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Ɋɚɜɟɧɫɬɜɨ ɷɧɟɪɝɢɣ ɤɨɧɟɱɧɨɝɨ ɢ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫ ɭɱɟɬɨɦ ɤɜɚɧɬɚ Z=
ɷɧɟɪɝɢɢ ɜɧɟɲɧɟɝɨ ɩɨɥɹ,
2
422
2
2
=
=
=
me
m
k
Z , (13)
ɩɪɢ ɭɫɥɨɜɢɢ
R
y!!Z= (ɧɚɩɨɦɧɢɦ, ɱɬɨ ) ɨɡɧɚɱɚɟɬ, ɱɬɨ
ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɧɚ, ɜɵɥɟɬɚɸɳɟɝɨ ɢɡ ɚɬɨɦɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɝɨ ɩɨɥɹ, ɨɱɟɧɶ
ɜɟɥɢɤɚ ɜ ɫɪɚɜɧɟɧɢɢ ɫ ɩɨɪɨɝɨɦ ɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ (
2224
2/2/ mameRy ==
Ry
). ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɫ ɩɨɦɨɳɶɸ
ɪɚɜɟɧɫɬɜɚ
=/2 Z| mk (14)
ɜɫɸɞɭ ɩɟɪɟɯɨɞɢɬɶ ɤ ɩɪɢɛɥɢɠɟɧɧɵɦ ɜɵɪɚɠɟɧɢɹɦ ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɩɪɟɧɟɛɪɟɱɶ ɜ
ɡɧɚɦɟɧɚɬɟɥɟ (12) ɜɟɥɢɱɢɧɨɣ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
2
a
2
k . ȼ ɪɟɡɭɥɶɬɚɬɟ, ɞɥɹ ɤɜɚɞɪɚɬɚ
ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ (9) ɩɨɥɭɱɢɦ:
T
¸
¸
¹
·
¨
¨
©
§
Z
S
222
5
2
cos
2
Ee
maV
M
if
=
. (15)
ɇɚɣɞɟɦ ɬɟɩɟɪɶ ɩɥɨɬɧɨɫɬɶ ɤɨɧɟɱɧɵɯ ɫɨɫɬɨɹɧɢɣ
f
U ɜ ɷɥɟɦɟɧɬɟ -
ɩɪɨɫɬɪɚɧɫɬɜɚ
p
p
p : ddppd
23
, (16)
ɝɞɟ – ɬɟɥɟɫɧɵɣ ɭɝɨɥ, ɜ ɤɨɬɨɪɨɦ ɧɚɯɨɞɢɬɫɹ ɢɦɩɭɥɶɫ
p
:d p
k
= ɷɥɟɤɬɪɨɧɚ,
ɜɵɥɟɬɟɜɲɟɝɨ ɢɡ ɚɬɨɦɚ. ɍɱɢɬɵɜɚɹ, ɱɬɨ
f
dpmdp H )/(,ɢ ɩɪɢɪɚɜɧɢɜɚɹ ɱɢɫɥɨ
ɫɨɫɬɨɹɧɢɣ ɜ ɷɥɟɦɟɧɬɟ (16) ɜɟɥɢɱɢɧɟ ,
ff
dHU
33
)2(/ =SpdV
S
HU
3
3
)2( =
pdV
d
ff f
dd
mkV
H:
S
p
3
)2( =
=
, (17)
ɩɨɥɭɱɢɦ:
U
f
p
:
S
d
mkV
23
8 =
. (18)
ɉɨɞɫɬɚɜɢɦ (15) ɢ (18) ɜ ɮɨɪɦɭɥɭ ɡɨɥɨɬɨɝɨ ɩɪɚɜɢɥɚ
2
2
if
Mwd
=
S
f
U . (19)
ȼɟɥɢɱɢɧɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ɷɥɟɤɬɪɨɧɚ ɡɚ ɟɞɢɧɢɰɭ
wd
140
ɜɪɟɦɟɧɢ ɢɡ ɚɬɨɦɧɨɝɨ
s
1-ɫɨɫɬɨɹɧɢɹ ɜ ɫɨɫɬɨɹɧɢɟ ɞɜɢɠɟɧɢɹ ɜɧɟ ɚɬɨɦɚ ɫ
ɢɦɩɭɥɶɫɨɦ ɜ ɬɟɥɟɫɧɨɦ ɭɝɥɟ
p
p
:d . ɉɨɥɶɡɭɹɫɶ ɪɚɜɟɧɫɬɜɨɦ (14), ɧɚɯɨɞɢɦ:
T
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
Z
Z
S
Z
:
2
2
2
2/9
00
cos
2
e
Ea
d
wd
p
, (20)
ɝɞɟ – ɯɚɪɚɤɬɟɪɧɚɹ ɚɬɨɦɧɚɹ ɱɚɫɬɨɬɚ,
34
0
/ =me Z
2
/
ae – ɩɨɪɹɞɨɤ ɜɟɥɢɱɢɧɵ
ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɹɞɪɚ, ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɷɥɟɤɬɪɨɧ ɜ ɚɬɨɦɟ
ɜɨɞɨɪɨɞɚ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɜɵɪɚɠɟɧɢɹ (20) ɩɨ ɭɝɥɚɦ ɜɵɥɟɬɚ ɷɥɟɤɬɪɨɧɚ ɢɡ ɚɬɨɦɚ
ɫ ɭɱɟɬɨɦ
3/4cos
2
S T:
³
p
d (21)
ɞɚɟɬ ɩɨɥɧɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ:
2
2
2/9
00
2/3
3
2
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§
Z
ZZ
e
Ea
w
. (22)
ɉɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɩɪɢɛɥɢɠɟɧɧɨɟ
ɨɩɢɫɚɧɢɟ
ɮɨɬɨɷɮɮɟɤɬɚ (ɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ ɫɜɟɬɨɦ) ɩɪɢ ɩɨɥɭɤɥɚɫɫɢɱɟɫɤɨɦ
ɩɨɞɯɨɞɟ ɤ ɡɚɞɚɱɟ ɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɜ ɫɢɫɬɟɦɟ «ɚɬɨɦ + ɩɨɥɟ ɢɡɥɭɱɟɧɢɹ». ȼ
ɩɨɥɭɤɥɚɫɫɢɱɟɫɤɨɦ ɩɨɞɯɨɞɟ ɭɱɢɬɵɜɚɟɬɫɹ ɤɜɚɧɬɨɜɚɧɢɟ ɥɢɲɶ ɚɬɨɦɧɨɣ ɫɢɫɬɟɦɵ, ɚ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɥɚɫɫɢɱɟɫɤɢɦ ɨɛɪɚɡɨɦ. ȼ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɦ ɠɟ ɤɜɚɧɬɨɜɨɦ ɩɨɞɯɨɞɟ ɭɱɢɬɵɜɚɸɬ ɢ ɤɜɚɧɬɨɜɚɧɢɟ ɩɨɥɹ (ɷɬɢ
ɜɨɩɪɨɫɵ ɢɡɭɱɚɸɬɫɹ ɜ ɤɭɪɫɚɯ ɤɜɚɧɬɨɜɨɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ ɢ ɤɜɚɧɬɨɜɨɣ ɨɩɬɢɤɢ).
ɉɪɢɦɟɧɢɦɨɫɬɶ ɩɨɥɭɤɥɚɫɫɢɱɟɫɤɨɝɨ
ɦɟɬɨɞɚ ɤ ɞɨɜɨɥɶɧɨ ɲɢɪɨɤɨɦɭ ɤɪɭɝɭ ɡɚɞɚɱ ɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɫɜɟɬɚ ɫ ɚɬɨɦɚɦɢ ɨɛɭɫɥɨɜɥɟɧɚ ɬɟɦ, ɱɬɨ ɡɨɥɨɬɨɟ ɩɪɚɜɢɥɨ
ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɢ ɞɥɹ ɚɬɨɦɧɵɯ ɩɟɪɟɯɨɞɨɜ
21 o ɜɤɥɸɱɚɟɬ ɡɚɤɨɧ
ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɜ ɬɚɤɨɣ ɮɨɪɦɟ, ɤɨɬɨɪɚɹ ɜ ɫɥɭɱɚɟ ɩɨɥɹ ɢɡɥɭɱɟɧɢɹ,
ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɝɨɫɹ ɨɩɪɟɞɟɥɟɧɧɨɣ ɱɚɫɬɨɬɨɣ 0!Z , ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɨɬɜɟɱɚɟɬ
ɩɪɟɞɫɬɚɜɥɟɧɢɸ ɨ ɞɢɫɤɪɟɬɧɵɯ ɩɨɪɰɢɹɯ ɷɧɟɪɝɢɢ ɩɨɥɹ Z= :
(ɩɨɝɥɨɳɟɧɢɟ ɮɨɬɨɧɚ),ZH H =
12
ZH =
21
H
(ɢɫɩɭɫɤɚɧɢɟ ɮɨɬɨɧɚ).
ɋ ɷɬɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɦɵ ɦɨɠɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɠɞɵɣ ɚɤɬ ɢɨɧɢɡɚɰɢɢ
ɚɬɨɦɚ ɩɟɪɢɨɞɢɱɟɫɤɢɦ ɩɨɥɟɦ )(
t
E ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɩɨɝɥɨɳɟɧɢɹ ɚɬɨɦɧɵɦ
ɷɥɟɤɬɪɨɧɨɦ ɮɨɬɨɧɚ ɫ ɷɧɟɪɝɢɟɣ Z= . ɉɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɮɨɬɨɧɨɜ ,
«ɩɚɞɚɸɳɢɯ» ɧɚ ɚɬɨɦ ɫɨ ɫɬɨɪɨɧɵ ɢɫɬɨɱɧɢɤɚ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɝɨ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɢɡɥɭɱɟɧɢɹ, ɪɚɡɭɦɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫɨɨɬɧɨɲɟɧɢɟɦ
ɞɚɩ
j
Z =/Ij
ɞɚɩ
, (23)
141
ɝɞɟ – ɭɫɪɟɞɧɟɧɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ
ɷɧɟɪɝɢɢ. Ɍɨɝɞɚ, ɪɚɡɞɟɥɢɜ (20) ɢ (22) ɧɚ (23), ɧɚɣɞɟɦ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɫɟɱɟɧɢɟ ɩɪɨɰɟɫɫɚ ɮɨɬɨɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ
S 8/
2
EcI
T
¸
¹
·
¨
©
§
Z
:
V
2
2/7
2
26
cos2
==
Ry
c
e
a
d
d
p
, (24)
ɚ ɬɚɤɠɟ – ɩɨɥɧɨɟ ɫɟɱɟɧɢɟ
2/7
2
2
8
3
2
¸
¹
·
¨
©
§
Z
S
V
==
Ry
c
e
a . (25)
ɗɬɢ ɮɨɪɦɭɥɵ ɩɨɥɭɱɟɧɵ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ
Ry
!!Z= , ɬɨ ɟɫɬɶ – ɞɥɹ
ɫɥɭɱɚɹ, ɤɨɝɞɚ ɷɧɟɪɝɢɹ ɮɨɬɨɧɚ ɧɚɦɧɨɝɨ ɩɪɟɜɵɲɚɟɬ ɤɪɚɫɧɭɸ ɝɪɚɧɢɰɭ
ɮɨɬɨɷɮɮɟɤɬɚ; ɩɪɢ ɦɟɧɶɲɟɦ ɡɧɚɱɟɧɢɢ Z= ɨɧɢ ɞɚɸɬ ɡɚɜɵɲɟɧɧɭɸ ɜɟɥɢɱɢɧɭ
ɜɟɪɨɹɬɧɨɫɬɢ. Ⱦɥɹ ɪɚɫɱɟɬɚ ɫɟɱɟɧɢɹ ɮɨɬɨɷɮɮɟɤɬɚ ɜɛɥɢɡɢ ɤɪɚɫɧɨɣ ɝɪɚɧɢɰɵ
ɧɟɨɛɯɨɞɢɦɨ ɤɨɧɟɱɧɵɟ ɫɨɫɬɨɹɧɢɹ ɷɥɟɤɬɪɨɧɚ ɨɩɢɫɵɜɚɬɶ ɧɟ ɩɥɨɫɤɢɦɢ ɜɨɥɧɚɦɢ, ɚ
ɬɨɱɧɵɦɢ ɜɨɥɧɨɜɵɦɢ ɮɭɧɤɰɢɹɦɢ ɧɟɩɪɟɪɵɜɧɨɝɨ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ
ɷɥɟɤɬɪɨɧɚ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ ɹɞɪɚ ɚɬɨɦɚ. Ɍɚɤɨɣ ɪɚɫɱɟɬ ɞɚɟɬ ɩɪɢ
Ry
tZ=
ɡɧɚɱɟɧɢɟ (ɫɦ., ɧɚɩɪɢɦɟɪ, [10])cea =/31
22
|V
ȿɳɟ ɨɞɢɧ ɚɫɩɟɤɬ ɩɪɢɛɥɢɠɟɧɢɣ, ɢɫɩɨɥɶɡɨɜɚɧɧɵɯ ɜ ɩɪɨɜɟɞɟɧɧɨɦ
ɪɚɫɫɦɨɬɪɟɧɢɢ, ɨɬɧɨɫɢɬɫɹ ɤ ɮɨɪɦɭɥɟ (6) ɞɥɹ ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ ɩɟɪɟɯɨɞɚ. ȼ
ɧɟɣ ɧɟ ɭɱɬɟɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɜɧɟɲɧɟɝɨ ɩɨɥɹ ɨɬ ɤɨɨɪɞɢɧɚɬ, ɤɨɬɨɪɚɹ ɞɨɥɠɧɚ ɛɵɥɚ
ɛɵ ɢɦɟɬɶ ɜɢɞ , ɝɞɟ – ɜɨɥɧɨɜɨɣ ɜɟɤɬɨɪ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ. ɗɬɨ
ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɨɞ ɢɧɬɟɝɪɚɥɨɦ ɜ (9) ɜɟɥɢɱɢɧɚ
rqi
e q
rqi
e ɡɚɦɟɧɟɧɚ ɟɞɢɧɢɰɟɣ – ɩɟɪɜɵɦ
ɱɥɟɧɨɦ ɟɟ ɪɹɞɚ Ɍɟɣɥɨɪɚ ɩɨ ɫɬɟɩɟɧɹɦ
rq . Ⱦɚɧɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɨɩɪɚɜɞɚɧɨ ɩɪɢ
ɢ ɧɚɡɵɜɚɟɬɫɹ ɞɢɩɨɥɶɧɵɦ ɩɪɢɛɥɢɠɟɧɢɟɦ. ɉɪɨɢɫɯɨɠɞɟɧɢɟ ɷɬɨɝɨ ɬɟɪɦɢɧɚ
ɦɨɠɧɨ ɩɨɹɫɧɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.
1qa
Ɉɛɨɡɧɚɱɢɦ ɧɨɦɟɪɚɦɢ 1 ɢ 2 ɧɚɱɚɥɶɧɨɟ ɢ ɤɨɧɟɱɧɨɟ ɫɨɫɬɨɹɧɢɹ ɷɥɟɤɬɪɨɧɚ,
ɹɜɥɹɸɳɢɟɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɞɥɹ ɧɟɜɨɡɦɭɳɟɧɧɨɝɨ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɚɬɨɦɚ .
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɜ ɧɭɥɟɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ ɜɧɟɲɧɟɦɭ ɩɨɥɸ
0
ˆ
H
]
ˆ
,
ˆ
[
ˆ
ˆ
0
r
r
p H
im
d
t
d
mi
=
= , (26)
ɢɦɟɟɦ ɪɚɜɟɧɫɬɜɨ
12
12
10212012
ˆ
)(
ˆ
ˆˆ
ˆ
ˆ
\\
H
H
\\\\ \\ rrrp
==
im
HH
im
,
ɬɨ ɟɫɬɶ
122112
ˆˆ
\\Z \\
rp mi , =/)(
1221
H
H
{Z . (27)
142
Ʉɚɤ ɧɚɦ ɢɡɜɟɫɬɧɨ, ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɩɟɪɟɯɨɞɚ 2ɱɚɫɬɨɬɚ ɜɧɟɲɧɟɝɨ
ɩɨɥɹ ɢ ɡɧɚɱɟɧɢɹ ɷɧɟɪɝɢɢ ɷɥɟɤɬɪɨɧɚ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ
1 o
HH Z =/)(
1221
Z .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ (27) ɞɥɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɪɚɠɟɧɢɹ (6),
ɩɨɥɭɱɢɦ:
\\
Z
1221
ˆ
2
pE
m
ie
M
2112
2
1
ˆ
2
1
dErE \\ e , (28)
ɝɞɟ – ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɨɩɟɪɚɬɨɪɚ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɞɢɩɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
ɚɬɨɦɧɨɣ ɫɢɫɬɟɦɵ ɞɥɹ ɩɟɪɟɯɨɞɚ 21 (ɫɦ. ɬɚɤɠɟ ɮɨɪɦɭɥɭ (29) ɜ ɡɚɞɚɱɟ 9.10).
21
d
o
Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢɦɟɧɹɬɶ ɜ ɩɨɞɨɛɧɵɯ ɪɚɫɱɟɬɚɯ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ
ɨɩɟɪɚɬɨɪɚ ɤɨɨɪɞɢɧɚɬ ɫɥɟɞɭɟɬ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɨɫɦɨɬɪɢɬɟɥɶɧɨɫɬɶɸ, ɬɚɤ ɤɚɤ
ɪɚɜɟɧɫɬɜɨ (27) ɫɩɪɚɜɟɞɥɢɜɨ ɥɢɲɶ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜ ɤɚɱɟɫɬɜɟ ɢ
1
\
2
\
ɬɨɱɧɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɚɬɨɦɚ .
0
ˆ
H
ɉɪɢ ɨɛɫɭɠɞɟɧɢɢ ɚɬɨɦɧɵɯ ɩɟɪɟɯɨɞɨɜ ɜ ɪɚɦɤɚɯ ɞɢɩɨɥɶɧɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ
ɩɨɥɟɡɧɨ ɢɦɟɬɶ ɜɜɢɞɭ ɩɪɚɜɢɥɚ ɨɬɛɨɪɚ, ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɟ ɜ ɡɚɞɚɱɟ 9.4.
ɇɚɩɪɢɦɟɪ, ɫɨɝɥɚɫɧɨ ɩɪɚɜɢɥɭ (11) ɭɤɚɡɚɧɧɨɣ ɡɚɞɚɱɢ ɫɥɟɞɭɟɬ ɨɠɢɞɚɬɶ, ɱɬɨ ɢɡ
ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
s
-ɬɢɩɚ (0
l
,0 m ) ɷɥɟɤɬɪɨɧ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɫ ɥɢɧɟɣɧɨɣ ɩɨɥɹɪɢɡɚɰɢɟɣ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɦɨɠɟɬ
ɩɟɪɟɣɬɢ ɥɢɲɶ ɜ ɫɨɫɬɨɹɧɢɟ ɫ
z
e
1
c
l
,0
c
m , ɤɨɬɨɪɨɟ, ɤɚɤ ɢɡɜɟɫɬɧɨ,
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɜɟɪɨɹɬɧɨɫɬɟɣ ɜɢɞɚ
T
2
2
10
cos~Y
. Ʉɚɤ ɪɚɡ
ɬɚɤɭɸ ɭɝɥɨɜɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɢɦɟɟɬ ɜɟɥɢɱɢɧɚ (15), ɚɫɧɟɣ – ɢ (20), (24).
9.12. ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ )(
t
E , ɜɪɚɳɚɸɳɟɟɫɹ ɫ ɱɚɫɬɨɬɨɣ ɜɨɤɪɭɝ ɨɫɢ
ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ:
Z z
ZZ tEtEt
yx
sincos)( eeE
Z ti
yx
eEi )(
2
1
ee
ti
yx
eEi
Z
)(
2
1
ee . (1)
Ɇɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɩɟɪɟɯɨɞɚ, ɫɨɩɪɨɜɨɠɞɚɸɳɟɝɨɫɹ ɩɨɝɥɨɳɟɧɢɟɦ
ɷɥɟɤɬɪɨɧɨɦ ɜ ɚɬɨɦɟ ɤɜɚɧɬɚ ɷɧɟɪɝɢɢ
if
M
Z= , ɫɜɹɡɚɧ ɫ ɩɟɪɜɵɦ ɫɥɚɝɚɟɦɵɦ ɜ (1). Ɉɧɨ
ɜɤɥɸɱɚɟɬ ɤɨɦɩɥɟɤɫɧɵɣ ɚɦɩɥɢɬɭɞɧɵɣ ɜɟɤɬɨɪ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
Ei
yx
)( eeE . (2)
ȼ ɞɢɩɨɥɶɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɞɨɥɠɧɵ ɜɵɩɨɥɧɹɬɶɫɹ ɫɨɨɬɧɨɲɟɧɢɹ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ
143
ifif
M \\ pE
ˆ
~
if
\\ dE
ˆ
~ . (3)
ɋ ɭɱɟɬɨɦ (2) ɷɬɢ ɫɨɨɬɧɨɲɟɧɢɹ ɩɪɢɧɢɦɚɸɬ ɜɢɞ
ifif
yixEM \wwww\ )//(~
if
yixE \\ )(~ . (4)
ȼɵɱɢɫɥɟɧɢɟ ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ
if
yix \wwww\ )//( ɞɥɹ ɫɥɭɱɚɹ,
ɤɨɝɞɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
i
\
s
-ɫɨɫɬɨɹɧɢɟ ɷɥɟɤɬɪɨɧɚ ɜ ɚɬɨɦɟ ɜɨɞɨɪɨɞɚ, ɚ
f
\
ɢɦɟɟɬ ɜɢɞ ɩɥɨɫɤɨɣ ɜɨɥɧɵ, ɩɪɢɜɟɞɟɬ, ɨɱɟɜɢɞɧɨ, ɤ ɜɵɪɚɠɟɧɢɸ, ɩɨɞɨɛɧɨɦɭ
ɪɟɡɭɥɶɬɚɬɭ (12) ɜ ɡɚɞɚɱɟ 9.11, ɧɨ ɜɦɟɫɬɨ ɦɧɨɠɢɬɟɥɹ ɨɧɨ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ
ɦɧɨɠɢɬɟɥɶ
z
k
)sinsincos(sin MTMT ikkik
yx
, (5)
ɝɞɟ ɢ – ɭɝɥɵ ɜɟɤɬɨɪɚ ɢɦɩɭɥɶɫɚTM
k
p = ɷɥɟɤɬɪɨɧɚ, ɜɵɥɟɬɚɸɳɟɝɨ ɢɡ ɚɬɨɦɚ
ɜɫɥɟɞɫɬɜɢɟ ɮɨɬɨɷɮɮɟɤɬɚ. ɗɬɢ ɭɝɥɵ ɨɬɫɱɢɬɵɜɚɸɬɫɹ ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ ɫ ɩɨɥɹɪɧɨɣ ɨɫɶɸ , ɩɚɪɚɥɥɟɥɶɧɨɣ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɫɤɨɦɚɹ ɭɝɥɨɜɚɹ ɡɚɜɢɫɢɦɨɫɬɶ
ɜɟɪɨɹɬɧɨɫɬɢ ɮɨɬɨɷɮɮɟɤɬɚ ɜ ɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ:
z
~~
2
if
M
d
d
p
:
V
T
2
2
sin~
yx
kik . (6)
ɇɚɣɞɟɧɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɦɚɤɫɢɦɚɥɶɧɚ ɞɥɹ ɧɚɩɪɚɜɥɟɧɢɣ , ɩɚɪɚɥɥɟɥɶɧɵɯ
ɩɥɨɫɤɨɫɬɢ, ɜ ɤɨɬɨɪɨɣ ɜɪɚɳɚɟɬɫɹ ɜɟɤɬɨɪ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ )
p
(
t
E ɰɢɪɤɭɥɹɪɧɨ
ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ. ɉɪɢ ɥɢɧɟɣɧɨɣ ɩɨɥɹɪɢɡɚɰɢɢ ɜɨɥɧɵ
ɜɟɪɨɹɬɧɨɫɬɶ ɮɨɬɨɷɮɮɟɤɬɚ ɦɚɤɫɢɦɚɥɶɧɚ ɩɪɢ , ɩɚɪɚɥɥɟɥɶɧɨɦ . Ɍɚɤɚɹ ɤɚɪɬɢɧɚ
ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɩɨɥɧɟ ɟɫɬɟɫɬɜɟɧɧɨɣ.
pE
Ɋɟɡɭɥɶɬɚɬ (6) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɚɜɢɥɭ ɨɬɛɨɪɚ (12), ɭɤɚɡɚɧɧɨɦɭ ɜ ɡɚɞɚɱɟ 9.4.
ɂɫɯɨɞɹ ɢɡ ɷɬɨɝɨ ɩɪɚɜɢɥɚ ɫɥɟɞɭɟɬ ɨɠɢɞɚɬɶ, ɱɬɨ ɷɥɟɤɬɪɨɧ ɢɡ ɧɚɱɚɥɶɧɨɝɨ
ɫɨɫɬɨɹɧɢɹ
s
-ɬɢɩɚ (0
l
,)ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɰɢɪɤɭɥɹɪɧɨ ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ
ɜɨɥɧɵ ɫ ɚɦɩɥɢɬɭɞɧɵɦ ɜɟɤɬɨɪɨɦ (2) ɦɨɠɟɬ ɩɟɪɟɯɨɞɢɬɶ ɥɢɲɶ ɜ ɫɨɫɬɨɹɧɢɟ ɫ
0 m
1
c
l
,,ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɜɟɪɨɹɬɧɨɫɬɟɣ ɜɢɞɚ1
c
m
T
2
2
11
sin~Y
. ȼ ɫɥɭɱɚɟ ɜɨɥɧɵ ɫ ɰɢɪɤɭɥɹɪɧɨɣ ɩɨɥɹɪɢɡɚɰɢɟɣ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ
ɨɪɢɟɧɬɚɰɢɢ ɫɩɪɚɜɟɞɥɢɜɵ ɫɨɨɬɧɨɲɟɧɢɹ
Ei
yx
)( eeE , ~
2
if
M T
2
2
sin~
yx
kik . (7)
ɉɪɢ ɷɬɨɦ ɫɨɝɥɚɫɧɨ ɩɪɚɜɢɥɭ ɨɬɛɨɪɚ (13) ɡɚɞɚɱɢ 9.4 ɨɠɢɞɚɟɦɵɦ ɤɨɧɟɱɧɵɦ
ɫɨɫɬɨɹɧɢɟɦ ɛɭɞɟɬ ɫɨɫɬɨɹɧɢɟ ɫ 1
c
l
,1
c
m ; ɞɥɹ ɷɬɨɝɨ ɫɨɫɬɨɹɧɢɹ ɭɝɥɨɜɨɟ
ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɛɧɚɪɭɠɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɜɧɟ ɚɬɨɦɚ ɢɦɟɟɬ ɩɪɟɠɧɢɣ
144
ɜɢɞ: T
2
2
1,1
sin~Y . ɉɟɪɟɱɢɫɥɟɧɧɵɟ ɩɪɚɜɢɥɚ ɨɬɛɨɪɚ ɞɥɹ ɤɜɚɧɬɨɜɨɝɨ ɱɢɫɥɚ
ɷɥɟɤɬɪɨɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɧɚɯɨɞɹɬɫɹ ɜ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɨ ɬɨɦ,
ɱɬɨ ɮɨɬɨɧ ɜ ɫɨɫɬɨɹɧɢɹɯ ɫ ɰɢɪɤɭɥɹɪɧɨɣ ɩɨɥɹɪɢɡɚɰɢɟɣ ɨɛɥɚɞɚɟɬ ɦɨɦɟɧɬɨɦ
ɢɦɩɭɥɶɫɚ (ɝɞɟ – ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɮɨɬɨɧɚ). Ɍɚɤɨɟ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɮɨɬɨɧɟ ɦɵ ɨɛɫɭɠɞɚɥɢ ɜ ɡɚɞɚɱɟ 3.9.
z
lm {
1r
z
J z
9.13. ɉɨɬɟɧɰɢɚɥ, ɫɨɡɞɚɜɚɟɦɵɣ ɞɜɭɦɹ ɨɞɢɧɚɤɨɜɵɦɢ ɫɢɥɨɜɵɦɢ ɰɟɧɬɪɚɦɢ,
ɧɚɯɨɞɹɳɢɦɢɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ
ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɟɫɬɶ
d
)()()(
00
drrr UUU , (1)
ɝɞɟ ) – ɩɨɬɟɧɰɢɚɥ ɨɞɧɨɝɨ ɰɟɧɬɪɚ. Ɉɛɨɡɧɚɱɢɦ ɤɨɦɩɨɧɟɧɬɭ Ɏɭɪɶɟ ɞɥɹ
ɮɭɧɤɰɢɢ )ɩɨɫɪɟɞɫɬɜɨɦ :
(
0
rU
(
0
rU )(
0
qu
)()(
0
3
0
rrq
rq
Uedu
i
³
. (2)
ȼɵɱɢɫɥɹɹ ɤɨɦɩɨɧɟɧɬɭ Ɏɭɪɶɟ ɞɥɹ )(
0
dr U , ɩɪɨɢɡɜɟɞɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ
:
drr o
³
)(
0
3
drr
rq
Ued
i
³
)(
0
)(3
rr
drq
Ued
i
)(
0
q
dq
ue
i
. (3)
Ɏɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɚ )ɫɭɦɦɚɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ (1) ɪɚɜɧɚ ɫɭɦɦɟ ɤɨɦɩɨɧɟɧɬ
Ɏɭɪɶɟ ɤɚɠɞɨɝɨ ɢɡ ɫɥɚɝɚɟɦɵɯ ɷɬɨɝɨ ɩɨɬɟɧɰɢɚɥɚ:
(
qu
)(
qu )(
0
qu )(
0
q
dq
ue
i
)(
0
qu )1(
dq
i
e . (4)
ɉɨɞɫɬɚɜɢɜ ɩɪɚɜɭɸ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ (4) ɜ ɮɨɪɦɭɥɭ Ȼɨɪɧɚ (ɫɦ. ɮɨɪɦɭɥɭ (XX)
ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ), ɧɚɣɞɟɦ ɢɫɤɨɦɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ:
)cos1(2)(
2
2
0
2
2
dqq
p
¸
¸
¹
·
¨
¨
©
§
S
:
V
u
m
d
d
=
, (5)
ɝɞɟ – ɢɦɩɭɥɶɫ, ɩɨɥɭɱɚɟɦɵɣ ɷɥɟɤɬɪɨɧɨɦ ɜ ɚɤɬɟ ɭɩɪɭɝɨɝɨ ɪɚɫɫɟɹɧɢɹ
( ɢ – ɜɟɤɬɨɪɵ ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɧɚ ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɫɨɫɬɨɹɧɢɹɯ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ).
ppq
c
=
p
c
p
ɂɦɟɸɳɟɟɫɹ ɜ (5) ɫɥɚɝɚɟɦɨɟ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɟ , ɩɪɢɞɚɟɬ
ɧɚɣɞɟɧɧɨɦɭ ɫɟɱɟɧɢɸ ɪɚɫɫɟɹɧɢɹ ɢɧɬɟɪɮɟɪɟɧɰɢɨɧɧɵɣ ɯɚɪɚɤɬɟɪ. ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ
ɧɚ ɤɚɱɟɫɬɜɟɧɧɨɦ ɭɪɨɜɧɟ ɭɠɟ ɛɵɥ ɩɨɥɭɱɟɧ ɧɚɦɢ ɪɚɧɟɟ ɫ ɩɨɦɨɳɶɸ ɧɚɝɥɹɞɧɵɯ
ɫɨɨɛɪɚɠɟɧɢɣ ɨ «ɜɨɥɧɨɜɵɯ ɫɜɨɣɫɬɜɚɯ» ɱɚɫɬɢɰ (ɫɦ. ɡɚɞɚɱɭ 2.3). Ɍɟɩɟɪɶ ɦɵ
ɭɛɟɠɞɚɟɦɫɹ ɜ ɟɝɨ ɫɩɪɚɜɟɞɥɢɜɨɫɬɢ ɧɚ ɨɫɧɨɜɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɣ ɤɜɚɧɬɨɜɨɣ
dq cos
145
ɬɟɨɪɢɢ. Ʉɪɨɦɟ ɬɨɝɨ, ɡɞɟɫɶ ɦɵ ɧɚɯɨɞɢɦ ɤɨɥɢɱɟɫɬɜɟɧɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ
ɢɫɩɨɥɶɡɨɜɚɧɧɨɦɭ ɜ ɡɚɞɚɱɟ 2.3 ɩɪɟɞɫɬɚɜɥɟɧɢɸ ɨɛ ɚɦɩɥɢɬɭɞɟ ɪɚɫɫɟɹɧɢɹ )(
q
f
(ɪɚɡɥɢɱɢɟ ɜ ɡɧɚɤɟ ɩɪɢ ɜɵɛɨɪɟ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɤɬɨɪɚ ɜ ɨɛɟɢɯ ɡɚɞɚɱɚɯ ɧɟ ɢɝɪɚɟɬ
ɩɪɢɧɰɢɩɢɚɥɶɧɨɣ ɪɨɥɢ). ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ Ȼɨɪɧɚ ɚɦɩɥɢɬɭɞɚ ɪɚɫɫɟɹɧɢɹ
q
)(q
f
ɫ
ɬɨɱɧɨɫɬɶɸ ɞɨ ɮɚɡɨɜɨɝɨ ɦɧɨɠɢɬɟɥɹ ɪɚɜɧɚ ɜɟɥɢɱɢɧɟ ), ɫɜɹɡɚɧɧɨɣ ɫ
ɪɚɫɫɟɢɜɚɸɳɢɦ ɩɨɬɟɧɰɢɚɥɨɦ )
()2/(
2
qum =S
(
r
U
ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ.
9.14. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɦɩɨɧɟɧɬɵ Ɏɭɪɶɟ )ɤɭɥɨɧɨɜɫɤɨɝɨ ɩɨɬɟɧɰɢɚɥɚ
ɩɪɢɦɟɧɢɦ ɮɨɪɦɭɥɭ (11) ɢɡ ɡɚɞɚɱɢ 9.11
(
qu
reZZrU /)(
2
21
J
³
ri
e
r
ed
1
3 rq
r
22
4
J
S
q
, (1)
ɩɟɪɟɣɞɹ ɜ ɧɟɣ ɤ ɩɪɟɞɟɥɭ ɩɪɢ 0oJ ; ɩɨɥɭɱɚɟɦ:
)(
qu
2
2
21
4
q
eZZS
. (2)
ɉɨɞɫɬɚɜɢɜ (2) ɜ ɮɨɪɦɭɥɭ Ȼɨɪɧɚ
2
2
2
)(
2
qu
m
d
d
¸
¸
¹
·
¨
¨
©
§
S
:
V
=
, (3)
ɧɚɯɨɞɢɦ:
:
V
d
d
4
222
21
)(
)(4
q
meZZ
=
. (4)
ȼɵɪɚɡɢɦ ɧɚɣɞɟɧɧɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ ɱɟɪɟɡ ɭɝɨɥ T ɦɟɠɞɭ ɧɚɱɚɥɶɧɵɦ
ɜɟɤɬɨɪɨɦ ɢɦɩɭɥɶɫɚ ɪɚɫɫɟɢɜɚɟɦɨɣ ɱɚɫɬɢɰɵ ɢ ɤɨɧɟɱɧɵɦ . ɋ ɷɬɨɣ ɰɟɥɶɸ
ɡɚɦɟɬɢɦ, ɱɬɨ
p
c
p
c
c
c
ppppppq 2)()(
2222
=
T
c
c
cos2
22
pppp . (5)
ɉɪɢ ɪɚɫɫɟɹɧɢɢ ɱɚɫɬɢɰɵ ɜ ɧɟ ɡɚɜɢɫɹɳɟɦ ɨɬ ɜɪɟɦɟɧɢ ɩɨɥɟ ɭɫɥɨɜɢɟ ɪɚɜɟɧɫɬɜɚ
ɷɧɟɪɝɢɣ ɧɚɱɚɥɶɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɫɨɫɬɨɹɧɢɣ (
pp
H H
c
) ɩɪɢɜɨɞɢɬ ɤ ɪɚɜɟɧɫɬɜɭ
. ɉɨɷɬɨɦɭ ɜɵɪɚɠɟɧɢɟ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (5) ɟɫɬɶpp
c
)2/(sin8)cos1(2)(
222
TH T mpq= , (6)
146
ɝɞɟ – ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ «ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ» (ɜ ɨɛɥɚɫɬɢ
ɩɪɢɛɥɢɠɟɧɧɨ ɫɜɨɛɨɞɧɨɝɨ ɞɜɢɠɟɧɢɹ). ɋ ɭɱɟɬɨɦ (6) ɮɨɪɦɭɥɚ (4) ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
mpmp 2/2/
22
c
H
:
V
d
d
)2/(sin
1
4
4
2
2
21
T
¸
¸
¹
·
¨
¨
©
§
H
eZZ
. (7)
Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ [1], ɱɬɨ ɩɪɢ ɪɚɫɫɟɹɧɢɢ ɱɚɫɬɢɰɵ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ
ɪɟɡɭɥɶɬɚɬ (7) ɫɩɪɚɜɟɞɥɢɜ ɢ ɜɧɟ ɪɚɦɨɤ ɛɨɪɧɨɜɫɤɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ, ɬɨ ɟɫɬɶ ɨɧ
ɹɜɥɹɟɬɫɹ ɬɨɱɧɵɦ. Ɏɨɪɦɭɥɚ (7) ɫɨɜɩɚɞɚɟɬ ɫɨ ɡɧɚɦɟɧɢɬɨɣ ɮɨɪɦɭɥɨɣ Ɋɟɡɟɪɮɨɪɞɚ,
ɤ ɤɨɬɨɪɨɣ ɩɪɢɜɨɞɢɬ ɬɚɤɚɹ ɠɟ ɡɚɞɚɱɚ ɨ ɪɚɫɫɟɹɧɢɢ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ.
9.15. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ
ɜɢɞ:
VHHH
pa
ˆˆˆˆ
, (1)
ɝɞɟ – ɝɚɦɢɥɶɬɨɧɢɚɧ ɚɬɨɦɚ (atom), ɫɨɞɟɪɠɚɳɟɝɨ
a
H
ˆ
Z
ɷɥɟɤɬɪɨɧɨɜ ɢ ɹɞɪɨ ɫ
ɡɚɪɹɞɨɦ
eZ , – ɝɚɦɢɥɶɬɨɧɢɚɧ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢɰɵ (particle) – ɷɥɟɤɬɪɨɧɚ,
ɩɚɞɚɸɳɟɝɨ ɧɚ ɚɬɨɦ ɢɡ ɜɧɟɲɧɟɝɨ ɢɫɬɨɱɧɢɤɚ, – ɷɧɟɪɝɢɹ ɤɭɥɨɧɨɜɫɤɨɝɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɚɞɚɸɳɟɝɨ ɧɚ ɚɬɨɦ ɷɥɟɤɬɪɨɧɚ ɫ ɢɦɟɸɳɢɦɢɫɹ ɜɧɭɬɪɢ ɚɬɨɦɚ
ɷɥɟɤɬɪɨɧɚɦɢ ɢ ɹɞɪɨɦ. ɗɬɭ ɷɧɟɪɝɢɸ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ,
p
H
ˆ
V
ˆ
rrr
2
1
2
ˆ
eZ
e
V
Z
i
i
¦
, (2)
ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜ ɤɚɱɟɫɬɜɟ ɜɨɡɦɭɳɟɧɢɹ. ȼ ɜɵɪɚɠɟɧɢɢ (2)
r – ɤɨɨɪɞɢɧɚɬɵ
ɩɚɞɚɸɳɟɝɨ ɧɚ ɚɬɨɦ ɷɥɟɤɬɪɨɧɚ, – ɤɨɨɪɞɢɧɚɬɵ ɷɥɟɤɬɪɨɧɨɜ ɜ ɚɬɨɦɟ; ɹɞɪɨ
ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɩɨɤɨɹɳɢɦɫɹ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ. ȼɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɫɩɢɧɨɜ ɦɵ
ɩɪɟɧɟɛɪɟɝɚɟɦ, ɬɚɤ ɱɬɨ ɫɩɢɧɨɜɵɟ ɫɨɫɬɨɹɧɢɹ ɚɬɨɦɚ ɢ ɪɚɫɫɟɢɜɚɟɦɨɝɨ ɷɥɟɤɬɪɨɧɚ
ɫɱɢɬɚɸɬɫɹ ɧɟ ɢɡɦɟɧɹɸɳɢɦɢɫɹ ɜ ɯɨɞɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ, ɢ ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɨ
ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɹɯ ɭɩɨɦɢɧɚɬɶ ɧɟ ɛɭɞɟɦ. Ʉɪɨɦɟ ɬɨɝɨ, ɩɪɟɧɟɛɪɟɠɟɦ
ɨɛɦɟɧɧɵɦɢ ɷɮɮɟɤɬɚɦɢ ɦɟɠɞɭ ɩɚɞɚɸɳɢɦ ɧɚ ɚɬɨɦ ɛɵɫɬɪɵɦ ɷɥɟɤɬɪɨɧɨɦ ɢ
ɚɬɨɦɧɵɦɢ ɷɥɟɤɬɪɨɧɚɦɢ (ɬɨ ɟɫɬɶ ɜ ɩɪɨɰɟɫɫɟ ɪɚɫɱɟɬɚ ɦɵ ɧɟ ɛɭɞɟɦ ɩɪɨɜɨɞɢɬɶ
ɚɧɬɢɫɢɦɦɟɬɪɢɡɚɰɢɸ ɦɧɨɝɨɷɥɟɤɬɪɨɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɜ ɰɟɥɨɦ); ɷɬɨ
ɨɩɪɚɜɞɵɜɚɟɬɫɹ ɬɟɦ, ɱɬɨ ɤɨɝɞɚ ɞɟɛɪɨɣɥɟɜɫɤɚɹ ɞɥɢɧɚ ɜɨɥɧɵ ɪɚɫɫɟɢɜɚɟɦɨɝɨ
ɷɥɟɤɬɪɨɧɚ ɦɚɥɚ ɜ ɫɪɚɜɧɟɧɢɢ ɫ ɪɚɡɦɟɪɨɦ ɚɬɨɦɚ, «ɨɛɦɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ» ɢɡ-ɡɚ
ɫɢɥɶɧɨ ɨɫɰɢɥɥɢɪɭɸɳɟɝɨ ɯɚɪɚɤɬɟɪɚ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɩɚɞɚɸɳɟɝɨ ɷɥɟɤɬɪɨɧɚ
ɞɚɸɬ ɜ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɨɜ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɵɣ ɜɤɥɚɞ
.
i
r
ɉɭɫɬɶ ɜ ɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ ɷɥɟɤɬɪɨɧ, ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɛɨɥɶɲɨɦ
ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɚɬɨɦɚ, ɞɜɢɠɟɬɫɹ ɫ ɢɦɩɭɥɶɫɨɦ
p
c
ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɤ ɚɬɨɦɭ,
ɩɪɟɛɵɜɚɸɳɟɦɭ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ n
c
. ɋɬɪɨɝɨ ɝɨɜɨɪɹ, ɬɚɤɨɟ ɫɨɫɬɨɹɧɢɟ
147
ɷɥɟɤɬɪɨɧɚ ɫɥɟɞɨɜɚɥɨ ɛɵ ɨɩɢɫɵɜɚɬɶ ɜɨɥɧɨɜɵɦ ɩɚɤɟɬɨɦ, ɨɞɧɚɤɨ ɩɪɢ
ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜɦɟɫɬɨ ɜɨɥɧɨɜɨɝɨ ɩɚɤɟɬɚ ɨɛɵɱɧɨɣ ɩɥɨɫɤɨɣ ɜɨɥɧɵ, ɤɚɤ
ɜɵɹɫɧɹɟɬɫɹ, ɬɟ ɠɟ ɫɚɦɵɟ ɪɟɡɭɥɶɬɚɬɵ ɞɥɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɯɨɞɨɜ ɛɭɞɭɬ
ɩɨɥɭɱɟɧɵ ɩɭɬɟɦ ɝɨɪɚɡɞɨ ɦɟɧɟɟ ɝɪɨɦɨɡɞɤɢɯ ɜɵɱɢɫɥɟɧɢɣ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
ɡɚɩɢɲɟɦ ɧɚɱɚɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ
i ɜɫɟɣ ɫɢɫɬɟɦɵ ɜ ɜɢɞɟ
ni
c
c
p , (3)
ɝɞɟ
p
c
– ɧɚɱɚɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ ɩɚɞɚɸɳɟɝɨ ɧɚ ɚɬɨɦ ɷɥɟɤɬɪɨɧɚ ɫ ɷɧɟɪɝɢɟɣ
, ɨɩɢɫɵɜɚɟɦɨɟ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɟɣ
mp 2/
2
c
H
c
p
=/2/1 rp
c
i
e
V
; n
c
– ɧɚɱɚɥɶɧɨɟ
ɫɨɫɬɨɹɧɢɟ ɚɬɨɦɚ, n
c
– ɩɨɥɧɵɣ ɧɚɛɨɪ ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ
ɫɬɚɰɢɨɧɚɪɧɨɟ ɫɨɫɬɨɹɧɢɟ ɷɥɟɤɬɪɨɧɧɨɣ ɫɢɫɬɟɦɵ ɚɬɨɦɚ ɫ ɷɧɟɪɝɢɟɣ ɢ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɟɣ .
n
E
c
),...,(
1 Zn
rr
c
\
ɉɨ ɩɪɨɲɟɫɬɜɢɢ ɧɟɤɨɬɨɪɨɝɨ ɜɪɟɦɟɧɢ ɪɚɫɫɟɢɜɚɟɦɵɣ ɷɥɟɤɬɪɨɧ ɢɫɩɵɬɵɜɚɟɬ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫ ɚɬɨɦɨɦ ɢ ɡɚɬɟɦ ɦɨɠɟɬ ɛɵɬɶ ɨɛɧɚɪɭɠɟɧ ɫɧɨɜɚ ɜɞɚɥɢ ɨɬ ɚɬɨɦɚ
ɤɚɤ ɱɚɫɬɢɰɚ, ɫɜɨɛɨɞɧɨ ɞɜɢɠɭɳɚɹɫɹ ɫ ɬɟɦ ɢɥɢ ɢɧɵɦ ɢɦɩɭɥɶɫɨɦ ɢ ɷɧɟɪɝɢɟɣ
. Ʉ ɷɬɨɦɭ ɠɟ ɜɪɟɦɟɧɢ ɚɬɨɦ ɦɨɠɟɬ ɛɵɬɶ ɨɛɧɚɪɭɠɟɧ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ
ɫɨɫɬɨɹɧɢɢ
p
mp 2/
2
H
p
n , ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɫɨɜɩɚɞɚɸɳɟɦ ɫ ɧɚɱɚɥɶɧɵɦ ɫɨɫɬɨɹɧɢɟɦ.
ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɧɚɱɚɥɶɧɵɦ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ ɚɬɨɦɚ, ɬɨ ɜ ɪɟɡɭɥɶɬɚɬɟ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɩɚɞɚɸɳɢɦ ɢɡɜɧɟ ɷɥɟɤɬɪɨɧɨɦ ɚɬɨɦ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ
ɜɟɪɨɹɬɧɨɫɬɶɸ ɩɟɪɟɣɞɟɬ ɜ ɨɞɧɨ ɢɡ ɜɨɡɛɭɠɞɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɢɥɢ ɛɭɞɟɬ
ɢɨɧɢɡɨɜɚɧ, ɢɥɢ ɬɚɤ ɢ ɨɫɬɚɧɟɬɫɹ ɜ ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ. Ⱥɦɩɥɢɬɭɞɵ ɜɟɪɨɹɬɧɨɫɬɢ
ɜɫɟɯ ɩɨɞɨɛɧɵɯ ɫɨɛɵɬɢɣ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɟɤɬɨɪɨɦ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ)(
,
tc
np
)(t\ . ȿɝɨ ɷɜɨɥɸɰɢɹ ɩɨɞɱɢɧɟɧɚ ɜɨɥɧɨɜɨɦɭ ɭɪɚɜɧɟɧɢɸ ɒɪɟɞɢɧɝɟɪɚ,
)(
ˆ
)( tHt
t
i \ \
w
w
=
ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (1), ɜɤɥɸɱɚɸɳɢɦ ɨɩɟɪɚɬɨɪ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ (2), ɩɪɢɱɟɦ
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ ɫɥɭɠɢɬ ɪɚɜɟɧɫɬɜɨ )(
ɱɚɧ
t\ ɧɚɱɚɥɶɧɨɦɭ ɜɟɤɬɨɪɭ
ɫɨɫɬɨɹɧɢɹ (3). Ⱥɦɩɥɢɬɭɞɵ ɜɟɪɨɹɬɧɨɫɬɢ )ɹɜɥɹɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
ɪɚɡɥɨɠɟɧɢɹ ɤɨɧɟɱɧɨɝɨ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ
(
,
tc
np
)(t\ ,
¦¦
\
p n
t)()(
,
tc
np
n
p ,
ɩɨ ɛɚɡɢɫɧɵɦ ɫɨɫɬɨɹɧɢɹɦ ɜɢɞɚ
f n
p . (4)
148
ɇɟ ɩɨɜɬɨɪɹɹ ɢɡɜɟɫɬɧɵɣ ɜɵɜɨɞ ɮɨɪɦɭɥ ɞɥɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɟɪɟɯɨɞɨɜ ɜ
ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ, ɫɪɚɡɭ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɡɨɥɨɬɵɦ
ɩɪɚɜɢɥɨɦ, ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɦɭ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ
f
i o ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ
ɟɫɬɶ
w
)(
2
2
ifif
M HU
S
=
, ɝɞɟ
fi
H
H
. (5)
ɍɫɥɨɜɢɟ ɪɚɜɟɧɫɬɜɚ ɷɧɟɪɝɢɣ ɤɨɧɟɱɧɨɝɨ ɢ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɣ ɜɫɟɣ ɫɢɫɬɟɦɵ
( – ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ) ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ ɢɦɟɟɬ ɜɢɞ:
if
H H
n
Emp 2/
2
n
Emp
c
c
2/
2
. (6)
Ɇɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɩɟɪɟɯɨɞɚ
f
i o ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɬɟɨɪɢɢ
ɜɨɡɦɭɳɟɧɢɣ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɜɨɥɧɨɜɵɦ ɮɭɧɤɰɢɹɦ ɫɨɫɬɨɹɧɢɣ (3) ɢ (4) ɫ ɭɱɟɬɨɦ
ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ (2):
iVfM
if
ˆ
³
r
3
1
d
V
³
¸
¸
¹
·
¨
¨
©
§
3
i
Z
i
d r
3
1
rq
rrrr
c
\\
i
Z
n
Zn
e),...,(),...,(
11
¸
¸
¹
·
¨
¨
©
§
¦
Z
i
i
r
Zee
1
22
rr
. (7)
ɝɞɟ
=/)( ppq
c
. (8)
Ɉɝɪɚɧɢɱɢɦɫɹ ɪɚɫɫɦɨɬɪɟɧɢɟɦ ɩɪɨɰɟɫɫɨɜ ɭɩɪɭɝɨɝɨ ɪɚɫɫɟɹɧɢɹ.
ɋɬɨɥɤɧɨɜɟɧɢɟ ɱɚɫɬɢɰ ɧɚɡɵɜɚɟɬɫɹ
ɭɩɪɭɝɢɦ, ɟɫɥɢ ɜ ɩɪɨɰɟɫɫɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɧɟ
ɩɪɨɢɫɯɨɞɢɬ ɩɪɟɜɪɚɳɟɧɢɹ ɢɫɯɨɞɧɵɯ ɱɚɫɬɢɰ ɜ ɧɨɜɵɟ ɱɚɫɬɢɰɵ ɢ ɧɟ ɦɟɧɹɟɬɫɹ
ɜɧɭɬɪɟɧɧɟɟ ɫɨɫɬɨɹɧɢɟ ɫɬɚɥɤɢɜɚɸɳɢɯɫɹ ɱɚɫɬɢɰ. ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɫɨɭɞɚɪɟɧɢɹɦ
ɷɥɟɤɬɪɨɧɚ ɫ ɚɬɨɦɨɦ ɷɬɨ ɨɩɪɟɞɟɥɟɧɢɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɚɤɬɟ ɭɩɪɭɝɨɝɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɷɥɟɤɬɪɨɧɚ ɢ ɚɬɨɦɚ ɦɟɧɹɟɬɫɹ ɬɨɥɶɤɨ ɜɟɤɬɨɪ ɢɦɩɭɥɶɫɚ
ɪɚɫɫɟɢɜɚɟɦɨɝɨ ɷɥɟɤɬɪɨɧɚ, ɚ ɤɜɚɧɬɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɚɬɨɦɚ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɵɦ:
nn
c
. (9)
ɂɡ ɪɚɜɟɧɫɬɜɚ (6) ɩɪɢ ɷɬɨɦ ɫɥɟɞɭɟɬ, ɱɬɨ ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɧɚ ɢ ɜɟɥɢɱɢɧɚ ɟɝɨ
ɢɦɩɭɥɶɫɚ ɧɟ ɦɟɧɹɸɬɫɹ ɩɪɢ ɭɩɪɭɝɨɦ ɪɚɫɫɟɹɧɢɢ:
pp
c
. (10)
ɂɡɦɟɧɹɟɬɫɹ ɥɢɲɶ ɧɚɩɪɚɜɥɟɧɢɟ ɢɦɩɭɥɶɫɚ ɪɚɫɫɟɢɜɚɟɦɨɝɨ ɷɥɟɤɬɪɨɧɚ.
Ɇɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ (7) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
VuM
if
/)(q , (11)
149