ t = 91510 6 32010 - 6
= 29310 3 W
q
( 3) =
vtb = 32010 - 6
3300485 = 20010 - 5 C/ m 2
E =
=
2 0
20010 - 5
288510 - 12 = 11310 6 V / m
49 q h ,
,
,
245

44 49
44 , q r a b,
, q a b
,
q
E a = E b =
4 0r
b , E b =
2 er
0,
b E b = - E a
, a
44 E a , E a = E b = E b , E a
E b , a
E a = E a + E a , 44 , E a ,
E a =
246
= -
2qh
2E a cos =
4 0( h 2 + r 2 ) 3/ 2
0E a = -
qh
2 ( h 2 + r 2 ) 3 / 2

410 q h
q ,
2h q - q , A = q 2 / ( 4 0
¡¤2h ) q F dr ,
? ?
q
F = -
q
A = - F ¡¤dz =
h
q 2
k
4 0( 2z ) 2
h
q 2
dz =
16 0z 2
q 2
16 0h
q , ,
,
411 45 q
?( )
45 411
45 , (
) ,
q
,
247

,
-
= -
q/ 2410
qh/ ( 2 ( r 2 + h 2 ) 3/ 2 ) ,
r 1
0 2
rdr = r 1
0
- qh2 rd r
= -
2 ( r 2 + h 2 ) 3/ 2
, q
q
r 1 = 3 h
412 , 1010 -
q
2
8 C/ m,
, 50m
??
,
E = 2
9
2 0h = 22910 1010 - 8
5
= -
= 72 V / m
0E = -
h = - 110 - 8
= - 6410 - 10 C/ m 2
5
F 1 =
2
2 0( 2h ) = 110 - 16 910 9
5
= 1810 - 7 N / m
413:
,
1cm 3 , 1g
,
= 1
18 60231023 1610 - 1 9 / ( 910 9 6410 6 )
248

10 6
V
414 R, q
46 ,
2 / ( 2 0) , , Rd
2 rsinx ,
dF x =
2
2 Rsin cos Rd
2 0
F x = dF x =
2 R 2
46414
0 / 2
0
sin cos d = R 2
2
=
2 0
q 2
32 0R 2
:
, ,
,
,
249

5
51H Cl , ( ) 0128nm,
?
HCl 3410 -
? ( 17e )
, HCl
30
Cm, H Cl
p 0 = el 0 = 1610 - 19 012810 - 9 = 2010 - 29 C ¡¤m
51 , H Cl
l
H Cl
p = - 18el + el 0
l = el 0 - p
18e
=
5151
( 20 - 34)10-
30
181610 - 19 = 5910 - 12 m
52 , R 1 =
002m R2= 006m ,
r 1= 6 r 2= 3 R=
004m Q= - 610 -
8 C :
( 1) D E , D r , E r ;
( 2) ;
( 3)
( 1) D
250

R 1: D = 0
r > R 1: D =
Q
4 r 2
E = D / 0 r,
r < R1: E = 0
R 1 < r < R: E =
Q
4 0 r 1r 2
R < r < R2: E =
Q
4 0 r 2r 2
r > R2: E =
Q
4 0r 2
D r Er 52
5252
( 2)
U =
R
= R R 1
R 2
Edr +
R 1
R
Q
4 0
= Q 1
-
4 0 r 1R1
E dr
R 2
dr +
r1r 2 R
Q
4 0
r2r 2 dr
1
r 1R + 1
r2R - 1
r2R 2
251

= 910 9 ( - 610 - 8 )
+
1
3004 - 1
3006
= - 3. 810 3 V
( 3) = P ¡¤en = - P n = - 0( r 1 -
= - 0( r 1 - 1)
Q
4 0 r 1r 2
1
6002 - 1
6004
1) E
= - ( 6 - 1) - 8
610-
460. 02 2
= 9. 910 - 6 C/ m 2
53 R 1 R2, R 2
< 2R 1, r 0
r1, r2, r 2= r 1/ 2
E m a x, ?
?
,
U , ( r= R 1
E 1 =
U
R1ln R 2 2
R 1r 0
( r= r 0 )
E 2 =
2U
r 0ln R 2 2
R1r 0
E 2/ E 1 = 2R1/ r 0
)
r 0< R2, R2< 2R1, E 2/ E 1> 0, ,
E ma xE 2= E m ax
252

U m ax =
E ma xr 0
ln R 2 2
2 R1r 0
54 r
Q:
Q = 1 -
1
Q
r
, E 0= / 0= Q/ ( 0S)
r , E
= E 0/ r= Q/ ( 0 rS ) ,
Q = S = S =
0( r - 1) ES = 1 -
1
Q
r
55 3kV / mm,
10cm , 10mm, 01mm
?
E b = m a x/ ( 2 0r ) , m ax = 2 0r E bE b = 3
10 3 kV / mm= 310 6 V/ m r
r 1 = 1. 0cm , ma x , 1 = 1. 710 - 6 C/ m
r 2 =
r 3 =
1. 0mm , ma x , 2 = 1. 710 - 7 C/ m
0. 1mm , ma x , 3 = 1. 710 - 8 C/ m
56
5210 - 9 m , 05210 - 3
C/ m 2 , 60,
: ( 1) ; ( 2)
( 1) E = / ( 0 r ) = 05210 - 3 / ( 88510 - 12 60) =
9810 6 V / m
( 2) U= E d = 9. 810 6 5. 210 - 9 = 5. 110 - 2 V = 51 mV
57E 0
253

, , 53
r,
53 ,
tan 2 = rt an 1 = rcot
,
E sin 2 = E 0sin 1 = E 0cos
E =
E 0cos / sin 2 = E 0
r
5357
sin 2 + rcos 2
= P cos 2 = 0( r - 1) E cos 2 = 0( r - 1)
r
58
,
(
54) ,
,
,
500mm 2 , 0600mm
E 0sin
54
58
0250pF,
?
254
C =
d 1 =
0S 1 d 1
-
0Sd 0
d 0 C +
1
d 0
0S

=
8. 8510 - 1 2 5010 - 6 0. 60010 - 3
0. 25010 - 12 0. 60010 - 3 + 8. 8510 - 12 5010 - 6
= 0. 44810 - 3 m = 0. 448 mm
d = d 0 - d 1 = 0. 600 - 0. 448 = 0. 152 mm
59 510 -
2 mm,
23 , 30 F,
?
S = Cd = 3010- 6 510 - 5
2. 38. 8510 - 12 = 7. 4 m 2
510 310 3 kV / m
50kV ,
?
E b= U/ d C= 0S
d
C/ S = 0E b
U = 8. 8510 - 1 2 310 6
= 5. 310 - 10 F / m 2
5010 3
51118cm
( 1) ?
( 2) 2010 5 V , ?
( 1) C= 4 0R= 0. 18/ ( 910 9 ) = 2. 010 - 1 1 F
( 2) Q= CU= 2. 010 - 1 1 2. 010 5 = 4. 010 - 6 C
512
2510 -
2 mm, 25mm,
100mm, (
)
C=
2 0l
ln R = 28.
8510 - 12 0. 1
= 8. 010 - 1 3 F
2
25
ln
R 1 2. 510 - 2
255

51355
, ,
,
S,
d
C =
( - 1) 0S
d
55513
, C1
= 0S / d ( - 1) , ,
C = ( - 1) C1 = ( - 1) 0S / d
514 002m 2 ,
05mm, ( 56)
025mm , ,
, ?
56514 ( )
57( a)
C = CA B +
CA K CBK
CA K + CB K
C= 0S/ d ,
256

C = 7. 0810 - 10 F
( B ) , 57( b)
,
C = CA K + CA B = 1. 0610 - 9 F
57514
( )
58515
515 ( 58) ,
a , b ,
d , l( l d )
58 x y , y= lx a + d
C =
a
0
0bdx
y
= a 0
0a b
l
0bdx
lx / a + d = 0a b ln 1 +
l
l
d - l 2
= 0a b
2d 2 d
516 ,
15cm 20cm
, ,
60% , ?
,
, E 0
E 0/ r
1 -
l d
l
2d
257

U = E 0( d 0 - d ) + E 0d / r = E 0d 0 1 -
= U 0 1 -
d
d 0
U = 0. 60U 0,
1 -
+
d
rd 0
d
d 0
1 -
d 0= 2. 0cm, d = 1. 5cm
r = 2. 1
1
= 0. 60
r
d
d 0
+
d
rd 0
517 , R1
R= ( R 2- R 1) R1
Q 1
-
4 0 R 1
R2,
Q, U =
1
R2
C = Q U = 4 R2R 1
0
R2 - R 1
R= ( R 2- R 1) R 1
, R 1R 2R 2 1,
R 2 1
C 4 0
R
4 R 2 1= S , R= d
,
C =
0S
d
51812 F 2 F
3 F 200V ,
?
3 F , 2 F
258

, 12 F U , 2 F
U/ 4, 12 F 3U/ 4
, 12 F U b , 3U / 4
U b,
U 4U b/ 3 = 4200/ 3 = 267 V
276V
519 S,
d ,
r 1 r2( 59)
0 r1 S
1
2 d 0 r2 S
C =
0 r 1S
0 r2S
d / 2 d / 2
510520
1
d ,
2
0 r1S
d / 2 +
0 r 2S =
d / 2
59519
2 0 r1 r2S
d ( r 1 + r2)
520
510
r , eff
(
)
a,
, r
( r = 1. 95) ( r= 33) ,
?
,
C =
b d
0 rhb + 0( a - h ) b = 0b h( r - 1) + a = 0
d d d
d
r , effa b
259

, e ff = 1 + ( r - 1)
r , , r, eff ,
521
r , ?
C = C- C0= 4 0R 1R 2
R 2- R 1
= 4 0R 1R2 r
R2- R 1 2 - 1
2
( r- 1) / 2
h
a
0R 1R2
1
2 + r
2 - 4
R2- R 1
= C0( r- 1) / 2
522 4 F 6 F
200V , ,
, ?
( 1) ;
( 2) ,
, Q 0,
Q 0 =
C1C2
C1 + C2
U 0 = 46 200 = 480
4 + 6 C
( 1) ,
, ,
( 2) , Q= 2Q 0
, C= C1+ C2= 10 F,
U = Q C = 2480
10
= 96 V
523 100pF 100V ,
, , 30V
? ? ?
260

,
C2 =
C1( U 1 - U 2)
U 2
C1U 1 = ( C1 + C2) U 2
=
- W = 1 2 C1U 2 1 -
100( 100 - 30)
30
1
2 ( C1 + C2) U 2 2
= 233 pF
= 1 2 10010 - 1 2 100 2
-
1
2 ( 100 + 233)10- 12 30 2
= 3. 510 - 7 J
524 , S , d (
511)
( 1) Q ,
b
,
?
( 2) , ( ) 511524
? ?
( 3) U , ( 1) , ( 2)
?
C0 = 0S/ d ,
b, C= 0S / ( d - b)
( 1) W= 1 Q 2
2 C - 1 Q 2
= Q 2
d - b
2 C0 2 0S - d
0S = - Q 2 b
2 0S
( 2) A e xt = W < 0, ,
,
261

( 3) U , ,
Q = ( C - C0) U
U ,
W = 1 2 CU 2 -
WS = Q ¡¤U = ( C - C0) U 2
1
2 C0U 2 = 1 2 ( C - C0) U 2 =
A ,
WS
+ A = W
0U 2 S b
2d ( d - b)
A = W - W S = 1 2 ( C - C0) U 2 - ( C - C0) U 2
= -
1
2 ( C - C0) U 2 = -
0U 2 S b
2d ( d - b)
525 512 ,
7cm ,
5cm ,
5kV ,
( )
512 x
C =
2 0x
ln( R2/ R1)
U ,
262
Q = CU =
2 0xU
ln ( R2/ R1)
512525

W C = 1 2
Q 2
C = 1 2
Q 2 ln ( R 2/ R 1)
2 0x
, ,
F = -
=
W C
x = 1 2
0U 2
ln( R 2/ R 1)
Q 2
ln( R 2/ R 1)
2 0x 2
= 8. 8510- 12 ( 510 3 ) 2
ln ( 7/ 5)
= 2. 110 - 3 N
F > 0, x
526 a , b,
( 513) Q
513526
( 1) , ?
( 2) b, , r
x ,
?
?
( 3) x ,
( 4) x 0 a ,
? ( )
263

( 5) U
, , ( 2) ( 4) ?(
)
( 1)
W = 1 2
Q 2
C =
Q2
2 0ab
( 2) x ,
C =
,
WC = 1 2
0b [ a + ( r - 1) x ]
Q 2
C =
Q 2
2 0b[ a + ( r - 1) x ]
( 3) ,
Q
F = -
dW C
dx = ( r - 1) Q 2
2 0b[ a + ( r - 1) x ] 2
0, F x , ,
( 4) ,
,
F dx = -
dW C
( 5) U , U
WC = 1 2 CU 2 = 1 2
0b[ a + ( r - 1) x ]
dx , ,
,
dQ =
UdC
U ,
,
264
dA E = UdQ = U 2 dC
U 2

dWC = 1 2 U 2 dC
dA E = dWC + F dx
F dx = dA E - dWC = 1 2 U 2 dC
F = 1 2 U 2 dC
dx =
0( r - 1) b
2 U 2
Q ,
F =
0( r - 1) b
2
Q 2
C 2 =
( r - 1) Q 2
2 0b[ a + ( r - 1) x ] 2
F > 0, F x , ,
F Q ( 3) ,
x , Q ,
527: ,
R 1, 2R1R 2/ ( R1 + R 2) , R1
R 2
?
Q ,
w e =
E =
0E 2
=
2
Q
4 0r 2
Q 2
32
2 0r 4
R ,
R
4
32
2 0r 4
R 1
Q 2
r 2 dr = 1 2 R 2
4 r 2 dr
32
2 0r 4
R 1
Q 2
265

1
R 1
-
1
R
= 1 2
1
R1
-
1
R2
R =
2R1R 2
R1 + R2
, R2,
R = 2R 1
528 S , y 0,
y = 0 , y = y 0
y ,
( 1) ?
r = 1 + 3 y
y 0
( 2) Q( + Q) ,
?
( 3)
( 4)
( 1) dy,
r ,
C ,
266
1
C = 1
dC
= y 0
0
dy
0
dC =
rS y =
0
0
0 rS
dy
dy
0S 1 + 3 y 0y
=
y 0
3 0S ln 4

C =
3 0S
y 0ln4
( 2) , 1 = P 1= - 0( r , y 0
- 1) E = - 0( r, y 0
- 1)
= -
r, y 0
- 1
= -
r, y 0
3Q
4S
, 2 = P 2= 0( r, 0- 1) E = 0( 1- 1) E= 0
D
0
( 3) y dy,
,
dq int
= dy =
= -
= -
Q
S d 1
r
Q
S
d
dy
0[ ( E + dE ) - E ] = 0dE =
1
= 3Q
r y 0S 1 + 3 y
y 0
y 0
0
Sdy = -
Q
r, y0
d
r, 0
( 4)
1 S = -
- 2
d( D / r)
1
= - Q 1 r
4 - 1 = 3Q 4
3
4 Q
52930 ,
2010 -
?
r, y 0
6
C,
Q1 Q 2
U 1 =
Q 1/ ( 4 0R) , U 2= Q 2/ ( 4 0 rR) ,
,
267

Q 1
4 0R = Q 2
4 0 rR
rQ 1 = Q 2
Q 1 + Q 2 = Q
Q 1 =
Q
r + 1
=
2. 010-
6
3 + 1
= 0. 5010 - 6 C
Q 2 = Q - Q1 = ( 2. 0 - 0. 5)10 - 6 = 1. 510 - 6 C
530 ,
,
;
( 1) 334 ,
( 2) r = 158,
?
( 1) ,
1/ r, 334 ,
W = -
r =
4 0 r
2 h 2
m ee 2
m ee 4
1
= W0
2( 4 0 r h) 2 2
2
r
W 0
( 2)
1/
2
r
268
W 1 = W01
= 13. 6 = 0. 054 eV
r 2 15. 8 2

6
61 240m
8mA ,
?
N 1 , I = N 1ev
N = l N 1 = lI
ev = 240810 - 3
1610 - 19 310 8 = 410 1 0 ( )
62 , 30cm
20cm/ s , ,
1210 6 V / m
, ?
I =
w v =
2 0Ew v
= 288510 - 12 1210 6 0302
= 1310 - 6 A
63 ,
1mm 30A , ?
, ,
20,
, ?
I
v =
Se = I M
4 D 2 N A e
M = 100g/ mol= 01kg/ mol, = 10g/ cm 3
= 10 4 k g/ m 3 ,
269

v =
43001
( 10 - 3 ) 2 10 4 610 23 1610 - 1 9
= 410 - 3 m/ s
v = 160
= 1110 5 m/ s
kT
m = 160 13810- 23 293
9110 - 31
64 20mm80mm, 2m,
50mV = 17510 - 8 ¡¤m,
8510 28 / m 3 : ( 1) ; ( 2)
; ( 3) ; ( 4) ; ( 5)
; ( 6)
( 1) R = l S = 2
8
17510-
208010 - 6
= 21910 - 5
( 2) I = U/ R = 5010 - 3 / ( 21910 - 5 ) = 22810 3 A
( 3) J = I / S = 22810 3 / ( 208010 - 6 )
= 14310 6 A/ m 2
( 4) E = J = 17510 - 8 14310 6 = 25010 - 2 V / m
( 5) P = I U = 22810 3 5010 - 3 = 11410 2 W
( 6) v = J / ( e) = 14310 6 / ( 8510 28 1610 - 1 9 )
= 10510 - 4 m/ s
65 , 20cm 25cm,
20A , ?
001¡¤m, 8710 -
I 1
I 2
I 1
= R 2
R 1
=
8 ¡¤m
I 2 ,
2r 2 1
1( r 2 2 - r 2 1) = 8710 - 8 2 2
001( 25 2 - 2 2 ) = 5
15510-
I 1 I 2 , I 2 20A , I 1 0
270

66 ,
( 61) 5m ,
AB , CD
30 , 70
P ,
AP
CP = R A B / 2
R CD / 2
= 30
70 = 3 7
AP + CP = 5 m
AP =
A 15m
15 m
6166
67
( 1) , 120V / m,
410 - 12 A / m 2 ?
( 2) 410 5 V ,
?
( 1) = E J = 120
= 310 13 ¡¤m
410 - 12
( 2) R = U U
410 5
=
=
I J4 R 2 E 410 - 1 2 4( 63710 6 ) 2
= 196
68 62 , r 1 = 05cm ,
62 68
, r 2 =
2cm , =
110 12 ¡¤
m
( 1) L = 1000m
;
271

( 2) 100V ,
?
( 1) R =
2 L ln r 2 2
r 2 1
=
110 12
22
ln = 2210 8
21000 05 2
( 2) I = U/ R = 100/ ( 2210 8 ) = 4510 - 7 A
69r 1 r 2,
R = r 2
r 1
dr
=
4 r 2
R =
1
4r 1
1
4r 1
-
610, , 200A
, ,
100m , 10m,
13m
-
1
r 2
,
, r J =
I / ( 2 r 2 ) , E =
U =
r 2
r 1
Edr =
=
I 1
2r 1
-
1
r 2
1
r 2
J =
= I r 2 - r 1
2r 2r 1
100200( 13 - 10)
21310
I / ( 2 r 2 ) ,
= 73 V
611 63 , E 1 = 30 V, r 1= 05 , E 2= 60 V , r 2 =
10 , R 1= 20 , R 2= 40 , R1 R2
63 , :
b: I 1- I 2+ I 3= 0
a bR 1a : - I 3r 3- E1+ I 1R 1= 0
a R 2ba : I 2r 2- E 1+ I 3r 1+ E 2+ I 2R 2= 0
272

, ,
I 1 = 3 4 A , I 2 = 2 3 A
63611
64612
61264 , E 1= 30 V , E 2= 10 V , r 1= 05 ,
r 2= 10 , R 1= 45 , R2= 190 , R 3= 100 , R 4= 50
64 ,
b: - I 1+ I 3+ I 2= 0
a R 1bR3a : - E 1+ I 1( r 1+ R1+ R 4) + I 3R 3= 0
a R 3bR2a : - I 3R 3+ I 2( R 2+ r 2) + E 2= 0
, ,
I 1 = I 4 = 016 A , I 2 = 002 A,
I 3 =
014 A
613 65 , I 1,
I 2, I g
3
, I g, R 1/ R 2 =
R 3/ R 4
I g = 0, 4
65 , :
R 1GR2 : I 1R 1 + I gR g - I 2R 2 = 0
65613
R 3R4G : ( I 1 - I g) R3 - ( I 2 + I g) R 4 - I gRg = 0
273

R 2R4E : - E + I 2R 2 + ( I 2 + I g) R 4 = 0
I g =
(R2R3 - R1R4)E
R1R3(R2 + R4) + R2R4(R1 + R3) + Rg( R1 + R3)(R2 + R4)
, R1/ R2 = R 3/ R4 , I g
= 04
614 66
E = 6 V, 0, U e c =
196V , U e b = 02V , I c = 2mA , I b =
20 A, I 2= 04mA , R c= 1kR 1,
R 2, R e
66 ,
,
,
66614
I 1 = I 2 + I b = 04 + 002 = 042 mA
I e = I b + I c = 002 + 20 = 202 mA
U bc = U ec - U e b = 196 - 02 = 176 V
R 1R c: I 1R1 - I cRc - U bc = 0
R 1
= 90 k
R eRcE: I cRc - E + I e Re + U e c = 0
Re = 10 k
R 2R e: I 2R 2 + U b c - I e Re = 0
R 2 = 56 k
615 : C , R
, t =
, Q =
0,
? R, C ,
274

U 0 ,
=
Q t / ( R C)
e-
RC
R
W = 2 Rdt =
0
0
Q 2
RC 2 e - 2t / ( RC ) dt = Q 2
- t/ ( R C)
q = Qe
1
2C = W C
t = RCln Q e = RCln CU e
C = 10 - 7 F, U = 10 2 V , R = 10 3 ,
t = 10 3 10 - 7 ln( 10 - 7 10 2 / 1610 - 1 9 ) = 310 - 3 s
616 2000 F
4000V ,
1A ,
, 05 ,
I m a x
Rm in = U / I m ax = 410 3 / 1 = 40 k
min = Rm inC = 410 3 210 - 3 = 80 s
275

I m a x = U R = 4103
05
= 8010 3 A
= RC = 05210 - 3 = 1010 - 3 s
617 400V 200A
, 20,
80, ( L/ min) ?
Q( L/ min ) ,
Q( kg/ min) ,
Q =
UI60 = cQ( t 2 - t 1)
UI60
( t 2 - t 1) c = 40020060
= 19 L/ min
( 80 - 60)4210 3
618( 615) :
, ( )
,
, ,
qin t =
0E ¡¤dS =
0 J ¡¤dS
, ,
qi nt
= 0 J ¡¤dS
, ,
qin t = 0
, ,
, qint 0,
276

7
71 0. 5g, 2510 -
8 C
6010 4 m / s,
,
?
, ,
B =
qvB = mg
mg
qv = 0. 510 - 3 9. 8
= 3. 3 T
2. 510 - 8 610 4
,
7. 2 7. 1, A
, v0= 110 7 m/ s
( 1) A C
, ;
7172
( 2) A C
( 1)
B =
ev0B = m v2 0
R
mv0
eR = 9. 1110- 31 110 7
1. 610 - 19 0. 05
= 1. 110 - 3 T
277

( 2)
t = T 2 = 2 R = 0. 05
= 1. 610 - 8 s
2v 0 110 7
7. 3 2. 010 3 eV , B =
0. 1T , B 89, ,
B T h r
v =
2E k
m = 2210 3 1. 610 - 1 9
9. 1110 - 31 = 2. 610 7 m/ s
T =
2 m
eB = 2
31
9. 1110-
1. 610 - 1 9 0. 1 = 3. 610 - 1 0 s
h = vcos89 T =
r =
= 1. 610 - 4 m
2. 610 7 cos893. 610 - 10
mvsin 89
= 9. 1110- 31 2. 610 7 sin 89
eB
1. 610 - 19 0. 1
= 1. 510 - 3 m
7. 4
2. 010 4 V , 0. 2m ,
0. 510 -
, ?
v =
4 T ?
2E k
m = 2eU
m = 21. 610 - 1 9 210 4
9. 110 - 31
= 8. 410 7 m/ s
278

R =
mv
eB = 9. 110 - 31 8. 410 7
1. 610 - 1 9 0. 510 - 4 = 9. 6 m
72 ,
x = R - R 2 - l 2 l 2
2R = 0. 2 2
29. 6
= 210 - 3 m = 2 mm
,
,
7. 5 7274
240m
1. 4910 -
R= mv/ ( eB) = p / ( eB)
18
kgm/ s,
B = p eR = 1. 4910 - 18
= 0. 244 T
1. 610 - 19 240/ ( 2 )
7. 6 10 -
10 -
1 6 kg m/ s,
8
T , ?
, 10 8 T ,
?
R 1=
p =
eB 1
10 - 16
= 610 10 m
1. 610 - 19 10 - 8
10 - 16
= 610 - 6 m
1. 610 - 19 10 8
R 2=
p =
eB 2
7. 7 , 20T ,
3. 5m
p = eRB = 1. 610 -
E =
p 2
2m 0, p
= 1. 1210 - 17 kg ¡¤m / s
=
19 3. 520
( 1. 1210 - 17 ) 2
21. 6710 - 2 7 = 3. 7510 - 7 J = 234 GeV
279

( 1GeV ) ,
,
E = c 2 p 2 - m 2 0, pc 4 cp = 310 8 1. 1210 - 1 7
= 3. 3610 - 9 J = 21 GeV
7. 80. 8010 8 m/ s
, 410 -
7 T
?
210 -
, ?
R 1= mv = 9. 110 - 31 0. 810 8
= 1. 110 3
eB 1 1. 610 - 1 410 9 - 7 m
5 T
R 2= mv = 9. 110 - 31 0. 810 8
= 23
eB 2 1. 610 - 1 210 9 - 5 m
7. 9 D 75cm,
1. 5T ( 7. 30)
3. 3410 - 2 7 kg, :
( 1) ?
( 2) MeV ?
( 1) = 1 T = eB = 1. 610- 19 1. 5
= 1. 110 7 Hz
2 m D 23. 3410 - 2 7
( 2) E k = 1 2 m Dv2 = p 2
2m D
= e2 B 2 R 2
2m D
= ( 1. 610- 19 ) 2 ( 1. 5) 2 ( 0. 75/ 2) 2
23. 3410 - 27
= 1. 210 - 12 J= 7. 6 MeV
7. 10 7. 3 m
q ( E
B) B ,
280

73710
( 1) l
m =
qBB l
2E
( 2) 16 O 29. 20cm ,
32. 86cm 16 O
16. 00u, ?
( 1)
qvB = qE
v = E / B
B
m =
R = l 2 = mv
qB
qB l
2v = qBB l
2E
( 2) m l ,
281

32. 86
16. 00 = 18. 01 u
29. 20
7. 117. 4 , d = 1. 0mm, B= 1. 5T
,
8. 4
10 22 ,
- e= - 1. 610 -
I = 200A ,
U a a ;
19 C,
( 1 )
( 2) b U a a ? ?
74711
( 1) U a a = I B
qd
2001. 5
=
8. 410 22 ( - 1. 610 - 19 )1. 010 - 3
a
= - 2. 2310 - 5 V
( 2) b U a a = U H U H = E H b= vb/ B
b , I , v= I / ( qbd )
b
7. 127. 5 , abc,
x I , z B
a = 0. 10cm , b= 0. 35cm , c= 1. 0cm, I = 1. 0mA, B = 3000G ,
U A A = 6. 55mV
( 1) ( P ) ( N ) ?
( 2)
( 1) A A
282

75712
I B
( 2) =
U A A qa = 1. 010 - 0. 3 3
6. 5510 - 1. 3 610 - 19 10 - 3
= 2. 8610 2 0 / m 3
7. 13 N ,
210 2 1 / m 3 , 1. 610 -
2 m
, , abc= 0. 5cm0. 2cm
0. 005 cm1V
, 0. 2cm
1. 05mV
a , b, c
B =
qU Hc
I
=
qU H c a =
Ubc
qU H a
Ub
= ( 210 21 1. 610 - 19 1. 0510 - 3
1. 610 - 2 0. 510 - 2 ) / ( 10. 210 - 2 )
= 1. 3410 - 2 T
7. 14,
7. 6 , l
B, ,
J
( 1) : l p =
283

J lB, ;
( 2) l 1. 00atm , ?
B= 1. 50T , l = 2. 00cm
76714
( 1) p = F S = I Bl
S = J Bl
( 2) J = p 1. 001. 013105
=
Bl 1. 502. 0010 - 2
= 3. 3810 6 A / m 2
= 338 A / cm 2
7. 157. 7
,
2. 0mm, 0. 080T , 0. 10mV ,
? ( )
,
284
77715

v =
E H
B =
qvB = qE H
U H
dB = 0. 1010 - 3
0. 080210 - 3 = 0. 63 m/ s
7. 167. 8 ,
,
B , l, B
I ,
; ,
m ,
( g = 9. 80
m / s 2 )
;
( 1) B
( 2 ) l = 10. 0 cm, = 5, I =
0. 10A , m= 8. 78g , B= ?
78716
( 1) M
,
,
Mg =
M g -
I Bl
( M + m) g = M g + I Bl
B =
mg
2I l
8. 789. 8010 - 3
( 2) B=
= 0. 860
250. 110. 010 - 2 T
7. 17 20mm, 10mm,
, 1000 , B= 1000G ,
100mA , 7. 9
,
285

( 7. 22)
79717
( 1) B ( 7. 9( a) ) ;
( 2) B ( 7. 9( b) )
( 1) ,
F 1
= F 1 = N I l 1B = 10 3 0. 10. 010. 1 = 0. 1 N
,
F 2
= F 2 = N I l 2B = 10 3 0. 10. 020. 1 = 0. 2 N
4 , ,
( 722) M = mBsin ,
m B 0 , M= 0,
( 2)
,
F 2
= F 2 = N I l 2B = 10 3 0. 10. 020. 1 = 0. 2 N
, ,
, F 2 F 1 2
M = F 2l 1 = N I l 2l 1B = 0. 20. 01 = 210 - 3 N ¡¤m
B
286

N I l 1l 2= N I S = m ,
B , m B 90 ,
( 722) M= mBsin , mB
7. 18 , 200
, 150mm, B= 4. 0T , I
= 8. 0A , :
( 1) m ;
( 2)
( 1) m= N I S = 2008. 0( 15010 - 3 ) 2 = 36 A m 2
( 2) M ma x= mB= 364. 0= 144 N m
7. 19 M R
,
, ©¦m©¦=
1 4 R 4 , m
L m=
q L, q
2M
710 dr
d©¦m©¦=
¡¤2 rdr r 2 = r 3 dr
T
©¦m©¦=d©¦m©¦= R r 3 dr
0
= R 4 / 4
©¦m©¦= 1
2M R 2 MR2
2
R 2 = q, MR 2 / 2= L,
©¦m©¦=
qL
2M
710719
287

m L ,
m =
q
2M L
7. 20
+ 2e/ 3 - e/ 3 ,
, ,
r v
, ,
;
( 1)
( 2) r= 1. 2010 - 15 m,
v m = 9. 6610 - 2 7
A m 2
( 1) 711 ,
m = m= v q
v + = - v -
2 r r 2 = r 2 v+ ¡¤ 2 3 e + 2v - ¡¤ - 1
3 e = 2 3 r ve
= v,
711720
( 2) v= 3m
2re = 39. 6610 - 27
21. 2010 - 15 1. 610 - 1 9 = 7. 5510 7 m / s
288

7. 210. 92810 - 2 3 J/ T
, R, e
, ?
10 -
18 m , ,
? ?
712 ,
m = ¡¤2 Rsin ¡¤
0
2 ¡¤ ( Rsin ) 2 Rd
=
0
R 4 sin 3 d
= 1 3 e R 2
e= 4 R 2 ,
712721
R= 10 -
v =
R =
18 m ,
3m
eR = 30. 92810 - 2 3
1. 610 - 1 9 10 - 1 8 = 1. 710 14 m / s
,
7. 22 ,
2 P 3/ 22. 3910 -
23 J/ T
289

,
, 39. 2 , 75 , 105 ,
140. 8
B= 2. 0T ,
?
Wm = - m B= - mBcos
, - 3. 7010 - 2 3
J, - 1. 2410 - 23 J, + 1. 2410 - 2 3 J, + 3. 7010 - 23 J
7. 23( 7. 29) :
( 1) S , B = 0,
S ,
B =
vE
c 2
( 2) S , E = 0,
S ,
E = -
( 1) ( 729) ,
B x = B x, B y = B y +
c E z
B = 0,
B x = 0, By = -
vB
, B z = B z -
c E y
v
c 2 E z , B z = v c 2 E y
v = vi, B
B = -
( 2) ( 729)
v
E zk + v E yj = vE
c 2 c 2 c 2
E x = E x, E y = ( E y - cBz ) , E z = ( E z + cBy )
E = 0,
E x = 0, E y = vBz , E z = - vBy
v = vi,
290

E = -
vB
7. 24S x ,
, , v 0 = v0i ,
- z z = a v = vi
( 1) S
F =
0 ev 0v
k
2 a
( 2) S, ,
+ = 1 - v 2 / c 2 , - = - 1 -
= v0v c 2 1 - v 2 / c 2
( 3) S
ev0v
F =
2 0a c 2 1 - v 2 / c 2
v 0v
c 2 1 - v 2 / c 2
( 4) ( 2. 16) F F ( 1)
( 1) S ( 713( a) )
I = - v - = v0, - x p
B= ( 0I / ( 2 a ) ) j ,
F = evB =
0 ev 0v
k
2 a
,
( 2) S ( 713( b) ) , l + , 0
l+ = l + , 0 1- v 2 / c 2 ,
+ = 1 - v 2 / c 2 291

713724
S l - l - , 0= l -
1- v 2 0/ c 2 ,
S ,
, l -
v - = v - , x =
v 0 - v
1 - v 0v/ c 2
, 0
l - = l - , 0 1 - v - 2
/ c 2 =
l -
1 - v 2 0/ c 2 1 -
v0 - v
1 - v0v/ c 2 2
c 2
=
l -
1 - v 2 0/ c 2 ¡¤
1 - v 2 0/ c 2 ¡¤ 1 - v 2 / c 2
1 - v 0v/ c 2
= l - 1 - v 2 / c 2
1 - v 0v/ c 2
, q= q ,
- l - = - l -
- = - l - / l - = - ( 1 - v0v/ c 2 ) 1 - v 2 / c 2
, S
= + + - = v 0v c 2 1 - v 2 / c 2
( 3) S p ,
292

z a ,
E =
2 0a k =
F = E e =
v 0v
2 0a c 2 1 - v 2 / c 2k
ev 0v
2 0ac 2 1 - v 2 / c k 2
( 4) [ ( 216 ] ,
F z =
F z /
1 + vv x / c 2
v x S S
, v x = 0, F z = Fz / F z = ev0v
2 0a c 2 1- v 2 / c 2
F z =
( 1)
ev0v 1 - v 2 / c 2
2 0a c 2 1 - v 2 / c = 0 ev 0v
2 2 a
293

8
8. 181 P B
8181
( a) P
, P
B = 1 2
0I
2 a =
0I
4 a
( b) P
, r , 0I /
( 2 r ) P ,
0I / ( 4r ) P ,
P =
0I
2 r +
0I
4r
( c) P a / ( 2 3 ) P
,
294

B =
0I
3
4 a / ( 2
3 ) ( cos30 - cos150
) = 9 0I
2 a
8. 2 25m , 1. 8
10 3 A
( 1) ?
( 2) , 0. 610 - 4 T ,
?
( 1) B= 0I
2 r = 410-
1. 7 810 3
= 1. 410 - 5 T
225
( 2) B/ B E= 1. 410 - 5 / ( 0. 610 - 4 ) = 0. 24
8. 3 , 0. 80m ,
20A
( 1) ? (
)
( 2) ( ) 0. 1810 - 4 T
,
, ?
,
( 1) B= 0I
2 r = 410-
20
7 = 5. 010 - 6 T
20. 80
( 2) B ,
1 = arctan B BE
= arctan
6
5. 010-
= 15 31
0. 1810 - 4
,
2 = arcsin B B E
= arcsin
6
5. 010-
= 16 8
0. 1810 - 4
8. 4 A , C ,
82 O ?
I 1
295

B 1 = 0I 1
2r
l 1
2 r =
0I 1l 1
4 r 2
I 2
B 2 = 0I 2
2r
l 1
I 1
I 2
l 2
2 r =
0I 2l 2
4 r 2
l 2 ,
= R2
R1
= l 2
l 1
I 1l 1= I 2l 2B1 B2 , ,
8284
8385
8. 5d = 40cm, I 1= I 2
= 20A , 83 :
;
( 1)
( 2) ( r 1= r 3= 10cm, l =
25cm )
( 1)
B 0 = 2
0I
2 d / 2 =
( 2)
296
24
7 10- 20
0. 4
= 4. 010 - 5 T

= 2B ¡¤dS = 2 r 1 + r 2
r 1
0I
2 r ldr =
= 4
7 10- 200. 25
ln
= 2. 210 - 6 Wb
0I l
ln r 1 + r 2
r 1
0. 10 + 0. 20
0. 10
8. 684 , I
x B
8486
84 , , I dl P
dB = 0I dlr
= 0I dl( x i - R)
4 r 3 4 r 3
= 0I [ ( dy j + dz k)x i - dlR]
4 r 3
= 0I [ - x dy k + x dz j - Rdli]
4 r 3
P
B = dB =
=
0I
4
r Ṟ - x dyk + 3 R
0 x dz j - 0
R
- 0I R
[ 2x k + Ri]
4 ( x 2 + R 2 ) 3 / 2
0 Rdli
297

: 84 , I dl P dB
dB= 0I dl/ ( 4 r 2 ) dB dBx
dB,
dBx = - dBsin , dB = dBcos
x
dBR dB dBy dBz , dBz =
- dB cos = - dBcos cos
z ,
By = dB y = 0
R
Bx = dB x = - dBsin= -
= -
B z = dB z
= -
0I R 2
4( x 2 + R 2 ) 3/ 2
0
0I
dlsin = -
4 r 2
R
0I dl
= - dBcos cos= - cos cos
0 4 r 2
/ 2
0I
cos
4 r 2
- / 2
Rdcos= -
P
B = -
0I Rx
2 ( x 2 + R 2 ) 3/ 2
0I R
[ Ri + 2x k]
4 ( x 2 + R 2 ) 3 / 2
0I sin 4 r R
dl
2
0
8. 7 , I = 5. 010 3 A
: 5. 0cm ,
7. 0cm, 9. 0cm (
)
(
) ,
, :
298
r 0. 05m, B =
0. 4r T

0. 05m r 0. 07m, B = 10 - 3 / r T
0. 07m r 0. 09m, B = 0. 31( 8. 110 - 3 - r 2 ) / r T
0. 09m r , B = 0
r 85
8587
8. 8( 84) , :
( 1) I ;
( 2) ( j )
( 1) 86( a ) , I = 2 Rj ,
P P d dI 1 =
j Rd 1= j r 1d / cos , dI 2 = j Rd 2= j r 2d / cos P
dB1 = 0dI 1
= 0j Rd 1
=
2 r 1 2 r 1
dB2 = 0dI 2
= 0j Rd 2
=
2 r 2 2 r 2
0j r 1d
2 r 1cos =
0j r 2d
2 r 2cos =
0j d
2 cos
0j d
2 cos
dB 1= dB2, , , dB1+ dB2=
0P
Bint = ( I )
( dB1 + dB2) = 0
299

300

86( b ) , P , r
d, dB 1= dB2= 0j d / ( 2 cos ) , dB1 dB2
OP P
OP dB1 dB2
OP P B
dB =
2dB1cos = 0j cos d / ( cos )
86( b) rsin = Rsin , rcos d = Rcos d
,
dB =
P
/ 2
B = dB =
- / 2
0j R d
r
0j R
d = 0j R =
r r
0I R
2 Rr =
0I
2 r
( 2) 87 , z j dy P
dB= 0j dy / ( 2 r ) x
, P x
P
B = By =dB y = + 0j dy
- 2 r
cos= + 0j x
dy
- 2
r 2
87 , y = xt an , dy = x d / cos 2 , r 2 = x 2 / cos 2 ,
y
+ / 2
B =
- / 2
0j
2
d=
0j
2
8. 984 I ( 85)
- +
[ ( 818) ] ?
301

8788( 2)
+
B
-
B ¡¤dx i =
+
= -
-
0I R 2 dx
= 0I
2( x 2 + R 2 ) 3 / 2
0I
2 0
sin d = 0I
2R + - dx
[ ( x / R) 2 + 1] 3/ 2
,
, dB= 0I dl/ ( 4 r 2 ) ,
, , r 2
+
,
-
+
B ¡¤dr
-
d©¦ r©¦ / r 2 0,
r =
B
8. 10 ( 88)
B= 0I
302
88810

, 0,
( b) ,
B ¡¤dr = B ¡¤a = 0aI
B = 0I
8. 11
( 89) ,
89811
( 1)
;
( 2) abc
;
( 3) , d ef
;
303

( 4) r 0
r =
2 r 0
( 1)
,
,
,
, ,
/ 2
( 2) , ( 3) , ,
, (
) ( ) , ,
( 4) r 2
r 2 0 , r 0
,
B= B0/ 2, r =
r 2 0B 0
2 r 0
= r 2 B
8. 12 ,
2. 0m, 1. 0m , 10km
7. 310 4 A ,
? ( )
B =
N = l/ ( 2 r )
B ¡¤2 R = 0N I =
0lI
2 r
0lI
4
2 rR = 4
7 10- 1010 3 7. 310 4
= 11. 6 T
4
2 1. 02. 0
8. 13810 ,
304

( R 1 R2, h,
) , N , I ,
?
810813
, , , B= 0
, r( R1< r < R 2) ,
B ¡¤dr = 2 rB = 0N I
, ,
B = 0N I / ( 2 r )
,
dr , h
= d=
d= Bhdr = 0N I h dr
2 r
0N
2 I h
R 2
1
R 1
r dr =
0N I h
2 ln R 2
R 1
8. 14 ,
j ,
, 0j / 2,
811 B, ,
305

, B= B1+ B 2= 0j ,
, B= B 1- B2= 0
811814
8. 15 I ,
, Rl
?
, r R, B= 0I r
2 R 2
, r R, B= 0I
2 r
,
=B ¡¤dS = R 0
0I r
¡¤ldr = 0I
2 R 2 4
l
8. 16, R
, r ,
8. 12
306

, J ,
d
812816
J
J J ( J = - J )
812 , J
P
B1 = 0
2 Jr 1
J P
B2 =
P
0
2 J r 2 = - 0
2 Jr 2
B = B1 + B2 = 0
J( r 1 - r 2) = 0
2 2 Jd
, , 0J d / 2,
8. 17( H elmholtz)
307

( 813)
R, R,
813817
( 1) z ;
( 2) z= 0 dB
2
dz d B
dz 2
( 1)
B =
2
0I R2
( 2) dB = -
dz z = 0
d 2 B
dz 2 z = 0
1
[ ( z + R/ 2) 2 + R 2 ] 3 / 2 +
+
= -
0I R 2
3( z + R/ 2)
2 [ ( z + R/ 2) 2 + R 2 ] 5 / 2
3( z - R/ 2)
= 0
[ ( z - R/ 2) 2 + R 2 ] 5/ 2 z = 0
+
3 0I R 2
2
1
[ ( z - R/ 2) 2 + R 2 ] 3/ 2
4( z + R/ 2) 2 - R 2
[ ( z + R/ 2) 2 + R 2 ] 7/ 2
4( z - R/ 2) 2 - R 2
[ ( z - R/ 2) 2 + R 2 ] 7 / 2 z = 0
= 0
8. 18 , R, q
: ,
308

B= 0 q
2 R
r ( r < R) , dr
dB =
0dI
2r
= 0
2r ¡¤2 rdr
2 =
B = dB =
R
0
0
2 dr
0
dr = 0 R = 0 q
2 2 2 R
8. 19814 ,
,
l, b,
a , I
I 1
,
?
?
814 ,
F l = BlI 1l = 0I I 1
2 a l
;
F r = BrI 1l =
0I I 1l
2 ( a + b)
814819
F = F l - F r =
,
0I I 1lb
2 a ( a + b)
,
309

8. 20, ,
j A / m
815 , dl dL d
Rd
dF =
0j Rd
j dldL =
2 a
,
0j Rd
4 Rsin( / 2) j dldL
815820
dF 1 =
dF
dldL =
0j 2 d
4 sin( / 2)
,
F 1 = F R 1 = dF R 1 = dF 1sin
2 2
= 0
0j 2 d
4 =
0j 2
2
8. 21
,
B1
B2( 816) ,
310

B l= Br= 0j
2 ,
B0
816 B B 2 >
B 1, Bl, Br
B0 , j
,
B 0 - B l= B 1, B0 + Br = B 2 B 0
= ( B1+ B 2) / 2, Bl = Br = ( B 2- B 1) / 2,
j = 2B l/ 0= ( B 2- B1) / 0
F = j B 0 = ( B 2 2 - B 2 1) / ( 2 0)
B1
816821
8. 22 5. 0cm , 30A
?
, , 30A
? 1. 0
10 -
3 m/ s
F m = 0I 1I 2
2 d = 410
- 7 3030
= 3. 610 - 3 N / m
2510 - 2
,
F e =
1 2
2 0d =
0c2 I 1I 2
2 d v 1v 2
= 3. 210 20 N / m
= c2
v 2 F m =
( 310 8 ) 2
( 110 - 3 ) 2 3. 610 - 3
,
8. 23817 , R ,
( z )
311

, I
( 1) ;
( 2)
(
I ) ,
, ?
( 1) 817 ,
j RdI
,
dF = dB ¡¤I =
0j Rd I
2 R
x
817823
, y
F = dF x
x
= dF sin = 0j I sin d = 0j I
20
=
0I 2
2 R
( 2) x , d
,
0I 2
2 R = 0I 2
2 d
d =
R/ 2
8. 24 (
10k m/ s) 8. 18 ,
I
, r ( ) , a
,
312

F =
0I 2
2 ln a + r
r
L = 5. 0m , a = 1. 2cm, r = 6. 7cm, m =
317g, 4. 2k m/ s
818824
( 1) ?
( )
?
( 2) ?
( 3) 40% ,
( B1
F =
2 a + r
r
I B1dr =
2 a + r
( 1)
r
0I 2
4 r dr =
)
0I
2
2 ln a + r
r
a = v 2 / ( 2L) = ( 4. 210 3 ) 2 / ( 25. 0)
= 1. 7610 6 m/ s 2
( 2) F = m a ,
a / g = 1. 7610 6 / 9. 8 = 1. 810 5
I =
0I 2
2 ln a +
r
r
= ma
2 ma
0ln [ ( a + r ) / r ]
1 / 2
313

=
231710 - 3 1. 7610 6
410 - 7 ln[ ( 1. 2 + 6. 7) / 6. 7]
= 4. 110 6 A
( 3)
P = 1 2 mv2 / ( 0. 4t) = 1 2 mv2 / ( 0. 42L/ v) = mv3
1. 6L
= 31710- 3 4. 2 3 10 9
1. 65. 0
= 2. 910 9 W = 2. 9 Mk W
8. 25B I ,
T( 819) ,
819825
( 1) : r = T / ( BI )
( 2) , P
p = qT / I - q ,
(
)
( 1) 819 , dl= r d
dF m= I Bdl = I Brd , T 1= T 2= T
T1 T2 - d , dF T =
314

T d , dF m = dF T
r = T / ( BI )
( 2) P ,
,
r =
p
qB =
qT
qBI =
,
8. 26( 816)
( v ) ,
:
( 1) ( ( 83) )
B =
0I cos 1
-
4 r ( 1 -
2 sin 2 1) 1 / 2
T BI
cos 2
( 1 -
2 sin 2
2) 1/ 2
( 2) [ ( 8. 4) ] ,
B =
0I
2 r
( 816) q
B1 =
0q( 1 -
) vr
4 r
( 1 -
sin 2 ) 3 / 2
r I dl
r dB
,
dB = B=
0( 1 -
2 ) vr
Sqdl
4 r
3 ( 1 -
2 sin 2 ) 3 / 2
Sqv= I v dl ,
dB =
0I ( 1 -
) dlr
4 r
( 1 -
sin 2 ) 3/ 2
315

( 1) ( 820) , P
B =
0I ( 1 -
2 ) r dlsin
4r
3 ( 1 -
2 sin 2 ) 3/ 2
l= - r cot , dl= rd / sin 2 , r = r / sin ,
B = 0I 2 ( 1 -
2 ) sin d
= -
4
1
( 1 -
2 sin 2 ) 3/ 2
r
= 0I cos 1
-
4 r ( 1 -
2 sin 2 1) 1/ 2
0I cos
4 r ( 1 -
sin 2
cos 2
( 1 -
2 sin 2
2) 1 / 2
2
) 1/ 2
1
820826( 1)
( 2) , 1= 0, 2=
B =
0I / ( 2 r )
8. 27811 ,
e ( 827) B
( 817) (
)
,
316

821 , ,
Rq R q
:
=
2 ( 1 - cos )
821827
( )
e =
t ,
q
0 4 =
d e
=
dt
q
( 1 - cos )
2 0
q
sin d
2 0 dt
821 , dt ,
d= vsin dt/ r
d
dt = vsin
r
B
B ¡¤dr =
B ¡¤2 R =
d e
( 827) B ¡¤dr = 0 0
dt
2 Br sin=
0q
2 sin 2 v r
2 Brsin
317

B = 0qvsin
4 r 2
( 817)
8. 28 5. 0cm
, , dE
dt = 1. 010 12 V/ ms
( 1) ;
( 2) B
dE
( 1) I d = 0 R 2
dt
= 8. 8510 - 1
2 0. 051. 2 010 - 12
= 7. 010 - 2 A
( 2) B= 0I d
2 R = 410-
710 7 - 2
= 2. 810 - 7 T
20. 05
8. 29 ( C= 1
10 - 1 2 F ) , 50H z, 1. 7410 5 V ,
e =
ES = S / 0 = Q/ 0 = CU / 0
d e
I d = 0 = C dU
dt dt
I d, ma x = 2CU m a x = 250110 - 12 1. 7410 5
= 5. 510 - 5 A
8. 30 c/ 3( c ) ,
110 -
1 0 m
,
F e = E e =
e 2
4 0r 2 ( 1 -
,
318
2 ) 1/ 2

F m = ev 1B2 = ev 1
=
e 2 v 2
4 0c 2 r 2 ( 1 -
E 2v2
c 2 =
2 ) 1 / 2
e 2 v 1v2
4 0c 2 r 2 ( 1 -
F e> F m , ,
F = F e - F m =
e 2
4 0r 2 ( 1 -
1 -
2 ) 1 / 2
= ( 1. 610- 19 ) 2 910 9
2 1/ 2 1 -
( 110 - 10 ) 2 1
1 -
3
= 2. 210 - 8 N
2 ) 1/ 2
8. 31a , 1
2, v 1 2
v2
? (
,
, )
822 , 1( p 1)
2( p 2)
F 1e = eE 2 =
e 2
4 0a 2 ( 1 -
2 2) 1/ 2
2
1 2
v 2
c 2
1
3
2
822831
F 1m = eB 2v1 =
e 2 v1v 2
4 0a 2 c 2 ( 1 -
2 2) 1/ 2
, v 2
1 2
319

F 1 = F 1e + F 1m
=
e 2
4 0a 2 ( 1 -
,
= arct an F 1m
F 1e
2 1
1 + v 2 1v 2 2
2 2) 1/ 2 c 4
= arct an v 1v 2
c 2
F 2e = eE 1 = e2 ( 1 -
2 1)
4 0a 2
1 2 , 2 2
F 2 F 2e, 1F 1 F 2
320

9
9. 1 , N ,
mB
: , Wm = - mB
: -
mB + mB (
) W
e -
:
W / ( kT ) B
/ ( kT )
M = N m emB - e
e mB / ( kT ) + e
( 1) mBkT ,
M = N m 2 B/ ( kT )
MB/ T
( 2) mBkT ,
M = N m
- mB / ( kT )
- mB / ( kT )
,
W- = - mB, N - = N e - mB / ( kT ) /
( e mB / ( kT ) + e - m B/ ( kT ) ) ; W+
= mB, N + = N e m B / ( kT ) / ( e mB / ( kT ) + e - mB / ( kT ) )
,
M = N + m + N - ( - m) = N m em B/ ( kT ) - e
e m B/ ( kT ) + e
( 1) mBkT ,
- mB / ( kT )
- mB / ( kT )
321

M =
N m
1 + mB
kT - 1 - mB
kT
1 + mB
kT + 1 - mB
kT
( 2) mBkT ,
= N m2 B
kT
M = N m emB / ( kT ) ( 1 - e - 2m B/ ( kT ) )
e mB / ( kT ) ( 1 + e - 2m B/ ( kT ) ) = N m
9. 291 , L B
( 1)
BeL sin ;
2m e
( 2) Be ,
2me
1T
9192
( 1) 91 ,
m =
e
r 2 =
ev
2 r r 2 = evr
2 = eL
2me
, L
M =
©¦mB©¦= mBsin= BeLsin / ( 2m e)
L B
dL
( 2) M= = Lsin , M ,
dt
B= 1T ,
322
=
= Be/ ( 2me )
11. 610-
19
29. 110 - 31 = 8. 810 10 rad/ s

9. 3, , r =
0. 5310 - 1 0 m, v= 2. 210 6 m / s
( 1) B ?
( 2) S = h/ 2= 0. 5310 - 3 4
Js, m= 1. 4110 -
26 Am 2 ,
, ?
( 1) B = 0ev = 410-
1. 7 610 - 19 2. 210 6
4 r 2 4( 0. 5310 - 10 ) 2
= 12. 5 T
( 2) M = mBsin , M =
S sin ,
=
mB
S = 1. 4110 - 2 6 12. 5
0. 5310 - 34 = 3. 310 9 rad / s
9. 4 ,
1. 0cm, 12cm,
7. 8g/ cm 3 ,
( ) 55. 85g/ mol
( 1) ?
( 2) ?
( 3) ?
( 4) ?
( 1) N = 120. 5
2 7. 86. 02310 23 2/ 55. 85
= 1. 610 2 4 ( )
( 2) m= N m B= 1. 610 24 9. 2710 - 24 = 15 A m 2
( 3) I = m/ S = 15/ ( ( 0. 005) 2 ) = 1. 910 5 A
( 4) B= 0I / l= 410 -
7 1. 910 5 / 0. 12= 2. 0 T
9. 5 ,
8. 5cm, 1. 0cm, 0. 02cm ,
dS
dt
=
323

7. 8g/ cm 3 ,
( ) 55. 85g/ mol
( 1) ?
( 2) , ?
0. 5210 -
?
4 T
( 3) ,
8. 51. 00. 027. 8
( 1) m = N m B= 6. 02310 2 3
55. 85
29. 2710 - 2 4 = 0. 27 Am 2
( 2) M = ©¦mBE©¦= mBE = 0. 270. 5210 - 4
= 1. 410 - 5 N m
( 3) Wm = - mBE = mBE = 0. 270. 5210 - 4 = 1. 410 - 5 J
9. 6 l = 10cm, N = 20,
I = 0. 1A
( 1) B0 H 0;
( 2) r = 4200 ,
B H ?
( 3) B0
B ?
( 1) B 0 = 0I = 0N I / l= 410 - 200. 7 1/ 0. 1
= 2. 510 - 5 T
H 0= I = N I / l= 200. 1/ 0. 1= 20 A / m
( 2) B= rB0= 42002. 510 -
H = H 0= 20 A / m
5 = 0. 11 T
( 3) B 0= 2. 510 - 5 T , B = B- B0= 0. 11 T
9. 7, 30cm, 1cm 2 ,
300 0. 032A ,
324

210 - 6 W b:
( 1) ( ) ;
( 2) ;
( 3) ( ) ;
( 4) ;
( 5)
( 1) B= / S = 210 - 6 / ( 110 - 4 ) = 210 - 2 T
( 2) H = I = N I / l= 3000. 032/ 0. 3= 32 A / m
( 3) J = M= B - H = 210- 2
- 32= 1. 610 4 A / m
0 410 - 7
( 4) = B/ H = 210 - 2 / 32= 6. 310 - 4 H / m
r= / 0= 6. 310 - 4 / ( 410 - 7 ) = 5. 010 2
( 5) M= J = 1. 610 4 A / m
9. 8,
1000 , 15. 0cm, 2. 0A ,
B= 1. 0T , :
( 1) H ;
( 2) r ;
( 3)
( 1) H = I = N I / ( 2 r) = 10002. 0/ ( 20. 15)
= 2. 110 3 A / m
( 2) = B/ H = 1/ ( 2. 110 3 ) = 4. 710 - 4 H / m
r= / 0= 4. 710 - 4 / ( 410 - 7 ) = 3. 810 2
( 3) J = M= ( r - 1) H = ( 3. 810 2 - 1)2. 110 3
= 8. 010 5 A / m
9. 99. 2
6. 0A , 1. 2T
, ,
325

?
9299
92 B = 1. 2T , H = 2. 2
10 2 A / m H = I 1, = H / I 1 ,
B,
I =
B
0 = BI 1
0H = 1. 26. 0
= 2. 610 4 A
410 - 7 2. 210 2
9. 10800
, 1. 0T , ? (
9. 2 BH )
BH B = 1. 0T , H = 1. 7
10 2 A / m
I = H / = 1. 710 2 / 800 =
0. 21 A
9. 11 ( )
93( a ) ,
9. 3( b)
, 0. 8mm , 0. 5mm ,
0. 3mm
326

, 9. 3( b ) , ,
? H c =
2A / m
93911
, ,
, H c=
m a x
= 2 r in tH c =
4
2
510- 2 =
2
m ax
r in t
,
2
3. 1 mA
9. 12 61cm, 1cm,
1000 1. 5A ,
B 0. 18T r (
)
r =
=
l
0N I / B -
0. 61
= 1.
410 - 7 10 3 3103
1. 5/ 0. 18 - 0. 01
9. 13 94
l 1= 500mm, l 2= 20mm,
r= 5000 B= 3T ,
N I
327

94913
N I = B l 1
+ l 2
0 r 1 = 3 0. 5
+ 0. 02
410 - 7 510 3
= 4. 910 4 ( )
9. 14 , 14. 4cm,
2. 0mm , 0. 60cm 2 ,
r= 16004. 810 - 6 W b,
N I 220V ,
20W , N
N I =
l 1
+ l 2
0S r 1
=
4. 810 - 10 6 - 2
14. 4- 0. 2
+ 0. 2
410 0. 7 610 - 4 1600
= 1. 3310 2 ( )
N = ( N I ) / I = 1. 3310 2 220/ 20 = 1. 4610 3 ( )
328

10
10. 1I = 5A ab,
l= 20cm, d = 10cm ( 101)
v= 10m/ s ,
? a , b ?
101101
101
Ea b = d E = ( vB) ¡¤dr = - vBdr = -
= -
0I v
2 ln d +
d
l
= -
= - 1. 110 - 5 V
E a b< 0, a
410 -
v
d + l
7
510
ln
2
d
0I
2 r dr
10 + 20
10
10. 2 12cm 410 3 , 0. 5G
30 ,
329

? , ?
E m a x= N BS
= 410 0. 3 510 -
4 0. 122
2 30
= 1. 7 V
,
10. 310. 2 ,
I = 5. 0A ,
110 3 , a = 10cm, L =
20cm, v = 2m/ s ,
d = 10cm
102 ,
102103
,
,
E = E 1 - E
2 = N ( B 1 - B 2) Lv = N Lv 0I 1
2d - 1
d +
= 110 3 0. 22 410-
7 5. 0
2
= 210 - 3 V
a
1
0. 1 - 1
0. 1 + 0. 1
10. 4 , =
5sin 100 t A , ?
= N = N B ¡¤dS =
E = -
d
= -
dt
N
d + a
0N L
2 ln d +
d
d
0
2 r ¡¤Ldr =
a
d
dt
0N L
2 ln d +
d
a
330

= -
410 - 7 110 3 0. 2
2
= - 4. 410 - 2 cos 100 t
ln
0. 1 + 0. 1
0. 1
d( 5sin 100 t)
dt
10. 5R , B
L
, 103
, dB ,
dt
,
103 , Oba
S , S
= BS , 103105
O
-
d
= - S dB
dt dt
= E idr =
b
= 0 + E ba + 0 = E ba
Eba = -
dB
S
dt
dB
= -
O E idr + a b E idr + O a E idr
1
2 L R 2 - L 2 / 4 dB
dt
> 0,
dt
E
ba < 0, b
: dl
E i = 1 2
dB
dt
, 103
Ea b b = E i ¡¤dl =
a b E idlcos =
a
b a
r cos dB
2 dt dl
= h dB
2 dt b dl = hL dB
= 1
a 2 dt 2 L R2 - L 2 / 4 dB
dt
E
a b> 0, b
10. 650 FBC1
331

12. 705m 0. 4210 - 4 T ,
M 1. 70( M , )
, ?
U = E= Blv= 0. 4210 -
= 0. 30 V
4 12. 7051. 70330
,
10. 7 ,
0. 7010 - 4 T , 200m ,
7. 010 -
3 V ,
v = E/ ( Bl ) = U/ ( Bl ) = 7. 010 - 3 / ( 0. 7010 - 4 200)
= 0. 50 m/ s
10. 8,
2. 010 -
2 T
10. 0cm20. 0cm, 120 ,
12. 0V ,
?
, Ema x= N BS = 2 N BS
=
E m ax
2 N BS = 12. 0
21202. 010 - 2 0. 10. 2
= 40 s - 1
10. 9 : ,
N , S ,
R,
, ,
qN , S , q, R
332

q = dt = 1 REdt = -
=
R =
N BS
R
B = qR/ ( N S )
1
R d
dt dt = - 1
R 0 d
10. 10, , b
, r a 2
B( 104)
1041010
,
E = Blv = Ba r
,,
a / ( a b) = / b
I = Eb/ = Bab r/
F = BI l= B 2 a 2 b r /
M = F r =
( Bar ) 2 b /
10. 11,
, B B
333

B ,
R ,
eE = mea t = me
evB = m ean = me
2 eB= m ev/ R( B ) , dv
dt =
1
dv
dt
v 2
R
eR
m e
,
dB
= E dt R
1 d
E = = R2 dB
2 R dt 2 R dt = R dB
( B
2 dt
) ,
dB
= 1 dB
dt 2 dt
B = B / 2
10. 12 1011( a)
, r :
v
F ( ) , B , v+ v
( )
v v+ v ,
F n =
me ( v + v) 2
r
-
m ev 2
B
334
r
v =
2m ev
v ( )
r
er
B
2me

B =
2m e v
er
F m = e( v + v) B 2mev v ( )
r
F m = F n
B , B
v ,
,
10. 13 l R
N 1, N 2
N 1 I 1
2 1 = N 2 1 = N 2 0N 1I 1S / l =
M =
21
= 0N 1N 2 R2
I 1 l
, N 2
0N 1N 2 R 2 I 1/ l
10. 14a 50 , 4. 0cm 2 ,
100 , 20. 0cm b
, :
( 1) ;
( 2) a 50A / s , b
;
( 3) b
( 1) b I b , a b
, a
a b =
N b
0I b
N aS a
2R b
335

M = a b
I b
= 0N a N bS a
2Rb
= 4
7 10- 501004. 010 - 4
20. 2
= 6. 310 - 6 H
( 2) d ba
= 1 d ba
= 1 M da
dt N b dt N b dt = 1
100 6. 310- ( 6 - 50)
= - 3. 110 - 6 Wb / s
( 3) E ba = - M da
dt = - 6. 310 - 6 ( - 50)
= 3. 110 - 4 V
a
I a I a a , a b
1, b 2 = 1+ 2,
b - 1= 2= ba
R b> R a , 2 I a
b a r B= 0m/
( 4 r 3 ) = 0I aS a N a / ( 4 r 3 )
ba =
2 = B ¡¤dS =
M =
R b
0I a S a N a
4 r 3
ba
= N b 0I aS a N a
I a 2R bI a
¡¤2 rdr =
= 0N aN bS a
2Rb
0I a S a N a
2R b
10. 15 2. 0cm , 30. 0cm,
1200 ,
( 1) ?
( 2) 3. 010 2 A / s ,
?
336

( 1) L = 0N 2 S/ l = 0N 2 R 2 / l
= 410 -
= 7. 610 - 3 H
( 7 1. 210 3 )
2 0. 02 2 / 0. 3
( 2) ©¦E©¦= L d = 7. 610- 3. 3 010 2 = 2. 3 V
dt
10. 16 10. 0A ,
20Wb; 4. 0A/ s
, 3. 2mV
L =
E d
dt = 3. 210- 3 / 4. 0 = 0. 810 - 3 H
L = N / I ,
N = LI / = 0. 810 - 3 10. 0/ ( 2010 - 6 ) = 400 ( )
10. 17105 , N
1051017
( 1) ;
( 2)
M 12M2 1, ?
( 1) I =
0N I h
2 ln R 2
R 1
337

L =
= N = 0N 2 h
I I 2 ln R 2
R 1
( 2) , 1
I 1
,
M2 1 = 21
I = 1
= 0N I 1h
1 I 1
I 2
R 2
12 = B2hdr =
R 1
M 12
2 ln R 2
R 1
/ I 1 = 0N h
2 ln R 2
R 1
, B 2 = 0I 2/ ( 2 r )
R 2
0I 2h
2
R 1
dr
r
= 12
I = N 12
= 0N h
2 I 2 2 ln R 2
R 1
M 12 = M2 1
= 0I 2h
2
ln
R2
10. 18 a , D ,
,
L 1 = 0
ln D -
a
,
I ,
= 2 D - a
a
B1dr ¡¤1 = 2 D -
a
a
a
0I
2 r dr =
L 1 = = 0
I
ln D -
a
a
R1
0I
ln D -
a
10. 19, ,
338
a

L 1 L 2, L
, :
( 1) ,
( 2) ,
L = L 1 + L 2
1
L = 1 L 1
+ 1 L 2
( 1) , E= E 1 + E 2, = 1 = 2 E=
- L d
dt , E 1 = - L 1
d
dt E 2 = - L 2
L = L 1 + L 2
d
dt
( 2) , = 1+ 2, E= E 1= E 2d dt = - E, d 1
L dt
= -
E 1
L 1
d2
dt = - E 2
L 2
d
dt = d1
dt + d2
dt
1
L = 1 + 1 L 1 L 2
, ,
,
,
10. 20L 1 L 2, M
( 106)
( 1) , 2, 3
, 1, 4 ,
L= L 1+ L 2+ 2M;
( 2) , 2, 4
, 1, 3 ,
L=
1061020
L 1+ L 2- 2M
( 1) ,
339

,
=
1 + 12 + 2 + 2 1
= LI , 1= L 1I 1, 12 = MI 2, 2= L 2I 2, 21 = MI 1, I
= I 1= I 2
L = L 1 + L 2 + 2M
( 2) ,
,
21 12 ,
,
L = L 1 + L 2 - 2M
10. 21 10 8 T ,
? ( , kg/ m 3
)
w m=
B 2
=
2 0c 2
44
( 10 8 ) 2
= 4. 410 4 kg/ m 3
2410 - ( 7 310 8 ) 2
10. 22 2. 0T ,
110 6 V / m ?
?
w m= B 2 / ( 2 0) = ( 2. 0) 2 / ( 2410 - 7 ) = 1. 610 6 J/ m 3
w e= 0E 2 / 2= 8. 8510 - 12 ( 10 6 ) 2 / 2= 4. 4 J/ m 3
,
10. 23
1kW h , 1. 0T , ?
500A ,
?
340

Vm = W / w m = 3. 610 6 2410 - 7 = 9. 0 m 3
L = 2W / I 2 = 23. 610 6 / 500 2 = 29 H
10. 24 5. 5mm, 20A
?
1. 6910 -
8 m
w m = B 2
= 1
2
0I
=
2 0 2 0 2 R
= 0. 21 J/ m 3
410 - 20 7 2
24 ( 2 5. 510 - 3 ) 2
,
E = J = I / ( R 2 ) ,
E = w e =
0
2 E 2 = 0( I )
2
2( R 2 ) 2
= 8. 8510 - 1 2 ( 1. 6910 - 8 20) 2
2[ ( 5. 510 - 3 ) 2 ] 2
= 5. 610 - 17 J/ m 3
10. 25,
R 1
R2, ,
r 1, I ( ,
) , ,
W m1 = B 2
0 R 1
0
= 1
2
dV
2 0
0I r
2 R 2 1
2
R 2
¡¤2 r dr ¡¤1 +
R 1
2
0I
2 r
= 0I 2
1 R2
+ ln
44 R1
W m1= L 1I 2 / 2,
L =
0 1
24 + ln R 2
R 1
¡¤2 r dr ¡¤1
341

10. 26 a ,
D 1a ( 107) I
;
( 1)
( 2) ,
D 2
A m ;
( 3)
Wm ;
( 4)
E
A E ;
( 5) A m, Wm A E
( 1)
L = =
I
2 D 1 -
= 0
ln D 1 - a
a
( 2)
a
a
0I
¡¤dr ¡¤1
2 r
I
0
ln D 1
a
A m = F m ¡¤dr =
D 2
( 3)
W m = 1 2 L 2I 2 -
= I 2
2
D 1
0I 2
dr ¡¤1 =
2 r
1
2 L 1I 2
0
ln D 2
a - 0
ln D 1
a
1071026
0I
2
2 ln D 2
D 1
= 0I 2
2 ln D 2
D 1
( 4) ,
E
342

A E = EI dt = -
= -
0I 2
ln D 2
D 1
( 5)
d
dt I dt = - I d= - I D 2
- A E = Wm + A m
2 0I dr ¡¤1
D 1
2 r
E ,
, ,
- A E , D 1 D 2 ,
,
10. 27 S ,
, 108 1
I 1
I 2
F m = 012SI 1I 2
2,
1081027
, ,
108 I 1 I 2
M =
01I 1x 2S
I 1
= 012Sx
Wm = 1 2 L 1I 2 1 + 1 2 L 2I 2 2 + MI 1I 2
,
343

I 1 I 2 , 2 x ,
W m = ( M) I 2I 2
, E 1 =
-
( MI 2)
= -
t
M
t I 2, E 2= -
M
t I 1I 1 I 2
,
x t ,
A e xt = - (E1I 1 t + E2I 2 t) = 2( M) I 1I 2
x , 2 F m
A m = F m x
A e xt = Wm + A m
2( M) I 1I 2 = ( M) I 1I 2 + F m x
F m = I 1I 2( M/ x ) = 012SI 1I 2
F m> 0, 2 x ,
344

11
11. 1
, : ,
( : ( 111) q= 0 e,
d e
, (
dt
L) )
S( 111) , B
LS , L
, ( 111)
L
B ¡¤dr = 1 c 2 d
dt S
E ¡¤dS
, L= 0,
L
B ¡¤dr = 0
111111
d
dt E ¡¤dS = 0
t
( 111) E ¡¤dS = qint / 0, dqin = 0,
dt
q ,
11. 2 : 112
(
) , B= 0
112 , r P
345

Br
, B
r S ,
S
B ¡¤dS =
4 r 2 B r
( 111) , S B ¡¤dS = 0 B r = 0
112112
Br P ,
S I S = J ¡¤dS = 0, e
= S
E ¡¤dS = S
S
J ¡¤dS = J ¡¤dS = 0 ( 111)
, ,
, B = 0
L
B ¡¤dr =
2 rB =
0I + 1 d e
= 0
c 2 dt
B , B= 0
P , , B= 0
11. 32. 010 24 m/ s 2
45, 20cm ,
? , ?
346

E = easin 1. 19 610-
=
4 0c 2 r
= 0. 11 V / m
t = r / c =
210 24 sin( / 4)910 9
( 310 8 ) 0. 2 2
0. 2
310 8 = 6. 710 - 10 s
11. 4X ,
( X ) 2
10 4 eV , 510 - 9 m
, 0. 3m
E = easin
4 0c 2 r = eE k
4 0c 2 msr
= 9109 1. 610 - 19 210 4 1. 610 - 19
( 310 8 ) 2 0. 9110 - 3 0 510 - 9 0. 3
= 3. 810 - 2 V/ m
11. 5( ) ,
v = v 0cos t, v0 = 8. 010 - 3 m / s, = 6. 0
10 6 rad/ s
( 1) ?
( 2) , 1. 0k m ,
?
1. 0km ?
( 1) a ma x= v0= 6. 0108. 6 010 - 3 = 4. 810 4 m / s 2
( 2) E , m a x=
ea ma xsin
4 0c 2 r
= 1. 19 610- 4. 810 sin 4 ( / 2)910 9
( 310 8 ) 10 2 3
= 7. 710 - 25 V / m
t = r/ c = 10 3 / ( 310 8 ) = 3. 310 - 6 s
11. 6 , 1. 110 1 4
m / s 2
347

( 1) 45 , 0. 50m
;
( 2) ( 1)
ea sin
( 1) E =
4 0c 2 r
= 1. 610 - 1 1. 9 110 14 sin ( / 4)910 9
( 310 8 ) 0. 2 5
= 2. 510 - 12 V / m
B = E / c = 2. 510 - 12 / ( 310 8 ) = 8. 310 - 2 1 T
( 2) a , E , B113
113116
11. 7 0. 26cm 3. 910 - 3
V / m , 12. 0A
( 1) ?
( 2) 4. 0m
?
( 3) 5. 0cm
, 4. 0m ?
( 4) 12. 0A ?
? ?
348

( 1) a = E e = 3. 910- 1. 3 610 - 19
= 6. 810 8 m/ s 2
me 9. 110 - 3 1
( 2) E 1= easin
4 0c 2 r
= 1. 19 610- 6. 810sin( 8 / 2)910 9
( 310 8 ) 4
2
= 2. 710 - 1 8 V / m
B 1=
E 1 / c = 2. 710 - 1 8 / ( 310 8 ) = 910 - 27 T
( 3)
( 4) B s
= D M N A = 8. 910 3
6. 310 - 3 6. 02310 23
= 8. 510 2 8 m - 3
E = E 1 r 2 l
= 2. 710 - 18 8. 510 28 0. 13 2 10 - 4
5. 010 - 2 = 6. 110 6 V / m
B= E / c = 6. 110 6 / ( 310 8 ) = 2. 010 - 2 T
= 0I l = 410-
12. 7 05. 010 - 2
4 r 2 44 2
= 3. 810 - 9 T
, ,
, , E 1
B 1,
11. 8 1. 3810 3 W/
m 2
E r ms=
I =
c 0E 2 rm s
I / ( c 0) = 1. 3810 3 / ( 310 8 8. 8510 - 12 )
= 7. 210 2 V / m
Br ms= E r ms / c = 7. 210 2 / ( 310 8 ) = 2. 410 - 6 T
349

11. 9 60 m,
300kW
?
S =
P =
R 2
E m = S / ( c 0)
30010 3
= 1. 110 14 W/ m 2
( 3010 - 6 ) 2
= 1. 110 1 4 / ( 310 8 8. 8510 - 12 )
= 2. 810 8 V / m
Bm = E m/ c = 2. 810 8 / ( 310 8 ) = 0. 93 T
11. 10 ( 514. 5nm ) 3. 8kW
0. 880 rad
3. 8210 5 km :
( 1) ;
( 2)
( 1) R = r / 2= 3. 82100. 8 88010 - 6 / 2
= 168 m
( 2) I = P / S = 3. 810 3 / ( 168 2 ) = 0. 043 W / m 2
11. 11 , l, a , ,
I ( 114) , :
( 1) ,
, ( :
)
( 2)
,
S ¡¤dA = I 2 R
dA , R
, ,
350

,
( 1) E
B 114 S= E
B/ 0 S
( 2) B = 0I / ( 2 a) , E = J =
I / ( a 2 ) , A= 2 a l
A
S ¡¤dA = - A
= -
E B
dA
0
0I I
¡¤2 al
2 a
0 a 2
= - I 2 l / ( a 2 ) = -
I 2 R
1141111
11. 12E R
r 1 r 2( 115)
1151112
( 1) r 1< r < r 2 E B S ,
S ;
( 2) S ,
E 2 / R, R
( 1) ,
U= E ,
115 S= E
351

B/ 0 S R
B =
0I
2 r =
, ( r 1 < r < r 2)
2 rR
0 E
l l,
r ( r 1< r < r 2) e= 2 rlE
, r E = e/
2 E l
r 2
( 2 rl ) E = U = Edr =
r 1
E
S r
I
S =
E =
e
2 l ln r 2
r 1
E
rln( r 2/ r 1)
EB = E 2
0 2 r 2 Rln( r 2/ r 1)
( 2) S
r 2
r 1
S 2
rdr =
r 2
r 1
E 2 dr
rRln ( r 2/ r 1) = E 2
R
, e =
ln( r 2/ r 1)
ln( r 2/ r 1) = E 2
R
ln( r 2/ r 1) ,
11. 133. 0 cm, E
30V / m , :
( 1) ?
( 2) ?
( 3) , 0. 5m 2
?
( 1) = c/ = 310 8 / ( 3. 010 - 2 ) = 1. 010 10 Hz
( 2) B m= E m / c= 30/ ( 310 8 ) = 1. 010 - 7 T
( 3) F = p A = 0E 2
A= 1 2 0E 2 mA
352

= 1 2 8. 8510- 12 30 2 0. 5= 210 - 9 N
11. 14 1. 110 3 W / m 2
? 0. 5m 2 ,
50%
p = S c ( 1 + 50% )
F = p A = SA c ( 1 + 50% )
= 1. 1103 0. 5
310 8 1. 5 = 2. 810 - 6 N
11. 15
,
1. 510 9 W , 1. 3mm 2
?
p m
= 2S m
c
= 2P m
cA
= 7. 710 6 Pa
=
21. 510 9
310 8 1. 310 - 6
11. 16 10m ,
10k W 100kg,
,
?
, I / c,
F = I / ca
= F / m= I / ( cm)
t =
2l
a =
2lcm
I
= 7. 7510 3 s = 2. 15 h
=
210310 8 100
1010 3
11. 17 ,
353

1. 0g/ cm 3 r ,
?(
6. 910 7 W/ m 2 ) ?
R ,
F g = GM S 4
R 2 3 r 3
F g= F r
F r = S c A = P SR2 S
r 2
cR 2
r =
=
3P SR 2 S
4GMS c
36. 910 7 ( 710 8 ) 2
46. 6710 - 1 1 210 30 310 8 1. 010 3
= 6. 310 - 7 m = 6. 310 - 4 mm
,
R ,
, ,
354

4
1
11,
;
x = 0. 05cos 8 t +
3
( 1)
( 2) t= 1s, 2s, 10s ;
( 3)
( 1) x = Acos ( t + )
= 8= 25. 1 s - 1 ;
T =
2 / = 0. 25 s;
A = 0. 05 m;
=
/ 3;
v m = A = 80. 05 = 1. 26 m/ s;
a m =
2 A = ( 8 ) 2 0. 05 = 31. 6m/ s 2
( 2) 1= t1+ = 81+ / 3= 25 / 3,
2= t2+ = 82+ / 3= 49 / 3,
3= t3+ = 810+ / 3= 241 / 3
355

( 3) x , v, a t 11
1111
12 , x A
t= 0 ,
: ( 1) x 0= - A; ( 2) x ; ( 3)
x = A / 2 , x
( 1) = ; ( 2) = - / 2; ( 3) = / 3
1212
1313
( 1) a , b, c, d , e ;
( 2) ;
( 3)
356

1313
( 1) x = A cos, x :
x a =
A, a = 0;
x b =
A 2 , b = / 3;
x c =
x d = -
x e = -
0, c =
/ 2;
A
2 , d = 2 / 3;
A
2 , e =
4 / 3
( 2) t= 0 , x = A 2
, = arccos( 1/ 2) = / 3 ;
v> 0, = - / 3 T = 1. 2s,
2
= 2 / ( T / 2) = 5 / 6
x =
0. 05cos
5
6 t -
3
( 3) 14
1413
14 , v m= 3cm/ s, A
= 2cm ,
( 1) ;
357

( 2) ;
( 3)
( 1) vm = A= 2 A / T ,
T =
2 A/ vm = 20. 02/ 0. 03 =
4. 2 s
( 2) a m=
2 A = v 2 m/ A = 0. 03 2 / 0. 02= 4. 510 - 2 m/ s 2
( 3) t = 0 , v= + v m, = - / 2, = vm = 0. 03/
A
0. 02= 1. 5s - 1 ,
x = A cos( t +
) = 0. 02cos( 1. 5t - / 2)
15 , A = 2. 010 - 2 m, T =
0. 50st= 0 ,
( 1) x = 1. 010 - 2 m , ;
( 2) x = - 1. 010 - 2 m ,
( 1) x = Acos
2 t+ = 2. 2 010-
T
( 2) x = 2. 010 - 2 cos( 4 t- 2 / 3)
cos 4 t+
3
1610g , 24cm,
4s, t= 0, + 24cm:
( 1) t= 05s , ;
( 2) x = 12cm
= 0,
( 1) t= 0. 5s ,
x = 0. 24cos
F = ma = -
358
x = Acos
= - 4. 210 - 3 N
2
t + = 0. 24cos
T 2 t
0. 5 = 0. 17 m
2
m
2 x = - 1010 - 3 ( / 2) 2 0. 17

( 2) x = 0. 12m , arccos( 1/ 2) = / 3
2 t m in=
3
tm in = 2 3 s
17
x 1= Acos( t + ) ,
,
( 1) ;
( 2) t= 0 , x 1= - A/ 2, x ,
x t
( 1) / 2 , ,
2- 1= 0- / 2, 2= 1- / 2= - / 2
x 2
= Acos( t +
2) = A cos( t +
- / 2)
( 2) t= 0 , x 1= A/ 2 v< 0, = 2 / 3
x 1
= Acos( t + 2 / 3) , x 2 = Acos( t +
x t 15
/ 6)
1517
359

18
, ,
,
16 ,
A
2
= Acos( t +
t +
1 = / 3, t +
1) = A cos( t +
©¦ ©¦= ©¦ 2- 1©¦= 0, 2 / 3, 2 / 3
2)
2 = / 3
1618
19 , k= 25N / m,
02J 06J , :
( 1) ?
( 2) , ?
( 3) , ?
( 1) A = 2E / k = 2( E k+ E p ) / k
= 2( 0. 2+ 0. 6) / 25= 0. 25m
( 2) E p= E k , E p= 1 2 kx 2 = 1 2 E= 1 2 kA 2 / 2
360
x = 2
2
2
A= 0. 25= 0. 18 m
2

( 3) E p= 1 2 k A 2
= 1 1
2 4 2 kA 2 = 1 4 ( 0. 2+ 0. 6) = 0. 2 J
110 k ,
m , b
,
A , 1 2 kA 2
mg = kb, k= mg / b
, x ,
mg = kb ,
mg - k( x + b) = m d 2 x
dt 2
m d 2 x
dt 2 = - kx
, T =
2m/ k ,
,
E p , e = 1 2 k( x + b) 2 -
,
E p , g = - mg x
1
2 kb2 = 1 2 kx 2 + kbx
E = E k + E p = E k + E p , e + E p, g
= 1 2 mv 2 + 1 2 kx 2 + ( kb - mg ) x
E = 1 2 mv2 + 1 2 kx 2
x = A , v= 0
361

E = 1 2 kA 2
111, 10g ,
49cm 80g ,
10cm , v0= 50cm/ s
m 1g = kx 1
k = m 1g / x 1 = 0. 019. 8/ ( 4. 910 - 2 ) = 2. 0 N / m
T =
2m2/ k = 20. 08/ 2. 0 = 1. 26 s
= k/ m 2 = 2. 0/ 0. 08 = 5. 0 s - 1
x , , x 0 = 0. 01m , v 0 =
- 0. 05m / s,
A = x 2 0 + v 2 0
= 0. 01 2 +
2
0 = arct an -
v 0
= arctan -
x 0
x 0> 0, v0< 0, 0= / 4
0. 052
( 5. 0) 2 = 2 10 - 2 m
x = 2 10 - 2 cos( 5t + / 4)
- 0. 05
5. 00. 01 =
4 , 5
4
112 k 1 k2
, , m ,
,
k
k = 1
1
k 1
+ 1 k2
362
=
k1k 2
k1 + k 2

T = 2m/ k = 2m( k1 + k2) / ( k 1k2)
113k 1 k2
m
, 17 ,
T = 2 m
k 1 + k 2
17113
17 , ,
x 1
x 2, k 1x 1= k 2x 2
x ,
k 1( x 1 + x ) - k 2( x 2 - x ) = - m d 2 x
dt 2
m d 2 x
dt
= - ( k 1 + k2) x
T = 2m/ ( k1 + k2)
114
005k g ( 18) 110 3 N / m
18 , , B ,
B x , A x ,
363

18114
2x B
m d 2 x
dt 2 = - k( 2x ) = - 2kx
B , A
= 1 2k
2m = 1 2110 3
2 0. 05
= 31. 8 Hz
115 11 ,
, m ,
, mv
,
( 1) : 1 2
m+ m
3
m
m+ v 2 ,
3
( 2) : k m+ m
1 / 2
3
( 1) L l,
dm = m dl/ L 1 2 dm v 2 l = 1 2
l 2 dl
L
E k =
0
364
m v 2
l 2 dl = 1 2L 3 6 m v2
E k = 1 2 m + m v 2
3
l
L v 2
= m v2
dm
2L 3

( 2)
E = E k + E p = 1 2 m + m v 2 + 1 3 2 kx 2 =
x ,
m +
m v dv
3 dx = m + m dv
3 dt = m + m a = -
3
= [ k/ ( m + m / 3) ] 1/ 2
116dT / T
dg / g g = 9800m/ s 2
kx
, 10s,
T = 2l/ g ,
dT
T = - dg
2g
T = 2s T = 286400/
( 86400- 10)
dT
T =
T - T
T
g = g + dg = g ( 1 - 2dT / T )
=
86400
86400 - 10 - 1 = 10
86390
= 9. 800( 1 - 210/ 86390) = 9. 798 m / s 2
117m, l, 0
=
0cos( t +
)
( )
19 ,
=
t
0cos( t +
)
365

E k = 1 2 mv2 = 1 2 m l d
2
= 1 dt 2 ml 2
2
2 0sin 2 ( t + )
( 0 )
E p = mg l( 1 - cos )
= mg l[ 1 - ( 1 -
2 / 2) ]
= 1 2
mg l
2 0cos 2 ( t +
)
2 = g / l,
E k = 1 2
mg l
2 0sin 2 ( t +
)
E = E k + E p = 1 2
mg l
2 0
19117
118 ( 110) ,
m , x
, ME , RE
110118
( 1) ( :
)
( 2) ,
366

( 3) ?
( 1) 110 ,
F =
Gm M
r 2 E
r 3
R 3 E
= GmME
R 3 E
( 2) x , F x = ma x
r
F x= - F sin = -
GmME
R 3 E
r x r ,
a x = -
GME
R 3 E
,
T =
2R 3 E/ ( GM E) = 2 R E RE / ( GM E)
( 3) ,
mv 2
RE
x
= GMEm
R 2 E
v = GME / R E
T =
2 RE / v =
2 R E R E/ ( GM E)
119 v r= dr
dt
( ) v= r d , m
dt
E =
1 2 m dr
dt
2
+ 1 2
2 mr d
2 -
dt
GmM S
r
M S , r
( 1) : L ,
367

E =
1 2 m dr
dt
2
+ 1 2
L 2
mr 2 -
GmM S
r
( 2) r= r 0 , r = r 0+ x , x r 0
:
E =
1 2 m dx
dt
2
+ 3 2
L 2
x 2 -
mr 4 0
GmM S
x 2 -
r 3 0
GmMS
2r 0
( 3) , ( 130) ,
, :
k =
GM Sm
r 3 0
( 4) :
2
d
( 1) L = mrv = mr 2 dt , d =
dt
,
E =
1 2 m dr
dt
2
+ 1 2
L 2
mr 2 -
GmM S
r
( 2) , v v 0, v 0 r 0
L 2
( mr 2 ) 2
L 0
L = m( r 0 + x ) vmr 0v0 = L 0
L 2 m 2 r 2 0v 2 0 = mr 3 0mv 2 0/ r 0 = mr 3 0GMSm/ r 2 0 = m 2 r 0GM S
r = r 0+ x ( 1) , x / r 0
E = 1 2 m d dt ( r 0 + x ) 2
= 1 2 m dx
dt
-
GmM S
r 0
368
2
+ 1 2
1 -
x
r 0
L 2
mr 2 0
+ 1 2
1 -
+ x 2
r 2 0
L 2
mr 2 0( 1 + x / r 0) 2 -
2x
r 0
+ 3x 2
r 2 0
2 ,
GmM S
r 0( 1 + x / r 0)

1 2
L 2
mr 2 0
1-
2x
r 0
m2 r 0GM S
2mr 2 0
= GmM S GmMS
-
2r 0 r 2 x
0
E x
E =
1 2 m dx
dt
2
+ 3 2
L 2
x 2 -
mr 4 0
( 3) L 2 m 2 r 0GM S, x 2
, E
1-
GmMS
x 2 -
r 3 0
2x
r 0
GmM S
2r 0
E = 1 2 mv2 x + 1 2
GmM S
x 2 -
r 3 0
GmMS
2r 0
E = 1 2 mv2 + 1 2 kx 2
, t
, E ,
m dvx
dt
= -
GmMS
x
r 3 0
,
k = GmM S/ r 3 0
( 4)
T = 2m/ k =
2 r 0 r 0/ ( GmM S)
111119
111 ( )
120 ,
050
369

( 1) 20H z ,
, ?
( 2) , 50cm,
, ?
( 1) ,
smg = ma m = mA
2 = mA( 2 ) 2
A = sg / ( 2 ) 2 = 0. 509. 8/ ( 22. 0) 2 = 3. 110 - 2 m
( 2) ,
mg = ma m = mA( 2 ) 2
= 1 g
2A = 1 9. 8
= 2. 2 Hz
20. 05
121112 , m,
l = 20cm ,
= 05
112121
112 ,
x ,
,
F = F 1 - F 2 = - mg l/ 2 +
l
= -
2mg x
l
x
+ mg l/ 2 -
l
x ,
,
370
x

T =
2 ml
2 mg = 2 0. 20
20. 59. 8 =
0. 90 s
122m= 121g U , S =
030cm 2 , T ?
136g/ cm 3
U , ,
,
y ,
E p
= ySg ¡¤ y 2 -
ySg - y
2
= gSy 2
E k=
1 2 mv 2 ,
E = 1 2 mv2 + g Sy 2
, U
,
T = 2m/ k =
2m/ ( 2 g S)
= 20. 121/ ( 213. 69. 80. 30)
= 0. 77s
123, R, Q
m, - q
,
- q
F = -
Qqx
4 0( R 2 + x 2 ) 3/ 2
x ©¦ x ©¦R ,
F = -
Qqx
4 0R 3
,
371

=
2 = 1 Qq
24
0R 3 m
124 a 0 =
0053nm , :
,
3810 15 Hz
r ,
E = - er
4 0a 3 0
F = -
e 2 r
4 0a 3 0
,
= 1 e 2
24
0a 3 0m p
m p ,
m p =
m em p
m e + m p
= me
= e 1
24
0m ea 3 0
=
1. 610-
19
2
= 6. 610 15 Hz
910 9
9. 110 - 31 ( 0. 05310 - 9 ) 3
3. 810 1 5 Hz ,
372

125H Cl 1310 -
E p ( r) = -
r
e 2
4 0r + B r 9
10 m,
( 1) HCl ( Cl
, Cl )
( 2) ( 135) , H Cl ( = 1) ,
,
,
dE p( r )
dr r= r 0
=
e 2
-
4 0r 2
9B
r 10 r = r 0
=
e 2
-
4 0r 2 0
9B
r 10
0
= 0
( 1)
B = e2 r 8 0
36 0
d 2 E p( r )
dr 2 r = r 0
= -
e 2
+ 90B =
2 0r 3 r 11 r = r 0
2e 2
> 0
0r 3 0
H Cl ,
= 1
2
d 2 E p
dr 2 m p = 1 2e 2
r = r 0
2
0r 3 0m p
= 1 8910 9 ( 1. 610 - 19 ) 2
= 1. 110 14 Hz
2( 1. 310 - 10 ) 3 1. 6710 - 27
( 2) ( 135) , = 1,
E = 3 2 h = 1 2 m
2 A 2
A =
3h
= 1 3h
m
2 2m
373

= 1 36. 6310 - 34
= 1. 710 - 11 m
21. 6710 - 27 1. 110 14
126 m
O , 113 C
O b, O I
O , OC
,
T =
2 I
mg b
113 ,
- mgbsin ,
I d 2
dt 2 = - mg bsin
, sin ,
I d 2
dt 2 = - mgb
,
T = 2I / ( mg b)
113126
127m, R,
, - mgRsin ,
2mR 2 , , ,
2mR 2
d 2
dt 2 = - mg Rsin - mg R
374
T =
2 2mR2
mg R =
22R g

128 1 14 , K ,
, , L
L = 0010H , C= 10 F , E= 14V , L
( )
114 ,
E = -
E L + u = 0
114128
L d
dt , u = q C , = - dq
dt ,
L d 2 q
dt 2 = -
, q ,
q =
q
C
Qcos t
= 1/ LC= 1 0. 01110 - 6 = 10 4 s - 1 , Q = CE= 1. 0
10 - 6 1. 4= 1. 410 - 6 C,
= -
dq
dt =
m ax = Q= 1. 410 - 2 A
Qsin t = 1. 410 - 2 sin ( 10 4 t)
129, 100m , 0=
5100s, 1= 4, 2
= 2 ? Q ?
1= 0e - t ,
2= 1e -
=
t =
ln 0 t = ln 5 1 4 100 = 2. 3 210- s - 1
t
, 1 2
ln 1 = ln 4 2 2 ( 2. 210 - 3 ) = 311 s
375

Q =
2 =
T =
g
2 l
=
1 g
2 l
=
1
9. 8
22. 210 - 3 1. 00 = 712
130 : ,
x = Acos( t+ )
v =
dx
dt
v ma x = A =
= - Asin( t + ) = - vm a xsin( t +
h
[ (
0 -
) 2 + 4
2 ] 1/ 2
)
dvma x
d =
h
[ (
0 -
) 2 + 4
]
[ (
2 0 -
2 ) 2 + 4
2
2 ] 1 / 2
-
- 2
2 (
2 0 -
2 ) + 4
2
2
[ (
2 0 -
2 ) 2 + 4
2
2 ] 1 / 2
=
=
h
{(
[ (
0 -
) 2 + 4
] 3 / 2 0 -
) 2 + 2
(
0 -
) }
h
(
[ (
0 -
) 2 + 4
] 3 / 2 0 - ) ( 0 + ) (
0 +
)
dvm ax
v ma x = 0,
d
= 0
131 ,
,
4 576kg,
1800r/ min ( ) ,
5 , 90% 5 ,
?
376

d= 1800r / min= 30 H z
4 ,
k 1 4 k= 4k1 d= 5 0
k 1
d = 5 0/ ( 2 ) = 5 k/ m/ ( 2 ) = 5 k1/ m
= 2 d
2 m/ 25 = 30 2
2 576/ 25 = 2. 0510 5 N/ m
1320994m, 12k g
( 1) , 150min ,
( 2) 8 ,
( 1) = 0e - t , = ln 0 t = ln 2 1
( 15. 060) = 7. 7
10 - 4 s - 1
( 2) P in= -
dE
= 2 E 0e -
dt t = 0
2 t = 2 E 0= 2
1
t= 0
2 mgl
2 0
= 27. 710 - 1 4 1. 29. 80. 994
2
( 8 / 180) 2 = 1. 7510 - 4 W
133 ,
x 1 = 0. 04cos 2t +
6
x 2 = 0. 03cos 2t -
A = A 2 1 + A 2 2 + 2A 1A 2cos( 2 - 1)
6
= 0. 04 2 + 0. 03 2 + 20. 040. 03cos( - / 6 - / 6)
377

= 0. 06 m
= arct an
= arct an
= 0. 08
A 1sin 1 + A 2sin 2
A 1cos 1 + A 2cos 2
0. 04sin ( / 6) + 0. 03sin( - / 6)
0. 04cos( / 6) + 0. 03cos( - / 6)
x = Acos( t + ) = 0. 06cos( 2t + 0. 08)
134 R,
L,
u
u 1= 90 2 cos100 t V
u 2= 200 2 cos 100 t +
u u 1 u 2
V
2
1= 0, 2= / 2
U = U 2 1 + U 2 2 + 2U 1U 2cos( 2 - 1) = U 2 1 + U 2 2
= ( 90 2 ) 2 + ( 200 2 ) 2 = 310 V
= arctan U 2
U 1
u = Ucos( t +
= arctan 200
90
= 0. 37
) = 310cos( 100 t + 0. 37 ) V
135
x 1 = 0. 08cos 314t +
6
x 2 = 0. 08cos 314t +
2
x 3 = 0. 08cos 314t + 5
6
: ( 1) ;
378

( 2) x =
115135
= 12. 5ms
2
2 A ( A )
( 1) 115
= 314s - 1 ,
A = A 1
2 + A 2 + A 3
2
= / 2,
x = 0. 16cos( 314t +
= 0. 16m,
/ 2)
( 2) 115 A
x =
2 A/ 2 ,
+ / 4 = 5 / 4, t 1 =
5 / ( 4 ) = 5 / ( 3144) = 0. 0125s
136 01kg
,
x =
y =
0. 06cos
0. 03cos
3 t +
3
3 t -
3
, ,
= x - y= / 3- ( - / 3) = 2 / 3
x 2
+ y 2
- 2 x y
A 2 B 2 AB cos = sin 2
x 2 + 2x y + 4y 2 = 2. 710 - 3
> 0, 116
379

116136
137
, 117 ,
27010 4 H z,
117 y / x = 23
y = 2 x / 3 = 22. 7010 4 / 3 =
138 118 C1
1. 8010 4 H z
100V , C2
K 1 K 2 C2 400V
117137
118 , LC1
118138
380

LC2
T 1 = 2LC1 = 2 10160010 - 6 = 0. 80 s
T 2 = 2LC2 = 21010010 - 6 = 0. 20 s
LC ,
/ 2K 1, t 1= T 1/ 4= 0. 20s
I m , K 1, K 2 I m C2 K 2 t 2=
T 2/ 4= 0. 05s, q2
K 2, C2 U 2 C2
C1 , 1 2 C2U 2 2= 1 2 C1U 2 1, U 2=
100= 400V ,
C1
C2
U 1=
1600
100
139119
( ) A B
, , ( A, B )
C ( )
L 1 L 2 S ,
( kHz) A , B C ,
50kH z,
10 -
5 m, ? A, B C
?
119139
381

A ( ) C
( ) / 2, L 1 L 2
, , ,
L 1
,
vm a x = r = 2 A =
L 2
25010 3 10 - 5 = 3. 14m/ s
382

2
21 740km/ h,
300km? 8000km
?
= u/ = 74010 3 / ( 30010 3600) 3 = 6910 - 4 Hz
t= s/ u= 8000/ 740= 108 h
2208m/ s x =
01m , y = 005sin ( 10
- 40t)
x = 01m
= 40s - 1
y = 005sin ( 10 - 40t)
= 2 u/ = 208/ 40 =
04m
x x = 01m
,
= 2 ( x - 01) / = 5x - 05
y = 005sin ( 10 - 40t + )
= 005sin ( 5x - 40t + 05)
= 005sin ( 40t - 5x + 264)
x
y = 005sin ( 10 - 40t - )
= 005sin ( 15 - 40t - 5x )
= 005sin ( 40t + 5x + 164)
383

x
23,
y = 210 - 2 sin2 ( 200t - 20x )
( 1) ;
( 2)
( 1)
,
y =
Asin2 t -
x
= 1/ 20 = 050 m, = 200 H z, u = = 100 m/ s
t x , x
( 2)
vm a x
v =
= 2 A =
dy
dt =
2 Acos2 t -
x
2200210 - 2 = 25 m / s
24, 1976 ,
2m , 1Hz,
3km/ s, ? ?
,
vm = 2gh
A = v m/ ( 2 ) = 2gh / ( 2 )
= 2982/ ( 21) =
= u/ = 3/ 1= 3 km
10 m
25t= 0 21
( 1) u= 008m/ s, ;
( 2) t= T / 8
384

2125
( 1) 21 , = 04m, u = 008m/ s, = u/ =
008/ 04= 02 Hz
, , t = 0, x = 0 , y = 0,
= / 2
y =
Acos 2 t -
x
+ = 004cos( 04
t -
5 x +
/ 2)
( 2) t= T / 8 x / 8=
005m , 22
2225( 2)
26y= Acos ( 4t+ 2x )
( 1) t= 42s ,
, ?
( 2) t= 42s
( 1) t = 42s y = A ,
( 4t + 2x ) = 2¡¤
385

t= 42s
x = - 84, = 0, 1, 2,
©¦ x ©¦, 8, x = - 04m
= 8 x = 0 t ,
( 4t + 20) = 28
t = 4 s
( 2) = 2 / ( 2 ) = 1m - 04m
23
2326( 2)
?
( 2)
2. 7500H z , 350m / s
( 1) , 60 ?
( 2) , 10 - 3 s ,
( 1) = u/ = 350/ 500= 07 m = 2 x /
x =
¡¤ / ( 2 ) = 0. 7
3 ( 2 ) = 012 m
= 2 t/ T = 2
t= 250010 -
3 =
2. 8 , 3. 17210 2
m / s, ?
u=
RT / M
= Mu 2
RT = 3210- 3 3172 2
831273
= 142
2. 9 5100m/ s, (
386

= 7. 810 3 k g/ m 3 )
u=
Y/
Y =
u 2 = 7. 810 3 ( 5110 3 ) 2 = 2. 0310 11 N/ m 2
2. 10 p
1 2 K
V V
2
, , VV
, p ,
S , V0, p = p 0= 0 , V 0 p = p -
p 0= p = - K ( V- V 0) / V0, V 0
r
r
A = - p dSdr = -
S
r 0
r 0
p Sdr = -
V
V ,
V 0
p dV
= K
V ( V - V0) dV = K ( V - V 0) 2
V 0 V 0
2V0
= 1 2 K ( V) 2
V 0
V Wp
VV 0,
w p = W p
V = 1 2 K V V
2. 11 ( 2. 17) , K = -
( 2. 23) ,
2
V dp
dV
: ,
u =
u =
V dp
dV
RT
M
387

u =
RT
M
332m/ s ? ( = 1. 4,
M= 29g/ mol )
, p V= C, ©¦dp / dV ©¦= p / V,
,
u T =
Vp
V =
p
=
, p V = C, ©¦dp / dV©¦= p / V, ,
RT
M
u S =
V p
V =
, ,
p
=
RT
M
uT =
831273
2910 - 3 = 280 m / s
, u S
uS =
14831273
2910 - 3 = 331 m / s
,
,
, ,
2. 12 u = 10 3 m/ s, A =
1. 010 - 4 m , = 10 3 Hz = 800kg/ m 3 ,
:
( 1) ;
( 2) S= 410 - 4 m 2
388

( 1) I =
1 2 ( 2 ) 2 A 2 u
= 1 800( 2
2 103 ) ( 2 1. 010 - 4 ) 10 2 3
= 1610 5 W / m 2
( 2) W = I S t= 1610 410 5 - 60= 4 3810 3 J
2. 13
2. 13, x S
, ,
( 230)
y =
Acos( t - kx )
, x S
F = -
Y dy
dx S = -
v =
dy
dt
P = F v =
YkA Ssin( t - kx )
= - Asin ( t - kx )
YkA 2 S sin 2 ( t - kx )
Y= u 2 , k= 2 / = 2 / u= / u,
P =
2 A 2 u Ssin 2 ( t - kx )
( 230)
2. 14 ,
, ?
, l
r 1 r 2
,
2 r 1lI 1 = 2 r 2lI 2
I 1r 1 = I 2r 2
389

I A 2
, A
r
2. 15P , Q ,
3 / 2, R P Q Q , P , Q
R
= 2
P R -
, P = Q , P Q=
2
QR + P - Q = 2
P Q + P - Q
3
2 ,
= 2
3 2 = 3
2. 16 A, B , ,
100Hz, , A, B 30m, 400m/ s, AB
AB A x , A B
= A -
= A -
2
x -
B - 2
( l - x )
B + 2
u ( l - 2x )
= + 2100
( l - 2x )
400
= + ( l - 2x ) / 2
= ( 2+ 1) ,
+
( l - 2x ) / 2 = ( 2+ 1)
x = l/ 2 - 2= 15 - 2
, 0< x < 30, x = 1, 3, 5, , 29 m
2. 17
y =
0. 02cos 20x ¡¤cos 750t
: ( 1) ?
390

?
( 2) ?
( 3) t = 2. 010 - 3 s , x = 5. 010 - 2 m
( 1) y= 2Acos kx cos t , A= 001 m
u =
/ k = 750/ 20 = 375 m/ s
( 2) = 2 / k,
x =
/ 2 = / k = / 20 = 0157 m
( 3) v = dy
dt = - 002750 cos 20x sin 750t
= - 002750cos( 20005) sin ( 7502010 - 3 )
= - 808 m/ s
2. 18 x , 2. 4 ,
A , , u
2. 42. 18
( 1) t = 0 , O x
, ;
( 2) ,
, x
( 1) O
y 0 = Acos( 2 t -
/ 2)
391

y i =
Acos 2 t -
2
u x -
2 , 0 x 3 4 = 3u
4
( 2) ,
y i y r
y r = Acos 2 t -
/ 4
= Acos 2 t +
2 3
u 4 -
2 + - 23
u 4 - x
2
u x -
2
, P , P / 2, x =
2. 19 ,
( 2. 5) ,
,
d = 2. 0mm, u= 5. 74
10 3 m/ s ,
?
2. 52. 19
= 2d
= u/ = u/ ( 2d ) = 57410 3 / ( 22010 - 3 )
= 1. 4410 6 Hz
2. 20 ,
210 4 W 12km
, 12k m
? 2. 3 ?
392

12km
L = 10lg I
I 0
= 10lg
= 10lg
P
I 04 r 2
210 4
= 70 dB
10 - 12 4( 1210 3 ) 2
2. 21, 115dB
?
I = I 010 L / 1 0 = 10 - 12 10 1 15/ 10 = 032 W/ m 2
2. 22,
,
( 2. 6) ,
; P S
, P
150m/ s, S 50m/ s
35ms, ?
2. 62. 22
393

d =
tuSuP
u P - u S
t = d
1
uS
-
1
u P
= 3. 510- 3 15050
150 - 50
= 0. 26 m
2. 23 ,
4. 0s, 60cm, 25m
( 1)
( 2) ,
( 1) u= / T = 25/ 4. 0= 6. 25 m/ s
( 2) v= 2 r / T = 2 A / T = 20. 6/ 4. 0= 0. 94 m/ s< u
2. 24 1080Hz, 30m/ s
65m/ s
331m/ s:
( 1) ;
( 2)
( 1)
b = ( u - v S) / S = ( 331 - 30) / 1080 = 0. 279 m
a = ( u + vS ) / S = ( 331 + 30) / 1080 = 0. 334 m
( 2)
u
R =
u -
=
394
R =
v R
R = u + vR
u - v S
S
u( u + vR )
( u - vS ) ( u - vR ) S
331( 331 + 65)
( 331 - 30) ( 331 - 65) 1080 = 177103 Hz

a = u/ R = 331/ ( 17710 3 ) = 0187 m
2. 25 120m, 10s
24m / s ?
? , ?
?
u= / T = 120/ 10= 12 m/ s
,
f = ( u + v) / = ( 24 + 12) / 120 = 0. 30 H z
T f = 1/ f = 1/ 0. 30 = 3. 3 s
,
a = ( v - u) / = ( 24 - 12) / 120 = 0. 10 H z
T a = 1/ a = 1/ 0. 10 = 10 s
2. 26 ,
, ?
= ( u- v S) T = ( u- vS) / S
u- vS
R=
= u - v S
u - v S= S
S
,
,
2. 27 ,
1. 810 4 Hz
220Hz, 1. 54
10 3 m / s
S,
S = u+ v
u- v S
= S - S =
u + v
u - v - 1 S =
2v
u -
v S
395

, uv, = 2v S / u,
v = u / ( 2 S ) = 1. 5410 3 220/ ( 21810 4 )
= 94 m / s
2. 280. 32m/ s
4. 0MHz ,
? 1. 5410 3 m/ s
= 1 -
=
u - v
u + v
= 2v u + v =
2v
u
20. 324. 0106
1. 5410 3 = 1. 6610 3 H z
2. 29 ,
5. 010 10 H z
, 1. 110 4 H z
100km/ h
= 2v / u
v = u / ( 2 ) = 310 8 1. 110 4 / ( 2510 10 )
= 33 m/ s = 119 km/ h > 100 km/ h
2. 30 ,
2. 3
5000m , 330m/ s
( 1)
( 2)
?
( 1) 27 ,
( 2) tan = h / s= h/ ( vt)
396
= arcsin u v = arcsin 1
2. 3
= 25. 8
t= h/ ( vt an ) = 5000/ ( 2. 3330tan 258 ) = 136 s

27230
, C
2. 31,
y 1 = Asin ( 5x - 10t)
y 2 = A sin( 4x - 9t)
: ( 1) , ;
( 2) ;
( 3)
( 1) y = y 1+ y 2= A sin( 5x - 10t) + Asin( 4x - 9t)
= 2Acos( 05x - 0. 5t) sin( 4. 5x - 9. 5t)
( 2) vg=
k = 10- 9
5- 4 = 1 m/ s
( 3) L= 1 2 2
= = 2
2- 1 k 1- k 2 5- 4 = 2 = 6. 3 m
2. 32 (
, )
=
k 2
,
u g= d
dk = d
dk ( k2 ) = 2 k 2
k
= 2
k = 2u
397

2. 33 ,
2 =
2 p + c 2 k 2
c , p
u u g, uu g= c 2
u =
k =
, uug= c 2
2 p+ c 2 k 2
k
, u g= d
dk = d
dk
2 p+ c 2 k 2 =
c 2 k
2 p + c 2 k 2
2. 34
3/ 2
=
c- v
c+ v S , =
c+ v
c- v S= 3 2 S
c+ v
c- v = 3 2 , v= 5
13 c= 12108 m/ s
2. 352. 39
x =
2
x k = 2 ( 2. 351)
t , , t
= x / u g,
t = 1 ( 2. 352)
( ) ,
( 2. 351) ( 2. 352) ,
k
, :
398

y = y 1 + y 2
= A sin( t - kx ) + Asin ( + ) t - ( k + k) x
= 2Acos
2 t - k
x sin + t - k + k x
2 2
2
cos ,
x
x = 1 2
2
k/ 2 = 2
k = 1
x k =
1
2
u g= / k, x = u g t=
k t
x k =
k t k = t =
t = 1
=
2
2
t = 2
2. 36 17 :
( = ) [ ( 236) ]
2. 8, Q 1 Q 2
( 1) 28 , A, B
Q 1Q1
, Q 1C= Q 1 C
t Q 1
A Q 2
=
Q 1A + AQ2
c
= Q 1 A + AQ2
c
Q 1 Q 2 B, Q 1 B Q 2
399

2. 82. 36 ( 1)
t Q 1
B Q 2
=
Q 1Q2
Q 1B + BQ2
c
, Q 1BQ 2
, =
= Q 1 B + BQ 2
c
= Q Q 2
< Q 1 A + AQ2
c
c
Q 1 Q 2 , t Q B Q 1 2
,
,
( 2) 29 , Q1 Q 2
400
29236 ( 2)

t =
Q 1A
u 1
+ AQ2
u 2
t , dt = 0,
dx
dt
dx = x
-
h 2 1 + x 2 u 1
=
h 2 1 + x 2
u 1
D -
x
+
h 2 2 + ( D - x ) 2 u
sin
sinr = u 1
u 2
h 2 2 + ( D - x ) 2
u 2
= sin -
u 1
sinr
u 2
= 0
401

3
31589. 3nm10 -
,
= c t/ = 310 8 10 - 8 / ( 589. 310 - 9 ) = 510 6
32
d = 0. 60mm, D = 2. 5m,
x = 2. 27mm ,
x = D d
= d x / D = 06010 - 3 22710 - 3 / 25
= 5. 4510 - 7 m = 545 nm,
33 31 , = 7. 2
10 - 7 m,
8 s
3. 133
x = 1 2
D
d = 1 30
20+
2 0. 22 7210- 7 m = 4. 510 - 5 m
402

34 015mm, 10m
1 10 36mm
= d 3
D 0. 3
1510-
x = 3610- 1. 0 10- 1
= 6010 - 7 = 060 m
35 30km ,
2010 5 Hz
/ 2(
)
32 ,
= 2 dsin
+
2
= 2 ,
3235
2 d sin +
c 2 = 2
sin = 2-
1
2
c
2d
403

1
2
310 8
= 1 23010 3 210 5 4 2- 1
2
= 2-
-
2 < < , - 1, 0, 1, 2, sin
2
- 5/ 8, - 1/ 8, 3/ 8, 7/ 8, - 39 ,
- 72 , 22 , 61
,
36 ( = 632. 8nm)
20m 1
14cm
( 1) ;
( 2) ?
( 1) d = D
9
2. 0632. 810-
= = 9. 010 - 6 m
x 0. 14
( 2) < = ,
2 2
k = d sin / = D / x = 2. 0/ 0. 14 = 14. 3
14 14
37 ( 35) ,
12cm, 50cm
45cm25cm/ s,
= x d
D
= u = uD
x d = 2550
4512 = 23 Hz
38d ,
( 1) :
d sin - d sin= k , k = 0, 1, 2,
( 2) ,
404

3338
( 1) 33 , d sin -
d sin,
d sin - d sin = k , = 0, 1, 2,
( 2) , sin ,
= k / d + sin
= k+ 1 - k = [ ( k + 1) / d + sin ] - [ k / d + sin ] = / d
, ( ) ,
39 ,
,
60m
, 98 MH z,
( ) ? (
)
,
2 h 2 + D 2 / 4 - D = m , m = 0, 1, 2, 3,
405

3439
D = 2h 2
m -
m
2
dD
dm = -
2h2 / ( m 2 ) - < 0, m , D D
2
m= 1 ,
D = 2h 2
-
2 = 2h2 c
-
c 2
= 260 2 9810 6
310 8 -
310 8
29810 6 = 2410 3 m
310 ,
, ( 35)
6010 7 H z, 25m
35310
35 ,
406

=
2 2hsin
+
= ( 2k+ 1) ,
k= 1
sin=
k = kc/ ( 2
2h h)
m in = arcsin c
2 h = arcsin 310 8
26010 7 25 = 57
311
=
2d
,
I = I m axcos 2
2
I = I m a x/ 2 , cos 2
=
= / 2
2 d sin /
I = I m a x/ 2 , / 2= 2 d sin 1/ 2/ = 2 d 1/ 2/ ,
1 / 2 = / ( 4d )
2 = 1 2
,
= 2 1 / 2 = / ( 2d )
312180m,
, ?
= k , k= 1, 2, 3,
407

= / k= 1. 80/ k m
077 m 0. 35 m , 060 m, 045 m,
036 m; 5010 1 4 H z, 6710 14 H z, 8310 14 H z
31336 M 1
, M 2 M 1,
G , M 1
M 2
45 , A 1A 2= 21
6328nm
3. 6313
36 ,
,
x =
V = I m a x - I m in
I m a x + I m in
/ cos45 = 632. 8 2 = 895 n m
= ( 2 + 1) 2 - ( 2 - 1) 2
( 2 + 1) 2 + ( 2 - 1) 2 = 08
314 = 6328nm,
( ) = 1310 9 H z
? ?
L =
2 / = c/
408
= 310 8 / ( 1310 9 ) = 023 m

= L/ c= 1/ = 1/ ( 1310 9 )
= 7710 - 10 s
315 10 -
2 rad,
550nm ,
?
d 0= / 0= 55010 - 9 / 10 - 2 = 5510 - 5 m = 55 m
316 ,
7
= 550n m, = 158,
t, S 2
7
r 1- r 2= 7,
r 1- ( t+ r 2- t) = 0
t = 7 / ( - 1) = 755010 - 9 / ( 158 - 1)
= 6610 - 6 m = 66 m
317, = 152 = 5893nm
, L= 50mm,
Lsin = / ( 2)
sin=
/ ( 2L)
= 589310 - 9 / ( 21525010 - 3 )
= 3910 - 5 rad = 8
318 ,
, ,
, Si 342, SiO 2
15, 5893nm, 7 ( 37
) SiO 2
e ?
2e= ( 2k+ 1) / 2
e = ( 2k +
1) / ( 4)
= ( 26 + 1)589310 - 9 / ( 415)
409

37318
= 12810 - 6 m = 128 m
319,
, ,
38 , G1 , G2 G1
,
5893nm ,
;
( 1) L = 050mm, G1, G2
( 2) G2 G1 ?
( 3) L 1 = 050mm, L 2 = 030mm ,
G 2 ? G2
?
,
38319
410

( 1) 38 , ,
G 1G 2
h = l = l / ( 2L)
= / ( 2L)
= 005589. 310 - 9 / ( 20. 5010 - 3 )
= 2. 9510 - 5 m
( 2) G2 , , G2 G 1
; , G2 G1
( 3) L 2< L 1, G2 2 G1 1,
G 2 , G2
, G 2 G1
, , 1= 2= 0,
320, 04 m, 150,
, , ?
?
=
2e -
/ 2 = k , k = 0, 1, 2,
4e
2k + 1 = 41500410 - 6
2k + 1
, k= 2, = 480nm ,
2e = k , k = 1, 2, 3,
=
2. 410-
6
2k + 1
= 2e/ k = 21500410 - 6 / k = 1. 210 - 6 / k
, k= 2, = 600nm ; k= 3, = 400nm ,
321 , ( = 15)
, ( = 20)
560nm , ?
,
m
411

2e+ / 2= kk= 1
emin
= / ( 4) = 56010 - 9 / ( 42. 0)
= 7. 010 - 8 m = 70 nm
322( = 15) ( = 132) ,
485nm ,
679nm ,
, ,
2e= ( 2k+ 1) / 2
=
2e = ( 2k + 1) 1/ 2 = ( 2k - 1) 2/ 2
2 1
2e( 2 - 1) = 679485
2132( 679 - 485)
= 643 nm
323 133 , 45
( 500n m) ,
, ?
,
k= 1
emi n =
2e 2 - sin 2 + / 2 = k , k = 1, 2, 3,
4 2 - sin 2 = 50010 - 9
4 1. 33 2 - sin 2 45 = 11110- 7 m
, 2e= / 2
= 4e = 41. 331. 1110 - 7
= 5. 910 - 7 m = 590 nm,
324 1= 152 2 =
138 MgF 2
?
, = 550nm ,
412
2e -
/ 2 = k , k = 0, 1, 2,

e = k + 1 2
2 = k + 1 - 9
2 55010 2138
= ( 1993k + 99. 6)10 - 9 m
e= 996 nm
32539
39325
, 162,
,
, ,
39
326 310( a) ( b)
,
( )
310( a ) , k= 0
/ 2, 11 / 4= 5 / 2+ / 4, 5 , k
= 1, 2, 3, 4, 5, 310( b ) ,
, k= 0, 5 , = 1, 2, 3, 4, 5,
, 310( b )
413

310326
327 ,
300mm , 5 460mm,
103m,
r 2 k= 2k- 1 Rr k 2 + 5=
2
= r 2 k+ 5 - r 2 k
5R
= d 2 k + 5 - d 2 k
20R
= 5. 9010 - 7 m = 590 nm
2( k+ 5) - 1
R
2
= ( 4. 6010 - 3 ) 2 - ( 3. 0010 - 3 ) 2
20103
328 , d
, ?
,
2( - 1) d
329 ,
L = 03220mm ,
1204 ,
L= N / 2,
= 2L/ N = 20322010 - 3 / 1204
= 5. 34910 - 7 m = 534. 9 n m
330 ( 329) ,
414

M 1 M2 ,
( G1
M1
) ,
,
M 1 , M 1
02073mm 15140,
M 1
,
, M1 M1
,
2l = 2( - 1) d
d = l/ ( - 1) = 0. 2073/ ( 1. 5140 - 1) = 0. 4033 mm
331
, 311 S , L , G1,
G 2 , , T 1, T 2
, d , ,
, E ,
, T 2
,
, E 98
= 5893nm , d = 20cm,
3. 11331
415

,
N ,
( - 1) d = N
= N
d + 1 = 98589310- 9
2010 - 2 + 1 = 100029
416

4
41 , a = 010mm, 50cm
, = 546. 1nm ,
x = 2f tan 12f sin 12f 1= 2f / a
= 25010 - 2 ( 546110 - 9 ) / ( 0110 - 3 )
= 5. 4610 - 3 m
42 = 632. 8nm ,
1 5 ,
a sin 1= ,
a = / sin 1 = 632. 810 - 9 / sin5 = 7. 2610 - 6 m
43, 3
600nm 2
,
a sin = ( 2k+ 1) / 2 ,
( 23 + 1) 1/ 2 = ( 22 + 1) 2/ 2
1 = 5 2/ 7 = 5600/ 7 = 429 nm
4420m 50m
?
a sin 1= ,
2 1 = 2arcsin( / a ) = 2arcsin( 20/ 50) = 47
45,
HeNe ,
417

6328nm, 265m
137mm , 10 ?
a sin k= k
l = 2D k = 2D arcsin k
a
= 2265arcsin 10632810- 9
13710 - 3
= 2. 4510 - 2 m
46,
( 1) 70mm, 550nm
?
?
( 2) 23mm
( 3) ( 025mm)
1510 5
( 1)
= 2 1 = 2122 / D
?
= 21. 2255010 - 9 / ( 7010 - 3 ) = 1910 - 4 rad
( 2)
D i = l = 1. 910 - 4 23 = 4. 410 - 3 mm
( 3)
N =
D 2 i / 4 =
( 4410 - 3 ) 2 1. 510 5 / 4 = 23( )
47 3510 4 k m
,
10cm,
15km?
?
418

sin 1= 1. 22 / D 1,
= 2 1 = 2arcsin( 1. 22 / D 1)
= 2arcsin [ 1. 2201/ ( 1510 3 ) ]
= 1. 610 - 4 rad
41
D 2 = r + D 1 = 3. 510 7 1610 - 4 + 1. 510 3
= 7. 110 3 m
4147
48( 414) ,
,
( ) ,
( 1)
,
( 2) 05
5 , ? 450n m
419

( 1) ,
( )
, ,
,
( 2) 1 ,
=
D = 2122 v/
2 1 = 2122 v/ D
= 212245010 - 9 / ( 505
/ 180)
= 2. 510 - 5 m
49, 120cm
, ?
50mm, 550nm ,
c= d / l= 1. 22 / D
l = d D / ( 1. 22 ) = 1. 200. 510 - 3 / ( 12255010 - 9 )
= 8. 910 3 m
410
( 1) 5cm,
160km ?
( 2) ? 500nm
( 1)
c = d / l = 510 - 2 / ( 16010 3 ) = 310 - 7 rad
( 2)
D = 1. 22 / c = 1. 2250010 - 9 / ( 310 - 7 ) = 2 m
411 (
420

425) 300m, 300m
4cm ,
?
c = 1. 22 / D = 122410 - 2 / 300= 1. 610 - 4 rad
= 0. 56
412 ,
( 10 4 km)
, (
)
10 4 km, ,
?
= 5cm,
c / d = 510 - 2 / 10 7 = 510 - 9 rad 110 - 3
413 750 7,
50mm
( 1) , ?
550nm
( 2) 70mm
, 7, ,
( 1) c =
1. 22 / D = 12255010 - 9 / ( 5010 - 3 )
= 1310 - 5 rad
( 2) c = 1. 22 / D = 12255010 - 9 / ( 710 - 3 )
= 9610 - 5 rad
c / 7= 13710 - 5 rad
c / 7> c, , ,
414 484
421

10 - 6 rad, 550nm
, ?
D = 1. 22 / c = 1. 2255010 - 9 / ( 48410 - 6 )
= 0139m
415
05
( 1) 01mm, 550nm
, ?
?
( 2 ) 001mm , ,
( 3) 01mm , ?
1= / d
( 1) 1 = / d = 55010 - 9 / ( 0110 - 3 ) = 5. 510 - 3 rad
= 03 < 05 , ;
( 2) 1 = / d = 55010 - 9 / ( 00110 - 3 ) = 5. 510 - 4 rad
= 3 > 05 , ;
( 3) 1 = re d/ d = 7510 - 7 / ( 0110 - 3 ) = 7. 510 - 3 rad
= 04 < 05 ,
416 , d = 0. 10mm, a = 002mm,
= 480n m ,
50cm , :
( 1) ;
( 2) ;
( 3)
( 1)
x = f / d = 5010 - 2 48010 - 9 / ( 01010 - 3 )
= 2410 - 3 m
( 2)
422

x = 2f / a = 25010 - 2 48010 - 9 / ( 00210 - 3 )
= 2410 - 2 m
( 3)
N =
x
x - 1 = 24
2. 4 - 1 = 9
41742 ,
:
42417
423

( 1) ? ?
( 2) a ? ( )
( 3) d / a ? ?
( 4) / d / a
( 1) ,
N , N - 1 , 42 , ( a )
; ( b ) ; ( c) ; ( d)
( 2) , sin a = / a , ( c)
( 3) ( a ) d / a = sin d/ sin = 2, 2 ;
( b ) d / a = 4, 4 ;
( d ) d / a = 3, 3
( 4) 42
418, 20cm, 6000
, ?
= arcsin( k / d ) = arcsin ( k l/ N )
= arcsin( k589310 - 9 2010 - 2 / 6000)
= arcsin( 01768k)
sin 1, k = 0, 1, 2, 3, 4, 5 0 ,
10 11 , 20 42 , 32
, 45 , 62 7
4196000 ,
1 20 , 2
?
= d sin 1 = 10 - 2 sin20 / 6000
424
= 5. 7010 - 7 m = 570 nm
2 = arcsin ( 2 / d )
= arcsin ( 25. 7010 - 7 6000/ 10 - 2 )

= 43. 2
4204000 ,
?
k k r= ( k+ 1) v
k =
v/ ( r - v) = 400/ ( 750 - 400) = 1. 1
k 1, 1
, ,
r =
arcsin( r/ d ) = arcsin( 75010 - 9 4000/ 10 - 2 ) = 17. 5
421 3000 , 555nm
30 ,
d ( sin - sin) = k , , = 0,
k = -
d sin/ = - 10 - 2 sin 30 / ( 55510 - 9 3000) = - 3
3
422 600nm , 2, 3
sin = 020 sin = 030 , 4
:
= 10
( 1) ;
( 2) ;
( 3)
( 1) d = 2 / sin 2= 260010 - 9 / 0. 2= 6. 010 - 6 m
( 2) d / a = 4, a = d / 4= 1510 - 6 m
( 3) m ax =
km ax = dsin ma x/
2 = 6. 6 010- / ( 60010 - 9 )
0, 1, 2, , 9( km ax )
423 = 656. 3nm ,
= 0. 18nm 1
425

,
/ = kN
N =
/ ( k ) = 656. 3/ ( 1018) = 3646
424 60cm, 6000 , 3
, = 500nm, ?
= / ( kN ) = 50010 - 9 / ( 3606000)
= 4. 610 - 12 m = 0. 0046 nm
425
28 ( 427, )
(
) ,
232MHz 6m,
192
?
?
?
,
dsin = m / N , m= 1
= arcsin( / ( N d ) ) = c/ ( N d )
= 310 8 / ( 19262. 3210 8 )
= 1. 1210 - 3 rad = 3. 85
d sin = m , m= 1
= arcsin( / d ) = arcsin( 310 8 / ( 62. 3210 8 ) )
= 0. 217 rad = 12. 4
426

426 420 , = 45 , X
0095nm 0130nm d =
0275nm, X ? ,
X ?
2d sin = k X
= 2dsin / k = 20275sin 45 / k = 0389/ k nm
3 = 0389/ 3 = 0130 nm
4 = 0389/ 4 = 0097 nm
4271927
( )
= 50 ( 43)
d = 0215nm,
43427
43 ,
= d sin
= = d sin = 0215sin50 = 0165 nm
4. 1444 ,
P
( A) , ( B,
427

) : P
(
)
44414
44 , ,
,
E 1 ,
,
E 2
, 44 , E 0
,
, ,
E 0 = E 1 + E 2
44 , a , (
) , E 0= 0,
428
E 1 = - E 2

I 1 = I 2
(
) 45 , ( a) ( b) , ( a1)
( b1) , ,
45
429

5
51 60 ,
I 1,
30, ?
I 0,
I 0/ 22 I 0cos 2 60 / 2= I 1
I 0 =
2I 1/ cos 2 60 = 8I 1
,
I = 1 2 I 0cos2 30 cos 2 30 = 1 2 8I 1 3 4 3 4 = 9 I 1 = 2. 25I 1
4
52
( 1) , ( 2) ,
?
( 1) , , ,
I 0/ 2( I 0/ 2) / 3 ,
I 0cos 2 / 2 = ( I 0/ 2) / 3
= arccos
( 2) I
1
3 = 54 44
I = I 0cos 2 / 2 = I 0/ 3
430
= arccos
2
3 = 35 16

53P 1 P 2 ( 51)
I 0
P 1, P 2 ,
P 2 ?
5. 153
52 , y , E 1
5253
431

P 1 , z OA P 2
, E 1
, P 2 E 1 OA ( xz
) ,
cos=
OC
OA
= OC
OC 2
+ OB
= cos
2
cos 2 + sin 2 cos 2
P 2
E 2
= E 1cos= E 1cos / cos 2 + sin 2 cos 2
I 2 = E 2 2 = E 2 1cos 2 / ( cos 2 + sin 2 cos 2 )
= 1 2 I 0cos2 / ( cos 2 + sin 2 cos 2 )
5453 ,
, 0
, 0,
53
5. 354
55133, 150,
432

, ?
, ? ?
,
0, 12 = arct an 2
,
t an 0 , 12 = 2
1
0, 21 = arct an 1
1
2
= arct an 1. 5
1. 33 = 48 26
= arct an
1. 33
1. 50 = 41 34
= ( tan 0 , 21 ) - 1 , 0, 12 + 0, 2 1 = / 2,
56 45 ,
?
sin = 2/ 1
0
= arctan 2
1
= arctan( sin ) = arctan( sin45 ) = 35 16
57
0= 58 ,
t an0= 2/ 1
2 = 1t an0 = 1. 00t an58 = 1. 60
58,
, ?
= arctan
, = 0 = arct an 2 1
1. 33
1. 00 = 53 4 = 90 - = 36 56
59
, 53 , e
433

, , o
e r e = 90 - 60 = 30e
, sin= esinr e
= arcsin ( esinr e) = arcsin ( 1. 4864sin30 ) = 48
r o o , osinr o= sin,
r o
= arcsin( sin/ o) = arcsin( sin48 / 16584) = 26 37
o o = 60 - r o= 33 23o
osino = sinr o
o 54
r o = 65 51
5459
510 ABCD 45
( 55
) , ABD AB, BCD
A B , o e
434
55510

55
511 56 , P 1, P 2
C , P 1 60
I i P 1
, P 2
( 1) , , ;
( 2)
( 1) 56 , P 1
, C
A 1
5. 6511( 1)
( 2) P 1 I 1= I i/ 2, A 1( 57) , C
A e = A 1cos , A o = A 1sin , / 2,
A 2
A 2 2 = A 2 e + A 2 o + 2A eAocos( / 2) = A 2 1cos 2 + A 2 1sin 2 = A 2 1
57511( 2)
435

P 2
I 2 = I 1 = I i / 2
A 3e= A e sin = A 1cos sin ,
A 3o= A ocos = A 1sin cos / 2+ ,
A 2 3 = A 2 3e + A 2 3o + 2A 3eA3ocos
= 2A 2 1sin 2 cos 2 =
2 +
2A 2 1sin 2 60 cos 2 60 = 3A 2 1/ 8
I 3 = 3I 1/ 8 = 3I i/ 16
512 6328n m o= 1. 66, e =
1. 49, ?
?
d mi n( o- e ) = / 4,
d mi n
= / [ 4( o - e ) ]
= 6328/ [ 4( 1. 66 - 1. 49) ] = 931 nm
513 o e
, 800nm 400nm
, 45
, ?
d ( o- e ) = 1/ 4= 2 2/ 4= 2/ 2, 800 nm( 1)
, 400nm ( 2)
,
45,
, ,
, ,
514 1823
436

(
155) , 58
,
,
? (
51 )
5. 8514
22 , o
r o =
arcsin
o
o
sin 22
= 13. 1
1. 658
= 90 - 131 = 76. 9
= arcsin
1. 55
1. 658
= 69. 2
> , o
e , , ,
e ( 1486) ,
437

5153968nm
R = 1. 55810, L = 1. 55821
= ( L - R) /
= ( 1. 55821 - 1. 55810) / ( 396. 810 - 9 )
= 0. 870910 3 rad/ m = 49. 9 / mm
516,
,
+ x - x , y z
x = 0 , y 45
+ x / 8 ,
, x
= / 8 , , + x / 4,
- x / 4 /
2( x ) x =
/ 4 , x = 3 / 8
x = / 2
, 59
59516
438

5
1
11
- 5, ,
?
M =
T 4 = 5. 6710 - 8 268 4 = 292 W / m 2
12 M m =
3. 410 14 H z ( 1) ( 2)
M
( 1) m = CT ,
T =
m / C= 3. 410 14 / ( 5. 8810 10 ) = 5. 810 3 K
( 2) M= T 4 = 5. 6710 - ( 8 5. 810 3 ) 4 = 6410 7 W / m 2
13, 1010 3 W / m 2
,
?
T =
, , I = T 4
4
I / =
4
1010 3 / ( 56710 - 8 ) = 364 K = 91
14P S = 3910 26 W ( 1) r
:
T
439

T 4 =
P S
16 r 2
( ) ( 2)
, r E= 1510 11 m,
r P = 5910 12 m
( 1) R , P a b=
P S
R 2 ,
4 r 2
P e j= T 4
4 R 2 P ab = P e j,
T 4 =
P S
16 r 2
P S
4 r 2 R 2 = T 4
4 R 2
( 2) T E=
3. 910 26
1/ 4
= 279
1656710 ( 8 1510 11 ) 2 K
T P =
3. 910 26
1 / 4
= 45
1656710 ( 8 5910 1 2 ) 2 K
15Procyon B 11 l. y. ,
1710 -
1 2 W / m 2 , 6600K
r = 9. 4610 15 11= 1. 0410 1 7 m
,
D = 2R = 2
= 2. 610 7 m
I 4 r 2 = T 4 4 R 2
I r 2
= 2 1. 710 - 12 ( 1. 0410 17 ) 2
T 4 5. 6710 - 8 ( 6600) 4
16
3K
( 1) M?
( 2) ?
( 1) m= CT = 58810 10 3= 17610 11 Hz
( 2) P = M4 R 2 E= 4 T 4 R 2 E
440

= 456710 -
8 3 4 ( 6410 6 ) 2 = 2. 3610 9 W
17[
( 13) ] ,
M=
2 hc2
e M ©¦ d ©¦= M d
d
M = M
d =
= 2 hc2
e 1
5 hc / (
1
5 hc / (
2 h
3
c 2 ( e h / ( kT ) - 1)
kT )
- 1
kT ) - 1
c
=
2
2 hc 3
c 2
3 hc / (
( e kT ) - 1)
c
2
18w
w M
M =
c 4 w
c
11
, dV dV
d dVd / ( 4 )
dS dV dS cos / r 2 , dV
dS dN = dVdScos / ( 4 r 2 ) dV =
1118
441

2 sin drd d
dN = r 2 sin dr d dScos
4 r 2
=
sin cos drd d dS
4
dt , rcdt dS dS
, dt dS
N = dN =
c dt
0
dr
2
2
d
0 0
dsin cos d
dS
4 =
c 4 dSdt
E ,
, ,
M =
E N
=
dS dt
c E
= c 4 4 w
w = E ,
19 [ ( 13) ]
m = CT
( : m/ T , C )
m ,
d 2 h
3
dc 2 ( e h / ( kT ) - 1)
=
= m
2 h
2 m
e h
c 2 ( e h m / ( kT ) - 1) 2 m / ( k T ) 3 -
h m
kT - 3 = 0
m m= 0,
e h m / ( k T ) 3 -
h m
kT - 3 = 0
m/ T C ,
m/ T = C
( h m/ ( kT ) = x , e x ( 3- x ) - 3= 0,
x = 2822C= 2822k/ h= 58810 10 H z/ K )
110[ ( 13) ] 0
442

M = T 4
( : M/ T 4 , )
M =
M 0 d
= 2 h 3 d
0 c 2 ( e h / ( kT ) - 1)
h / ( kT ) = x ,
M =
2 k 4
T 4
c 2 h 3
0
x 3 dx
e x - 1
,
M = T 4
k, c, h
(
4 / 15 = 2
5 k 4 / ( 15c 2 h 3 ) = 5. 6710 - 8
W / m 2 K 4 )
, :
111 4. 2eV, 200nm
( 1) ;
( 2) ;
( 3)
( 1) E k, m = h - A= h c
- A
= 6. 6310- 34 310 8
20010 - 9 1610 - 19 - 42= 2. 0 eV
( 2) U c= E k, m / e= 2. 0/ 1= 2. 0 V
( 3) 0 = c = hc
0 A
= 6. 6310 - 3 4 310 8
4. 21610 - 19 = 29610 - 7 m= 296 nm
11210 - 1 5 J/ m 3 ,
? 500nm
443

N = w h
= w
hc = 10 - 15 50010 - 9
= 2. 510 3 / m 3
66310 - 34 310 8
113 10W 10m
( 225eV)
, ?
1310 -
10 m( ) ? (
, 10 -
,
t =
A =
P
r 2 t
4 R 2
9 s )
4R2 A
P r 2 = 41 2 2. 251. 610 - 19
1. 0( 1. 310 - 1 0 ) 2 = 85 s
1145eV c/ 2
,
?
S O ,
+ x , 0x > 0 + x u
= c/ 2 , - x u= c/ 2
,
=
1 - u/ c
1 + u/ c 0
h =
1 - u/ c
1 + u/ c h 0 = 1 - 1/ 2
1 + 1/ 2 5 = 5
3
= 2. 9 eV
115X 060MeV
20%
, =
444

12 0,
E = hc/ = hc/ ( 1. 2 0) = h 0/ 1. 2 = E 0/ 1. 2
,
E e = E 0 - E 0/ 1. 2 = E 0/ 6 = 0. 60/ 6 = 0. 10 MeV
1164010 3 eV ,
?
,
= 0+ 2h/ ( mec) ,
E = hc = hc
0 + 2h/ ( m ec) = hc
hc/ E 0 + 2h/ ( mec) = E 0me c 2
m ec 2 + 2E 0
, ,
E e = E 0 - E = E 0 1 -
= 4. 010 3 1 -
m ec 2
mec 2 + 2E 0
0. 51110 6
051110 6 + 2410 3 = 62 eV
117 :
,
,
m 0c 2
+ h = mc 2
mv = h / c
h ,
m 0 = m( 1 - v/ c) =
m0( 1 - v/ c)
1 - v 2 / c 2 = 1 - v/ c
1 - v 2 / c 2
v= 0 v= c,
445

118 5010 4 eV 20
10 4 eV
?
E p , E e p e
,
E 0 + E e 0 = E + E e
p 0 - p e 0 = - p + p e
p 0
= E 0/ c, p = E / c
p e 0 = 1 c
E 2 e0 - ( m ec 2 ) 2 , p e = 1 c
E 2 e - ( m ec 2 ) 2
, ,
E e =
2E 0 + E e 0 - E 2 e0 - m 2 e c 4 2
+ m
2
e c 4
2 2E 0 + E e 0 - E 2 e0 - m 2 e c 4
mec 2 = 5. 1110 5 eV , E e 0= E k0 + mec 2 = 2. 010 4 + 5. 1110 5
= 5. 3110 5 eV , E 0= 510 4 eV ,
E e = {[ 25. 010 4 + 5. 3110 5
- ( 5. 3110 5 ) 2 - ( 5. 1110 5 ) 2 ] 2 + ( 5. 1110 5 ) 2 }
/ {2[ 25. 010 4 + 5. 3110 5
- ( 5. 3110 5 ) 2 - ( 5. 1110 5 ) 2 ] }
= 51210 5 eV
E ek = E e - m ec 2 = 51210 5 - 51110 5 = 0110 4 eV
E = E 0 + E e 0 - E e = 510 4 + 53110 5 - 51210 5
= 6910 4 eV
446

119 020nm
?
E e = ( pc) 2 + ( m0c 2 ) 2
p = h/ = 6. 6310 - 34 / ( 0. 2010 - 9 )
= 3. 3210 - 24 kg ¡¤m/ s
= ( 3. 3210 - 24 310 8 ) 2 + ( 0. 91110 - 30 910 16 ) 2
= 8. 1910 - 14 J = 5. 1210 5 eV
E = p c = 3. 3210 - 24 310 8
= 9. 910 - 16 J = 6. 1910 3 eV
120( 300 K )
E k = 3 2 kT = 3 2 13810 - 2 3 300 = 62110 - 21 J
E 0 = mnc 2 = 16710 - 27 910 1 6 = 15010 - 10 J
E kE 0,
=
h
2mnE k
=
66310 - 34
216710 - 2 7 62110 - 21
= 14610 - 10 m = 0. 146 nm
12140keV
?
40k eV 511k eV ,
447

=
h
2me E k
=
= 6110 - 12 m
66310 - 34
209110 - 30 410 4 1610 - 1 9
122 m =
m 0/
1- v 2 / c 2 = E / h = mc 2 / h = h/ p = h/ ( mv) ,
v g v
v g= d
dk = d
d( 1/ )
d = d( mc 2 / h) = c2
h dm
d
1
= d mv
h
vg =
=
= 1 h
d
d( 1/ ) = c2
h
dm
c 2
dv
v dm
dv + m
m = m0/ 1 - v 2 / c 2
dm
dv =
m 0v
c 2 ( 1 - v 2 / c 2 ) 3/ 2
v g= v
( vdm + mdv)
dm/ [ ( vdm + mdv) / h ]
123r
,
L = mer v = h, = 1, 2, 3,
448

2 r = , = 1, 2, 3,
= h/ ( mev) ,
m evr = h/ ( 2 ) = h
L mevr,
L = m evr = h
124
10 -
1 2 m050kg
, , ?
v =
x = 210 -
p
m =
12 m
h
2m x = 1. 0510 - 34
= 5. 310 - 23 m/ s
20. 5210 - 12
125 9kV ,
050mm , 030m
?
y = 0. 50mm
p x = 2meE
d = 2 p y
l =
=
p x
h l
yp x
=
y
p y = h / ( 2 y )
h l
2meE
10510 -
34 030
0510 - 3 29110 - 31 910 3 1610 - 19
= 1. 210 - 9 m = 1. 2 nm
,
d = 2 l =
1. 22
2 l =
y
2. 44hl
yp
=
2. 442h l
y
2meE
449

h l
y 2meE
126 77MeV
6710 -
10 -
?
2 7 kg ?
10
m , ,
E 0 = mc 2 = 6710 - 27 910 1 6 = 3. 810 4 MeV
E E 0, ,
=
h
2mE = 66310 - 34
26710 - 27 7710 6 1610 - 1 9
= 5. 210 - 1 5 m
10 -
1 0 m,
127 ,
22GeV
? 10 -
?
15 m
( 22GeV )
( 051MeV) , , p = E / c
= h/ p = hc/ E
10 -
= 66310 - 3 4 310 8 / ( 2210 8 1610 - 1 9 )
= 5. 710 - 17 m
15
m, ,
128 : ,
h 2 / ( 8m ed 2 ) d =
0091nm, ?
450

= 2dsin
, p =
2me E , = h 2m eE
,
2d sin= h
2meE
E =
h 2 / ( 8m ed 2 sin 2 )
sin1,
E min = h 2 / ( 8m ed 2 )
d = 0091n m ,
E min =
( 66310 - 34 ) 2
89. 1110 - 31 ( 009110 - 9 ) 2 1610 - 19
= 45. 5 eV
129 7210 -
15 m( 1) ( m =
6. 710 - 27 kg) ? ( 2)
MeV ?
( 1)
p h / ( 2 x ) = 1. 0510 - 34 / ( 27210 - 15 )
p p ,
= 7. 2910 - 21 kg ¡¤m/ s
p 72910 -
21 k g ¡¤m / s
E k , = p 2 / ( 2m )
( 7. 2910 - 21 ) 2 / ( 26. 710 - 27 1. 610 - 19 )
= 2. 4810 4 eV
( 2) , p 7. 2910 -
21 kgm/ s
451

E e, min = p 2 e / ( 2me )
p e p = 7. 2910 - 2 1 kg ¡¤m/ s
= ( 7. 2910 - 21 ) 2 / ( 29. 110 - 31 1. 610 - 1 9 )
= 1. 8210 8 eV = 182 MeV
0511MeV ,
E e, ,
E e, min = (p ec) 2 + ( mec 2 ) 2
= ( 7. 2910 - 21 310 8 ) 2 + ( 9. 1110 - 31 910 16 ) 2
= 2. 1910 - 12 J = 137 MeV
Ek, e = E e - mec 2 = 13. 7 - 0. 5 = 13. 2 MeV
130: m a
, ( )
E m in = 3 h 2
8ma 2
x = a , p x h/ ( 2 x ) = h/ ( 2a )
p x p x,
,
p x h / ( 2a )
p y h / ( 2a ) , p z h / ( 2a )
E m in = p 2 m in
2m = p 2 x , min + p 2 y , m in + p 2 z , min
2m
= 3 h 2
8ma 2
452

2
21 10 - 5 m, 10 - 14 g
, 1= 100 2 = 101
?
E 1
E 2
=
2 h 2
2
2ma 2 1 = 2 ( 10510 - 34 ) 2
100 2
210 - 17 10 - 10
= 5410 - 3 7 J
=
2 h 2
2
2ma 2 2 = 2 ( 10510 - 34 ) 2
101 2
210 - 17 10 - 10
= 5510 - 3 7 J
E = E 2 - E 1 = ( 55 - 54)10 - 37
= 110 - 3 8 J
22
, 10 cm
( 1) ?
(2) T = 300K
3kT / 2, ? + 1
?
( 1) E 1 =
2 h 2
=
2ma 2
m= 3210- 3
6. 0210 23 = 5. 310 - 26 k g
= 1010 - 4 0 J
( 2 10510 - 34 ) 2
25310 - 26 01 2
453

( 2)
3
2
2 h 2
kT = E = 2 = E 1 2
2ma 2
=
3kT
2E 1
=
= 7810 9
313810 - 2 3 300
21010 - 40
E = E 1[ ( + 1) 2 - 2 ] = E 1( 2+ 1)
= 1010 - 4 ( 27810 9 + 1)
= 1. 610 - 3 0 J
2321
= 5
E k= E - U, = h/ p = h 2mE ,
k
, E k , E k , ,
, ,
, , 23
, x - 1 ( )
, 21
2123
24
1/ 4 ?
454

( 233) ,
=
2
a sin
a x
x = 0 x = a / 4
P =
a / 4
0
= 1 4 - 1
2 =
2
dx = 2 a a / 4
0
0091
sin 2
a x dx
25 ,
( 233) x x 2
x 2
x
=
a
x
x 2
0
=
a
x
2 dx = 2 a a
0xsin 2 a x dx = a 2
0
x 2
2 dx =
a 2
a
0x 2 sin 2 a x dx
1
= a 2 3 - 1
2 2
2
26 : m( x , t) ( x , t)
,
a 0 m ( x , t) ( x , t) dx = 0
( 233) ,
a 0 m ( x , t) ( x , t) dx
= a 0
2
a
E m t / h ei2 sin
m
a x 2
a e-
i2 E t/ h sin
455
a x dx

= 2 i( E a e2 m - E
) t/ h
a sin
0
m
a x
sin
a x dx
= 2 sin ( ( m - ) x / a )
i( E
a e2 m - E ) t/ h 2( m - ) / a
-
sin( ( m + ) x / a )
2( m + ) / a
m, ,
a
0
a 0 m ( x , t) ( x , t) dx = 0
27, E ,
?
E =
2 h 2 2 / ( 2ma 2 ) ,
F = -
dE
da =
2 h 2 2
ma 3
28
, a
E =
2 h 2
2
2ma 2
= 2a / , = 1, 2, 3,
p = h/ = h/ ( 2a ) = h / a
E =
p 2
2m =
2 h 2
2
2ma 2
29 , a
E =
2 h 2
( 2
2ma 2 x + 2 y + 2 z )
x, y , z
, 3 ,
456

x = 2a / x, y = 2a / y, z = 2a / z
p x = h
a
E = p 2
2m = p 2 x + p 2 y + p 2 z
2m
, x, y, z
h
x, p y = y, p z =
h
a
a z
=
2 h 2
( 2
2ma 2 x + 2 y + 2 z )
210, 0= A e - a x 2 , A , a
( 251)
x
( 251)
d 2
+ 2m E -
dx 2 h 2
0= Ae - a x 2 ,
4a 2 -
E 0 = 1 2 h
m 2
2 x 2 = 2 a -
h 2
1
2 m
2 x 2 = 0
x , x 2
4a 2 -
m 2
2 = 0, a -
h 2
a ,
211H 2
E 0
= h
2 = h
2
m
h 2 E 0
,
m
h 2 E 0 = 0
,
k= 11310 3 N / m, m= 16710 -
2 7 kg
( eV ) ?
457

, ?
= 1 k
2m
E = + 1 2 h = + 1 2
h k
2m
= + 1 2
= + 1 2
66310 - 3 4
2
054 eV
11310 3
16710 - 27 1610 - 1 9
E = E - E - 1 = + 1 2 - [ ( - 1) + 1 2 ] h = 054 eV
= hc
34 E = 66310- 310 8
0541610 - 1 9
= 23010 - 7 m = 2310 3 nm
458

3
31
m in = c =
ma x
ch
E - E 1
= 91410 - 8 m = 914 nm
m ax = c =
m in
ch
=
E 2 - E 1
= 12210 - 7 m = 122 nm
=
310 8 66310 - 34
0 - ( - 1361610 - 1 9 )
310 8 66310 - 3 4
( 1/ 4 - 1)( - 1361610 - 19 )
32= 5
? ?
p ,
p 2
2m H
+ h = E 5 - E 1
p =
h / c
h h
2m H c 2 + 1 = E 5 - E 1
h©¦E 1©¦= 13. 6eV , mH c 2 = 939MeV , m H c 2 h ,
h = E 5- E 1
= c = ch
E 5 - E 1
= 095210 - 7 m = 95. 2 nm
=
310 8 66310 - 34
( 1/ 25 - 1)( - 1361610 - 1 9 )
459

v =
=
p mH
=
h
mH
66310 - 3 4
16710 - 27 095210 - 7 = 4. 17 m/ s
33:
E = -
E = -
h 2
2m ea 2 0
1
2
e 2
1
8 0a 0 2
a 0= 4 0 h 2 / ( mee 2 ) ,
E = -
E = -
= -
m ee 4
1
= -
2( 4 0) 2 h 2 2
h 2
- 2m ea 2 0
1
2
mee 4
1
= -
2( 4 0 ) 2 h 2 2
h 2
1
2m e( 4 0 h 2 / ( mee 2 ) ) 2 2
e 2
8 0 [ 4 0 h 2 / ( m e e 2 ) ]
1
2 = -
e 2
1
8 0 a 0 2
34 = 1 , ( 34)
a 0
a 0
E k=
e 2
E = E k + E p =
E p= -
8 0a 0
e 2
-
8 0a 0
e 2
e 2
4 0a 0
= -
4 0a 0
e 2
= -
8 0a 0
mee 4
2( 4 0) 2 h 2
351884
( 1/ ) ,
460

1 = R
4 - 1
2
, = 3, 4, 5,
R ,
( R =
1097373177( 83)10 7 m - 1 )
E = -
m ee 4
1
2( 4 0) 2 h 2 2
3 = 2
,
1 =
=
c =
,
R =
=
m ee 4
2 ( 4 0) 2 h 3 c
E
ch = m ee 4
2 ( 4 0) 2 h 3 c
1
2 2 -
9. 1110 - 31 ( 16010 - 19 ) 4
2 ( 488510 - 12 ) 2 ( 1. 0510 - 34 ) 3 30010 8
= 1. 1110 7 m - 1
36
,
L = h
? = 2 = 1 ,
?
r ,
e 2
= m
4 0( 2r ) 2 e
v 2
r
L = 2mev r = h
1
2
461

, m in
r = 4 0 h 2
2
me e 2
= 4 0 h 2
1 2
me e 2
=
( 10510 - 34 ) 2
910 9 9. 1110 - 31 ( 1610 - 19 ) 2
= 5310 - 1 1 m
E = E k, +
E p ,
= 2 1 2 m ev2 + -
= -
mee 4
1
4( 4 0) 2 h 2 2
e 2
4 0 ¡¤2r
= 2 = 1 ,
= E 2 - E 1
2h
=
m ee 4
1
-
8 ( 4 0) 2 h 3 1 2
= 9. 1110 - 3 1 ( 1610 - 19 ) 4 ( 910 9 ) 2
3 8( 10510 - 34 ) 3 4
= 12510 15 Hz
37, : ,
, ,
, L = h,
( 1) ;
( 2) 15010 11 m
?
( 3)
?
( 1) , ,
462
1
2 2

G Mm
R 2
mvR =
R =
= mv 2
R
h
h 2
GMm 2 2
( 2) =
GMm 2 R
h
= 5981024 66710 - 11 19910 30 15010 11
1. 0510 - 34
= 2. 5410 74
( 3) R = R + 1- R
=
=
h 2
[ ( + 1) 2 - 2 h 2
] = 2
GMm 2 GMm 2
( 1. 0510 - 34 ) 225410 2 74
66710 - 1 19910 1 3 ( 0 59810 24 ) 2
= 1. 1810 - 63 m
38,
4117nm 4357nm
( 1)
, ?
?
( 2) , ?
( 1) ,
= 411. 7nm, = 435. 7nm
,
463

=
=
1
4 + ch
E 1
= c = ch
E =
- 1/ 2
ch
E 1( 1/ 2 - 1/ 4)
1
4 + 310 8 66310 - 34
411710 - 9 ( - 1361610 - 1 9 )
= 599 6
0 =
=
ch
E 1( 1/ 6 2 - 1/ 4)
310 8 66310 - 34
1361. 610 - 19 ( 1/ 4 - 1/ 36)
= 4. 11310 - 7 m = 411. 3 nm
m =
/ 0 = 411. 7/ 411. 3 = 1. 00097
> , < , = 5 = 2
- 1/ 2
ch
0 =
E 1( 1/ 5 2 - 1/ 4) = 310 8 66310 - 34
1361. 610 - 1 9 ( 1/ 4 - 1/ 25)
= 435. 3 nm
m = / 0 = 435. 7/ 435. 3 = 1. 00092 m
( 2) ,
464
1 - u / c
= 0
1 + u / c 0( 1 - u/ c)
u = ( 1 -
/ 0) c = ( 1 - 0/ ) c

= ( 1 - 411. 3/ 411. 7)310 8 = 2. 910 5 m / s
39
t
[ ( 134) ] , E
E
t 110 -
8
s
,
, E 1= 0
= E 2 - E 1
h
=
= E 2 - E 1
h
= E 2
= 1 h h
1
= 810 6 H z = 8 MH z
4110 - 8
h
2 t = 1
4 t
310 ,
( ) , ,
300K, H
( 45610 14 Hz) ?
c v
= 0
c v
vc,
= 0 1 v c
T = 300K ,
v =
kT
m H
=
RT
M H
465

, H
2v
= 0
c =
2 0 RT
c M H
=
24. 5610
14
310 8
8. 31300
110 - 3 = 4. 810 9 Hz
311: , [
( 312) ] r 0 1
?
P 1, 0, 0dr = 4
0
a 3 0
0
r 2 e -
2r / a 0 dr
= [ 1 - e - 2r / a 0 ( 1 + 2r / a 0 + 2r 2 / a 2 0) ]
0 = 1
1
312, r
r =
0 rP 1, 0, 0dr = 4 a 3 0
0 r 3 e - 2r / a 0 dr
= 3 2 a0[ 1 - e- 2r/ a 0
( 1 + 2r/ a0 + 2r 2 / a 2 0 + 4r 3 / a 3 0) ]
0
= 3 2 a 0
313 , ,
,
?
E
- e
p1
2
- e2 4r
= P 1 , 0, 0dr = e - 2r/ a 0
dr = -
0 4 0r 40
= -
466
0
a 3 0
e 2
4 0a0
m ee 4
= 2E
( 4 0) 2 h 2 1 = 2( - 136) = - 272 eV

E
k = E 1 - E
p 1 = - 13. 6 - ( - 272) = 136 eV
v 2 = 2E k/ m e
= 21361610 - 19 / ( 091110 - 30 )
= 21810 6 m/ s
314 = 2, l = 1 m= 0, + 1, - 1
2, 1, 0( r , , ) = 1/ ( 4 2 ) ( a - 0 3 / 2
) ( r/ a0) e - r / ( 2a 0 ) cos
2, 1, 1( r , , ) = 1/ ( 8 ) ( a - 0 3/ 2
) ( r / a 0) e - r/ ( 2a 0 ) sin e i
2 , 1, - 1( r , , ) = 1/ ( 8 ) ( a - 0 3/ 2
) ( r / a 0) e - r/ ( 2a 0 ) sin e -
( 1) P 2, 1 , 0, P 2, 1, 1P 2, 1 , -
36( b ) ( c)
( 2)
( 3) P 2, 1 , 01,
P =P 2 , 1, 0 = 2
( 1) P 2, 1, 0= ©¦
0
0
2
2, 1 , 0©¦
2
P 2, 1, 1= ©¦ 2, 1 , 1©¦
P 2, 1, -
0 ©¦
=
2
2 , 1, 0©¦
1 r 2
32 a 5 0
e -
r 2 sin drd d = 1
r / a 0
cos 2
= 1/ ( 8 ) ( a - 3/ 2 ) ( r / a 0) e - r / ( 2 a 0 ) sin 2
e e i -
i
= 1 r 2
e - r / a 0
sin 2
64 a 5 0
1= ©¦ 2 , 1, - 1©¦
2 = 1 r 2
64
e -
a 5 0
r/ a 0
sin 2
i
1
467

36( b ) ( c)
( 2) P 2, 1, 0+ P 2, 1 , 1+ P 2, 1 , - 1= 1 r 2
e r/ a 0
( cos 2 + sin 2 )
32 a 5 0
= 1 r 2
e r/ a
32 a 5 0
0
, ,
2
0
0 0
( 3) P = P 2, 1, 0 =
1
r 2 e - r / a 0
cos 2 r 2 sin drd d
32 a 5 0
1
=
32
2
3 24 = 1
= 2, l = 1, m = 0 ,
315l= 1 ,
31 , , j 1= l + 1/ 2= 3/ 2, j 2 =
l - 1/ 2= 1/ 2
31315
j 1= 3 2
468
, S=
3
4 h, L= 2 h, J 1= 15
4
h,

15/ 4 = 3/ 4 + 2 - 2 3 2 cos 1, 1 = 65. 9
4
j 2= 1 2 , S=
3
4 h , L= 2 h, J 2=
3
4 h ,
3
4 = 3 4 + 2 + 2 3 2 cos 2,
4 2 = 1447
316 , 2P 3 / 2 2P 1/ 2
4510 -
?
5 eV
( 1)
( 2) = 2, l = 1 ,
( 1) h = E 2p- E 1s
E 2 p - E 1 s
=
h
=
E 2p - 0
h
= 4510- 5 1610 - 19
66310 - 3 4 = 1. 110 10 Hz
©¦ ©¦= c = c =
2
2 c
= ( 12210- 7 ) 2
310 8 1. 110 10
= 5410 - 13 m = 0. 54 pm
( 2) E 2p= 2 BB,
B =
E 2p
= 4510 - 5 1610 - 19
= 039 T
2 B 292710 - 24
317 ,
B= 12T
32 S 1 = S 2
469

= 3 h/ 2, S z = h/ 2,
cos = 1/ 3 , 1 = 547 , 2 = 125. 3
32
E = BB = 92710 - 24 12 = 1110 - 2 3 J
1
318, 40cm,
15 T / mm2500K
( 1)
( 2)
dB
( 1) F = B
dz = 92710 - 2 1510 4 3
= 1410 - 2 0 N
( 2) z = Bd 2
dB
24
3kT dz = 92710- 0. 041510 2 3
313810 - 2 2500
3
= 2. 110 - 4 m= 0. 21 mm
319160T
14110 -
470
26 J/ T

,
? ?
2 z B,
h = 2 z B
= 2 zB/ h = 21. 4110 - 26 1. 60/ ( 6. 6310 - 34 )
= 0. 68110 8 H z = 68. 1 MH z
= c/ = 310 8 / ( 0. 68110 8 ) = 4. 41 m
320:
( 1) , l 2( 2l+ 1)
( 2) 2 2
( 1) , l , m l 0, 1, 2, , l,
( 2l+ 1) m l 1/ 2 m s ,
2( 2l+ 1) , , 2( 2l+ 1)
( 2) , l 0, 1, 2, , - 1 ,
- 1
2( 2l + 1) =
l = 0
2 + 2( 2- 2 + 1)
= 2 2
2
321 ( B, Z= 5) , ( A r, Z = 18) , ( Cu , Z = 29) ,
( Br , Z= 35)
B: 1s 2 2s 2 2p 1 ;
A r: 1s 2 2s 2 2p 6 3s 2 3p 6 ;
Cu: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 1 0 4s 1 ;
Br: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 1 0 4s 2 4p 5
322 30k eV X
0041nm
471

E k= eV= h = hc/ cu t
h = cu teV/ c
= 004110 - 9 1. 610 - 19 3010 3 / ( 310 8 )
= 6610 - 34 J ¡¤s
3230100nm X , X
?
V m in= hc e = 66310 - 34 310 8
= 1. 2410 4 V
010010 - 1610 9 - 19
32440k eV 4
3 , X
?
4
, 20keV, 10keV , 5keV 5k eV = ch/ E
X 0062nm, 0124nm , 0248n m
0248nm
Z =
325X K 316nm
( 1) L K ?
( 2) ?
( 1) E = h = hc/
= 66310 - 34 310 8 / ( 3. 1610 - 9 )
= 6. 2910 - 17 J= 393 eV
( 2) = 4. 9610 7 ( Z- 1)
c/
4. 9610 7 + 1 =
Z 7, ( N )
310 8 / ( 31610 - 9 )
4. 9610 7 + 1 = 7. 21
326 K L 8979keV
0951keV X N aCl
741
472

?
d =
2dsin = m , = ch/ E L K m= 1
=
ch
2sin E LK
310 8 66310 - 34
2sin741( 8979 - 0951)10 3 1610 - 19
= 0. 8010 - 1 0 m
327CO 2
106 m
( 1) CO 2 ?
( 2) 300K , CO 2
?
( 3) CO 2
1% ,
?
( 1) E = h = ch / = 310 66310 8 - 3 4 / 10. 610 - 6
= 18810 - 20 J= 0117 eV
( 2) N 2/ N 1 = exp ( - E/ ( kT ) )
( 3) T = -
= -
= exp [ - 1. 8810 - 20 / ( 1. 3810 - 2 3 300) ]
= 1. 07%
hc
k ln N 2
N 1
66310 - 34 310 8
ln
10. 610 - 13810 6 - 23
= - 1. 310 5 K
1. 01
1
328
10fs( 1fs= 10 -
500nm
1 5 s) ?
N = ct/ = 31010 8 - 1 10/ 5 ( 50010 - 9 ) = 6
329 6944nm ,
473

12ps, 0150J: ( 1) ; ( 2)
; ( 3)
( 1) l = ct= 310 8 1210 - 12
= 3610 - 3 m = 036 mm
( 2) P = W/ t= 0150/ ( 1210 - 1 2 ) = 1. 2510 10 W
= 12. 5 GW
( 3) N = W/ ( h ) = W / ( hc)
= 0150694410 - 9 / ( 66310 - 34 310 8 )
= 5210 1 7
330 GaA lA s 200 m 3 ( = 2
10 - 7 mm 3 ) 50mW 080 m
?
N = P / ( h ) = P / ( hc)
= 5010 - 3 08010 - 6 / ( 66310 - 3 4 310 8 )
= 2. 010 16 / s
331 300mm,
500W, 515nm
350cm ,
( 1) ? ( 2)
? ( 3) 84% ,
?
( 1)
I 0 = P =
r 2
( 2)
474
500
= 70710 5 W / m 2
( 1510 - 3 ) 2
= sin=
1. 22
d

R = f =
1. 22 f = 1. 2251510 - 9 35010 - 2
d
3. 0010 - 3
= 7. 3310 - 6 m = 733 m
( 3)
I F = P84% =
R 2
500084
= 24910 1 0 W / m 2
( 73310 - 6 ) 2
3328610 1 0
H z266
10 - 2 6 kg
= h
2 I = h
mr 2 0
r 0 =
h
m = 1. 0510 - 34
26610 - 2 6 8610 10
= 1210 - 1 0 m = 012 nm
333 ,
?
I = 2 5 ma 2 02= 4 5 ma2 0
( j = 1)
E 1 = h 2
I
=
5 h
4ma 2 0
= 5410 - 1 8 J = 34 eV
=
5( 10510 - 34 ) 2
4091110 - 3 0 ( 05310 - 10 ) 2
T = E 1/ k = 5410 - 1 8 / ( 1. 3810 - 23 ) = 3910 5 K
,
,
334CO 64210 13 Hz
475

= 1 K
2m m= m1m 2/ ( m1+ m 2)
m 1m
K = ( 2 ) 2 2
m 1 +
m2
, m
= ( 64210 13 ) 2 1216
27
16610-
12 + 16
= 1. 8510 3 N / m
476

4
41 193g/ cm 3 ,
?
,
h 2
E F = ( 3
2 ) 2/ 3 2/ 3
2m e
( 10510 - 34 ) 2
19310 3 60210 23 2/ 3
= ( 3
2 ) 2/ 3 291110 - 31 19710 - 3
= 8. 8010 - 19 J = 5. 50 eV
v F = 2E F / m e = 28. 8010 - 19 / ( 9. 1110 - 31 )
= 1. 3910 6 m/ s
T F = E F/ k = 8. 8010 - 19 / ( 1. 3810 - 23 )
= 6. 3810 4 K
=
h
2m e E F
=
= 52410 - 10 = 0524 nm
6. 6310 - 34
29. 1110 - 31 8. 8010 - 19
42 0K
E t ot al = E F
E g ( E ) dE = E F
0
= 1 5 ( 2me ) E 5/ 2
3/ 2 F
= 3
2 h 3 5 E F
E
= E t ot al / = 3E F / 5
0
E ( 2m e ) 3/ 2
E 1 / 2 dE
2
2 h 3
477

430K
, vF
v = 1 E F
0
vg ( E ) dE
= 1 E F
0
2E
me
1
2
( 2me ) 3/ 2
E 1/ 2 dE
2
2 h 3
= 2m e 1
2 h 3 2 E F 2 = 3 4 vF
v 2 =
1
E F
0
v 2 g ( E ) dE
1/ 2
=
1
E F
0
2E
me
( 2me ) 3/ 2
E 1/ 2 dE
2
2 h 3
1 / 2
=
3
5 2 E F
me
1/ 2
=
3
5 v F
44 5
10 16 kg/ m 3
E F
h 2
= ( 3
2 ) 2/ 3 3
2m n 2/
( 1. 0510 - 34 ) 2
2 / 3
= ( 3
2 ) 2/ 3 51016
216710 - 27 16710 - 27
= 30. 510 - 1 3 J= 19 Mev
v F = 2E F/ mn = 230. 510 - 13 / ( 16710 - 27 )
= 6010 7 m/ s
45 ,
10% ? 5eV
478
2 kT R
2E F
= 0. 1 3 2 R

T =
0. 3E F
=
2 k
1 9
0. 351610-
= 1810 3 K
2 1. 3810 - 2 3
46 ,
T 3
T , ,
,
Cm = ( 20810 - 3 T + 25710 - 3 T 3 ) J/ ( mol ¡¤K )
( 1)
( 2) ?
( 1)
E F
2 Rk
2E F
= 20810 - 3
= 2 83113810 - 2 3
220810 - 3
= 27210 - 19 J = 1. 70 eV
20810 - 3 T = 25710 - 3 T 3
T =
20810 - 3
25710 - 3 = 0900 K
47 10510 3 k g/ m 3 , 1610 - 8
m( )
( 1)
( 2)
( 3)
( 4) ( 2) ( 3)
( 1) = m e/ ( e 2 ) , =
D N A / MA g= 10510 3 602310 2 3 / ( 10810 - 3 ) = 5. 8610 2 8
m -
1 ,
479

=
m e
e 2 =
= 3810 - 14 s
9. 1110 - 3 1
1610 - 8 58610 28 ( 1610 - 19 ) 2
( 2)
c la = v =
8kT
me
813810 - 2 300
3
=
3810 - 1 4
91110 - 3 1
= 40910 - 9 m= 4. 09 nm
( 3)
F er= v F = 139103810 6 - 14
= 5310 - 8 m= 53 nm
( 4) a = 3 1/ = ( 58610 2 8 ) - 1/ 3 = 2610 - 1 0 m= 026 nm
481000K , 01eV
? 010eV
?
+ =
E -
- =
E + = E F + 010eV
1
e ( E + - E F ) / ( kT ) + 1 = 1
e 011 610 - 19 / ( 13 81 0 - 23 10 00)
= E F - 010eV
e ( E -
1
- E F
) / ( kT )
+ 1 = 1
e - 0 1161 0- 19 / ( 13810 - 23 1 000 )
4955eV
+ 1 = 024
+ 1 = 076
( 1) ?
300K
?
( 2)
( 1) N u p
= e - E / ( kT ) = e - 5 5161 0- 19 / ( 13810 - 23 300 )
N be
480
= 4. 910 - 93

,
( 2) m a x=
ch
E = 3108 66310 - 34
551610 - 19
= 22610 - 7 m= 226 nm
410 0 10 1 6 m - 3
10 6 , ( 1)
? 233g/ cm 3
( 2) 10g ?
( 1)
= 10 6 0 - 0 = 10 6 0 = 10 22 m - 3
10 2 2 , 10 22 1m 3
S i =
D S i
MS i
N A =
( 2) mp
2. 3310 3
28110 - 3 60210 23 = 50010 28 m - 3
=
Si
= m S i
Mp =
D S iN A
10 2 2
50010 28 =
1
510 6
10 - 3 10 2 2 3110 - 3
23310 - 3 602310 23
= 2210 - 1 0 kg= 0. 22 g
41112eV,
E D = 0045eV
m ax =
ch = 310 66310 8 - 34
E D 00451610 - 19
= 27610 - 5 m= 276 m
412 CdS P bS 242eV
030eV ? ?
CdS,
481

ma x = ch = 3108 66310 - 34
E g 2421610 - 19
PbS,
= 51310 - 7 m = 513 nm
ma x = ch = 3108 66310 - 34
E g 0301610 - 19
= 414 m
413Ga A sP 19eV
?
m ax =
ch = 310 66310 8 - 34
E g 191610 - 1 9
= 65410 - 7 m= 654 nm
414KCl 76eV
140nm , ?
140nm
E = hc = 66310 - 3 4 310 8
14010 - 9
= 14210 - 1 9 J = 8. 9 eV
K Cl ,
482

5
51 6MeV ( 1 97 Au )
, ? ( 14 N ) ?
?
?
6MeV ( 410 3 MeV ) ,
6MeV E k M ,
2Ze 2
E k= E k + E kM + = 1 4 0r mi n 2
E k =
mv =
m
( m + M) v
2 +
( m + M) v
m + M E k + 2Ze 2
4 0r m in
r m in = m + M 2Ze2
ME k 4 0
, M= 197m , ,
r min =
2Ze 2
4 0r m in
2Ze 2
= 2791610- 19 910 9
E k 4 0
610 6
= 3810 - 14 m
, M= 14, ,
r mi n = 4 + 14 2Ze2 = 1871610 - 1 9 910 9
14E k 4 0
7610 6
= 43210 - 15 m
r N = 12A 1/ 3 10 - 15 = 28910 - 15 m
483

6MeV
52 1 6 N , 1 6 O 1 6 F 16006099u,
15994915u 16011465u
BE = Mc 2 = [ Zm p + N m n - ( M - Zm e) ] c 2
16
N : BE = 1180 MeV
16
O: BE = 1277 MeV
16
F : BE = 1115 MeV
53 , Z
A
56
Fe 238 U ( MeV )
h 2
h 2
E F = ( 3
2 ) 2/ 3 2/ 3
p = ( 3
2 ) 2/ 3
2mp
2mp
=
=
2/
9
3 h 2
4 2m p R 2 0
Z
A
2/ 3
Z3
4 R 3 0A
2/ 3
9
10510 - 34
4 216710 - 27 ( 1210 - 15 ) 2 1. 610 - 13
2/ 3
Z = 53 Z A A
2/ 3
MeV
2/ 3
E F , 1 = 3 5 E F = 32 Z A
2 / 3
MeV
25 Fe , Z= 26, A= 56, E F = 32MeV , E F , 1= 19MeV
238 U , Z= 92, A = 238, E F = 28MeV , E F , 1= 17MeV
54
1 1
C 11 B, 15 O 15 N , 21 N a 2 1 N e
279MeV, 348MeV
484

430MeV A 1 / 3 ?
?
, ,
1,
E p =
3e 2
54 0r [ Z2 - ( Z - 1) 2 ] =
r = 3e2 ( 2Z - 1)
54 0 E p
11 C 1 1 B, Z= Z C= 6, r 11= 341fm
15 O 15 B, Z= Z O = 8, r 15= 372fm
21 Na 21 N e, Z= Z N a = 11, r 2 1= 422fm
r = R0A 1 / 3 ,
R 0, 1 1 = r 1 1/ 11 1/ 3 = 153 fm
R 0, 1 5 = r 1 5/ 15 1/ 3 = 151 fm
R 0, 2 1 = r 2 1/ 21 1/ 3 = 153 fm
3e 2
[ 2Z - 1]
54 0r
, r A 1/ 3 , R0= 15fm
55
, 12 C
, 16 O
12 C 16 O
, 242MeV
,
BE O = ( 81007825 + 81008665 -
= 127. 7 MeV
15994915)9315
BE C = ( 61007825 + 61008665 - 12000000)9315
= 9216 MeV
BE = ( 21007825 + 21008665 - 4002603)93149
485

= 28. 3 MeV
BE O = 4BE + E C 2 4 = 4283 + 2426
= 1277 MeV
BE C = 3BE + E C 2 3 = 3283 + 2423
= 922 MeV
56 23 2 T h
Z = 90, A = 232
E = 2BE A / 2, Z/ 2- BE A , Z
= - 4. 6A 2/ 3 + 026 Z 2
A 1 / 3
= 169 MeV
57Z, A 2Z, 2A
12
C
E = BE 2A , 2Z- 2BE A , Z
= 15. 7532A - 17804( 2A ) 2/ 3 - 07103 ( 2Z) 2
2
( 2A- 4Z)
- 2369
2A
( 2A) 1/ 3
- 215. 753A - 217804A 2/ 3 - 207103 Z 2
2
( A- 2Z)
- 22369
A
= - 08345 Z 2
+ 73458A 2 / 3
A 1 / 3 MeV
12 C, Z= 6, A = 12
486
E =
254MeV
A 1 / 3

58
? ?
©¦ dN ©¦= N 0e -
t / dt/ ,
1
t 2 = N t 2 ©¦ dN ©¦= N 0
0
N 0
t = ,
t
2
e- t / dt = 2
2
( t - ) 2 = t 2 - 2t+
2 = 2
2 - 2
2 +
2 =
,
0
59 40 K 1210 -
4
1310 9 a ,
037%
70kg ,
A = N =
=
0. 693
t 1/ 2
mN A
M
0. 693
700. 0037
1. 310 9 3. 1510 7 39. 110 - 3
1210 - 4 60210 2 3
= 8. 1 k Bq
51010kg ( U 3O 8) 226 Ra 2 31 Pa
2 38 U 9927% , 2 35 U 072% ; 226 Ra
1600a, 231 Pa 32710 4 a
226 Ra 23 1 Pa , 22 6 Ra 238 U , 231 Pa
235
U 226 Ra 2 38 U ( 44610 9 a) , 23 1 P a
23 5 U ( 70410 8 a )
N A A = N B B N A / t 1/ 2 , A = N B / t 1/ 2, B
226 Ra
487

N 238 U =
N 226 Ra
m 226 Ra =
10 4
816 + 3238 36023102 3 09927
= 21310 25
= 21310 25 t 1/ 2, 226 R a
t 1/ 2, 238 U
= 76410 18
= 2. 1310 25 1600
4. 4610 9
76410
18
60210 23 226 = 2. 8710 - 3 g = 2. 87 mg
231 Pa
N 235 U =
10 4
816 + 3235 360210 23 00072
= 15210 23
N 231 P a
= 15210 23 t 1/ 2, 231 P a
t 1/ 2, 235 U
= 70610 18
4
3. 2710
= 1. 5210 23
7. 0410 8
m 231 P a =
70610
18
60210 23 231 = 2. 7110 - 3 g = 271 mg
511 30 Ci 12 3 I 24 ,
4 Ci 12 3 I 131h
24 12 3 I (
15% 40% )
A 0 = 30 Ci, A 0, t h =
A 0p 24 ,
4= A t h = A 0, t he - t = A 0p e - t =
p = 0. 48 = 48%
30 p e -
0. 6 9324
1 31
512 24 N a 300k Bq
10 30Bq
24 Na 1497h
488

V = A 0
e -
A 1
t = 30010 3
30
A 1V = A 0e -
e -
t
0. 6 9310
1 49 7
= 62910 3 cm 3 = 629 L
513
03mol CO 2 14 C
9
t =
N 0= 031310 - 12 60210 23 = 2. 3510 1 1
A = 9/ 60 s - 1 , = 82703. 1510 7 s
A = A 0e - t/ = 1 N 0e - t /
ln N 0
A = 8270ln 2. 3510 11 60
827031510 7 9 =
15104 a
51403g 238 U 012g 206 Pb
238 U ,
2 38 U 2 06 Pb 2 38 U
, 20 6 Pb ,
2 38 U 20 6 P b 2 38 U
N U , 0 = N U + N P b =
0. 36. 0210
23
238
= 75910 20 + 35110 2 0 = 11110 2 1
N U = N U , 0e -
t =
t
1
ln N U , 0
N U
=
9
4. 4610
ln
0693
+ 012602102 3
206
1. 11102
1
75910 2 0 = 24510 9 a
515 2 26 Ra 47825MeV
489

?
47825MeV ( 410 3 MeV )
, mM d
,
m v = M dvd
E kd = 1 2 Mdv 2 d = 1 2
m
v 2 =
M d
m
E k
Md
= 4 47825 = 00862 MeV
222
E = E k + E kd = 4. 7825 + 0. 0862 = 4. 8707 MeV
516
64
Cu 64 Zn,
+ 64 Ni
Q 057MeV 066MeV 64 Zn
64 Ni , u
64 Cu
64 6
Cu
4 Zn +
- +
e
64
Cu
+
Q
- = [ M Cu - ( M Zn + me ) ] c 2
64
Cu
64
N i +
+ + e
Q
+ = [ M Cu - ( M N i + m e) ] c 2
Q
+ - Q
- = ( MZn - MN i) c 2
E = M Zn - MN i = Q + - Q
-
=
c 2
=
066 - 057
9315
= 9710 - 5 u
0. 66 - 0. 57
c 2
MeV
517 60 Co ( 527a)
490

, 60 Co
6 0 Co
59 Co
59
Co+ n
60
Co+
120b2cm (
59
Co 100% ) 210 1 2 cm - 2 s - 1
6h 8858g/ cm 3
59 Co
N 59 = 8885860210 23 / 59 = 7. 2310 23
, ,
60 Co
R = N 59I = 12010 - 2 4 72310 23 210 1 2
= 1. 7410 1 4 s - 1
6h527a, 6h 6 0 Co
, 60 Co Rt, 60 Co
A =
N 60 = Rt = 0. 6931. 741014 63600
5. 273. 1510 7 3. 710 1 0
= 042 Ci
518Cd 8 ,
Cd 4000b,
95% , Cd Cd 1124g/ mol,
864g / cm 3
S d Cd ,
Cd ,
R = 0. 95I S =
d =
=
0. 95
=
N I =
0. 95M
DN A
Sd I
0. 95112. 4
400010 - 28 8. 6410 3 6. 0210 23 491

= 5. 110 - 5 m = 51 m
519Q , :
13
C( p , ) 10 B, 13 C( p , d ) 1 2 C, 13 C( p , ) 14 N
13 C( p , ) 10 B
Q = ( 13003355 + 1007825 - 4002603 - 10012937)9315
= - 406 MeV
13 C( p , d ) 12 C
Q = ( 13003355 + 1007825 - 2014102 - 120000)9315
= - 272 MeV
13 C( p , ) 14 N
Q = ( 13003355 + 1007825 -
= 755 MeV
14003074)9315
520 13 C( p , ) 10 B , ,
13 C
,
E 1 = Q 1 + m p
mC
= 4. 06 1 + 1
13
= 437 MeV
13 C ,
r = r c + r p = r 0A 1/ 3
C + r 0
= 1210 - 15 ( 13 1/ 3 + 1) = 4010 - 15 m
E 2 = 6e2 1 + m p
4 0r mC
492
= 61. 610- 19 910 9
4010 - 15 10 6 1 + 1 13
= 23 MeV

E t h = E 1 + E 2 = 4. 37 + 2. 3 = 6. 7 MeV
521 1510 3 0 kg ,
3910 26 W ,
?
4 1 H
4
H e+ 2e + + 2 e+ 24. 67 MeV
N = 1510 30 / ( 16710 - 27 ) = 0. 9010 5 7
t = 0901057 246710 6 1610 - 19
43910 26
= 2. 2810 18 s = 7. 210 1 0 a
522
, :
1
H +
12 C
13
N
1
H + 13 C
1 3
N + ( 1)
1 3
C + e + + e ( 2)
1 4 N + ( 3)
1
H + 14 N
1 5 O + ( 4)
1
H +
1 5 O
15
N
1 5 N + e + + e ( 5)
1 2
C + 4 H e ( 6)
( 1) ,
( 2) ( 3)
?
( 1) ,
,
4 1 H
4
H e + 2e + + 3+ 2 e
( 2)
493

E 1 = ( 1007825 + 12000000 - 13005738)9315
= 1944 MeV
E 2 = ( 13005738 - 13003355)9315 - 20511
= 1198 MeV
E 3 = ( 1007825 + 13003355 - 14003074)9315
= 7551 MeV
E 4 = ( 1007825 + 14003074 - 15003065)9315
= 7297 MeV
E 5 = ( 15003065 - 15000109)9315 - 20511
= 1732 MeV
E 6 = ( 1007825 + 15000109 - 12000000 - 4002603)
9315 =
4966 MeV
( 3)
E =
6
E = 2469 MeV
= 1
494