5H5=8O 7040G.
10 :;0AA 425=04F0B8;5B=59 H:>;K
040=85 10(12)-1. « 07<8=:0
1.1 0:>= 87<5=5=8O 25@B8:0;L=>9 :>>@48=0BK :0<CH:0 8<55B 284
2
gt
y = v0t sinÄ… - . (1)
2
>;030O :>>@48=0BC @02=>9 2KA>B5 h , ?>;CG8< :204@0B=>5 C@02=5=85, 4;O >?@545;5=8O
2@5<5=8
gt2
- v0t sinÄ… + h = 0 . (2)
2
8A:@8<8=0=B MB>3> C@02=5=8O @025=
2
D = (v0 sinÄ…) - 2gh . (3)
;O 2KA>BK h1 = 2,5 < 53> G8A;5==>5 7=0G5=85 @02=>
<2
2 2
D1 = (v0 sinÄ…) - 2gh1 = (15 Å" 0,5) - 2 Å"9,8Å" 2,5 = 7,25 .
A2
!;54>20B5;L=>, C@02=5=85 (2) 8<55B 420 :>@=O
v0 sinÄ… Ä… D 15 Å" 0,5 Ä… 7,25
t1,2 = = Ò! t1 = 0,49c; t2 = 1,0c . (4)
g 9,8
10 :>@=O 8<5NB D878G5A:89 A<KA;, B0: :0: :0<CH5: 1C45B =0E>48BLAO =0 C:070==>9
2KA>B5 42064K  ?@8 ?>4J5<5 8 ?@8 A?CA:5.
;O 2B>@>3> 7=0G5=8O 2KA>BK h2 = 3,0 < 48A:@8<8=0=B :204@0B=>3> C@02=5=8O (2)
<2
2 2
D2 = (v0 sinÄ…) - 2gh2 = (15 Å" 0,5) - 2 Å"9,8 Å"3,0 = -2,6
A2
>B@8F0B5;L=K9, ?>MB><C C@02=5=85 :>@=59 =5 8<55B. -B> >7=0G05B, GB> :0<CH5: =0 MBC
2KA>BC =5 ?>4=8<5BAO.
1.2 1>7=0G8< =0?@O65=85 =0 ;0<?>G:5 U . 7 D>@<C;K
?@82545==>9 2 CA;>288 I = a U , A;54C5B, GB> MB> =0?@O65=85
A2O70=> A 8A:><>9 A8;>9 B>:0 A>>B=>H5=85<
1
2
U = I . (1)
a2
0?@O65=85 =0 @578AB>@5, ?> 70:>=C <0, @02=>
UR = IR0 . (2)
!C<<0 MB8E =0?@O65=89 @02=0 =0?@O65=8N 8AB>G=8:0
1
2
I + IR0 = U (2)
0
a2
8A:@8<8=0=B MB>3> :204@0B=>3> C@02=5=8O
U0
2
D = R0 + 4 (3)
a2
?>;>68B5;L=K9, ?>MB><C C@02=5=85 (2) 8<55B 420 :>@=O
a2 ëÅ‚ U0 öÅ‚
2
ìÅ‚- ÷Å‚
I = R0 Ä… R0 + 4 .
ìÅ‚
2 a2 ÷Å‚
íÅ‚ Å‚Å‚
1
4=0:>, >B@8F0B5;L=K9 :>@5=L C@02=5=8O (2) A<KA;0 =5 8<55B (=8 <0B5<0B8G5A:>3>  >=
?>O28;AO ?@8 2>72545=88 8AE>4=>9 D>@<C;K 2 :204@0B, =8 D878G5A:>3>), ?>MB><C >B25B><
40==>9 7040G8 O2;O5BAO D>@<C;0
a2 ëÅ‚ 2 U0 öÅ‚
ìÅ‚
I = R0 + 4 - R0 ÷Å‚ . (4)
ìÅ‚
2 a2 ÷Å‚
íÅ‚ Å‚Å‚
0<5B8<, GB> ?@8 R0 0 , >=0 ?5@5E>48B 2 D>@<C;C, ?@82545==CN 2 CA;>288.
@0D8: ?>;CG5==>9 7028A8<>AB8
<>65B 1KBL ?>AB@>5= =0 >A=>20=88
C@02=5=8O (2): 7028A8<>ABL U0(I) -
87>1@0605BAO 10=0;L=>9 ?0@01>;>9.
;O ?>;CG5=8O >1@0B=>9 7028A8<>AB8
=5>1E>48<> 2K1@0BL 55 =C6=K9
CG0AB>:, 0 70B5< 53> ?>25@=CBL 8
>B@078BL.
1.3 @09=5 =57=0G8B5;L=0O @07=>ABL B5?;>B 2>7=8:05B 87-70 @07=>3> ?>=865=8O F5=B@0
<0AA ;L40 ?@8 53> ?;02;5=88, 87-70 G53> C<5=LH05BAO ?>B5=F80;L=0O M=5@38O 2>4K. -B0
M=5@38O 70B@0G8205BAO =0 ?;02;5=85 ;L40.
KA>B0 C@>2=O 2>4K (A=0G0;0 2 B25@4><, 0 70B5< 2 684:>< A>AB>O=88) <>65B 1KBL
=0945=0 87 >G5284=>3> 2K@065=8O
m
m = ÁV = ÁSh Ò! h = .
ÁS
7<5=5=85 2KA>BK C@>2=O ?@8 ?;02;5=88 @0AAG8BK205BAO ?> D>@<C;5
ëÅ‚ öÅ‚
m 1 1
ìÅ‚ - ÷Å‚
"h = . (1)
ìÅ‚
S Á;L40 Á2>4K ÷Å‚
íÅ‚ Å‚Å‚
!;54>20B5;L=>, C<5=LH5=85 ?>B5=F80;L=>9 M=5@388 (8 @02=>5 59 :>;8G5AB2> B5?;>BK)
?@8 ?;02;5=88 ;L40 @02=>
ëÅ‚ öÅ‚
1 m2g 1 1
ìÅ‚ - ÷Å‚
Q = mg"h = . (2)
ìÅ‚
2 S Á;L40 Á2>4K ÷Å‚
íÅ‚ Å‚Å‚
"5?5@L 2848<, GB> MB0 B5?;>B0 7028A8B >B ?;>I048 A>AC40 8 2 1>;55 C7:>< A>AC45 >=0
1>;LH5. !;54>20B5;L=>, 8A:><0O @07=>ABL B5?;>B @02=0
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
m2g 1 1 m2g 1 1 m2g 1 1
ìÅ‚ - ÷Å‚ - ìÅ‚ - ÷Å‚ ìÅ‚ - ÷Å‚
´Q = = .
ìÅ‚ ìÅ‚ ìÅ‚
2S Á;L40 Á2>4K ÷Å‚ 4S Á;L40 Á2>4K ÷Å‚ 4S Á;L40 Á2>4K ÷Å‚
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
2
040=85 10(12)-2 . «"@C10 45;>
-;5:B@8G5A:>5 A>?@>B82;5=85 <5B0;;8G5A:>9 B@C1K @02=>
Å‚ L Å‚ L
M M
R = = = 2,71Å"10-4<. (1)
SM 2Ä„ rh
>I=>ABL, :>B>@0O 2K45;O5BAO ?@8 ?@>B5:0=88 M;5:B@8G5A:>3> B>:0 2 B@C15 ?> 70:>=C
6>C;O-5=F0 @02=0
2 2
U U 2Ä„ rh
P = = =179B. (2)
R Å‚ L
M
@5<O, 70 :>B>@>5 @0AB05B 25AL ;54 2 B@C15, >?@545;8< 87 C@02=5=8O B5?;>2>3> 10;0=A0.
"5?;>B0, 2K45;82H0OAO ?@8 ?@>B5:0=88 M;5:B@8G5A:>3> B>:0, ?>H;0 =0 =03@520=85 ;L40
=0 "T = 103@04 8 53> ?;02;5=85
PÄ1 = Q , (3)
PÄ1 = Ä„ r2LÁ ( + c "T ), (4)
rL2Á ( + c "T )Å‚
M
Ä1 = = 22, 2A. (5)
2
2U h
A;8 =0 =03@520=85 ;L40 2 B@C15 B@0B8BAO G0ABL Ä… B5?;>2>9 <>I=>AB8 P , B> ?>B@51C5BAO
1>;LH55 2@5<O 4;O ?;02;5=8O ;L40:
Ä… PÄ2 = Ä„ r2LÁ ( + c "T ), (6)
rL2Á ( + c "T )Å‚
M
Ä2 = = 44, 4A. (7)
2
2Ä…U h
!B>8B >B<5B8BL, GB> MB> 2@5<O ?@>?>@F8>=0;L=> @048CAC B@C1K
Ä = br, (8)
L2Á ( + c "T )Å‚
M A
345 b = = 222 .
<
2
2Ä…U h
AB02H0OAO <>I=>ABL (1-Ä… )P ?>945B =0 =03@520=85 >:@C60NI59 ?>G2K 8 ?;02;5=85
;L40 2 =59. 0 2@5<O t >BB05B ?>G20 2 ?@545;0E F8;8=4@0 @048CA>< x , :>B>@K9 =0945< 87
C@02=5=8O B5?;>2>3> 10;0=A0
(1-Ä… )Pt = Ä„ (x2 - (r + h)2)L[²Á ( + c "T ) + (1- ² )Á?c?"T ], (9)
1>7=0G8< 2K@065=85 2 :204@0B=KE A:>1:0E  B5?;>BC >BB0820=8O 1<3 ?>G2K
B = ²Á ( + c "T ) + (1- ² )Á?c?"T = 50,5Å"106 6 / <3. (10)
B>3>
(1-Ä…)Pt
x = (r + h)2 + . (11)
Ä„ LB
@8<5@=K9 3@0D8: MB>9 7028A8<>AB8 87>1@065= =0 @8AC=:5.
!>?@>B82;5=85 >BB0O2H59 ?>G2K @02=>
Å‚ L Å‚ L Å‚ L2B
? ? ?
R? = = = .
S? Ä„ (x2 - (r + h)2) (1-Ä… )Pt
(12)
@8 MB>< ?> 70:>=C 6>C;O-5=F0 2 ?>G25
2K45;O5BAO 4>?>;=8B5;L=0O <>I=>ABL
2 2
U U (1-Ä… )Pt
P? = = .
R? Å‚ L2B
?
(13)
3
>?>;=8B5;L=0O B5?;>20O <>I=>ABL =5 ?>AB>O==0, 0 ?@O<>
P
?@>?>@F8>=0;L=0 2@5<5=8, B>340 :>;8G5AB2> B5?;>BK, :>B>@>5
2K45;8BAO 70 2@5<O t <>6=> >?@545;8BL, :0: ?;>I04L ?>4
3@0D8:>< <>I=>AB8 >B 2@5<5=8.
2
U (1-Ä…)P t2
Q? = . (14)
Å‚ L2B 2
?
t
0 2@5<O Ä2 2 ?>G25 2K45;8BAO :>;8G5AB2> B5?;>BK
2 2
U (1-Ä…)P Ä2
Q?(Ä2) = , (15)
Å‚ L2B 2
?
?@8 MB>< >B B@C1K ?>G20 ?>;CG8B B5?;>BC
Q(Ä2) = (1-Ä…)PÄ2. (16)
2
Q?(Ä2) U Ä2
B=>H5=85 = = 0,01, MB> >7=0G05B, GB> 4>?>;=8B5;L=> 2K45;82H59AO 2
Q(Ä2) Å‚ L2B 2
?
?>G25 B5?;>B>9 <>6=> ?@5=51@5GL.
A;8 x AB0=>28BAO @02=> H , ?@>8AE>48B «7075<;5=85 8 =03@520=85 B@C1K ?@5:@0I05BAO.
>A:>;L:C 25AL ;54 2 B@C15 4>;65= @0AB>?8BLAO, B> 2@5<O Ä3 , 70 :>B>@>5 MB> ?@>87>945B,
@02=> ?> D>@<C;5 (8)
Ä3 = brmax. (17)
0:A8<0;L=K9 @048CA B@C1K >?@545;O5BAO 87 A>>B=>H5=8O
2
(1-Ä…)Pbrmax U 2Ä„ hrmax (1-Ä…)brmax
2
H = (rmax + h)2 + = (rmax + h)2 + . (18)
Ä„ LB Å‚ L Ä„ LB
M
>;CG05BAO :204@0B=>5 C@02=5=85 >B=>A8B5;L=> rmax :
2
ëÅ‚ öÅ‚
2U h(1-Ä…)b
2 2
rmax + 2hrmax + h2 - H = 0, (19)
ìÅ‚1+ Å‚ L2B ÷Å‚
íÅ‚ M Å‚Å‚
:>B>@>5 =5A:>;L:> C?@>I05BAO, CG8BK20O <0;>ABL h ?> A@02=5=8N A H 8 rmax :
2
ëÅ‚ öÅ‚
2U h(1-Ä…)b
2 2
rmax - H = 0, (20)
ìÅ‚1+ Å‚ L2B ÷Å‚
íÅ‚ M Å‚Å‚
H
rmax = = 0,37<. (21)
2
ëÅ‚ öÅ‚
2U h(1-Ä…)b
ìÅ‚1+ Å‚ L2B ÷Å‚
íÅ‚ M Å‚Å‚
@>87>945B MB> 70 2@5<O
Ä3 = brmax = 82A. (22)
4
040=85 10(12)-3. «>;>==0 02B><>18;59
1.1 0AAG8B05< 8=B5@20; 2@5<5=8 Ä1 70 :>B>@K9 02B><>18;L 4>AB8305B B@51C5<>9
A:>@>AB8 ?>AB>O==>3> 42865=8O, 8A?>;L7CO 70:>= @02=>CA:>@5==>3> 42865=8O
<
v0 20 A
v0 = a0Ä1 Ò! Ä1 = = = 10c . (1)
a0 2,0 <
A2
0 MB> 2@5<O >= ?@>945B ?CBL, @02=K9
2 2
aÄ v0
s = = = 100< . (2)
2 2a0
-B> @0AAB>O=85 1>;LH5 G5< @0AAB>O=85 l1 , ?@8 :>B>@>< B@>305BAO A;54CNI89 02B><>18;L.
!;54>20B5;L=>, A;54CNI89 02B><>18;L B@>=5BAO @0=LH5, G5< ?5@2K9 4>AB83=5B A:>@>AB8
v0 , 0 8<5==> 2 <><5=B 2@5<5=8 Ä1 , :>B>@K9 =0945< 87 70:>=0 42865=8O
2
a0Ä 2(l1 - l0 )
l1 - l0 = Ò! Ä = = 5,0c (3)
2 a0
0<5B8<, GB> 2@5<O @073>=0 A2O70=> A ?@><56CB:>< 2@5<5=8 <564C =0G0;0<8 42865=8O
02B><>18;59 ?@>ABK< A>>B=>H5=85< Ä1 = 2Ä .
"0:8< >1@07><, ?5@2K9 02B><>18;L 2
B5G5=85 ?@><56CB:0 2@5<5=8 2Ä 1C45B 42830BLAO
@02=>CA:>@5==>, A CA:>@5=85< a0 (3@0D8:
7028A8<>AB8 A:>@>AB8 >B 2@5<5=8  =0:;>==0O
?@O<0O, 3@0D8: 70:>=0 42865=8O  ?0@01>;0), 0
70B5< @02=><5@=> (3@0D8: 70:>=0 7028A8<>AB8
A:>@>AB8 >B 2@5<5=8  3>@87>=B0;L=0O ?@O<0O;
3@0D8: 70:>=0 42865=8O - =0:;>==0O ?@O<0O
;8=8O). @0D8: 42865=8O 2B>@>3> 02B><>18;O 0=0;>38G5= ?5@2><C, B>;L:> A<5I5= =0
8=B5@20; 2@5<5=8 Ä 8 =0 @0AAB>O=85 - l0 ; B@5B89 A<5I5= ?> >A8 2@5<5=8 =0 2Ä 8 =0
- 2l0 ?> >A8 :>>@48=0B.
5
1.2 0AAB>O=85 <564C 02B><>18;O<8 (?>A;5 B>3>, :0: 704=89 ?>;=>ABLN =01@0; A:>@>ABL),
?@>I5 2A53> >?@545;8BL 87 3@0D8:0 7028A8<>AB8 A:>@>AB59 >B 2@5<5=8. >6=> 70<5B8BL,
GB> ?;>I04L <564C 3@0D8:0<8 7028A8<>AB59 A:>@>AB59 42CE 02B><>18;59 G8A;5==> @02=0
87<5=5=8N @0AAB>O=89 <564C =8<8 (=0 @8AC=:5 70HB@8E>20=0). 53:> >?@545;8BL, GB>
87<5=5=85 @0AAB>O=8O <564C 42C<O A>A54=8<8 02B><>18;O<8 @02=> "l = v0Ä = 100< .
!;54>20B5;L=>, 2 ?@>F5AA5 42865=8O @0AAB>O=85 <564C 02B><>18;O<8 1C45B @02=K<
l2 = l0 + v0Ä = 110< . (4)
1.3 '8A;> 8=B5@20;>2 <564C N 02B><>18;O<8 5ABL (N -1), ?>MB><C 4;8=0 :>;>==K @02=0
L = (N -1)l2 H" 5,4 :< . (5)
1.4 "0: :0: 02B><>18;8 B@>30NBAO G5@57 @02=K5 ?@><56CB:8 2@5<5=8 Ä , 8 =0E>4OBAO =0
@0AAB>O=88 l0 , B> A:>@>ABL 42865=8O «2>;=K @07@O65=8O @02=0
l0 <
V = - = -2,0 . (6)
Ä A
'0ABL 2. «AB0=>2:0
2.1 7>1@078< AE5<0B8G5A:8 7028A8<>AB8 A:>@>AB59 42CE
A>A54=8E 02B><>18;59 >B 2@5<5=8 2 ?@>F5AA5 8E
B>@<>65=8O. 0: 8 @0=55, ?;>I04L <564C 3@0D8:0<8 @02=0
87<5=5=8N @0AAB>O=8O <564C 02B><>18;O<8, B> 5ABL
"x = v0Ä . "0: :0: @0AAB>O=85 4>;6=> 87<5=8BLAO =0
25;8G8=C "l = l1 - l0 , B> 8=B5@20; 2@5<5=8 <564C =0G0;0<8
B>@<>65=89 4>;65= 1KBL @025=
"l
Ä = = 5,0c , (7)
2
v0
?@8G5< =5 7028A8<> >B CA:>@5=8O!
2.2 @5<5=0 <564C >AB0=>2:0<8 02B><>18;59 @02=K Ä , :064K9 A;54CNI89 02B><>18;L
2
>AB0=02;8205BAO =0 @0AAB>O=88 l0 = 10 < , ?>MB><C 2>;=0 A60B8O ?@>15305B A> A:>@>ABLN
l0 <
V = - = -2,0 . (8)
Ä A
2
6
10 :;0AA >48==04F0B8;5B=59 H:>;K.
040=85 10(11)-1. « 07<8=:0
1.1 C45< AG8B0BL, GB> ?@8 42865=88 ?5AG8=:8 =5 <5H0NB 4@C3 4@C3C. 0AA<>B@8< >4=C
?5AG8=:C, =0E>4OICNAO 2=CB@8 2@0I0NI59AO AD5@K. >A:>;L:C B@5=85 2 A8AB5<5 A;01>5,
B> ?@8<5<, GB> 42865=85 ?5AG8=:8 ?@>8AE>48B B>;L:> ?>4 459AB285<
r
r
A8;K BO65AB8 mg 8 A8;K @50:F88 >?>@K N (A<. @8A).
 MB>< A;CG05 >A=>2=>9 70:>= 48=0<8:8 ?@8<5B 284
r
r r
m a = mg + N (1)
CABL 2 CAB0=>282H5<AO ?>;>65=88 ?5AG8=:8 25:B>@ A8;K
r
@50:F88 N A>AB02;O5B C3>; Ä… A 25@B8:0;LN. 5AG8=:0, 2@0I0OAL
2<5AB5 A> AD5@>9, >?8AK205B >:@C6=>ABL, @048CA :>B>@>9 r =0945< 87
@8AC=:0
r = RsinÄ… .
@8 42865=88 ?> >:@C6=>AB8 55 F5=B@>AB@5<8B5;L=>5 CA:>@5=85 <>65B 1KBL
=0945=>, :0:
a = aF.A. = É2r = É2RsinÄ… . (2)
@>5F8@CO (1) =0 >A8 Ox 8 Oy ?>;CG05< A8AB5<C C@02=5=89
ma = N sinÄ…
Å„Å‚
òÅ‚mg = N cosÄ… ,
ół
7 ?5@2>3> C@02=5=8O ?>;CG5==>9 A8AB5<K A CG5B>< (2) =0945<
mÉ2RsinÄ… = N sinÄ… ,
>B:C40, A>:@0I0O =0 sinÄ… ( sinÄ… `" 0 ), ?>;CG8<
N = mÉ2R . (3)
>4AB02;OO (3) 2> 2B>@>5 C@02=5=85 A8AB5<K, =0945<
g
cosÄ… = . (4)
É2R
0: A;54C5B 87 (4) CAB0=>282H55AO ?>;>65=85 ?5AG8=:8
>4=>7=0G=> >?@545;O5BAO C3;>2>9 A:>@>ABLN É 2@0I5=8O AD5@K,
A;54>20B5;L=>, ?5AG8=:8 ?@8 2@0I5=88 AD5@K @0A?@545;OBAO
?@8<5@=> @02=><5@=> 8 >1@07CNB :>;LF>, 2848<>5 87 55 F5=B@0 AD5@K
?>4 C3;>< Ä… (A<. @8A).
-B> ?>;>65=85 ?5AG8=>: O2;O5BAO CAB>9G82K< ?@8 1>;LH8E
C3;>2KE A:>@>ABOE 2@0I5=8O, ?>A:>;L:C ?@8 ;N1KE <0;KE A<5I5=8OE
?5AG8=:0 AB@5<8BAO 2>AAB0=>28BL =0G0;L=>5 7=0G5=85 C3;0 Ä… .
Ä„
"0: ?@8 É " cosÄ… 0 , B.5. Ä… , 8 2A5 ?5AG8=:8 A>15@CBAO «=0 M:20B>@5
2
AD5@K. @8 MB>< @048CA ?5AG0=>3> :>;LF0 AB0=5B @02=K< @048CAC
AD5@K R .
@8 C<5=LH5=88 C3;>2>9 A:>@>AB8 2@0I5=8O AD5@K 4>
g
7=0G5=8O É" = :>A8=CA C3;0 ?@8=8<05B A2>5 <0:A8<0;L=>5
R
7=0G5=85, @02=>5 1.
!>>B25BAB25==>, ?@8 É < É " 2A5 ?5AG8=:8 A>15@CBAO =0 4=5
AD5@K (Ä… = 0 ), >1@07CO =51>;LHCN :CG:C (A<.@8A).
-B> B0:65 1C45B CAB>9G82>5 ?>;>65=85 @02=>25A8O ?5AG8=>: =0 4=5 AD5@K ?@8
<0;KE C3;>2KE A:>@>ABOE 55 2@0I5=8O.
7
r r
1.2 >A:8 8 F8;8=4@ =0E>4OBAO 2 @02=>25A88 ?>4 459AB285< A8; BO65AB8 ( Mg,mg ),
r r r r
B@5=8O ( FB@, - FB@ ) 8 @50:F89 ( N, - N ), 87>1@065==KE =0
@8AC=:0E. !8;K @50:F88 2 >A8  , =0 @8AC=:5 =5 ?>:070=K,
?>A:>;L:C 8E <><5=BK >B=>A8B5;L=> B>G:8  @02=K =C;N.
0?8H5< 2B>@>5 CA;>285 @02=>25A8O (?@028;> <><5=B>2)
4;O >4=>9 87 4>A>: >B=>A8B5;L=> B>G:8 
L L
mg sinÄ… = N . (1)
2 2
r
><5=B A8;K B@5=8O FB@ B0:65 =5 2E>48B 2 (1), ?>A:>;L:C
;8=8O 55 459AB28O ?@>E>48B G5@57 >AL 2@0I5=8O.
>A:>;L:C F8;8=4@ B0:65 =0E>48BAO 2 @02=>25A88, B> AC<<0 A8;, 459AB2CNI8E =0
=53> 4>;6=0 1KBL @02=0 =C;N (@8A. 01). !>>B25BAB25==>, ?5@2>5 CA;>285 @02=>25A8O,
70?8A0==>5 4;O F8;8=4@0 2 ?@>5:F88 =0 25@B8:0;L=CN >AL, 8<55B 284
Mg + 2N sinÄ… = 2FB@ cosÄ… . (2)
@8 70?8A8 (2) <K CG;8, GB> A8;K B@5=8O 8 @50:F88, 459AB2CNI85 =0 F8;8=4@ A>
AB>@>=K 4>A>:, A>3;0A=> III 70:>=C LNB>=0 @02=K ?> <>4C;N 8
?@>B82>?>;>6=K ?> =0?@02;5=8N A>>B25BAB2CNI8< A8;0<,
459AB2CNI8< =0 4>A:8 A> AB>@>=K F8;8=4@0.
7 C@02=5=89 (1)  (2) ?>;CG05<
N = mg sinÄ… ,
Mg + 2mg sin2 Ä…
FB@ = .
2cosÄ…
>A:>;L:C ?@545;L=>5 7=0G5=85 A8;K B@5=8O ?>:>O
>?@545;O5BAO 7=0G5=85< A8;K B@5=8O A:>;L65=8O (O2;5=85< 70AB>O ?@5=51@565<), B>
FB@ d" µN . (3)
0: A;54C5B 87 (3), A8AB5<0 A<>65B =0E>48BLAO 2 @02=>25A88 B>;L:> ?@8
2K?>;=5=88 CA;>28O
FB@
µ e" . (4)
N
7 @8AC=:0 =5A;>6=> 2K@078BL, GB>
2R L
sinÄ… = , cosÄ… = .
4R2 + L2 4R2 + L2
7 (4) A CG5B>< 2K@065=89 4;O A8;K B@5=8O 8 @50:F88, 0 B0:65 7=0G5=89 A8=CA0 8
:>A8=CA0, ?>;CG05< >:>=G0B5;L=K9 >B25B
(L2 + 4R2)M 2R
µ e" + .
4mRL L
0AG5B 405B
µ e" 1,0 .
>;CG5==>5 G8A;5==>5 7=0G5=85 :>MDD8F85=B0 B@5=8O 4>AB0B>G=> 1>;LH>5, >4=0:>
A;54C5B 70<5B8BL, GB> A>2@5<5==K5 A8=B5B8G5A:85 <0B5@80;K <>3CB >1;040BL 4065
1Ì;LH8<8 :>MDD8F85=B0<8 B@5=8O.
8
1.3 ;O 2KG8A;5=8O <>4C;O 25:B>@0 8=4C:F88 <03=8B=>3> ?>;O =0 >A8 :>;LF0 A B>:><
8A?>;L7C5< 70:>= 8>-!020@0-0?;0A0, A>3;0A=> :>B>@><C
r
<0;K9 M;5<5=B B>:0 (I"l ) A>7405B =0 >A8 :>;LF0 8=4C:F8N
k
µ0 (I"l)
k
"Bk = .
4Ä„ rk2
r
r
5:B>@ "Bk =>@<0;5= 25:B>@C rk , ?@>2545==><C >B M;5<5=B0
B>:0 2 B>G:C @0AA<>B@5=8O A (A<.@8A).
r
>A:>;L:C 2A5 M;5<5=BK B>:0 I"lk =0E>4OBAO =0
>48=0:>2>< @0AAB>O=88 rk = R2 + z2 >B F5=B@0 :>;LF0 8
A>AB02;ONB A 25@B8:0;LN >48=0:>2K9 C3>; Õ , B>
AC<<8@>20=85 2K?>;=8BLAO =5A;>6=>
µ0 (I"l) µ0 I cosÕ µ0 I cosÕ µ0IR
k
B = = cosÕ = = 2Ä„R = cosÕ .
""Bzk " ""lk
2 2 2 2
4Ä„ r 4Ä„ r 4Ä„ r 2r
k k k
R
7 @8AC=:0 A;54C5B, GB> cosÕ = , A CG5B>< 2K@065=8O 4;O @0AAB>O=8O r , ?>;CG8<
r
>:>=G0B5;L=>5 2K@065=85 4;O <>4C;O 25:B>@0 8=4C:F88 <03=8B=>3> ?>;O =0 @0AAB>O=88
z >B F5=B@0 :>;LF0.
µ0IR µ0IR2 µ0I R2
B(z) = cosÕ = = Å" . (1)
3
2r2 2r3 2
2
(R2 + z2)
0 1>;LH8E @0AAB>O=8OE z ( z >> R ) >B :>;LF0 A;0305<K< R2 2 7=0<5=0B5;5 (1)
<>6=> ?@5=51@5GL ?> A@02=5=8N A> A;0305<K< z2 .  @57C;LB0B5 ?>;CG8<
µ0I R2 µ0I R2 µ0I R2 a
B(z) = Å" = {z "}H" Å" = Å" = . (2)
3 3
2 2 2 z3 z3
2 2
(R2 + z2) (z2)
!>>B25BAB25==>, 2K@065=85 4;O 8A:><>3> :>MDD8F85=B0 a ?@8=8<05B 284
µ0I R2
a = . (3)
2
0: A;54C5B 87 (2) ?@8 ?045=88 :>;5G:0 8 ?@81;865=88 53> : :>;LFC <>4C;L
<03=8B=>9 8=4C:F88 ?>;O 2>7@0AB05B, B.5. <03=8B=K9 ?>B>: G5@57 :>;5G:> C25;8G8205BAO.
@8@0I5=85 <03=8B=>3> ?>B>:0 G5@57 :>;5G:> ?@8 A<5I5=88 53> =0 25;8G8=C "z
2
"$ = -Ä„ r2B (z)"z .
8DD5@5=F8@CO (2) ?> z , ?>;CG8<
3a
2
B (z) = - .
z4
!>3;0A=> 70:>=C M;5:B@><03=8B=>9 8=4C:F88 $0@045O 2 :>;5G:5 2>7=8:=5B -!
8=4C:F88
"$ 3Ä„ r2a "z 3Ä„ r2a
µi = - = - Å" = Å"Å (z) . (4)
"t z4 "t z4
!>>B25BAB25==>, A8;0 8=4C:F8>==>3> (=02545==>3>) B>:0 2 :>;5G:5 >:07K205BAO
?@>?>@F8>=0;L=>9 53> A:>@>AB8 42865=8O
µi 3Ä„ r2a
Ii = = Å"Å (z) . (5)
R0 R0 z4
9
 >A:>;L:C 2 :>;5G:5 ?>O28;AO 8=4C:F8>==K9 B>:, B> =0 =53> A> AB>@>=K
<03=8B=>3> ?>;O :>;LF0 1C4CB 459AB2>20BL A8;K <?5@0.
45AL A;54C5B 70<5B8BL, GB> 25@B8:0;L=0O (>A520O) A>AB02;ONI0O <03=8B=>3>
r r
?>;O B(z) A>7405B M;5<5=B0@=K5 A8;K <?5@0 FAi , :>B>@K5
;8HL 45D>@<8@CNB (A68<0NB 8;8 @0ABO3820NB) :>;5G:>, =>
=5 <>3CB 70<54;8BL 8;8 CA:>@8BL 53> 42865=8O, ?>A:>;L:C =5
8<5NB 25@B8:0;L=>9 ?@>5:F88.
r
 2>B @0480;L=K5 A>AB02;ONI85 Bri 8=4C:F88 <03=8B=>3>
?>;O, 2>7=8:0NI85 2A;54AB285 @0AA5O=8O ;8=89 <03=8B=>9
8=4C:F88 >B >A8 Oz , 2 A8;C ?@028;0 5=F0 ?@8B>@<06820NB
:>;5G:>, ?>A:>;L:C 8<5NB 25@B8:0;L=K5 ?@>5:F88 (A<. @8A).
;O =0E>645=8O <>4C;O Br 21;878 >A8 Oz ?@8<5=8<
B5>@5<C 0CAA0 4;O <03=8B=>3> ?>;O, :>B>@0O CB25@6405B, GB>
?>B>: 25:B>@0 <03=8B=>9 8=4C:F88 ?> ?@>872>;L=>9
70<:=CB>9 ?>25@E=>AB8 @025= =C;N.
0AA<>B@8< F8;8=4@ 2KA>B>9 "z , @048CA :>B>@>3> A>2?0405B A @048CA>< :>;5G:0
(A<. @8A). C45< AG8B0BL, GB> 2 A8;C <0;>AB8 @048CA0 r :>;5G:0 >A520O A>AB02;ONI0O
B (z) <03=8B=>3> ?>;O 2 53> ?@545;0E <5=O5BAO =57=0G8B5;L=>. ">340 @07=>ABL <03=8B=KE
?>B>:>2 G5@57 =86=55 8 25@E=55 >A=>20=8O F8;8=4@0
2
"$ = Ä„ r2B (z)"z (6)
2 A8;C B5>@5<K 0CAA0 4>;6=0 1KBL @02=0 ?>B>:C 25:B>@0
8=4C:F88 <03=8B=>3> ?>;O G5@57 1>:>2CN ?>25@E=>ABL
F8;8=4@0
$1>: = Br 2Ä„ r"z . (7)
025=AB20 (6)-(7) 8<5NB =03;O4=K9 35><5B@8G5A:89
A<KA;: =5@07@K2=K5 ;8=88 <03=8B=>9 8=4C:F88, 2>H54H85 2 F8;8=4@ G5@57 =86=55
>A=>20=85, <>3CB 2K9B8 87 =53> 8;8 G5@57 25@E=55 >A=>20=85, 8;8 G5@57 53> 1>:>2CN
?>25@E=>ABL.
@8@02=820O (5) 8 (6) ?>;CG8<
2
B (z) 3 ar
Br = - r = Å" . (8)
2 2 z4
!>>B25BAB25==>, 4;O AC<<0@=>9 :><?>=5=BK A8;K <?5@0, B>@<>7OI59 ?045=85
:>;LF0, ?>;CG8< 2K@065=85
9Ä„ r3a2
FA = Ii Br 2Ä„r = {(5),(8)}= Å (z) . (9)
2R0 z8
>A:>;L:C, A>3;0A=> CA;>28N, CA:>@5=85 :>;LF0 <0;> (3>@074> <5=LH5 CA:>@5=8O
A2>1>4=>3> ?045=8O), B> <>6=> AG8B0BL, GB> :>;LF> 2 ;N1>9 <><5=B 2@5<5=8 42865BAO
@02=><5@=>. >4>1=K5 ?@>F5AAK =07K20NBAO :2078AB0F8>=0@=K<8.
;O ?>4>1=KE ?@>F5AA>2 E0@0:B5@=>5 2@5<O CAB0=>2;5=8O @02=>25A=>3> A>AB>O=8O
A8AB5<K (2@5<O @5;0:A0F88) 4>;6=> 1KBL 4>AB0B>G=> <0;>.
 @0<:0E 40==>9 <>45;8 <>65< 70?8A0BL
9Ä„ r3a2
mg = FA = Å (z) .
2R0 z8
7 ?>;CG5==>3> C@02=5=8O =0945< 8A:><CN 7028A8<>ABL A:>@>AB8
CAB0=>282H53>AO ?045=8O :>;LF0 =0 2KA>B5 z
2mgR0 z8 8mgR0
Å (z) = = Å" z8 . (10)
2 2
9Ä„ r3a2 9Ä„r3µ0 I R4
10
040=85 10(11)-2. «@028B0F8>==K9 48?>;L
@8=F8?80;L=>5 @07;8G85 <564C ?>2545=85< M;5:B@8G5A:>3> 8 3@028B0F8>==>3>
48?>;O 70:;NG05BAO 2 B><, GB> A8;K, 459AB2CNI85 =0 H0@8:8 3@028B0F8>==>3> 48?>;O
=0?@02;5=K 2 >4=C AB>@>=C, 0 4;O M;5:B@8G5A:>3> - 2 ?@>B82>?>;>6=K5. > 2A5E G0ABOE
l
7040G8 «?@>A<0B@8205BAO <0;K9 157@07<5@=K9 ?0@0<5B@ , ?>MB><C A;54C5B
R
2=8<0B5;L=> A;548BL 70 ?@028;L=K< 8A?>;L7>20=85< ?@81;865==KE D>@<C;.
3.1- >4C;L A8;K ?@8BO65=8O @025=
@07=>AB8 <>4C;59 A8;, 459AB2CNI8E =0
:064K9 87 H0@8:>2.  A2>N >G5@54L, MB8
A8;K >?@545;ONBAO 70:>=>< C;>=0
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
qQ qQ qQ ìÅ‚ 1 1 ÷Å‚
F = F1 - F2 = - = - =
2 2
4Ä„µ R2 ìÅ‚ ëÅ‚1- l öÅ‚2 ëÅ‚1+ l öÅ‚2 ÷Å‚
l l
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚ 0
ìÅ‚ ÷Å‚
4Ä„µ R - ÷Å‚ ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
4Ä„µ R +
ìÅ‚
0 0
ìÅ‚ ÷Å‚
2 2 2R 2R .
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
qQ l 2
=
2
2
4Ä„µ R2 R
0 ëÅ‚ öÅ‚
l
ìÅ‚
ìÅ‚1- 4R2 ÷Å‚
÷Å‚
íÅ‚ Å‚Å‚
 40==>< 2K@065=88 A;54C5B ?@5=51@5GL A;0305<K<8 2B>@>3> ?>@O4:0 <0;>AB8. >MB><C
>:>=G0B5;L=0O D>@<C;0 4;O A8;K ?@8BO65=8O 48?>;O 8<55B 284
Qql Qq 2l
F = = Å" . (1)
2Ä„µ R3 4Ä„µ R2 R
0 0
@8<5G0=85. -B>B @57C;LB0B <>65B 1KBL ?>;CG5= 87 8725AB=>9 D>@<C;K 4;O A8;K, 459AB2CNI59
=0 48?>;L,
dEx
Fx = p ,
dx
345 p = ql -48?>;L=K9 <><5=B 48?>;O.
3.1 ;O 3@028B0F8>==>3> 48?>;O <>4C;8 A8;
=5>1E>48<> ?@>AC<<8@>20BL
2
ëÅ‚ öÅ‚
l
ìÅ‚
2ìÅ‚1+ ÷Å‚
4R2 ÷Å‚
mm0 mm0 mm0 íÅ‚
Å‚Å‚
F = F1 + F2 = G + G = G .
2 2
R2 ëÅ‚ l 2 öÅ‚2
l l
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
R R +
ìÅ‚ - ÷Å‚ ìÅ‚ ÷Å‚ ìÅ‚
ìÅ‚1- 4R2 ÷Å‚
÷Å‚
2 2
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
 MB>< 2K@065=88 B0:65 A;54C5B ?@5=51@5GL <0;K<8 25;8G8=0<8, ?>MB><C AC<<0@=0O
A8;0, 459AB2CNI0O =0 «3@028B0F8>==K9 48?>;L @02=0
2mm0
F = G . (2)
R2
-B>B @57C;LB0B >G52845=, B0: :0: ?@8 2K?>;=5=88 CA;>28O R >> l , «48?>;L <>65B
@0AA<0B@820BLAO :0: <0B5@80;L=0O B>G:0.
11
3.2- @8 ?>2>@>B5 48?>;O =0 <0;K9 C3>; Ä… 2>7=8:05B <><5=B A8;, @02=K9
l
M = (F1 + F2 ) sinÄ… .
2
8 2>72@0I0NI89 53> 2 8AE>4=>5
A>AB>O=85. 0<5B8<, GB> 2 40==><
2K@065=88 D83C@8@C5B AC<<0 <>4C;59 A8;, ?>MB><C 8E <>6=> AG8B0BL >48=0:>2K<8 8
@02=K<8
Qq
F = .
4Ä„µ R2
0
"0: :0: :>;510=8O O2;ONBAO <0;K<8, B> <>6=> AG8B0BL, GB> sinÄ… H" Ä… . ;O >?8A0=8O
42865=8O 48?>;O <>6=> 8A?>;L7>20BL C@02=5=85 48=0<8:8 2@0I0B5;L=>3> 42865=8O (0
<>6=> 8A?>;L7>20BL 8 4@C385 ?>4E>4K). 7 MB>3> C@02=5=8O A;54C5B C@02=5=85
30@<>=8G5A:8E :>;510=89
2
l Qql Qq
2mëÅ‚ öÅ‚ ² = - Ä… Ò! ² = - Ä… , (1)
ìÅ‚ ÷Å‚
2 4Ä„µ0R2 2Ä„µ0R2ml
íÅ‚ Å‚Å‚
87 :>B>@>3> =0E>48< ?5@8>4 :>;510=89
2Ä„µ0R2ml
T = 2Ä„ . (2)
Qq
3.2  MB>< A;CG05 A8;K =0?@02;5=K 2 >4=C
AB>@>=C, ?>MB><C 8E <><5=BK ?@>B82>?>;>6=K.
!C<<0@=K9 <><5=B A8;, 2>72@0I0NI89 53> 2
8AE>4=>5 A>AB>O=85, >?@545;O5BAO @07=>ABLN
<>4C;59 A8;, ?>MB><C O2;O5BAO <0;>9 25;8G8=>9
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
l Gmm0 ìÅ‚ 1 1 ÷Å‚ l Gmm0 l
M = -(F1 - F2 ) sinÄ… = - - sinÄ… H" - Å" Å" lÄ… .
2 R2 ìÅ‚ ëÅ‚1- l öÅ‚2 ëÅ‚1+ l öÅ‚2 ÷Å‚ 2 R2 R
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
2R 2R
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
!>>B25BAB2CNI55 C@02=5=85 48=0<8:8 2@0I0B5;L=>3> 42865=8O, B0:65 O2;O5BAO
C@02=5=85< 30@<>=8G5A:8E :>;510=89
2
2
l Gmm0l 2Gm0
2mëÅ‚ öÅ‚ ² = - Ä… Ò! ² = - Ä… .
ìÅ‚ ÷Å‚
2 R3 R3
íÅ‚ Å‚Å‚
5@8>4 <0;KE :>;510=89 2 MB>< A;CG05 @025=
3
R
T = 2Ä„ . (3)
2Gm0
=B5@5A=> 8 =5>6840==>  MB>B ?5@8>4 =5 7028A8B >B D878G5A:8E E0@0:B5@8AB8: 48?>;O!
# ?>25@E=>AB8 5<;8 CA:>@5=85 A2>1>4=>3> ?045=8O @02=>
m0
g = G .
R2
!;54>20B5;L=>, ?5@8>4 @0AA<0B@8205<KE :>;510=89 @025=
R3 R 6,35Å"106
T = 2Ä„ = 2Ä„ H" 2Ä„ = 3,6 Å"103 c H" 1G0A .
2Gm0 2g 2 Å" 9,8
12
3.3- 'B>1K >AL 48?>;O 1K;0 =0?@02;5=0 2A5
2@5<O =0 F5=B@ 70@O65==>3> H0@0 =5>1E>48<>,
GB>1K >10 H0@8:0 48?>;O 2@0I0;8AL A
>48=0:>2K<8 C3;>2K<8 A:>@>ABO<8. >GB8
>G5284=>, GB> AB5@65=L @0ABO=CB, ?>MB><C C@02=5=8O 2B>@>3> 70:>=0 LNB>=0 2 ?@>5:F88
=0 @0480;L=>5 =0?@02;5=85, 70?8AK20NBAO 2 2845
l Qq 1
öÅ‚
2
mÉ RëÅ‚1- ÷Å‚
= Å" - N
ìÅ‚
2R 4Ä„µ R2 ëÅ‚1- l öÅ‚2
íÅ‚ Å‚Å‚
0
ìÅ‚ ÷Å‚
2R
íÅ‚ Å‚Å‚
. (4)
l Qq 1
öÅ‚
2
mÉ RëÅ‚1+ = - + N
ìÅ‚ ÷Å‚
2R 4Ä„µ R2 ëÅ‚1+ l öÅ‚2
íÅ‚ Å‚Å‚
0
ìÅ‚ ÷Å‚
2R
íÅ‚ Å‚Å‚
!:;04K20O MB8 C@02=5=8O, 2848<, GB> F5=B@>AB@5<8B5;L=>5 CA:>@5=85 O2;O5BAO <0;>9
25;8G8=>9 ?5@2>3> ?>@O4:0
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
Qq ìÅ‚ 1 1 ÷Å‚ 2Qq l
2
2mÉ R = - = .
4Ä„µ0R2 ìÅ‚ ëÅ‚1- l öÅ‚2 ëÅ‚1- l öÅ‚2 ÷Å‚ 4Ä„µ0R2 R
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
2R 2R
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
5@8>4 2@0I5=8O ;53:> 2K@078BL G5@57 C3;>2CN A:>@>ABL
2Ä„ 4Ä„µ0R4
T = = 2Ä„ . (5)
É mQql
!;>68< B5?5@L C@02=5=8O A8AB5<K (4) 4;O @0AG5B0 A8;K @50:F88 AB5@6=O
2
Qq Qq Qq l 2Qq
ëÅ‚ öÅ‚
2
2N = mÉ l + + = +
ìÅ‚ ÷Å‚
2 2
4Ä„µ0R2 R 4Ä„µ0R2
íÅ‚ Å‚Å‚
l l
öÅ‚ öÅ‚
4Ä„µ0 ëÅ‚ R + 4Ä„µ0 ëÅ‚ R - ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚
2 2
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
848<, GB> A8;0 =0BO65=8O AB5@6=O >?@545;O5BAO :C;>=>2A:8<8 A8;0<8 (F5=B@>156=K5
A8;K  <0;K)
Qq
N = . (6)
4Ä„µ R2
0
3.3  MB>< A;CG05 C@02=5=8O 2B>@>3> 70:>=0
LNB>=0 4>;6=K 1KBL 70?8A0=K A;54CNI8<
>1@07><
ëÅ‚1- l mm0 1
öÅ‚
2
mÉ R = G - N
ìÅ‚ ÷Å‚
2
2R R2
íÅ‚ Å‚Å‚
ëÅ‚1- l
öÅ‚
ìÅ‚ ÷Å‚
2R
íÅ‚ Å‚Å‚
(7) .
l mm0 1
ëÅ‚1+ öÅ‚
2
mÉ R = G + N
ìÅ‚ ÷Å‚
2
2R R2
íÅ‚ Å‚Å‚
ëÅ‚1- l
öÅ‚
ìÅ‚ ÷Å‚
2R
íÅ‚ Å‚Å‚
;O >?@545;5=8O F5=B@>AB@5<8B5;L=>3> CA:>@5=8O 8 ?5@8>40 >1@0I5=8O A;>68< MB8
C@02=5=8O
13
2
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
l
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
2ìÅ‚1+
4R2 ÷Å‚
mm0 ìÅ‚ 1 1 ÷Å‚ mm0 íÅ‚
2 Å‚Å‚
2mÉ R = G + = G . (8)
R2 ìÅ‚ ëÅ‚1+ l öÅ‚2 ëÅ‚1+ l öÅ‚2 ÷Å‚ R2 ëÅ‚ l 2 öÅ‚2
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚ ìÅ‚
ìÅ‚ ÷Å‚
ìÅ‚1- 4R2 ÷Å‚
÷Å‚
2R 2R
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
@5=51@530O <0;K<8 25;8G8=0<8, =0E>48< ?5@8>4 >1@0I5=8O
mm0 R3
2
mÉ R = G Ò! T = 2Ä„ , (10)
R2 Gm0
:>B>@K9 A>2?0405B A ?5@8>4>< >1@0I5=8O <0B5@80;L=>9 B>G:8.
;O @0AG5B0 A8;K =0BO65=8O AB5@6=O 70?8H5< @07=>ABL C@02=5=89 (8):
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
mm0 ìÅ‚ 1 1 ÷Å‚
2
2N = mÉ l + G - .
R2 ìÅ‚ ëÅ‚1- l öÅ‚2 ëÅ‚1+ l öÅ‚2 ÷Å‚
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
2R 2R
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
7 MB>3> 2K@065=8O A;54C5B, GB> 8 F5=B@>156=K5 A8;K (A CG5B>< (10)) 8 @07=>ABL
3@028B0F8>==KE A8; 8<5NB >48=0:>2K9 ?5@2K9 ?>@O4>: <0;>AB8, ?>MB><C 4>;6=K 1KBL
CGB5=K
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
1 l 1 mm0 ìÅ‚ 1 1 ÷Å‚
2
N = mÉ R + G - =
2 R 2 R2 ìÅ‚ ëÅ‚1- l öÅ‚2 ëÅ‚1+ l öÅ‚2 ÷Å‚
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
2R 2R
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
íÅ‚ Å‚Å‚
1 mm0 l 1 mm0 2l 3 mm0 l
= G + G = G
2 R2 R 2 R2 R 2 R2 R
040=85 10(11)-3. « 07@O4:0 :>=45=A0B>@0
q0
1. >A:>;L:C =0G0;L=>5 =0?@O65=85 =0 :>=45=A0B>@5 U0 = , B>
C
A>3;0A=> 70:>=C <0 A@07C ?>A;5 70<K:0=8O :;NG0  2 F5?8
U0 q0
2>7=8:=5B B>: A8;>9 I0 = = .
R RC
> <5@5 @07@O4:8 :>=45=A0B>@0 =0?@O65=85 =0 =5<, 0 A;54>20B5;L=> 8 A8;0 B>:0 2 F5?8,
1C4CB <>=>B>==> C1K20BL A B5G5=85< 2@5<5=8.
!B@>3>5 <0B5<0B8G5A:>5 @5H5=85 40==>9 7040G8 ?@82>48B : =5>6840==><C @57C;LB0BC 
?>;=0O @07@O4:0 :>=45=A0B>@0 ?@>8AE>48B «15A:>=5G=> 4>;3>,
?>A:>;L:C 3@0D8: 2@5<5==>9 7028A8<>AB8 70@O40 :>=45=A0B>@0
q(t) 0A8<?B>B8G5A:8 AB@5<8BAO : =C;N (A<. @8A).
;O >F5=:8 2@5<5=8 @07@O4:8 :>=45=A0B>@0 8A?>;L7C5<
B@048F8>==K9 ?@85<: 1C45< AG8B0BL, GB> A8;0 B>:0 2 F5?8
q0
«A>E@0=O5B A2>5 <0:A8<0;L=>5 7=0G5=85 I0 = 2 B5G5=85 2A53>
RC
2@5<5=8 @07@O4:8 :>=45=A0B>@0.  B0:>< A;CG05 4;O ?>;=>9 @07@O4:8 :>=45=A0B>@0
?>B@51C5BAO 2@5<O
q0
Ä = = RC . (1)
I0
@0D8G5A:8 40==0O >F5=:0 A>>B25BAB2C5B B>G:5 ?5@5A5G5=8O :0A0B5;L=>9 : 3@0D8:C q(t) ,
?@>2545==>9 2 =0G0;L=>9 B>G:5, A >ALN 01AF8AA.
14
2. A;8 A8;0 B>:0 2 F5?8 >AB05BAO ?>AB>O==>9 8 @02=>9 I , B> 70@O4 :>=45=A0B>@0
;8=59=> C1K205B A> 2@5<5=5< ?> 70:>=C
q(t) = q0 - I Å" t .
0?@O65=85 =0 :>=45=A0B>@5, ?@>?>@F8>=0;L=>5 53>
<3=>25==><C 70@O4C q(t) , 4>;6=> 1KBL @02=> ?045=8N =0?@O65=8O
=0 @578AB>@5, A>?@>B82;5=85 :>B>@>3> <5=O5BAO A> 2@5<5=5< (A<.
@8A). !;54>20B5;L=>, 2 40==>< A;CG05 A?@0254;82> @025=AB2>
q(t) q0 - I Å" t q0 - I Å" t
= = I Å" R(t) Ò! R(t) = . (2)
C C IC
!>3;0A=> (2) 4;O B>3>, GB>1K A8;0 B>:0 2 F5?8 ?@8 @07@O4:5 :>=45=A0B>@0
>AB020;0AL ?>AB>O==>9, A>?@>B82;5=85 @5>AB0B0 =C6=>
C<5=LH0BL A> 2@5<5=5< ?> ;8=59=><C 70:>=C (A<. @8A).
q0
>A:>;L:C 2 =0G0;L=K9 <><5=B 2@5<5=8 I = , B> 8A:><CN
R0C
7028A8<>ABL R(t) <>6=> ?@54AB028BL 2 2845
t
R(t) = R0(1- ) ,
Ä
345 Ä = R0C .
0: =8 AB@0==>, => 2@5<O Ä ?>;=>9 @07@O4:8 :>=45=A0B>@0 ?@8 ?>AB>O==>9 A8;5
q0
B>:0 (Ä = ) B0:65 @02=>
I
Ä = R0C .
>;8G5AB2> B5?;>BK, 2K45;82H55AO =0 @578AB>@5 70 2A5 2@5<O @07@O4:8, A>3;0A=>
70:>=C A>E@0=5=8O M=5@388 @02=> =0G0;L=>9 M=5@388 :>=45=A0B>@0
2
q0
Q = .
2C
0<5B8<, GB> MB>B 65 @57C;LB0B <>6=> ?>;CG8BL 8 8=K< A?>A>1>< 
?@>AC<<8@>20BL 8 CA@54=8BL <>I=>ABL B5?;>2K45;5=8O 6>C;O-5=F0.
3. @8 ?>2>@>B5 >4=>9 87 ?;0AB8= =0 C3>; Ä… >B=>A8B5;L=> 4@C3>9 ?;0AB8=K ?;>I04L
?5@5:@KB8O ?;0AB8= C<5=LH8;0AL =0 25;8G8=C ?;>I048 A5:B>@0 C3;>2>9
25;8G8=>9 Ä… (A<. @8A)
Ä…r2 Ä…
S = S0 - = S0(1- ) .
2 Ä„
!>>B25BAB25==>, M;5:B@>5<:>ABL =>2>3> ?;>A:>3> :>=45=A0B>@0
AB0;0 @02=>9
µ0S Ä…
! = = C0(1- ) . (3)
d Ä„
?OBL 65, 5A;8 A8;0 B>:0 2 F5?8 >AB05BAO ?>AB>O==>9 8 @02=>9 I , B>
70@O4 :>=45=A0B>@0 ;8=59=> C1K205B A> 2@5<5=5< ?> 70:>=C
q(t) = q0 - I Å" t .
 A8;C @025=AB20 <3=>25==KE =0?@O65=89 =0 @578AB>@5 8 :>=45=A0B>@5 2 ;N1>9 <><5=B
2@5<5=8 <>65< 70?8A0BL
q(t) q0 - I Å" t q0 - I Å" t
= = I Å" R Ò! C(t) = (4)
C(t) C(t) IR
!>3;0A=> (4) 4;O B>3>, GB>1K A8;0 B>:0 2 40==>9 F5?8 ?@8 ?>2>@>B5 ?;0AB8=K >AB020;0AL
?>AB>O==>9, M;5:B@>5<:>ABL :>=45=A0B>@0 =C6=> C<5=LH0BL A> 2@5<5=5< ?> ;8=59=><C
15
70:>=C (A<. 3@0D8: ?@54K4CI53> ?C=:B0). -B> 7=0G8B, GB> 2@0I0BL >48= 87 48A:>2 =C6=> A
Ä… Ä„ Ä„
?>AB>O==>9 C3;>2>9 A:>@>ABLN É = = = .
t Ä RC0
q0
>A:>;L:C 2 =0G0;L=K9 <><5=B 2@5<5=8 I = , B> @025=AB2> (3) <>6=> ?5@5?8A0BL 2
C0R
2845
t
!(t) = !0(1- ) , (5)
Ä
q0
345 Ä = RC0 . @8 MB>< 2@5<O @07@O4:8 :>=45=A0B>@0 Ä = ?>-?@56=5<C (C65 <>6=>
I
A:070BL «B@048F8>==>) @02=>
Ä = RC0 .
>A:>;L:C A8;0 B>:0 2 F5?8 >AB05BAO ?>AB>O==>9, B> 2 40==>< A;CG05 2K45;8BAO
:>;8G5AB2> B5?;>BK
2
2
ëÅ‚ öÅ‚
q0 q0
2
ìÅ‚ ÷Å‚
Q = I RÄ = R(RC0) = . (6)
ìÅ‚ ÷Å‚
C0R C0
íÅ‚ Å‚Å‚
0: A;54C5B 87 (6) :>;8G5AB2> 2K45;5==>9 =0 @578AB>@5 B5?;>BK 2 420 @070
?@52KH05B :>;8G5AB2> M=5@388, 70?0A5==>9 2 :>=45=A0B>@5. -B> «?@>B82>@5G85
>1JOA=O5BAO B5<, GB> ?@8 ?>2>@>B5 ?;0AB8=K 2=5H=85 A8;K A>25@H8;8 =04 A8AB5<>9
2
q0
?>;>68B5;L=CN @01>BC 2= = , GB> 8 ?@825;> : C25;8G5=8N
2C0
M=5@388 A8AB5<K 2 :>=5G=>< A>AB>O=88.
4. @8 C25;8G5=88 @0AAB>O=8O x <564C ?;0AB8=0<8 (A<. @8A)
M;5:B@>5<:>ABL :>=45=A0B>@0 C<5=LH05BAO ?> 70:>=C
µ0S µ0S x0 x0
C = = Å" = C0 Å" .
x x0 x x
0?8AK20O CA;>285 @025=AB20 =0?@O65=89 =0 @578AB>@5 8 :>=45=A0B>@5, ?>;CG8<
q(t) q0 - I Å" t x0
= Å" x(t) = I Å" R Ò! x(t) = . (7)
t
C(t) !0x0
1-
Ä
q0
@8 70?8A8 (7) >?OBL 65 CGB5=>, GB> I = , Ä = RC0 .
C0R
0: A;54C5B 87 ?>;CG5==>3> @025=AB20, 2 40==>< A;CG05 42865=85
>1:;04:8 1C45B =5@02=><5@=K<, ?>A:>;L:C ?@8 t Ä DC=:F8O (7)
=5>3@0=8G5==> 2>7@0AB05B (A<. @8A).
 40==>< A;CG05 :>=45=A0B>@ ?>;=>ABLN @07@O48BAO ?@8 x " , GB>
?@>87>945B G5@57 :>=5G=>5 2@5<O, «=587<5==> @02=>5
t = Ä = RC0 .
0AG5B 2K45;82H53>AO :>;8G5AB20 B5?;>BK Q ?@8 @0742830=88 ?;0AB8= ?>;=>ABLN
0=0;>38G5= ?@54K4CI5<C ?C=:BC
2
q0
Q = .
C0
=5H=85 A8;K ?@8 MB>< >?OBL 65 A>25@H0B @01>BC ?@>B82 A8; :C;>=>2A:>3> ?@8BO65=8O
?;0AB8=, GB> ?>72>;8B 2K45;8BLAO =0 @578AB>@5 :>;8G5AB2C B5?;>BK, 2 420 @070
?@52KH0NI5<C =0G0;L=CN M=5@38N :>=45=A0B>@0.
16
11 :;0AA >48==04F0B8;5B=59 H:>;K.
040=85 11(11)-1. « 07<8=:0
1.1 0?8H5< C@02=5=8O 48=0<8:8 4;O
:064>3> F8;8=4@0 2 ?@>5:F88 =0 >AL
AB5@6=O (MB0 A8AB5<0 =5 8=5@F80;L=0O)
2
m1a1 = m1É x1 - N
. (1)
2
m2a2 = m2É x2 - N
"0: :0: 4;8=0 =8B8 =5 87<5=O5BAO, B> x2 = (l - x1), ?>MB><C <>6=> 70?8A0BL
2
m1a1 = m1É x1 - N
.
2
- m2a1 = m2É (l - x1)- N
KG8B0O 87 ?5@2>3> C@02=5=8O 2B>@>5, ?>;CG8< C@02=5=85, >?8AK20NI55 42865=85
>4=>3> F8;8=4@0
2 2
(m1 + m2 )a1 = (m1 + m2 )É x1 - m2É l . (2)
>>@48=0B0 ?>;>65=8O @02=>25A8O >?8AK205BAO D>@<C;>9
m2
a1 = 0 Ò! x1 = l . (3)
m1 + m2
5 A;>6=> ?>:070BL, GB> MB> ?>;>65=85 @02=>25A8O O2;O5BAO =5CAB>9G82K<: 87 C@02=5=8O
(2) A;54C5B, GB> ?@8 x1 > x1 CA:>@5=85 MB>3> F8;8=4@0 a1 > 0 , ?>MB><C >= 1C45B A<5I0BLAO
40;LH5; 8, =0>1>@>B, ?@8 x1 < x1 CA:>@5=85 a1 < 0 . !;54>20B5;L=>, ?@8 =0G0;L=>9
:>>@48=0B5 <5=LH59 :>>@48=0BK ?>;>65=8O @02=>25A8O x10 < x1 MB>B F8;8=4@ A<5AB8BAO
: >A8 2@0I5=8O, 2 ?@>B82=>< A;CG05 : >A8
2@0I5=8O A<5AB8BAO 2B>@>9 F8;8=4@.
!;54>20B5;L=>, :>>@48=0B0 ?5@2>3> F8;8=4@0
AB0=5B @02=>9 4;8=5 =8B8.  B>< 65 A;CG05, :>340
x10 = x1 :>=5G=>5 ?>;>65=85 F8;8=4@0
>4=>7=0G=> =5 >?@545;O5BAO, >=> 7028A8B >B
=0?@02;5=8O A;CG09=>3> =0G0;L=>3> A42830
F8;8=4@>2. 0==0O 7028A8<>ABL-ABC?5=L:0
?>:070=0 =0 @8AC=:5.
1.2 @8 2:;NG5=88 M;5:B@8G5A:>3> ?>;O =0
H0@8:0E 8=4CF8@CNBAO M;5:B@8G5A:85 70@O4K. "0:
:0: H0@8:8 8 ?@C68=:0 ?@>2>4OI85, B> @07=>ABL
?>B5=F80;>2 <564C H0@8:0<8 4>;6=0 1KBL @02=>9
=C;N. #G8BK20O, GB> l >> r , CA;>285
M:28?>B5=F80;L=>AB8 <>6=> 70?8A0BL 2 2845
q (- q)
- = El . (1)
4Ä„µ0r 4Ä„µ r
0
!;54>20B5;L=>, 25;8G8=K 70@O4>2 @02=K
q = 2Ä„µ rlE . (2)
0
708<>459AB285 MB8E 70@O4>2 A
M;5:B@8G5A:8< ?>;5< ?@82545B : A<5I5=8N
H0@8:>2 8 45D>@<0F88 ?@C68=K. CABL
:064K9 H0@8: A<5AB8;AO =0 @0AAB>O=85 x
>B=>A8B5;L=> F5=B@0 ?@C68=K. ">340
17
C@02=5=85 2B>@>3> 70:>=0 LNB>=0 4;O H0@8:0 ?@8<5B 284
q2
ma = qE - 2kx - . (3)
2
4Ä„µ0(l + 2x)
>4AB02;OO 7=0G5=85 70@O4>2 (A CG5B>< 8E 7028A8<>AB8 >B @0AAB>O=8O), ?>;CG8<
C@02=5=85
2
(2Ä„µ0r(l + 2x)E)
2
ma = 2Ä„µ0r(l + 2x)E - 2kx - .
2
4Ä„µ0(l + 2x)
>A;54=8< A;0305<K< <>6=> ?@5=51@5GL 2 284C <0;>AB8 r . "0:8< >1@07><, ?@8E>48< :
C@02=5=8N 42865=8O H0@8:0
2 2
ma = -(2k - 4Ä„µ rE )x + 2Ä„µ rlE . (4)
0 0
7 MB>3> C@02=5=8O A;54C5B, GB> :>;510=8O 2>7<>6=K, B>;L:> ?@8 CA;>288, GB>
k > 2Ä„µ0rE2 , (5)
2 ?@>B82=>< A;CG05, ?@C68=0 =5 A<>65B A45@60BL A8;K M;5:B@8G5A:>3> 2708<>459AB28O.
@8 2K?>;=5=88 CA;>28O (5), C@02=5=85 (4) O2;O5BAO C@02=5=85< 30@<>=8G5A:8E
:>;510=89, A ?5@8>4><
m
T = 2Ä„ . (6)
2k - 4Ä„µ0rE2
7 C@02=5=8O (4) A;54C5B, GB> ?>;>65=8N @02=>25A8O A>>B25BAB2C5B 7=0G5=85
2Ä„µ0rlE2
x0 = . (7)
2k - 4Ä„µ0rE2
"0: :0: 4> 2:;NG5=8O ?>;O x = 0 , B> D>@<C;0 (7) >?@545;O5B 8 0<?;8BC4C :>;510=89
:064>3> H0@8:0.
1.3  A>>B25BAB288 A 70:>=>< $8:0, A:>@>ABL 87<5=5=8O G8A;0 <>;5:C; 2>74CE0 2=CB@8
?C7K@O >?8AK205BAO C@02=5=85<
"N n - n0
= -D S . (1)
"t h
5;8G8=K, 2E>4OI85 2 MB>9 C@02=5=85, =5>1E>48<> 2K@078BL G5@57 @048CA ?C7K@O.
07=>ABL 402;5=89 2>74CE0 2=CB@8 8 A=0@C68 ?C7K@O @02=0 0?;0A>2A:><C 402;5=8N:
4Ã
P - P0 = . (2)
R
07=>ABL A>>B25BAB2CNI8E :>=F5=B@0F89 <>6=> 2K@078BL 87 C@02=5=8O A>AB>O=8O
P - P0 4Ã
(n - n0) = = . (3)
kT kTR
;O 70?8A8 D>@<C;K 4;O G8A;0 <>;5:C; 2=CB@8 ?C7K@O B0:65 =5>1E>48<> 8A?>;L7>20BL
D>@<C;C 0?;0A0 8 C@02=5=85 A>AB>O=8O
4 4 P 4Ä„R3 4Ã
ëÅ‚ öÅ‚
N = nV = n Ä„R3 = Ä„R3 = P0 + . (4)
ìÅ‚ ÷Å‚
3 3 kT 3kT R
íÅ‚ Å‚Å‚
$>@<C;0 4;O ?;>I048 ?>25@E=>AB8 E>@>H> 8725AB=0:
S = 4Ä„R2 . (5)
0:>=5F, B>;I8=0 ?;5=:8 2K@0605BAO 87 CA;>28O ?>AB>O=AB20 55 >1J5<0:
2
R0
2
h Å" 4Ä„R2 = h0 Å" 4Ä„R0 Ò! h = h0 . (6)
R2
!>18@0O 2A5 D>@<C;K 2>548=>, ?>;CG8< C@02=5=85
18
ëÅ‚
1 4Ä„R3 4Ã 4Ã R2
öÅ‚öÅ‚
ìÅ‚
"ìÅ‚ ëÅ‚ P0 + = -D Å" 4Ä„R2 Ò!
ìÅ‚ ÷Å‚÷Å‚
"t 3kT R h0R0
íÅ‚ Å‚Å‚÷Å‚ kTR 2
íÅ‚ Å‚Å‚
"(P0R3 + 4ÃR2) 4ÃD
= - R3
2
"t h0R0
KG8A;5=85 ?@>872>4=>9 >B A;>6=>9 DC=:F88, ?@82>48B : >:>=G0B5;L=><C 2K@065=8N
4;O A:>@>AB8 87<5=5=8O @048CA0 ?C7K@O:
4ÃD "R 4ÃD R3
(3P0R2 + 8ÃR)"R = - R3 Ò! = - . (7)
2 2
"t h0R0 "t h0R0 3P0R2 + 8ÃR
B<5B8<, GB>, 5A;8 ?@5=51@5GL ;0?;0A>2A:8< 402;5=85<, ?> A@02=5=8N A 0B<>AD5@=K<
402;5=85<, B> 40==0O A:>@>ABL ?@>?>@F8>=0;L=0 @048CAC ?C7K@O, B> 5ABL @048CA ?C7K@O C1K205B
M:A?>=5=F80;L=>.
"0: :0:, ?> CA;>28N 7040G8 87<5=5=8O @048CA0 <0;>, B> 4;O 2KG8A;5=8O 2@5<5=8
C<5=LH5=8O @048CA0 <>6=> ?@5=51@5GL 87<5=5=85< @048CA0 ?C7K@O 8 ?>;>68BL 53>
@02=K< R0 . ">340 8A:><>5 2@5<O >?@545;O5BAO D>@<C;>9
2 2
h0R0 3P0R0 + 8ÃR0 h0(3P0R0 + 8Ã ). (8)
"t = "R Å" = "R
3
4ÃD R0 4ÃD
040=85 11(11)-2. «0;L20=><03=8B=K5 O2;5=8O
!G8B05< M;5:B@>= ?>;>68B5;L=> 70@O65==>9 G0AB8F59.
1. -;5:B@>= 428305BAO A CA:>@5=85<
eEX
aX = (1)
m
2 B5G5=85 2@5<5=8 Ä . 0 MB> 2@5<O M;5:B@>= =018@05B =0?@02;5==CN A:>@>ABL
eEX
v = aXÄ = Ä (2).
m
!@54=OO A:>@>ABL 4@59D0
eÄ
vX = EX (3).
2m
>MDD8F85=B 1 2 =5 8<55B ?@8=F8?80;L=>3> 7=0G5=8O.
F
2. >4AB02;OO 7=0G5=85 = EX 2 D>@<C;C (3), ?>;CG8<:
e
eÄ
µ = (4).
2m
3. A;8 M;5:B@>=K 42830NBAO 24>;L ?>;O A 4@59D>2>9 A:>@>ABLN vX = µEX , B>
A>740205<0O MB8< =0?@02;5==K< 42865=85< ?;>B=>ABL B>:0:
jX = nevX = neµEX = Ã0EX (5),
>B:C40
Ã0 = neµ (6).
4. 2865=85 M;5:B@>=0 24>;L >A8 OX ?@82545B : ?>O2;5=8N A8;K >@5=F0,
459AB2CNI59 2 >B@8F0B5;L=>< =0?@02;5=88 >A8 OY.  A2>N >G5@54L 42865=85 24>;L >A8
OY ?@8254QB : ?>O2;5=8N A8;K >@5=F0, 459AB2CNI59 2 ?>;>68B5;L=>< =0?@02;5=88
>A8 OX. (0@O4 M;5:B@>=0 ?>;>68B5;L=K9.) !;54>20B5;L=>:
FX = eEX + evY BZ
Å„Å‚
òÅ‚F = eEY - evX BZ (7).
ół Y
19
 57C;LB8@CNI0O A8;0 ?@82545B : ?>O2;5=8N 4@59D>2>9 A:>@>AB8, A>AB02;ONI85 :>B>@>9
A2O70=K 0=0;>38G=K<8 A>>B=>H5=8O<8:
vX = µ(EX + vY BZ ) (8).
Å„Å‚
òÅ‚v = µ(EY - vX BZ
)
ół Y
7 A8AB5<K (8) 2K@078< vX 8 vY .
µ(EX + µEY BZ )
Å„Å‚v =
X
ôÅ‚ 2
ôÅ‚ 1+ µ2BZ
(9).
òÅ‚
µ(EY - µEX BZ )
ôÅ‚
vY =
2
ôÅ‚ 1+ µ2BZ
ół
5. A;8 ?>;5 EY = 0 , B> 8AE>4O 87 (9):
µEX
Å„Å‚
vX =
ôÅ‚ 2 2
1+ µ BZ
ôÅ‚
(10).
òÅ‚ 2
ôÅ‚vY = - µ BZ EX
2 2
ôÅ‚ 1+ µ BZ
ół
A;8 2 >BACBAB285 <03=8B=>3> ?>;O ?;>B=>ABL B>:0:
jX = nevX = neµEX = Ã0EX (11).
"> B5?5@L:
neµEX
2
jX = nevX = H" neµEX (1- µ2BZ )= ÃEX (12).
2
1+ µ2BZ
!;54>20B5;L=>, >B=>A8B5;L=>5 87<5=5=85 C45;L=>9 ?@>2>48<>AB8:
2 2
"Ã neµ - neµ(1- µ BZ )
2
= = (µBZ ) (13).
à neµ
0
Å‚ = 2 (130).
6. ".:. C3>; Ä… << 1 (?>;5 A;01>5), B>:
vY
Ä… H" tgÄ… = = µBZ (14).
vX
7. 0@O4K ?5@5AB0=CB =0:0?;820BLAO =0 3@0=8F0E, 5A;8 vY = 0. @8 MB><, :0:
A;54C5B 87 (9):
EY = µBZ EX (15).
H = 1 (16).
8. >4AB0282 (15) 2 2K@065=85 4;O A:>@>AB8 vX (9), ?>;CG8<:
2
µ(EX + µ2BZ EX )
vX = = µEX (17),
2
1+ µ2BZ
B.5. ?>;CG05BAO B0:>5 65 7=0G5=85, :0: 8 2 >BACBAB285 ?>;O BZ . 03=5B>A>?@>B82;5=8O
=5B.
9. >;=K9 B>: 2 A;CG05 42CE B8?>2 M;5:B@>=>2:
jX = j1X + j2 X = nev1X + nev2 X = neµEX + ne Å" 2µEX = 3neµEX (18).
Ã0 = 3neµ (19).
10. ;O :064>3> A>@B0 M;5:B@>=>2 <>6=> 70?8A0BL A8AB5<C C@02=5=89 (8).
;O ?5@2>3> B8?0 ( µ1 = µ ):
v1X = µ(EX + v1Y BZ )
Å„Å‚
òÅ‚v = µ(EY - v1X BZ
) (20).
ół 1Y
;O 2B>@>3> B8?0 ( µ2 = 2µ ):
20
v2 X = 2µ(EX + v2Y BZ )
Å„Å‚
òÅ‚v = 2µ(EY - v2 BZ (21).
)
ół 2Y X
5H0O MB8 A8AB5<K >B=>A8B5;L=> A:>@>AB59, ?>;CG8<:
µ(EX + µEY BZ )
Å„Å‚
ôÅ‚v1X = 1 + µ 2 2
ôÅ‚ BZ
(22)
òÅ‚
)
ôÅ‚v2Y = µ(EY - µEX BZ
2 2
ôÅ‚ 1 + µ BZ
ół
2µ(EX + 2µEY BZ )
Å„Å‚v =
2 X
ôÅ‚ 2
ôÅ‚ 1 + 4µ2BZ
(23)
òÅ‚
2µ(EY - 2µEX BZ )
ôÅ‚
v2Y =
2
ôÅ‚ 1 + 4µ2BZ
ół
">: 24>;L >A8 OY ?@5:@0B8BLAO, 5A;8 (:>=F5=B@0F88 M;5:B@>=>2 :064>3> B8?0
>48=0:>2K):
v1Y = -v2Y (24).
".5.:
µ(EY - µEX BZ ) 2µ(EY - 2µEX BZ )
= - (25).
2 2
1 + µ2BZ 1 + 4µ2BZ
5H0O C@02=5=85 (25) >B=>A8B5;L=> EY , ?>;CG8<:
8
2
1+ µ2BZ
5
EY = µBZ EX 5 2 (26).
3 1+ 2µ2BZ
Ä…
A?>;L7CO D>@<C;C ?@81;86Q==>3> 2KG8A;5=8O (1+ x) H" 1+ Ä…x ?@5>1@07C5< 4@>1L 2
C@02=5=88 (26).
8
2
1+ µ2BZ
8 2
ëÅ‚1
2 2 2
3
H" + µ2BZ öÅ‚ Å"(1+ 2µ2BZ )H" 1- µ2BZ (27).
ìÅ‚ ÷Å‚
2
1 + 2µ2BZ íÅ‚ 5 5
Å‚Å‚
">340:
5 2 5
3
EY = µBZ EX - µ3BZ EX H" µBZ EX (28)
3 3 3
3
>6=> ?@5=51@5GL, A;0305<K< ?>@O4:0 µ3BZ :
5
H = (29).
3
11. >4AB0282 (28) 2 2K@065=8O 4;O v1X (22) 8 v2 X (23), ?>;CG8<:
5
2
µEX ëÅ‚1 + µ2BZ öÅ‚
ìÅ‚ ÷Å‚
3
íÅ‚ Å‚Å‚
v1X = (30),
2
1+ µ2BZ
10
2
2µEX ëÅ‚1+ µ2BZ öÅ‚
ìÅ‚ ÷Å‚
3
íÅ‚ Å‚Å‚
v2 X = (31).
2
1+ 4µ2BZ
;>B=>ABL B>:0:
ëÅ‚ 10 öÅ‚
2
2
ìÅ‚1 + 5 µ2BZ 2ëÅ‚1+ µ2BZ öÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
3
íÅ‚ Å‚Å‚ ÷Å‚
jX = ne(v1X + v2 X ) = neµEX ìÅ‚ 3 2 + (32).
2
ìÅ‚
1 + µ2BZ 1 + 4µ2BZ ÷Å‚
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
21
A?>;L7CO D>@<C;C ?@81;86Q==>3> 2KG8A;5=8O, C?@>AB8< 2K@065=85, AB>OI55 2
2
A:>1:0E, A>E@0=OO B>;L:> 25;8G8=K ?>@O4:0 µ2BZ .
10
öÅ‚
5 2
2
ìÅ‚ ÷Å‚
1+ µ2BZ 2ëÅ‚1+ µ2BZ
3
íÅ‚ Å‚Å‚
3
+ =
2 2
1+ µ2BZ 1+ 4µ2BZ
5 10
ëÅ‚1+ 2 ëÅ‚1+ µ2BZ öÅ‚
2 2 2
= µ2BZ öÅ‚ Å"(1- µ2BZ)+ 2 Å" Å"(1- 4µ2BZ )= (33).
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
3 3
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
2 2 2
2 2 2
= 1+ µ2BZ + 2ëÅ‚1- µ2BZ öÅ‚ = 3ëÅ‚1- µ2BZ öÅ‚
ìÅ‚ ÷Å‚ ìÅ‚ ÷Å‚
3 3 9
íÅ‚ Å‚Å‚ íÅ‚ Å‚Å‚
">340 4;O ?;>B=>AB8 B>:0 ?>;CG8<:
2
2 2
jX = 3neµEX ëÅ‚1- µ BZ öÅ‚ (34).
ìÅ‚ ÷Å‚
9
íÅ‚ Å‚Å‚
B=>A8B5;L=>5 87<5=5=85 C45;L=>9 ?@>2>48<>AB8:
"Ã 2
2
= (µBZ ) (35).
Ã0 9
2
´ = (36).
9
12. 5=55 ?>4286=K5 M;5:B@>=K 1C4CB C2;5:0BLAO ?>;5< EY 24>;L
?>;>68B5;L=>3> =0?@02;5=8O >A8 OY. 0 1>;55 ?>4286=K5 M;5:B@>=K 2;8O=85 A8;K
>@5=F0 1C45B A8;L=55, ?>MB><C 24>;L >A8 OY >=8 1C4CB 42830BLAO 2 >B@8F0B5;L=><
=0?@02;5=88.
;O >?@545;5=8O C3;>2, ?>4AB028< 7=0G5=85 ?>;O EY (28) 2 (22) 8 (23). >A;5
=5:>B>@KE ?@5>1@07>20=89 ?>;CG8<:
Å„Å‚ 2
ëÅ‚1+ 2 öÅ‚
1X
ôÅ‚v = µEX ìÅ‚ 3 µ2BZ ÷Å‚
ôÅ‚
íÅ‚ Å‚Å‚
(37),
òÅ‚
2
ôÅ‚
v2Y = µ2BZ EX
ôÅ‚
ół 3
Å„Å‚
ëÅ‚1- 2
öÅ‚
2
2 X
ôÅ‚v = 2µEX ìÅ‚ 3 µ2BZ ÷Å‚
ôÅ‚
íÅ‚ Å‚Å‚
(38).
òÅ‚
2
ôÅ‚
v2Y = - µ2BZ EX
ôÅ‚
ół 3
3
@5=51@530O 25;8G8=0<8 ?>@O4:0 (µBZ ) , ?>;CG8<:
v1Y 2
Ä…1 H" tgÄ…1 = H" µBZ (39).
v1X 3
v2Y 1
Ä…2 H" tgÄ…2 = H" - µBZ (40).
v2 X 3
22
040=85 11(11)-3. «1@07>20=85 >1;0:>2
'0ABL 1. !B0F8>=0@=0O 0B<>AD5@0.
1.1 7 D>@<C;K (1) A;54C5B, GB>
"T = T0a"z (1)
!;54>20B5;L=>,
"T
"z = (2)
T0a
1.2 @8 ?>4J5<5 =0 <0;CN 2KA>BC "z , 402;5=85 C<5=LH05BAO =0 25;8G8=C
"P = -Ág"z , (3)
345 Á - ?;>B=>ABL 2>74CE0, 7028AOI0O >B 53> 402;5=8O 8 B5<?5@0BC@K. -B0 7028A8<>ABL
2K@0605BAO 87 C@02=5=8O A>AB>O=8O 8450;L=>3> 3070
m m PM
PV = RT Ò! Á = = . (4)
M V RT
7 C@02=5=89 (3)-(4) A;54C5B, GB> ?@>872>4=0O 402;5=8O ?> 2KA>B5 @02=0
"P PM PMg
= -Ág = - g = - . (5)
"z RT RT0(1- az)
Ä…
! 4@C3>9 AB>@>=K, 87 D>@<C;K P = P0(1- az) A;54C5B, GB>
Ä… -1
2
P = -Ä…aP0(1- az) .
>4AB02;OO MB8 2K@065=8O 2 C@02=5=85 (5) ?>;CG8<
Ä…
P0(1- az) Mg P0Mg
Ä… -1 Ä… -1
- Ä…aP0(1- az) = - = - (1- az) . (6)
RT0(1- az) RT0
7 MB>3> 2K@065=8O A;54C5B, GB> D>@<C;0 (2), ?@82545==0O 2 CA;>288, C4>2;5B2>@O5B
C@02=5=8N (5), ?@8
Mg
Ä… = . (7)
aRT0
1.3 $>@<C;0, >?8AK20NI0O 7028A8<>ABL ?;>B=>AB8 >B 2KA>BK A;54C5B 87 2K@065=8O (4) 8
=0945==>9 10@><5B@8G5A:>9 D>@<C;K
Ä…
PM MP0(1- az) MP0
Ä… -1
Á(z) = = = (1- az) . (8)
RT RT0(1- az) RT0
"0:8< >1@07><, ?>:070B5;L
Mg
² = Ä… -1 = -1.
aRT0
(9)
1.4 !E5<0B8G5A:85 3@0D8:8 7028A8<>AB59
B5<?5@0BC@K, 402;5=8O 8 ?;>B=>AB8 >B 2KA>BK
?>:070=K =0 @8AC=:5. 06=> >B<5B8BL, GB>
B5<?5@0BC@0 ?0405B ?> ;8=59=><C 70:>=C, 402;5=85
C1K205B 1KAB@55 2A53>, 0 ?;>B=>ABL 70=8<05B
?@><56CB>G=>5 7=0G5=85.
23
'0ABL 2. >AE>4OI85 ?>B>:8.
2.1 "0: :0: 402;5=85 2 ?>4=8<0NI59AO ?>@F88 2>74CE0 A;54C5B AG8B0BL @02=K< 402;5=8N
>:@C60NI53> 2>74CE0 =0 ;N1>9 2KA>B5, B> 4;O >?@545;5=8O 7028A8<>AB8 55 B5<?5@0BC@K
A;54C5B 70?8A0BL C@02=5=85 048010B=>3> ?@>F5AA0 2 «:>>@48=0B0E (T, P). 7 C@02=5=8O
A>AB>O=8O
PV
= const
T
2K@078<
T
V = const Å"
P
8 ?>4AB028< 2 C@02=5=85 048010B=>3> ?@>F5AA0:
1-Å‚
Å‚
T
Å‚ Å‚
P = P1-Å‚T = const Ò! TP = const .
PÅ‚
>=AB0=BC 2 MB>< C@02=5=88 <>6=> 2K@078BL 87 CA;>289 =0 ?>25@E=>AB8 75<;8
1-Å‚
1-Å‚
Å‚
Å‚
TP = T0P0 (10)
>4AB02;OO =0945==CN 7028A8<>ABL 402;5=8O >B 2KA>BK, =0945< 7028A8<>ABL
B5<?5@0BC@K ?>4=8<0NI53>AO 2>74CE0 >B 2KA>BK:
1-Å‚
Å‚ -1
ëÅ‚
P0 öÅ‚ Å‚
Ä…
÷Å‚
Å‚
T1(z) = T0 ìÅ‚ = T0(1- az) . (11)
Ä…
ìÅ‚ ÷Å‚
P0(1- az)
íÅ‚ Å‚Å‚
"0:8< >1@07><, 2 MB>9 D>@<C;5 ?>:070B5;L AB5?5=8 @025=
Å‚ -1 Å‚ -1 Mg
´ = Ä… = . (12)
Å‚ Å‚ aRT0
2.2 ;>B=>ABL 2>74CE0 2K@0605BAO 87 C@02=5=8O A>AB>O=8O
Ä…
PM MP0(1- az) MP0
Ä… -´
Á1(z) = = = (1- az) . (13)
´
RT1 RT0
RT0(1- az)
>:070B5;L AB5?5=8 2 MB>9 D>@<C;5 @025=
Mg Å‚ -1 Mg 1 Mg
µ = Ä… - ´ = - = . (14)
aRT0 Å‚ aRT0 Å‚ aRT0
2.3 >4J5< 2>74CE0 1C45B ?@>4>;60BLAO, 5A;8 =0 ;N1>9 2KA>B5 53> ?;>B=>ABL <5=LH5,
G5< ?;>B=>ABL >:@C60NI53> 2>74CE0
Á1(z) < Á(z). (15)
-B> CA;>285 1C45B 2K?>;=OBLAO, 5A;8 ?>:070B5;L AB5?5=8 2 D>@<C;5 (13) 1C45B 1>;LH5,
G5< 2 D>@<C;5 (8), B> 5ABL ?@8 Ä… - ´ > Ä… -1, 8;8 ?@8 ´ < 1. A?>;L7CO =0945==>5
7=0G5=85 MB>3> ?0@0<5B@0 (12), =0945< B@51C5<>5 7=0G5=85 ?0@0<5B@0 a :
Å‚ -1 Mg Å‚ -1 Mg
< 1 Ò! a > H" 3,3Å"10-5 <-1 . (16)
Å‚ aRT0 Å‚ RT0
@8 MB>< 7=0G5=88 ?0@0<5B@0 B5<?5@0BC@0 4>;6=0 ?>=860BLAO =0 >48= 3@04CA =0 2KA>B0E
<5=LH8E, G5<
"T 1
"z = = H" 100< . (17)
T0a 300 Å" 3,3Å"10-5
24
'0ABL 3. >=45=A0F8O.
;O =0G0;0 :>=45=A0F88 =5>1E>48<>, GB>1K B5<?5@0BC@0 ?>4=8<0NI53>AO 2>74CE0 AB0;0
@02=>9 B5<?5@0BC@5 B>G:8 @>AK. B0:, ?CABL ?@8 B5<?5@0BC@5 T0 402;5=85 =0AKI5==>3>
?0@0 @02=> P=0A.(T0 ), B>340 ?0@F80;L=>5 402;5=85 2>4O=KE ?0@>2 @02=> ÕP=0A.(T0 ). -B>
402;5=85 5ABL 402;5=85 =0AKI5==KE ?0@>2 ?@8 B5<?5@0BC@5 B>G:8 @>AK Tx :
ÕP=0A.(T0 ) = P=0A.(Tx ). (18)
A?>;L7CO C@02=5=85 7028A8<>AB8 402;5=8O =0AKI5==KE ?0@>2 >B B5<?5@0BC@K, =0945<
B5<?5@0BC@C B>G:8 @>AK:
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
P=0A.(Tx ) ÕP=0A.(T0 ) qM1 1 1 qM1 T0 ÷Å‚
ln = ln = lnÕ = - ìÅ‚ - ÷Å‚ - ìÅ‚ -1÷Å‚ Ò!
=
ìÅ‚
P=0A.(T0 ) P=0A.(T0 ) R Tx T0 ÷Å‚ RT0 ìÅ‚ Tx Å‚Å‚
íÅ‚ Å‚Å‚ íÅ‚
.
T0
Tx =
RT0
1- lnÕ
qM1
"5?5@L ?>4AB028< 2K@065=85 4;O 7028A8<>AB8 B5<?5@0BC@K ?>4=8<0NI53>AO 2>74CE0 >B
2KA>BK T1(z)
T0
´
. (19)
T0(1- az) =
RT0
1- lnÕ
qM1
;O @5H5=8O MB>3> C@02=5=8O >F5=8< G8A;5==>
RT0 8,3Å" 300 Å" ln 0,7
- lnÕ = - H" 0,0224 ,
qM1 2,2 Å"106 Å"18Å"10-3
-B0 25;8G8=0 O2;O5BAO <0;>9 (?> A@02=5=8N A 1), ?>MB><C <0;>9 O2;O5BAO 8 25;8G8=0 az ,
?>MB><C C@02=5=85 (19) <>6=> C?@>AB8BL:
T0 RT0
´
T0(1- az) = Ò! 1- ´az = 1+ lnÕ .
RT0
qM1
1- lnÕ
qM1
0:>=5F, 8A?>;L7CO 2K@065=85 4;O ?>:070B5;O AB5?5=8 ´ , ?>;CG8< 2K@065=85 4;O
2KA>BK >1@07>20=8O >1;0:>2
RT0 Å‚ -1 Mg RT0
´az = - lnÕ Ò! az = - lnÕ Ò!
qM1 Å‚ aRT0 qM1
.
2 2
Å‚ (RT0 ) 1,4 (8,3Å"300) Å" ln 0,7
z = - lnÕ = - H" 0,69 Å"103 <
Å‚ -1 qM1Mg 0,4 2,2 Å"106 Å" 29 Å"10-3 Å"18Å"10-3 Å" 9,8
25