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Файл:Функции Грина и проводимость. Учебное пособие
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ɋɬɚɪɬɭɟɦ ɫ ɮɨɪɦɭɥɵ (4.65) ɢ ɫɪɚɡɭ ɩɟɪɟɩɢɲɟɦ ɟɟ ɞɥɹ ɭɫɪɟɞɧɟɧɧɵɯ ɜɟɥɢɱɢɧ
(ɡɞɟɫɶ ݀ൌ͵, ɢ ɦɵ ɫɦɟɧɢɥɢ ɨɛɨɡɧɚɱɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɧɚ ɛɨɥɟɟ ɩɪɢɜɵɱɧɵɟ ɞɥɹ
ɬɪɟɯɦɟɪɧɨɝɨ ɫɥɭɱɚɹ)
ఈ
r
൯ൈ
భ
ሺ
ͷǤ͵ͻ
ሺ
ͷǤͶͲ
ሻ
ۧ
ߪሺߗሻൌ
ଶ
݁
ʹͶߨ݉
ߪሺߗሻൌۦߪሺߗǡr
Ԝන݀
ଶ
Ԝൈۦܩ
ଷ
r
Ԝන ݀
ଵ
ିஶ
ோ
ሺ
߱ߗǡrǡr
ஶ
߱Ԝቆ
݀݊
ଵ
݀߱
ሻ
ܩ
ி
ሺ߱ሻ
ሺ
߱ǡr
ǡ
ቇሺ
ଵ
ǡr
ఈ
r
െ
ሻ
ۧ
ఈ
ሻ
r
ǡ
ɝɞɟ ɩɨɫɥɟ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚɦɢ ɧɭɠɧɨ ɩɨɥɨɠɢɬɶ rൌr ɢ r
ఈ
൫
r
ଵൌrଵ
భ
െ
.
Ɋɟɡɭɥɶɬɚɬ ɭɫɪɟɞɧɟɧɢɹ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ ɝɪɚɮɢɤɨɜ, ɢɦɟɸɳɢɯ ɜɢɞ
ɩɟɬɥɢ, ɩɨɤɚɡɚɧɧɨɣ ɧɚ ɪɢɫ. 5.7. ȼ ɷɬɨɣ ɩɟɬɥɟ ɜɟɪɯɧɟɣ ɥɢɧɢɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɮɭɧɤ-
ɰɢɹ Ƚɪɢɧɚ ܩ
ோ
ሺ
߱ߗǡrǡr
ሻ
ɚ ɧɢɠɧɟɣ ܩ
ଵ
ሺ
߱ǡr
ሻ
ǡr
(ɦɵ ɜɵɩɢɫɚɥɢ ɤɨɨɪɞɢɧɚɬɵ ɩɨ-
ଵ
ɫɥɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ).
Ɋɢɫ. 5.7
ሻ
ሻ
Ƚɪɚɮɢɤɢ ɩɨɥɭɱɚɸɬɫɹ ɢɡ ɷɬɨɣ ɩɟɬɥɢ ɩɪɢɤɪɟɩɥɟɧɢɟɦ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ ɜɫɟɜɨɡɦɨɠɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ɪɢɫ. 5.8.
Ɋɢɫ. 5.8
Ɍɚɤ ɠɟ ɤɚɤ ɢ ɩɪɢ ɭɫɪɟɞɧɟɧɢɢ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ, ɩɪɢ ɭɫɥɨɜɢɢ ݒ
ɤɚɦɢ ɫ ɩɟɪɟɫɟɤɚɸɳɢɦɢ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɬɚɤ ɤɚɤ ɢɯ
ɜɤɥɚɞ ɦɚɥ ɜ ɨɬɧɨɲɟɧɢɢ
ሺ
ݒ
߬
ିଵ
ሻ
ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɝɪɚɮɢɤɚɦɢ
߬
بͳ ɝɪɚɮɢ-
ɛɟɡ ɩɟɪɟɫɟɱɟɧɢɣ.
Ɂɚɦɟɬɢɦ, ɱɬɨ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɫɪɟɞɧɟɟ
ோ
ɪɚɜɧɨ ɩɪɨɢɡɜɟɞɟɧɢɸ ɫɪɟɞɧɢɯ
ۦ
ܩ
ሺ
߱ߗ
ۦ
ሻ
ۧ
ோ
ܩ
ۦ
ሺ
߱ߗሻܩ
ܩ
ሺ߱ሻ
ۧ
ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɧɟ
ሺ߱ሻ
ۧ
. ȼ ɩɪɢɛɥɢɠɟɧɢɢ ɧɟɩɟɪɟ-
ɫɟɤɚɸɳɢɯɫɹ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ ɜ ɪɚɡɧɨɫɬɶ ɷɬɢɯ ɫɪɟɞɧɢɯ ɞɚɸɬ ɜɤɥɚɞɵ ɞɢɚɝɪɚɦɦɵ,
51
ɫɨɞɟɪɠɚɳɢɟ ɩɪɢɦɟɫɧɵɟ ɥɢɧɢɢ, ɫɨɟɞɢɧɹɸɳɢɟ ܩ
ோ
ሺ
߱ߗሻ ɢ ܩ
ሺ߱ሻ
, ɩɨɞɨɛɧɵɟ
ɜɬɨɪɨɣ ɢ ɞɢɚɝɪɚɦɦɟ ɧɚ ɪɢɫ. 5.8. ɇɚɡɨɜɟɦ ɭɫɥɨɜɧɨ ɬɚɤɢɟ ɥɢɧɢɢ «ɩɨɩɟɪɟɱɧɵɦɢ».
Ɉɞɧɚɤɨ ɩɨ ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɨɛɪɚɠɟɧɢɹɦ, ɫɧɚɱɚɥɚ ɜɵɱɢɫɥɢɦ ɩɪɨɜɨɞɢɦɨɫɬɶ,
ɩɪɟɧɟɛɪɟɝɚɹ ɷɬɨɣ ɪɚɡɧɨɫɬɶɸ, ɢ ɩɪɨɫɥɟɞɢɦ, ɤɚɤ ɩɨɥɭɱɚɟɬɫɹ ɮɨɪɦɭɥɚ Ⱦɪɭɞɟ ɛɟɡ
ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɠɧɟɧɢɣ. Ⱥ ɭɠɟ ɩɨɫɥɟ ɷɬɨɝɨ ɭɱɬɟɦ ɝɪɚɮɢɤɢ ɫ «ɩɨɩɟɪɟɱɧɵɦɢ» ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ
ߪሺߗ
݁
ሻ
ൌ
ʹͶߨ݉
ൈۦܩ
ଶ
Ԝන݀
ଶ
ଷ
r
Ԝන ݀
ଵ
ோ
ሺ
߱ߗǡrǡr
ஶ
ିஶ
߱Ԝቆ
ଵ
ሺ߱ሻ
݀݊
ி
ቇሺ
݀߱
ሻ
ሺ
ۦ
ۧ
ܩ
߱ǡr
ఈ
ఈ
ሻ
൫
െ
r
r
ሻ
ۧ
ǡr
Ǥ
ଵ
ఈ
ఈ
െ
r
൯ൈԜ
r
భ
భ
ሺ
ͷǤͶͳ
ɍɫɪɟɞɧɟɧɧɵɟ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɬɪɚɧɫɥɹɰɢɨɧɧɨ-ɢɧɜɚɪɢɚɧɬɧɵ, ɢ ɩɨɷɬɨɦɭ
ଷ
p
݀
ଵ
ሺ
ሻ
rିr
ோ
ሺ
ۦ
߱ߗǡrǡr
ܩ
ሺ
ۦ
ܩ
ோ
൫߱ߗǡpଵ൯ ɢ ܩ൫߱ǡpଶ൯ ɞɚɸɬɫɹ ɜɵɪɚɠɟɧɢɹɦɢ (5.36) ɢ (5.37). ɉɨɞɫɬɚɜɥɹɟɦ
ɝɞɟ ܩ
߱ǡr
ሻ
ۧ
ൌන
ଵ
ሻ
ۧ
ǡr
ଵ
ൌන
ሺʹߨሻ
ሺʹߨሻ
ɜ (5.41), ɨɩɟɪɚɬɨɪɵ ɞɚɸɬ ɦɧɨɠɢɬɟɥɶ െ൫p
ɩɨ r
ଵ
ߪሺߗሻൌെ
ɞɚɟɬ
ߜ൫pଵെpଶ൯, ɜ ɢɬɨɝɟ ɩɨɥɭɱɢɦ
ஶ
ଶ
݁
ߨ݉
Ԝන ݀
ଶ
ൌെ
߱Ԝቆ
ିஶ
ߨ݉
݁
ଶ
ଶ
ሺʹߨሻ
݀݊
ி
݀߱
ሺ߱ሻ
Ԝ
ଷ
ቇන
ସ
Ͷߨ
ݒ
ଷ
ሺʹߨሻ
ൈන
ቀ߱ߗെߦ
ଷ
ଷ
݀
p
ଶ
ଷ
݀
ሺʹߨሻ
න݀
ିஶ
݀ߦ
݅
ቁቀ߱െߦെ
ʹ߬
p
భ
݁
p
మ
݁
ଵpଶ
ଷ
p
ଷ
ஶ
߱Ԝቆ
భ
ܩோ൫߱ߗǡpଵ൯ǡ
ሺ
ሻ
r
ିr
భ
ܩ൫߱ǡpଶ൯ǡ
൯ఈ൫pଵpଶ൯
ோ
݀݊
ி
ሺ
ሺ߱ሻ
ȁpȁଶԜԜܩ
݀߱
݅
ʹ߬
ఈ
ɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ
߱ߗǡpሻܩ
ቇൈ
Ǥ
ቁ
ሺ
߱ǡpሻൌ
ሺ
ͷǤͶʹ
ሺ
ͷǤͶ͵
ሺ
ͷǤͶͶ
ɉɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥ ɪɚɜɟɧ
ʹߨ߬
ͳെ݅߬ߗ
ǡ
ሺ
ͷǤͶͷ
ɤɪɨɦɟ ɬɨɝɨ, ɢɦɟɟɦ
ஶ
න݀
ିஶ
ɢ
Ȁ݉ݒൌͳ, ɜ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ
߱Ԝቆ
ሺ߱ሻ
݀݊
ி
݀߱
ߪሺߗሻൌ
ቇൌ݊
ଶ
݁
ߨଶ݉
ி
ሺλሻ
ଷ
െ݊
߬
Ԝ
ͳെ݅߬ߗ
ሺെλሻ
ி
Ǥ
ൌെͳǡ
ሺ
ͷǤͶ
ሺ
ͷǤͶ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
52
ɂɦɟɹ ɜɜɢɞɭ ɱɬɨ ɞɥɹ ɷɥɟɤɬɪɨɧɨɜ ɛɟɡ ɭɱɟɬɚ ɫɩɢɧɚ
ଶ
ߪሺߗሻൌ
ߪ
ͳെ݅߬ߗ
ǡԜԜԜߪൌ
݁
݊߬
݉
Ȁߨଶൌ݊
ǡ
, ɩɨɥɭɱɚɟɦ
ሺ
ͷǤͶͺ
ሻ
ଷ
ɷɬɨ ɟɫɬɶ ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɚɹ ɮɨɪɦɭɥɚ Ⱦɪɭɞɟ.
ȼɨɨɛɳɟ-ɬɨ ɮɨɪɦɭɥɭ Ⱦɪɭɞɟ «ɲɤɨɥɶɧɵɦ» ɦɟɬɨɞɨɦ ɦɨɠɧɨ ɜɵɜɟɫɬɢ ɜ ɞɜɟ
ɫɬɪɨɱɤɢ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ ɧɚ ɷɥɟɤɬɪɨɧ ɞɟɣɫɬɜɭɟɬ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ȿ, ɬɨ-
ɝɞɚ ɭɫɤɨɪɟɧɢɟ, ɤɨɬɨɪɨɟ ɩɪɢɨɛɪɟɬɚɟɬ ɷɥɟɤɬɪɨɧ, ɟɫɬɶ ݁ࡱȀ݉, ɭɫɤɨɪɹɹɫɶ ɷɥɟɤɬɪɨɧ ɡɚ
ɜɪɟɦɹ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ߬ ɩɪɢɨɛɪɟɬɚɟɬ ɫɤɨɪɨɫɬɶ ࢜
ൌ݁ࡱ߬Ȁ݉, ɛɭɞɟɦ ɫɱɢɬɚɬɶ
ఛ
ɷɬɭ ɫɤɨɪɨɫɬɶ ɫɤɨɪɨɫɬɶɸ ɞɪɟɣɮɚ ɷɥɟɤɬɪɨɧɨɜ ɜ ɩɨɥɟ, ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ ɫɜɹɡɚɧɚ
ɫɨ ɫɤɨɪɨɫɬɶɸ ɞɪɟɣɮɚ ൌ݁݊
ଶ
ൌሺ݁
݊߬Ȁ݉ሻࡱൌߪࡱ, ɨɬɤɭɞɚ ɩɨɥɭɱɚɟɦ ɮɨɪɦɭɥɭ (5.48).
࢜ఛ
, ɝɞɟ ݊
ɤɨɧɰɟɧɬɪɚɰɢɹ ɷɥɟɤɬɪɨɧɨɜ, ɜ ɢɬɨɝɟ
Ȼɨɥɟɟ ɫɬɪɨɝɢɣ ɜɵɜɨɞ ɮɨɪɦɭɥɵ Ⱦɪɭɞɟ ɨɫɧɨɜɚɧ ɧɚ ɤɢɧɟɬɢɱɟɫɤɨɦ ɭɪɚɜɧɟɧɢɢ.
Ɇɵ ɷɬɨɬ ɜɵɜɨɞ ɡɞɟɫɶ ɜɨɫɩɪɨɢɡɜɨɞɢɬɶ ɧɟ ɛɭɞɟɦ, ɨɬɦɟɬɢɦ ɬɨɥɶɤɨ, ɱɬɨ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɢɧɬɟɝɪɚɥɚ ɫɬɨɥɤɧɨɜɟɧɢɣ ɜ ɤɢɧɟɬɢɱɟɫɤɨɦ ɭɪɚɜɧɟɧɢɢ ɫɱɢɬɚɟɬɫɹ ɡɚɞɚɧɧɵɦ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ ɷɥɟɤɬɪɨɧɚ ɧɚ ɩɪɢɦɟɫɢ
ɦɟɧɬ ɬɟɥɟɫɧɨɝɨ ɭɝɥɚ ݀ߗൌʹߨߠ݀ߠ, ɝɞɟ ߠ
ɭɝɨɥ ɪɚɫɫɟɹɧɢɹ. ɉɪɢ ɪɟɲɟɧɢɢ ɤɢɧɟ-
ሺ
݀ߪȀ݀ߗሻሺߠ
ሻ
ɜ ɷɥɟ-
ɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɨɫɩɪɨɢɡɜɨɞɢɬɫɹ ɮɨɪɦɭɥɚ (5.48), ɨɞɧɚɤɨ ɜɟɥɢɱɢɧɚ ߬, ɜɯɨɞɹɳɚɹ ɜ ɷɬɭ ɮɨɪɦɭɥɭ, ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ ߬
ͳ
ൌ݊ݒߪ௧ǡ
߬
௧
ɝɞɟ ߪ
ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ ɩɨɥɧɨɟ ɬɪɚɧɫɩɨɪɬɧɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ ɷɥɟɤɬɪɨɧɚ ɧɚ
௧
, ɬɚɤɨɣ, ɱɬɨ
௧
ሺ
ͷǤͶͻ
ሻ
ɩɪɢɦɟɫɢ, ɪɚɜɧɨɟ
ߪ
௧
ൌන൬
݀ߪ
൰ሺͳെߠሻ݀ߗǤ
݀ߗ
ሺ
ͷǤͷͲ
ሻ
5.3. Ɏɨɪɦɭɥɚ Ⱦɪɭɞɟ ɫ ɬɪɚɧɫɩɨɪɬɧɵɦ ɜɪɟɦɟɧɟɦ
ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ
ɂɫɩɨɥɶɡɭɹ ɬɟɯɧɢɤɭ ɭɫɪɟɞɧɟɧɢɹ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɩɨ ɛɟɫɩɨɪɹɞɤɭ, ɦɵ ɩɨɥɭɱɢɥɢ
ɮɨɪɦɭɥɭ Ⱦɪɭɞɟ ɫ ɜɟɥɢɱɢɧɨɣ ߬, ɨɩɪɟɞɟɥɹɟɦɨɣ ɮɨɪɦɭɥɨɣ (5.31), ɝɞɟ ɫɟɱɟɧɢɟ (5.29),
ɜɯɨɞɹɳɟɟ ɜ ɧɟɟ, ɟɫɬɶ ɨɛɵɱɧɨɟ, ɚ ɧɟ ɬɪɚɧɫɩɨɪɬɧɨɟ ɫɟɱɟɧɢɟ. Ɉɤɚɡɵɜɚɟɬɫɹ, ɷɬɨ ɩɪɨɢɡɨɲɥɨ ɨɬ ɬɨɝɨ, ɱɬɨ ɦɵ ɧɟ ɭɱɥɢ ɞɢɚɝɪɚɦɦɵ, ɫɨɞɟɪɠɚɳɢɟ ɩɪɢɦɟɫɧɵɟ ɥɢɧɢɢ, ɫɨ-
ɟɞɢɧɹɸɳɢɟ ܩ
ȼɟɪɧɟɦɫɹ ɤ ɮɨɪɦɭɥɟ (5.40), ɭɱɢɬɵɜɚɹ ɩɪɢɨɛɪɟɬɟɧɧɵɣ ɨɩɵɬ, ɟɟ ɦɨɠɧɨ ɢɡɦɟ-
ɧɢɬɶ, ɜɨ-ɩɟɪɜɵɯ, ɩɨɫɥɟ ɭɫɪɟɞɧɟɧɢɹ ɤɨɦɩɥɟɤɫ ɢɡ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɮɚɤɬɢɱɟɫɤɢ ɧɟ
ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɨɬ ɱɚɫɬɨɬɵ ߱, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɩɨ ɧɟɣ ɫɪɚɡɭ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ, ɷɬɨ
ɞɚɫɬ ɦɧɨɠɢɬɟɥɶ െͳ.
ߪሺߗሻൌെ
ோ
ଶ
݁
ʹͶߨ݉
ሺ
߱ߗሻ ɢ ܩ
ଷ
Ԝන݀
r
ଶ
ଵ
Ԝሺ
ሺ߱ሻ
. ɉɪɨɞɟɥɚɟɦ ɷɬɨ ɫɟɣɱɚɫ.
ఈ
r
െ
ఈ
ሻ
൫
r
ఈ
r
భ
െ
ఈ
r
భ
൯Ԝۦܩ
ோ
ሺ
ߗǡrǡr
ሻ
ሺ
ܩ
ଵ
Ͳǡr
ሻ
ۧ
ǡr
ଵ
ǡሺͷǤͷͳ
ሻ
53
ɜɨ-ɜɬɨɪɵɯ, ɯɨɬɹ ɬɟɧɡɨɪɧɚɹ ɫɬɪɭɤɬɭɪɚ ɩɪɨɜɨɞɢɦɨɫɬɢ ɬɪɢɜɢɚɥɶɧɚ, ɦɟɬɨɞɢɱɟɫɤɢ
ɭɞɨɛɧɟɟ ɛɭɞɟɬ ɟɟ ɜɨɫɫɬɚɧɨɜɢɬɶ:
ଶ
݁
ߪ
ఈఉ
ሺߗሻ
ൌെ
ͺߨ݉
ଷ
ଶ
Ԝන݀
ఈ
r
Ԝሺ
െ
ଵ
r
ఉ
ఈ
ሻ
ቀ
r
ఉ
ோ
െ
r
భ
ሺ
ߗǡrǡr
Ԝۦܩ
ቁ
r
భ
ሻ
ሺ
Ͳǡrଵǡr
ሻۧ
ǤሺͷǤͷʹ
ܩ
ଵ
ሻ
ɉɟɪɟɣɞɟɦ ɜ ɷɬɨɣ ɮɨɪɦɭɥɟ ɤ ɢɦɩɭɥɶɫɧɨɦɭ ɩɪɟɞɫɬɚɜɥɟɧɢɸ, ɩɪɢ ɷɬɨɦ ɭɱɬɟɦ,
ɱɬɨ ɫɪɟɞɧɟɟ
ൌන
ோ
ۦ
ܩ
ଷ
݀
ሺʹߨሻ
p
ሺ
߱ߗǡrǡr
ۦܩ
ଷ
݀
p
ଵ
ଵ
ଷ
ଷ
ሺʹߨሻ
ൈ݁
ଵ
ோ
ሺ
ଷ
݀
p
ሺʹߨሻ
pభrାp
ሻ
ሺ
ܩ
߱ǡr
߱ߗǡrǡr
ଷ
݀
p
ଷ
ଶ
ଷ
ଷ
ሺʹߨሻ
r
ାpయrభାp
భ
మ
ሻ
ۧ
ǡr
ɬɪɚɧɫɥɹɰɢɨɧɧɨ ɢɧɜɚɪɢɚɧɬɧɨ
ଵ
ሻ
ሺ
ܩ
ଵ
ଷ
݀
p
ସ
ሺʹߨሻ
ଷ
ሺʹߨሻ
r
ర
Ԝߎఆ൫pଵǡpଶǡpଷǡpସ൯ǡ
ሻ
߱ǡr
ۧ
ǡr
ଵ
ଷ
ൌ
Ԝߜ൫pଵpଶpଷpସ൯ൈ
ሺ
ͷǤͷ͵
ɩɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ (5.52), ɫɧɢɦɚɹ ɥɢɲɧɢɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɫ ɩɨɦɨɳɶɸ ߜ ɮɭɧɤɰɢɣ,
ɢ ɩɟɪɟɨɛɨɡɧɚɱɚɹ ɢɦɩɭɥɶɫɵ, ɩɨɥɭɱɢɦ
ଷ
ଷ
p
݀
݀
p
ଶ
ሺʹߨሻ
ଷ
ሺʹߨሻ
ଵ
Ԝන
ఉ
ఈ
p
Ԝߎఆ൫pଵǡെpǡpǡെpଵ൯Ǥ
ଷԜpଵ
ሺ
ͷǤͷͶ
ߪ
ఈఉ
ሺߗሻ
ൌ
ଶ
݁
ʹߨ݉
Ⱦɥɹ ɞɢɚɝɪɚɦɦɵ ɛɟɡ «ɩɨɩɟɪɟɱɧɵɯ» ɥɢɧɢɣ ɥɟɝɤɨ ɭɜɢɞɟɬɶ
ߎ
ሺሻ
൫pଵǡെpǡpǡെpଵ൯ൌሺʹߨ
ఆ
ଷ
ሻ
Ԝߜ൫pଵെp൯ܩ
ோ
ሺ
ߗǡpሻܩ
ሺ
Ͳǡpሻǡ
ሺ
ͷǤͷͷ
ɩɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ (5.54), ɩɨɥɭɱɢɦ ɫɬɚɪɵɣ ɪɟɡɭɥɶɬɚɬ (5.44):
ଶ
݁
ൌ
ʹߨ݉
ሺሻ
ሺߗሻ
ߪ
ఈఉ
ଷ
݀
ఈpఉԜܩோ
Ԝන
ଶ
ሺʹߨሻ
Ԝp
ଷ
ሺ
ߗǡpሻܩ
ሺ
ͲǡpሻǤ
ሺ
ͷǤͷ
Ⱦɢɚɝɪɚɦɦɚ ɫ ɨɞɧɨɣ «ɩɨɩɟɪɟɱɧɨɣ» ɥɢɧɢɟɣ ɞɚɟɬ
݊
ሺଵሻ
ߎ
൫pଵǡെpǡpǡെpଵ൯ൌ
ఆ
หݑො൫pଵെp൯หଶܩோ൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯ܩ
ଶ
ோ
ሺ
ߗǡpሻܩ
ሺ
ͲǡpሻǡሺͷǤͷ
ሻ
ሻ
ሻ
ሻ
ሻ
ɟɫɥɢ ɩɨɞɫɬɚɜɢɬɶ ɷɬɨ ɜ (5.54), ɬɨ ɦɨɠɧɨ ɨɬɧɟɫɬɢ ɜɨɡɧɢɤɲɢɣ ɜ ɢɧɬɟɝɪɚɥɟ ɛɥɨɤ
ଷ
p
݀
න
ሺʹߨሻ
ɤɚɤ «ɨɞɟɜɚɧɢɟ» ɢɦɩɭɥɶɫɚ p
݊
ଵ
ఈ
หݑො൫pଵെp൯หଶܩோ൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯
ଷpଵ
ଶ
ఈ
ɜ ɞɢɚɝɪɚɦɦɟ ɛɟɡ «ɩɨɩɟɪɟɱɧɵɯ» ɥɢɧɢɣ (5.56) ɩɨ ɩɪɚ-
ሺ
ɜɢɥɭ
ଷ
݀
p
݊
p
ఈ՜pఈ
න
ሺʹߨሻ
ଵ
ଷpଵ
ఈ
หݑො൫pଵെp൯หଶܩோ൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯Ǥ
ଶ
ሺ
ɗɬɨɦɭ ɜɵɪɚɠɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɞɜɟ ɞɢɚɝɪɚɦɦɵ ɧɚ ɪɢɫ. 5.9.
54
ͷǤͷͺ
ͷǤͷͻ
ሻ
ሻ
Ɋɢɫ. 5.9
Ⱦɨɛɚɜɥɹɹ ɬɟɩɟɪɶ ɬɚɤɢɟ ɠɟ ɞɢɚɝɪɚɦɦɵ ɫ ɞɜɭɦɹ, ɬɪɟɦɹ ɢ ɬ. ɞ. ɩɨɩɟɪɟɱɧɵɦɢ ɥɢ-
ɧɢɹɦɢ, ɦɵ ɩɨɥɭɱɢɦ ɜɟɥɢɱɢɧɭ ߁
ɩɭɥɶɫ p
ఈ
:
ሺߗሻ
ߪ
ఈఉ
ɍɪɚɜɧɟɧɢɟ ɞɥɹ ߁
ൌ
ఈ
ʹߨ݉
ሺ
ߗǡp
݁
ఈ
ሺ
ሻ
ߗǡp
, ɤɨɬɨɪɚɹ ɜ ɜɵɪɚɠɟɧɢɢ (5.56) ɡɚɦɟɧɢɬ ɢɦ-
ଶ
ଷ
݀
ఈ
න
ଶ
ሺʹߨሻ
ሻ
ɥɟɝɤɨ ɜɵɩɢɫɚɬɶ, ɬɚɤ ɤɚɤ ɪɹɞ ɞɥɹ ɧɟɟ ɮɚɤɬɢɱɟɫɤɢ ɹɜ-
ሺ
Ԝ߁
ଷ
ߗǡpሻԜp
ఉԜܩோ
ሺ
ߗǡpሻܩ
ሺ
ͲǡpሻǤ
ɥɹɟɬɫɹ ɫɭɦɦɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ. Ɉɧɨ ɢɦɟɟɬ ɜɢɞ
ఈ
ሺ
ߗǡpሻൌp
߁
ఈ
ଶ
න
ଵ
หݑො൫pଵെp൯หଶܩோ൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯Ԝ߁ఈ൫ߗǡpଵ൯ǡሺͷǤͳ
ଷ
ሺʹߨሻ
ଷ
p
݀
݊
ɜ ɞɢɚɝɪɚɦɦɧɨɦ ɜɢɞɟ ɨɧɨ ɢɡɨɛɪɚɠɟɧɨ ɧɚ ɪɢɫ. 5.10.
Ɋɢɫ. 5.10
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ ߁
ɥɹɟɦ ɷɬɨ ɜ (5.61) ɢ ɭɦɧɨɠɚɟɦ ɫɤɚɥɹɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚ pȀ
ଷ
p
ߛሺߗሻൌͳ
݊
ଶ
ȁpȁ
݀
ଶ
න
ሺʹߨሻ
ଵ
ఈ
ఈ
൫p
p
ଷ
൯หݑො൫pଵെp൯หଶܩோ൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯ԜߛሺߗሻǡሺͷǤʹ
ଵ
ఈ
ሺ
ߗǡpሻൌp
ȁpȁ
ߛሺߗ
ఈ
ଶ
, ɩɨɥɭɱɢɦ
ሻ
, ɩɨɞɫɬɚɜ-
ሺ
ͷǤͲ
ሻ
ሻ
ሻ
ɦɨɞɭɥɢ ɢɦɩɭɥɶɫɨɜ p ɢ p
ɛɥɢɡɤɢ ɤ , ɩɪɢ ɷɬɨɦ ൫p
ଵ
ଶ
݉
ݒ
ଷ
p
݀
ଵ
ൌ
Ԝʹߨߠ݀ߠ݀ߦǤ
ଶ
55
ఈ
p
ଶ
ఈ
ȁpȁ
൯Ȁ
ଵ
ൌߠ, ɚ ɬɚɤɠɟ
ሺ
ͷǤ͵
ሻ
గ
Ԝන݀ߠ
ଶ
ோ
൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯Ǥ
ʹߨԜߠԜԜ݀ߦߠหݑො൫pଵെp൯หଶൈ
ߛሺߗሻൌͳߛሺߗ
ሻ
ͺߨଷ
݊
ൈܩ
݉ଶݒ
ଶ
ɍɱɬɟɦ, ɱɬɨ
గ
ଶ
݉
ൌ
ߪ
௧
ʹߨ
න݀ߠ
ସ
ԜߠԜሺͳെߠሻหݑො൫pଵെp൯ห
ଶ
ɢ ɩɨɷɬɨɦɭ
గ
ଶ
݉
න݀ߠ
ʹߨ
ସ
ԜߠԜߠหݑො൫pଵെp൯หଶൌߪെߪ௧ǡ
ɚ ɬɚɤɠɟ
න݀ߦܩ
ோ
൫ߗǡpଵ൯ܩ൫Ͳǡpଵ൯ൌ
ʹߨ߬
ͳെ݅߬ߗ
ǡ
ɩɨɥɭɱɢɦ:
Ɉɛɨɡɧɚɱɚɹ ͳȀ߬
ߛሺߗሻൌͳߛሺߗ
௧ൌ݊ݒߪ௧
ߛሺߗሻൌͳߛሺߗ
, ɬɨɝɞɚ
݊
ሻ
ሺ
ݒ
ͳെ݅߬ߗ
ͳെ߬Ȁ߬
ሻ
ͳെ݅߬ߗ
ߪെߪ
ሻ
߬
௧
Ǥ
௧
ǡ
ɨɬɤɭɞɚ
߬
ͳെ݅߬ߗ
݊߬
௧
߬
ͳ
Ԝ
ͳെ݅߬ߗ
ͳെ݅߬
ൌ
Ǥ
ߗ
௧
ଶ
݁
݊߬
݉
௧
Ԝ
ͳെ݅߬
ͳ
ߗ
௧
ɉɨɞɫɬɚɜɥɹɹ ߁
ఈ
ሺ
ߗǡpሻൌp
ߪሺߗሻൌߛሺߗ
ߛሺߗሻൌ
ߛሺߗሻ ɜ (5.60), ɩɨɥɭɱɢɦ ɞɥɹ ɩɪɨɜɨɞɢɦɨɫɬɢ ߪሺߗ
ఈ
ଶ
݁
ሻ
݉
ɗɬɨ ɮɨɪɦɭɥɚ Ⱦɪɭɞɟ ɫ ɬɪɚɧɫɩɨɪɬɧɵɦ ɜɪɟɦɟɧɟɦ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ.
ሺ
ሻ
ͷǤͶ
ሺ
ሻ
ͷǤͷ
ሺ
ሻ
ͷǤ
ሺ
ሻ
ͷǤ
ሺ
ሻ
ͷǤͺ
ሺ
ሻ
ͷǤͻ
ሺ
ሻ
ͷǤͲ
ሻ
ሺ
Ǥ
ͷǤͳ
ሻ
Ʌɢɬɟɪɚɬɭɪɚ
Ɍɟɯɧɢɤɚ ɭɫɪɟɞɧɟɧɢɹ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɩɨ ɛɟɫɩɨɪɹɞɤɭ ɪɚɫɫɦɨɬɪɟɧɚ ɜ [5], § 39,
ɬɚɦ ɠɟ ɞɚɧ ɞɢɚɝɪɚɦɦɧɵɣ ɜɵɜɨɞ ɮɨɪɦɭɥɵ (5.71).
Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
1. ɉɨɱɟɦɭ ɜ ɬɪɟɯɦɟɪɧɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɧɟ ɭɱɢɬɵɜɚɬɶ ɞɢɚɝɪɚɦɦɵ ɫ ɩɟɪɟɫɟɤɚ-
ɸɳɢɦɢɫɹ ɩɭɧɤɬɢɪɧɵɦɢ ɥɢɧɢɹɦɢ?
2. ȼ ɱɟɦ ɮɢɡɢɱɟɫɤɚɹ ɩɪɢɱɢɧɚ ɬɨɝɨ, ɱɬɨ ɜ ɤɪɟɫɬɨɜɨɣ ɞɢɚɝɪɚɦɦɧɨɣ ɬɟɯɧɢɤɟ
ɩɪɢɦɟɫɧɚɹ ɩɭɧɤɬɢɪɧɚɹ ɥɢɧɢɹ ɧɟ ɭɫɥɨɠɧɹɟɬɫɹ ɷɥɟɤɬɪɨɧɧɵɦɢ ɩɟɬɥɹɦɢ?
56
Ƚɥɚɜɚ 6
ɄȼȺɇɌɈȼɕȿ ɉɈɉɊȺȼɄɂ Ʉ ɉɊɈȼɈȾɂɆɈɋɌɂ
6.1. Ʉɭɩɟɪɨɧ ɢ ɤɜɚɧɬɨɜɚɹ ɩɨɩɪɚɜɤɚ ɤ ɩɪɨɜɨɞɢɦɨɫɬɢ
ɜ ɬɪɟɯɦɟɪɧɨɦ ɫɥɭɱɚɟ
ɇɚ ɩɟɪɜɵɣ ɜɡɝɥɹɞ ɦɨɠɟɬ ɜɨɡɧɢɤɧɭɬɶ ɜɨɩɪɨɫ, ɚ ɫɬɨɢɬ ɥɢ ɝɨɪɨɞɢɬɶ ɜɟɫɶ ɷɬɨɬ
ɨɝɨɪɨɞ ɫ ɭɫɪɟɞɧɟɧɢɟɦ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ ɢ ɩɪɨɜɨɞɢɦɨɫɬɢ, ɟɫɥɢ ɮɨɪɦɭɥɭ Ⱦɪɭɞɟ
ɦɨɠɧɨ ɛɟɡ ɨɫɨɛɨɝɨ ɬɪɭɞɚ ɩɨɥɭɱɢɬɶ ɢɡ ɤɢɧɟɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ. ɇɚ ɷɬɨ ɦɨɠɧɨ
ɜɨɡɪɚɡɢɬɶ, ɩɨɠɚɥɭɣ ɬɚɤ.
ȼɨ-ɩɟɪɜɵɯ, ɜɵɜɨɞ ɩɪɨɜɨɞɢɦɨɫɬɢ ɞɥɹ ɤɜɚɧɬɨɜɨɣ ɫɢɫɬɟɦɵ ɢɡ ɩɟɪɜɵɯ ɩɪɢɧɰɢɩɨɜ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɛɨɥɶɲɨɣ
ɩɪɨɫɥɟɞɢɬɶ ɡɚ ɬɟɦ, ɤɚɤ ɜɨɡɧɢɤɚɸɬ ɯɚɪɚɤɬɟɪɧɵɟ ɩɚɪɚɦɟɬɪɵ, ɫɜɹɡɚɧɧɵɟ ɫ ɤɢɧɟɬɢɤɨɣ ɢɡ ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɫɢɫɬɟɦɵ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɜ ɤɢɧɟɬɢɱɟɫɤɨɦ ɭɪɚɜɧɟɧɢɢ ɨɧɢ
ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɡɚɞɚɧɧɵɦɢ ɚɩɪɢɨɪɢ.
ȼɨ-ɜɬɨɪɵɯ, ɢɡɥɨɠɟɧɧɵɣ ɩɨɞɯɨɞ ɩɨɡɜɨɥɹɟɬ ɜɵɱɢɫɥɢɬɶ ɩɨɩɪɚɜɤɢ ɤ ɩɪɨɜɨɞɢɦɨɫɬɢ, ɢɦɟɸɳɢɟ ɤɜɚɧɬɨɜɭɸ ɩɪɢɪɨɞɭ, ɤɨɬɨɪɵɟ, ɜ ɩɪɢɧɰɢɩɟ, ɧɟɜɨɡɦɨɠɧɨ ɧɚɣɬɢ ɢɡ
ɤɢɧɟɬɢɱɟɫɤɨɝɨ
ɭɪɚɜɧɟɧɢɹ. Ʉɚɤ ɦɵ ɭɜɢɞɢɦ ɧɢɠɟ, ɜ ɬɪɟɯɦɟɪɧɨɦ ɷɥɟɤɬɪɨɧɧɨɦ ɝɚɡɟ
ɷɬɚ ɩɨɩɪɚɜɤɚ ɨɤɚɡɵɜɚɟɬɫɹ ɦɚɥɨɣ ɜ ɨɬɧɨɲɟɧɢɢ
ɨɫɧɨɜɧɨɦ ɥɢɲɶ ɦɟɬɨɞɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ. Ɉɞɧɚɤɨ ɜ ɧɢɡɤɨɪɚɡɦɟɪɧɵɯ ɫɢɫɬɟɦɚɯ ɫɢɬɭɚɰɢɹ ɤɚɪɞɢɧɚɥɶɧɨ ɦɟɧɹɟɬɫɹ. ȼ ɞɜɭɦɟɪɧɨɦ ɷɥɟɤɬɪɨɧɧɨɦ ɝɚɡɟ, ɤɚɤ ɦɵ ɭɜɢɞɢɦ,
ɤɜɚɧɬɨɜɚɹ ɩɨɩɪɚɜɤɚ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ ɱɚɫɬɨɬɟ ɩɨɥɹ, ɫɬɪɟɦɹɳɟɣɫɹ ɤ ɧɭɥɸ, ɱɬɨ ɫɜɢɞɟɬɟɥɶɫɬɜɭɟɬ ɨ ɩɪɨɹɜɥɟɧɢɢ ɫɭɳɟɫɬɜɟɧɧɨ ɤɜɚɧɬɨɜɨɝɨ ɹɜɥɟɧɢɹ ɥɨɤɚɥɢɡɚɰɢɢ ɷɥɟɤ-
ɬɪɨɧɧɵɯ ɫɨɫɬɨɹɧɢɣ. ɑɬɨ ɤɚɫɚɟɬɫɹ ɨɞɧɨɦɟɪɧɨɝɨ ɷɥɟɤɬɪɨɧɧɨɝɨ ɝɚɡɚ,
ɱɚɟ ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɜɨɨɛɳɟ ɧɟɩɪɢɦɟɧɢɦɨ ɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɨɥɶɤɨ
ɤɜɚɧɬɨɜɵɣ ɩɨɞɯɨɞ.
ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ ɦɵ ɩɪɨɞɟɦɨɧɫɬɪɢɪɭɟɦ ɬɟɯɧɢɤɭ ɜɵɱɢɫɥɟɧɢɹ ɤɜɚɧɬɨɜɨɣ ɩɨɩɪɚɜɤɢ ɤ ɩɪɨɜɨɞɢɦɨɫɬɢ ɞɥɹ ɬɪɟɯɦɟɪɧɵɯ ɷɥɟɤɬɪɨɧɨɜ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɪɚɫɫɦɨɬɪɢɦ
ɜɤɥɚɞɵ ɜ ɩɪɨɜɨɞɢɦɨɫɬɶ ɨɬ ɧɟɤɨɬɨɪɨɣ ɝɥɚɜɧɨɣ ɫɟɪɢɢ ɞɢɚɝɪɚɦɦ ɫ ɩɟɪɟɫɟɤɚɸɳɢɦɢɫɹ ɩɨɩɟɪɟɱɧɵɦɢ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦ. ɗɬɨ ɬɚɤ
ɝɪɚɦɦɵ.
ɑɬɨɛɵ ɧɟ ɡɚɝɪɨɦɨɠɞɚɬɶ ɮɨɪɦɭɥɵ ɧɟɩɪɢɧɰɢɩɢɚɥɶɧɵɦɢ ɭɫɥɨɠɧɟɧɢɹɦɢ ɜ ɷɬɨɦ
ɪɚɡɞɟɥɟ, ɦɵ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɪɚɫɫɟɹɧɢɟ ɧɚ ɩɪɢɦɟɫɹɯ ɢɡɨɬɪɨɩɧɨ, ɚ ɷɬɨ ɡɧɚɱɢɬ,
ɱɬɨ ɮɭɪɶɟ-ɨɛɪɚɡ ɩɨɬɟɧɰɢɚɥɚ ݑ
ɫɬɜɟɧɧɨɣ), ɤɨɬɨɪɭɸ ɨɛɨɡɧɚɱɢɦ ݑ
ɦɟɫɢ, ɫɨɝɥɚɫɧɨ (5.29), ɪɚɜɧɨ
ଶ
݉
ߪൌ
ʹߨ
ɂɦɟɹ ɜɜɢɞɭ ɫɨɨɬɧɨɲɟɧɢɟ ɞɥɹ ɜɪɟɦɟɧɢ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɷɥɟɤɬɪɨɧɚ ߬
ൌ݊
ߪ, ɫɪɚɡɭ ɜɵɪɚɡɢɦ ɦɧɨɠɢɬɟɥɶ ݃ൌ݊ݑ
ݒ
ɦɟɫɧɚɹ ɥɢɧɢɹ ɜ ɞɢɚɝɪɚɦɦɵ ɱɟɪɟɡ ɜɪɟɦɹ ߬:
ɦɟɬɨɞɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ. ɗɬɨ ɜɵɜɨɞ ɩɨɡɜɨɥɹɟɬ
ିଶ
ሻ
ሺ
߬
ݒ
ɢ ɬɨɠɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɜ
ɧɚɡɵɜɚɟɦɵɟ ɜɟɟɪɧɵɟ ɞɢɚ-
ሺkሻ
ɹɜɥɹɟɬɫɹ ɤɨɧɫɬɚɧɬɨɣ (ɫ ɧɟɨɛɯɨɞɢɦɨɫɬɶɸ ɜɟɳɟ-
. ɉɪɢ ɷɬɨɦ ɩɨɥɧɨɟ ɫɟɱɟɧɢɟ ɪɚɫɫɟɹɧɢɹ ɧɚ ɩɪɢ-
గ
න݀ߠߠ
ସ
หݑ൫p െpଵ൯หଶൌ
ଶ
ଶ
Ȁ
ଶ
݉
ଶ
ݑ
Ǥ
ସ
ߨ
, ɤɨɬɨɪɵɣ ɜɧɨɫɢɬ ɤɚɠɞɚɹ ɩɪɢ-
ɬɨ ɜ ɷɬɨɦ ɫɥɭ-
ሺ
Ǥͳ
ିଵ
ൌ
ሻ
57
ଶ
݃ൌ
ߨ
݉ଶݒ߬
Ǥ
ሺ
Ǥʹ
Ⱦɥɹ ɢɡɨɬɪɨɩɧɨɝɨ ɪɚɫɫɟɹɧɢɹ ɜɫɟ ɞɢɚɝɪɚɦɦɵ ɫ ɧɟɩɟɪɟɫɟɤɚɸɳɢɦɢɫɹ ɩɨɩɟɪɟɱɧɵɦɢ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ ɪɚɜɧɵ ɧɭɥɸ. Ʉɚɤ ɦɵ ɜɢɞɟɥɢ ɜ ɤɨɧɰɟ ɩɪɟɞɵɞɭɳɟɝɨ
ɪɚɡɞɟɥɚ ɜɫɟ ɬɚɤɢɟ ɞɢɚɝɪɚɦɦɵ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ɪɚɡɧɨɫɬɢ ߪെߪ
, ɚ ɞɥɹ ɢɡɨɬɪɨɩ-
௧
ɧɨɝɨ ɪɚɫɫɟɹɧɢɹ ɷɬɢ ɫɟɱɟɧɢɹ ɪɚɜɧɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɞɢɚɝɪɚɦɦɵ, ɩɪɢɜɟɞɟɧɧɵɟ ɧɚ ɪɢɫ. 6.1, ɨɧɢ ɩɪɟɞɫɬɚɜɥɹɸɬ
ɫɨɛɨɣ ɞɢɚɝɪɚɦɦɵ ɫ ɧɚɢɛɨɥɟɟ ɫɢɥɶɧɨ ɩɟɪɟɫɟɤɚɸɳɢɦɢɫɹ ɩɨɩɟɪɟɱɧɵɦɢ ɩɪɢɦɟɫɧɵɦɢ ɥɢɧɢɹɦɢ.
Ɋɢɫ. 6.1
Ɉɧɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɬɟɦ, ɱɬɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɬɨɱɟɤ ɩɪɢɤɪɟɩɥɟɧɢɹ ɩɪɢɦɟɫɧɵɯ ɥɢɧɢɣ ɧɚ ɜɟɪɯɧɟɣ ɱɚɫɬɢ ɩɟɬɥɢ ɨɛɪɚɬɧɚ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɧɚ ɧɢɠɧɟɣ,
ɤɚɤ ɜ ɪɚɫɤɪɵɬɨɦ ɜɟɟɪɟ ɫ ɬɨɱɤɨɣ ɤɪɟɩɥɟɧɢɹ ɩɟɪɶɟɜ ɩɨɫɟɪɟɞɢɧɟ,
ɩɨɷɬɨɦɭ ɬɚɤɢɟ ɞɢɚɝɪɚɦɦɵ ɧɚɡɵɜɚɸɬɫɹ «ɜɟɟɪɧɵɦɢ» ɞɢɚɝɪɚɦɦɚɦɢ. ȼɤɥɚɞ ɜ ɩɪɨ-
ሺሻ
ɜɨɞɢɦɨɫɬɶ ɨɬ ɫɨɜɨɤɭɩɧɨɫɬɢ ɜɟɟɪɧɵɯ ɞɢɚɝɪɚɦɦ ɦɵ ɨɛɨɡɧɚɱɢɦ ɤɚɤ ߜߪ
ɨɬ ɫɥɨɜɚ
fan, ɱɬɨ ɨɡɧɚɱɚɟɬ ɜɟɟɪ.
ɉɪɢɦɟɫɧɵɟ ɥɢɧɢɢ ɜ ɤɚɠɞɨɣ ɬɚɤɨɣ ɞɢɚɝɪɚɦɦɟ ɦɨɠɧɨ «ɪɚɫɩɚɪɚɥɥɟɥɢɬɶ», ɞɟɮɨɪɦɢɪɨɜɚɜ ɧɢɠɧɸɸ ɥɢɧɢɸ ɩɟɬɥɢ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.2.
ሻ
Ɋɢɫ. 6.2
Ɍɚɤ ɦɵ ɩɨɥɭɱɢɦ ɛɥɨɤ ɜɧɭɬɪɢ ɩɟɬɥɢ, ɤɨɬɨɪɵɣ ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ «ɥɟɫɬɧɢɱɧɵɯ»
ɞɢɚɝɪɚɦɦ 6.3, ɩɪɢ ɷɬɨɦ ɜ ɤɚɠɞɨɣ ɬɚɤɨɣ «ɥɟɫɬɧɢɱɧɨɣ» ɞɢɚɝɪɚɦɦɟ ɜ ɜɟɪɯɧɸɸ ɥɢɧɢɸ
ɜɯɨɞɢɬ ɢɦɩɭɥɶɫ pଵ, ɚ ɜɵɯɨɞɢɬ pଶ, ɚ ɜ ɧɢɠɧɸɸ ɧɚɨɛɨɪɨɬ. ɉɪɢ ɷɬɨɦ ɟɫɥɢ
ɮɭɧɤɰɢɹ Ƚɪɢɧɚ ܩ
ɧɢɠɧɹɹ ɮɭɧɤɰɢɹ ܩ
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɤɚɠɞɚɹ ɬɚɤɚɹ ɞɢɚɝɪɚɦɦɚ ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɨɬ ɫɭɦɦɵ q ൌp
ɢ ɨɬ ɱɚɫɬɨɬɵ ߗ, ɤɨɬɨɪɭɸ ɧɟɫɭɬ ɮɭɧɤɰɢɢ ܩ
ோ
ɧɚ ɜɟɪɯɧɟɣ ɥɢɧɢɢ ɢɦɟɟɬ ɚɪɝɭɦɟɧɬ k, ɬɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɟɣ
ɢɦɟɟɬ ɚɪɝɭɦɟɧɬ p
െk, ɢ ɩɨ k ɢɞɟɬ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ.
pଶ
ோ
ɧɚ ɜɟɪɯɧɟɣ ɥɢɧɢɢ.
ଵpଶ
58
Ɋɢɫ. 6.3
ሺqሻ
ɋɭɦɦɭ ɬɚɤɢɯ ɞɢɚɝɪɚɦɦ ɨɛɨɡɧɚɱɢɦ ࣝ
, ɟɟ ɧɚɡɵɜɚɸɬ ɤɭɩɟɪɨɧɨɦ, ɩɨɬɨɦɭ ɱɬɨ
ఆ
ɬɨɱɧɨ ɬɚɤɚɹ ɠɟ ɜɟɥɢɱɢɧɚ ɨɩɢɫɵɜɚɟɬ ɤɭɩɟɪɨɜɫɤɢɟ ɩɚɪɵ ɜ ɫɜɟɪɯɩɪɨɜɨɞɧɢɤɟ. Ⱥɧɚɥɨɝɢɹ ɦɟɠɞɭ ɤɭɩɟɪɨɧɨɦ ɜ ɬɟɨɪɢɢ ɤɜɚɧɬɨɜɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ ɢ ɚɦɩɥɢɬɭɞɨɣ ɤɭɩɟɪɨɜɫɤɢɯ ɩɚɪ ɨɤɚɡɵɜɚɟɬɫɹ ɜɟɫɶɦɚ ɫɨɞɟɪɠɚɬɟɥɶɧɨɣ, ɩɨɫɤɨɥɶɤɭ ɜ ɮɢɡɢɱɟɫɤɢɯ
ɫɜɨɣɫɬɜɚɯ ɷɬɢɯ ɜɟɥɢɱɢɧ ɢɦɟɟɬɫɹ ɧɟɦɚɥɨ ɨɛɳɟɝɨ.
ɋ ɩɨɦɨɳɶɸ ࣝ
ሺqሻ
ɜɤɥɚɞ ɜ ɩɪɨɜɨɞɢɦɨɫɬɶ ɨɬ ɜɫɟɣ ɫɭɦɦɵ ɜɟɟɪɧɵɯ ɞɢɚɝɪɚɦɦ
ఆ
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ:
ߜߪ
ሺሻ
ሺߗሻ
ൌ
ൈܩ
ଶ
݁
ߨ݉
ሺ
Ԝන
ଶ
Ͳǡpሻܩ
ଷ
݀
p
ଷ
ሺʹߨሻ
ோ
ሺ
ߗǡq െpሻܩ
ଷ
݀
ሺʹߨሻ
q
Ԝp
ଷ
ఈ
ሺ
q െp
ሺ
Ͳǡq െpሻǤ
ఈ
ሻ
Ԝࣝ
ோ
ሺqሻ
ሺ
Ԝܩ
ఆ
ߗǡpሻൈ
ሺ
Ǥ͵
ሻ
ɉɪɢ ɷɬɨɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɜɨɝɨ ɱɥɟɧɚ ɢɦɟɟɬ ɜɢɞ (ɩɟɪɜɚɹ ɞɢɚɝɪɚɦɦɚ ɧɚ
ɪɢɫ. 6.1):
ଷ
݀
ሺሻ
ሺqሻ
ࣝ
ఆ
ൌ݃
ଶ
න
ሺʹߨሻ
p
ோ
ሺ
ܩ
ଷ
ߗǡpሻܩ
ሺ
Ͳǡq െpሻǡ
ሺ
ǤͶ
ሻ
ɚ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɤɭɩɟɪɨɧɚ ɥɟɝɤɨ ɩɨɥɭɱɚɟɬɫɹ ɞɢɚɝɪɚɦɦɧɵɦ ɫɭɦɦɢɪɨɜɚɧɢɟɦ
(ɪɢɫ. 6.4) ɢ ɢɦɟɟɬ ɜɢɞ
ଷ
݀
ൌࣝ
ఆ
ሺሻ
ሺqሻ
݃න
ሺqሻ
ࣝ
ఆ
ሺʹߨሻ
p
ோ
ሺ
ܩ
ଷ
ߗǡpሻܩ
ሺ
Ͳǡq െpሻࣝ
ሺqሻ
Ǥ
ఆ
ሺ
Ǥͷ
ሻ
Ɋɢɫ. 6.4
ȼɵɱɢɫɥɢɦ ɢɧɬɟɝɪɚɥ ɜ (6.4), ɨɛɨɡɧɚɱɢɜ ɟɝɨ ࣤ ɢ ɫɱɢɬɚɹ
ൎȁq െpȁൎ
59
ȁqȁ
ɦɚɥɵɦ, ɬɚɤ ɱɬɨ ȁp
ȁ
ൎ
ൌ
ሺʹߨሻ
ଶ
න݀ߗp݀݇
ଷ
ࣤൌන
ଷ
p
݀
ሺʹߨሻ
ቀߗെݒ
ோ
ሺ
ܩ
ଷ
݇
ߗǡpሻܩ
ሺ
Ͳǡq െpሻൌ
ͳ
݅
ቁቀെݒ݇െ߂ሺpǡqሻെ
ʹ߬
ʹ߬
ሺ
ǡ
݅
ቁ
Ǥ
ሻ
ɝɞɟ ߂ሺpǡqሻൌߦሺq െpሻെߦ
ሺpሻ
, ɚ ݀ߗ
ɷɥɟɦɟɧɬ ɬɟɥɟɫɧɨɝɨ ɭɝɥɚ, ɫɜɹɡɚɧɧɵɣ ɫ
p
ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɟɤɬɨɪɚ p. Ⱦɚɥɟɟ ɢɧɬɟɝɪɢɪɭɟɦ ɩɨ k
ሺʹߨሻ
ଶ
ͺߨଶ߬න
ଷ
ݒ
݀ߗ
p
Ͷߨ
ͳെ݅߬൫ߗ߂ሺpǡqሻ൯
ͳ
ɫɱɢɬɚɹ ߬ߗاͳ ɢ ߬߂ሺpǡqሻاͳ, ɪɚɡɥɨɠɢɦ ɡɧɚɦɟɧɚɬɟɥɶ ɜ ɪɹɞ
ሺʹߨሻ
ɩɪɢ ɦɚɥɵɯ ݍൌ
ଶ
ͺߨଶ߬න
ଷ
ݒ
ȁqȁ
ɢɦɟɟɦ ߂ሺpǡqሻൌݒ
݀ߗ
p
ቀͳ݅߬൫ߗ߂ሺpǡqሻ൯െ߬ଶ߂
Ͷߨ
ݍߠ, ɝɞɟ ߠ ɭɝɨɥ ɦɟɠɞɭ ɜɟɤɬɨɪɚɦɢ p ɢ q.
݀ߗ
p
න
߂ሺpǡqሻൌͲǡ
Ͷߨ
݀ߗ
p
ଶ
Ͷߨ
߂
ሺ
pǡqሻൌ
න
ͳ
ݒ
͵
ଶ
ݍଶൌ
ͳ
߬
ܦݍଶǡ
ɝɞɟ
ଶ
ܦൌݒ
߬Ȁ͵
െ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ. Ɍɨɝɞɚ
ሺʹߨሻ
ଶ
ͺߨଶ߬ሺͳെ݅߬ߗെ߬ܦݍ
ଷ
ݒ
ଶ
݉
ଶ
ݒ߬
ሻ
ൌ
ߨ
ሺ
ଶ
ͳ݅߬ߗെ߬ܦݍ
ǡ
ଶ
ሺ
pǡqሻቁǡ
ሺ
ሻ
Ǥ
ሺ
ሻ
Ǥͺ
ሺ
ሻ
Ǥͻ
ሺ
ሻ
ǤͳͲ
ሺ
ሻ
Ǥͳͳ
ଶ
ሻ
ǡ
ሺ
Ǥͳʹ
ሻ
ȼ ɢɬɨɝɟ ɞɥɹ ɜɵɱɢɫɥɹɟɦɨɝɨ ɢɧɬɟɝɪɚɥɚ ɫ ɭɱɟɬɨɦ (6.2) ɩɨɥɭɱɚɟɦ
ࣤൌ
ͳ
ሺ
ͳ݅߬ߗെ߬ܦݍ
݃
ଶ
ሻ
Ǥ
ሺ
Ǥͳ͵
ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ (6.4), ɩɨɥɭɱɢɦ
ሺሻ
ሺqሻൌ݃ሺ
ࣝ
ఆ
ͳ݅߬ߗെ߬ܦݍ
Ɍɟɩɟɪɶ ɩɪɢɫɬɭɩɢɦ ɤ ɪɟɲɟɧɢɸ ɭɪɚɜɧɟɧɢɹ (6.5). ɂɫɤɨɦɚɹ ɮɭɧɤɰɢɹ ࣝ
ଶ
ሻ
Ǥ
ሺ
ǤͳͶ
ሺqሻ
ɜɵ-
ఆ
ɧɨɫɢɬɫɹ ɢɡ-ɩɨɞ ɢɧɬɟɝɪɚɥɚ ɢɡ-ɡɚ ɦɚɥɨɫɬɢ ݍ, ɩɨɫɥɟ ɱɟɝɨ ɨɧ ɫɨɜɩɚɞɚɟɬ ɫ ࣤ, ɜ ɢɬɨɝɟ
ൌࣝ
ሺሻ
ఆ
ሺqሻ
ࣝ
ሺqሻሺ
ͳ݅߬ߗെ߬ܦݍ
ఆ
ሺqሻ
ࣝ
ఆ
ଶ
ሻ
Ǥ
ሺ
Ǥͳͷ
60
ሻ
ሻ
ሻ
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