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Файл:Основы классической электродинамики. Учебное пособие
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ɛɵɫɬɪɟɟ ɦɟɧɹɟɬɫɹ ɜɨ ɜɪɟɦɟɧɢ ɢɧɞɭɤɰɢɹ B ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɬɟɦ ɛɨɥɶɲɟ
E
ɧɚɩɪɹɠɟɧɧɨɫɬɶ
ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ, ɢ ɧɚɨɛɨɪɨɬ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ
ɨɛɪɚɡɨɜɚɧɢɹ ɢɧɬɟɧɫɢɜɧɵɯ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɧɟɨɛɯɨɞɢɦɨ ɜɨɡɛɭɞɢɬɶ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɤɨɥɟɛɚɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɜɵɫɨɤɨɣ ɱɚɫɬɨɬɵ.
Ʉɨɥɟɛɚɧɢɹ ɜɵɫɨɤɨɣ ɱɚɫɬɨɬɵ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫ ɩɨɦɨɳɶɸ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ
ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɤɨɧɬɭɪɚ, ɟɫɥɢ ɭɦɟɧɶɲɚɬɶ ɟɦɤɨɫɬɶ ɤɨɧɞɟɧɫɚɬɨɪɚ ɢ ɢɧɞɭɤɬɢɜ-
ɧɨɫɬɶ ɤɚɬɭɲɤɢ
§
¨
0
¨
©
·
1
¸
. Ɉɞɧɚɤɨ ɛɨɥɶɲɚɹ ɱɚɫɬɨɬɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ
¸
LCʌȞ2
¹
ɤɨɥɟɛɚɧɢɣ ɟɳɟ ɧɟ ɝɚɪɚɧɬɢɪɭɟɬ ɢɧɬɟɧɫɢɜɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ
ɜɨɥɧ. ȼ ɨɛɵɱɧɨɦ (ɡɚɤɪɵɬɨɦ) ɤɨɥɟɛɚɬɟɥɶɧɨɦ ɤɨɧɬɭɪɟ ɩɨɱɬɢ ɜɫɟ ɦɚɝɧɢɬɧɨɟ
ɩɨɥɟ ɫɨɫɪɟɞɨɬɨɱɟɧɨ ɜ ɤɚɬɭɲɤɟ ɢɧɞɭɤɬɢɜɧɨɫɬɢ, ɚ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ – ɦɟɠɞɭ
ɨɛɤɥɚɞɤɚɦɢ ɤɨɧɞɟɧɫɚɬɨɪɚ, ɬɨ ɟɫɬɶ ɜ ɭɞɚɥɟɧɢɢ ɨɬ ɤɨɧɬɭɪɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ
ɩɨɥɟ ɩɪɚɤɬɢɱɟɫɤɢ ɨɬɫɭɬɫɬɜɭɟɬ. Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɤɨɥɟɛɚɬɟɥɶɧɵɣ ɤɨɧɬɭɪ
ɨɛɥɚɞɚɥ ɜɵɫɨɤɨɣ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ,
ɧɟɨɛɯɨɞɢɦɨ ɜɚɠɧɨɟ ɭɫɥɨɜɢɟ – ɩɟɪɟɦɟɧɧɵɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɹ ɞɨɥɠɧɵ ɡɚɧɢɦɚɬɶ ɨɞɢɧ
ɨɛɴɟɦ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɷɬɨɬ ɨɛɴɟɦ ɞɨɥɠɟɧ ɛɵɬɶ ɧɚɢɛɨɥɶɲɢɦ. ɗɬɨɦɭ ɭɫɥɨɜɢɸ
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɬɤɪɵɬɵɣ ɤɨɥɟɛɚɬɟɥɶɧɵɣ ɤɨɧɬɭɪ.
Ʉ ɨɬɤɪɵɬɨɦɭ ɤɨɥɟɛɚɬɟɥɶɧɨɦɭ ɤɨɧɬɭɪɭ ɦɨɠɧɨ ɩɟɪɟɣɬɢ, ɟɫɥɢ ɩɨɫɬɟɩɟɧɧɨ
ɪɚɡɞɜɢɝɚɬɶ ɨɛɤɥɚɞɤɢ ɤɨɧɞɟɧɫɚɬɨɪɚ, ɨɞɧɨɜɪɟɦɟɧɧɨ ɭɦɟɧɶɲɚɹ ɢɯ ɩɥɨɳɚɞɶ
ɢ ɱɢɫɥɨ ɜɢɬɤɨɜ ɤɚɬɭɲɤɢ (ɪɢɫ. 6.19). Ɍɚɤɨɣ ɜɢɞɨɢɡɦɟɧɟɧɧɵɣ ɤɨɧɬɭɪ ɨɛɥɚɞɚɟɬ
ɦɚɤɫɢɦɚɥɶɧɨ
ɜɨɡɦɨɠɧɨɣ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ ɢ, ɩɨ ɫɭɬɢ, ɹɜɥɹɟɬɫɹ
ɢɡɥɭɱɚɸɳɟɣ ɚɧɬɟɧɧɨɣ.
+
+
-
1
Ɋɢɫ. 6.19
-
ɉɪɢ ɢɡɭɱɟɧɢɢ ɭɫɥɨɜɢɣ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɛɨɥɶɲɭɸ ɪɨɥɶ ɫɵɝɪɚɥ ɜɢɛɪɚɬɨɪ (ɢɥɢ ɞɢɩɨɥɶ) Ƚɟɪɰɚ. Ɉɧ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɜɚ ɫɬɟɪɠɧɹ ɫ ɲɚɪɢɤɚɦɢ ɧɚ ɤɨɧɰɚɯ; ɫɬɟɪɠɧɢ ɩɨɞɤɥɸɱɚɸɬɫɹ
ɤ ɢɫɬɨɱɧɢɤɭ ɜɵɫɨɤɨɝɨ ɧɚɩɪɹɠɟɧɢɹ (ɢɧɞɭɤɰɢɨɧɧɨɣ ɤɚɬɭɲɤɟ). Ʉɨɝɞɚ ɧɚɩɪɹɠɟɧɢɟ ɦɟɠɞɭ ɫɬɟɪɠɧɹɦɢ ɫɬɚɧɨɜɢɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɦ, ɦɟɠɞɭ ɲɚɪɢɤɚɦɢ
ɩɪɨɫɤɚɤɢɜɚɟɬ ɢɫɤɪɚ (ɪɢɫ. 6.20).
131
ȼɢɛɪɚɬɨɪ Ƚɟɪɰɚ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɨɬɤɪɵɬɵɣ ɤɨɥɟɛɚɬɟɥɶɧɵɣ
E
E
J
ɤɨɧɬɭɪ. ȿɦɤɨɫɬɶɸ ɜ ɬɚɤɨɦ ɤɨɧɬɭɪɟ ɹɜɥɹɟɬɫɹ ɟɦɤɨɫɬɶ ɦɟɠɞɭ ɫɬɟɪɠɧɹɦɢ, ɩɪɟɢɦɭɳɟɫɬɜɟɧɧɨ ɦɟɠɞɭ ɢɯ ɤɨɧɰɚɦɢ, ɧɚ ɤɨɬɨɪɵɯ ɢ ɧɚɤɚɩɥɢɜɚɸɬɫɹ ɡɚɪɹɞɵ ɩɪɢ
ɤɨɥɟɛɚɧɢɹɯ. ɋɚɦɢ ɫɬɟɪɠɧɢ ɨɛ-
+
w
tEw
.
w
tBw
w
tEw
.
-
ɩɟɪɟɦɟɧɧɨɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɜɨɡɧɢɤɚɟɬ ɢɡɦɟɧɹɸɳɟɟɫɹ ɜɨ
ɜɪɟɦɟɧɢ ɜɢɯɪɟɜɨɟ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ, ɨɧɨ ɜɧɨɜɶ ɩɨɪɨɠɞɚɟɬ ɬɚɤɠɟ ɜɢɯɪɟɜɨɟ
ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɢ ɬ.ɞ.
ɚɥɶɧɨɟ ɨɬɥɢɱɢɟ ɨɬ ɦɟɯɚɧɢɱɟɫɤɢɯ ɜɨɥɧ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɡɞɟɫɶ ɤɨɥɟɛɥɸɬɫɹ ɧɟ ɱɚɫɬɢɰɵ ɫɪɟɞɵ, ɚ ɜɟɤɬɨɪɵ ɧɚɩɪɹɠɟɧɧɨɫɬɢ
ɷɬɨɦɭ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɜɨɡɦɨɠɧɵ ɢ ɜ ɜɚɤɭɭɦɟ, ɝɞɟ ɨɬɫɭɬɫɬɜɭɸɬ ɱɚɫɬɢɰɵ ɜɟɳɟɫɬɜɚ.
ɋ ɩɨɦɨɳɶɸ ɨɩɢɫɚɧɧɨɝɨ ɜɢɛɪɚɬɨɪɚ Ƚ. Ƚɟɪɰ ɩɨɥɭɱɢɥ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ
ɜɨɥɧɵ ɞɥɢɧɨɣ ɩɪɢɦɟɪɧɨ 3 ɫɦ (Ȟa100 ɆȽɰ). ɉɨɡɠɟ, ɜ 1895 ɝ., ɉ. ɇ. Ʌɟɛɟɞɟɜɭ
ɭɞɚɥɨɫɶ ɩɨɥɭɱɢɬɶ ɜɨɥɧɵ ɞɥɢɧɨɣ 6 ɦɦ, ɚ Ⱥ. Ⱥ. Ƚɥɚɝɨɥɟɜɚ-Ⱥɪɤɚɞɶɟɜɚ ɜ 1923 ɝ.
ɫ ɩɨɦɨɳɶɸ ɬɚɤ
ɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɞɥɢɧɨɣ ɞɨ 85 ɦɤɦ.
ɂɫɫɥɟɞɭɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ, Ƚ. Ƚɟɪɰ ɭɫɬɚɧɨɜɢɥ, ɱɬɨ ɨɧɢ:
– ɨɬɪɚɠɚɸɬɫɹ ɢ ɩɪɟɥɨɦɥɹɸɬɫɹ ɜɟɳɟɫɬɜɨɦ;
– ɨɛɥɚɞɚɸɬ ɫɜɨɣɫɬɜɚɦɢ ɢɧɬɟɪɮɟɪɟɧɰɢɢ ɢ ɞɢɮɪɚɤɰɢɢ;
– ɹɜɥɹɸɬɫɹ ɩɨɩɟɪɟɱɧɵɦɢ ɜɨɥɧɚɦɢ, ɬɨ ɟɫɬɶ ɤɨɥɟɛɚɧɢɹ ɜɟɤɬɨɪɨɜ
ɩɪɨɢɫɯɨɞɹɬ ɜɨ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɩɥɨɫɤɨɫɬɹɯ.
ɂɡɜɟɫɬɧɵɟ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɪɚɡɥɢɱɚɸɬɫɹ ɩɨ
ɫɩɨɫɨɛɚɦ ɝɟɧɟɪɚɰɢɢ, ɱɚɫɬɨɬɟ (ɞɥɢɧɟ ɜɨɥɧɵ) ɢ ɫɜɨɣɫɬɜɚɦ. ɉɨɷɬɨɦɭ ɨɧɢ
ɞɟɥɹɬɫɹ ɧɚ ɧɟɫɤɨɥɶɤɨ ɜɢɞɨɜ – ɪɚɞɢɨɜɨɥɧɵ, ɫɜɟɬɨɜɵɟ ɜɨɥɧɵ, ɪɟɧɬɝɟɧɨɜɫɤɨɟ
-ɢɡɥɭɱɟɧɢɹ (ɬɚɛɥ. 6.1).
ɢ
ɧɚɡɵɜɚɟɦɨɝɨ ɦɚɫɫɨɜɨɝɨ ɢɡɥɭɱɚɬɟɥɹ ɩɨɥɭɱɢɥɚ ɷɥɟɤɬɪɨ-
6.5. ǟǼǬǮǹDZǹǴȋ ȉǷDZǶǾǼǺǸǬǯǹǴǾǹǺǵ ǮǺǷǹȇ
Ɉɫɧɨɜɭ ɬɟɨɪɢɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɫɨɫɬɚɜɥɹɸɬ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ
(ɷɬɢ ɭɪɚɜɧɟɧɢɹ ɜ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɟ ɢɝɪɚɸɬ ɬɚɤɭɸ ɠɟ ɪɨɥɶ, ɱɬɨ ɢ ɡɚɤɨɧɵ
ȼɨɡɧɢɤɚɟɬ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɚɹ ɜɨɥɧɚ. ɉɪɢɧɰɢɩɢ-
ɥɚɞɚɸɬ ɢɧɞɭɤɬɢɜɧɨɫɬɶɸ. ȼ ɨɬɥɢɱɢɟ ɨɬ ɨɛɵɱɧɨɝɨ ɤɨɧɬɭɪɚ
ɩɨɥɟ ɜɢɛɪɚɬɨɪɚ ɧɟ ɥɨɤɚɥɢɡɨɜɚɧɨ ɜ ɨɝɪɚɧɢɱɟɧɧɨɦ ɨɛɤɥɚɞɤɚɦɢ
ɤɨɧɞɟɧɫɚɬɨɪɚ ɨɛɴɟɦɟ, ɚ ɢɦɟɟɬ
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɯɚɪɚɤɬɟɪ.
ɉɪɢ ɜɨɡɛɭɠɞɟɧɢɢ ɜ ɜɢɛɪɚɬɨɪɟ ɢɦɩɭɥɶɫɨɜ ɛɵɫɬɪɨɩɟɪɟɦɟɧɧɨɝɨ ɬɨɤɚ ɜɨɡɧɢɤɚɸɬ ɤɨɥɟɛɚɧɢɹ ɨɝɪɨɦɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ
ɷɥɟɤɬɪɨɧɨɜ, ɞɜɢɠɭɳɢɯɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ, ɬɨ ɟɫɬɶ ɜ ɨɤɪɭɠɚɸɳɟɦ
ɜɢɛɪɚɬɨɪ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɨɡɧɢɤɚɟɬ
ɢ ɢɧɞɭɤɰɢɢ B . ɉɨ-
ɢ B
132
ɇɶɸɬɨɧɚ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ). ɍɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ ɨɬɪɚɠɚɸɬ ɬɨɬ
O
Q
J
E
E
ɮɚɤɬ, ɱɬɨ ɢɫɬɨɱɧɢɤɚɦɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɦɨɝɭɬ ɛɵɬɶ ɥɢɛɨ ɷɥɟɤɬɪɢɱɟɫɤɢɟ
ɡɚɪɹɞɵ, ɥɢɛɨ ɢɡɦɟɧɹɸɳɢɟɫɹ ɜɨ ɜɪɟɦɟɧɢ ɦɚɝɧɢɬɧɵɟ ɩɨɥɹ. Ɇɚɝɧɢɬɧɵɟ ɩɨɥɹ
ɦɨɝɭɬ ɜɨɡɛɭɠɞɚɬɶɫɹ ɥɢɛɨ ɞɜɢɠɭɳɢɦɢɫɹ ɷɥɟɤɬɪɢɱɟɫɤɢɦɢ ɡɚɪɹɞɚɦɢ (ɬɨɤɚɦɢ),
ɥɢɛɨ ɩɟɪɟɦɟɧɧɵɦɢ ɷɥɟɤɬɪɢɱɟɫɤɢɦɢ ɩɨɥɹɦɢ.
ǞǬǭǷǴȂǬ 6.1
ȼɢɞ ɢɡɥɭɱɟɧɢɹ
Ɋɚɞɢɨɜɨɥɧɵ 103 – 10-4 3·105 - 3·1012
ɋɜɟɬɨɜɵɟ ɜɨɥɧɵ:
– ɢɧɮɪɚɤɪɚɫɧɨɟ
ɢɡɥɭɱɟɧɢɟ
– ɜɢɞɢɦɵɣ ɫɜɟɬ
– ɭɥɶɬɪɚɮɢɨɥɟɬɨɜɨɟ
ɢɡɥɭɱɟɧɢɟ
Ɋɟɧɬɝɟɧɨɜɫɤɨɟ
ɢɡɥɭɱɟɧɢɟ
-ɢɡɥɭɱɟɧɢɟ
5·10
8·10
4·10
2·10
, ɦ
-4
– 8·10-7
-7
– 4·10-7
-7
– 10-9
-9
– 6·10
< 6·10
, Ƚɰ
-12
1,5·1017 – 5·1019 Ɋɟɧɬɝɟɧɨɜɫɤɢɟ ɬɪɭɛɤɢ
-12
> 5·1019
6·10
4·10
7,5·10
11
– 4·1014
14
– 7,5·1014
14
– 3·1017
ɂɫɬɨɱɧɢɤ ɢɡɥɭɱɟɧɢɹ
Ʉɨɥɟɛɚɬɟɥɶɧɵɣ ɤɨɧɬɭɪ,
ɜɢɛɪɚɬɨɪ Ƚɟɪɰɚ,
ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɣ
ɝɟɧɟɪɚɬɨɪ
Ʌɚɦɩɵ, ɥɚɡɟɪɵ
əɞɟɪɧɵɟ ɪɟɚɤɰɢɢ,
ɤɨɫɦɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ
ɍɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ ɩɪɢɜɨɞɢɥɢɫɶ ɜ ɝɥɚɜɟ 5. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɞɚɥɶɧɟɣɲɟɝɨ
ɢɡɥɨɠɟɧɢɹ ɡɚɩɢɲɟɦ ɷɬɢ ɭɪɚɜɧɟɧɢɹ ɟɳɟ ɪɚɡ.
ɉɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ Ɇɚɤɫɜɟɥɥɚ ɜ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɨɪɦɟ
w
B
³³
LS
³³
LS
³³
SV
ɗɬɢ ɭɪɚɜɧɟɧɢɹ ɫɜɹɡɵɜɚɸɬ ɡɧɚɱɟɧɢɹ
ɫɨ ɡɧɚɱɟɧɢɹɦɢ
B (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ D ) ɜ ɬɨɱɤɚɯ ɨɩɢɪɚɸɳɟɣɫɹ ɧɚ ɤɨɧɬɭɪ
ldE (6.25)
³
S
§ww
jldH
¨
¨
©
,Sd
w
t
(6.26)
,SdB 0
·
D
,Sd
(6.27)
¸
¸
t
¹
(6.28)
.dVȡqSdD
ɢɥɢ H ɜɞɨɥɶ ɧɟɤɨɬɨɪɨɝɨ ɤɨɧɬɭɪɚ
ɩɨɜɟɪɯɧɨɫɬɢ. ɋ ɩɨɦɨɳɶɸ ɬɟɨɪɟɦ ɜɟɤɬɨɪɧɨɝɨ ɚɧɚɥɢɡɚ (ɬɟɨɪɟɦɵ ɋɬɨɤɫɚ
ɢ ɬɟɨɪɟɦɵ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ) ɨɬ ɭɪɚɜɧɟɧɢɣ (6.25) – (6.28) ɦɨɠɧɨ
ɩɟɪɟɣɬɢ ɤ ɭɪɚɜɧɟɧɢɹɦ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɮɨɪɦɟ, ɤɨɬɨɪɵɟ ɫɜɹɡɵɜɚɸɬ ɡɧɚ-
ɢɥɢ H ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫ B (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ D )
ɜ ɬɨɣ ɠɟ ɫɚɦɨɣ ɬɨɱɤɟ.
133
ɉɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ Ɇɚɤɫɜɟɥɥɚ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɮɨɪɦɟ
t
A
Arot
A
t
t
t
t
B
w
Erot
(6.29)
,
t
w
,Bdiv 0 (6.30)
D
w
jHrot
(6.31)
,
w
.ȡDdiv (6.32)
Ɇɟɠɞɭ ɜɟɥɢɱɢɧɚɦɢ, ɜɯɨɞɹɳɢɦɢ ɜ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ, ɫɭɳɟɫɬɜɭɟɬ
ɫɥɟɞɭɸɳɚɹ ɫɜɹɡɶ:
,EİİD0
(6.33)
,HȝȝB0 (6.34)
.EȖj
(6.35)
ɋɨɜɨɤɭɩɧɨɫɬɶ ɭɪɚɜɧɟɧɢɣ (6.29) – (6.35) ɨɛɪɚɡɭɟɬ ɨɫɧɨɜɭ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ
ɩɨɤɨɹɳɢɯɫɹ ɫɪɟɞ.
(
ɉɨɞ ɪɨɬɨɪɨɦ ɜɟɤɬɨɪɚ
) ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ ɜɟɤɬɨɪ ɫ ɫɨɫɬɚɜɥɹɸ-
ɳɢɦɢ:
A
w
½
y
,
°
z
w
°
°
A
w
°
z
(6.36)
,
¾
x
w
°
°
A
w
x
.
°
y
w
°
¿
ɉɨɞ ɞɢɜɟɪɝɟɧɰɢɟɣ ɜɟɤɬɨɪɚ
w
A
z
)Arot(
x
w
y
A
w
)Arot(
)Arot(
x
y
z
w
A
w
y
z
x
w
(Adiv ) ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɫɤɚ-
ɥɹɪɧɚɹ ɜɟɥɢɱɢɧɚ:
w
x
Adiv
(6.37)
x
w
w
A
w
y
z
.
z
y
w
A
w
A
Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ ɞɥɹ ɨɞɧɨɪɨɞɧɨɣ ɧɟɣɬɪɚɥɶɧɨɣ (0 ȡ )
ɢ ɧɟɩɪɨɜɨɞɹɳɟɣ (
0 j
) ɫɪɟɞɵ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɩɪɨɧɢɰɚɟɦɨɫɬɹɦɢ H ɢ P.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ
134
H
w
,
ȝȝ
0
w
E
w
,
İİ
0
w
B
w
w
D
w
w
,Hdiv ȝȝBdiv0
H
E
E
H
x
E
¹
·
©
§
t H
t Erot
t
P P
.Ediv İİDdiv0
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɪɚɜɧɟɧɢɹ (6.29) – (6.32) ɩɪɢɦɭɬ ɜɢɞ
w
rotE ȝȝ
w
ww
HH
xz
divH
divE
www
xyz
rotH İİ
w
ww
EE
xz
www
xyz
,
0
H
0
E
(6.38)
w
t
y
w
w
t
y
0,
,
(6.40)
0.
ɉɪɢɦɟɧɢɦ ɤ ɭɪɚɜɧɟɧɢɸ (6.38) ɨɩɟɪɚɰɢɸ rot:
w
ro
) (
ro
0
. (6.42)
¸
¨
w
ɋɢɦɜɨɥ rot ɨɡɧɚɱɚɟɬ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ. Ɇɟɧɹɹ ɩɨɪɹɞɨɤ
ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ ɢ ɜɪɟɦɟɧɢ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
rot
w
§
¨
©
w
H
·
¸
w
w
tt
¹
).( Hrot
ɉɪɨɢɡɜɟɞɹ ɜ ɭɪɚɜɧɟɧɢɢ (6.42) ɷɬɭ ɡɚɦɟɧɭ ɢ ɩɨɞɫɬɚɜɢɜ ɜ ɩɨɥɭɱɢɜɲɟɟɫɹ
ɜɵɪɚɠɟɧɢɟ ɮɨɪɦɭɥɭ (6.40), ɩɨɥɭɱɢɦ
2
w
rot rotE İİȝȝ
00
.
2
t
w
ɉɪɢɦɟɧɢɜ ɨɩɟɪɚɰɢɸ rot ɤ ɭɪɚɜɧɟɧɢɸ (6.40) ɢ ɩɪɨɢɡɜɟɞɹ ɚɧɚɥɨɝɢɱ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
2
w
rot rotH İİȝȝ
00
.
(6.44)
2
t
w
ɇɚɣɞɟɦ ɩɪɨɟɤɰɢɸ ɜɟɤɬɨɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (6.43) ɧɚ ɨɫɶ ɯ:
2
ªº
rot rotE İİȝȝ
¬¼
x
00
w
.
(6.45)
2
t
w
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɪɨɬɨɪɚ (ɫɦ. ɮɨɪɦɭɥɵ (6.36))
ªº
rot rotE rot E rot E
¬¼
ww
x
ww
yz
zz
.
(6.39)
(6.41)
(6.43)
ɧɵɟ
135
ɉɨɞɫɬɚɜɥɹɹ ɜ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɡɧɚɱɟɧɢɹ
x
E
x
x
x
E
x
E
E
x
H
x
E
x
¹
·
©
§
¹
·
©
§
zEyEx
E z
E y E xEx
E
z y x
z
y
x
x
E
w
§·
wwwww
ªº
rot rotE
¬¼
ɉɪɢɛɚɜɥɹɹ ɢ ɜɵɱɢɬɚɹ
>@
)(
rot rot
x
2
E
w
xy xy
ww ww
22
ww
2
w
w
w
¨
¨
w
y
¨¸
xy zx
ww w ww w
©¹
yz
222
EEE
www
y
xxx
22
yz
ww
E
x
, ɩɪɢɜɟɞɟɦ ɞɚɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɤ ɜɢɞɭ
x
ww
w
w
w
EEE
w
w
)Erot( ɢ
z
xxz
§·
¨¸
©¹
, ɩɨɥɭɱɢɦ
)Erot(
y
.
2
w
¨
¸
¸
¨
w
2
w
2
w
2
w
2
2
w
.
¸
¸
ɍɱɢɬɵɜɚɹ (6.41), ɭɪɚɜɧɟɧɢɟ (6.45) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
222
www
www w
Ⱥɧɚɥɨɝɢɱɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɸɬɫɹ ɞɥɹ
2
E
w
222 2
yz t
EE
y
zx
İİȝȝ
00
E ɢ
y
.
(6.46)
. Ɉɛɴɟɞɢɧɢɜ ɭɪɚɜɧɟ-
E
z
ɧɢɹ ɞɥɹ ɜɫɟɯ ɬɪɟɯ ɤɨɦɩɨɧɟɧɬ, ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ
ɜɟɤɬɨɪɚ
:
222 2
www w
www w
EE E
222 2
yz t
İİȝȝ
00
(6.47)
.
ɋɯɨɞɧɵɦ ɨɛɪɚɡɨɦ ɭɪɚɜɧɟɧɢɟ (6.44) ɦɨɠɧɨ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɤ ɜɢɞɭ
222 2
www w
www w
HH H
222 2
yz t
İİȝȝ
00
(6.48)
.
ɍɪɚɜɧɟɧɢɹ (6.47) ɢ (6.48) ɧɟɪɚɡɪɵɜɧɨ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ, ɬɚɤ ɤɚɤ ɨɧɢ
ɩɨɥɭɱɟɧɵ ɢɡ ɭɪɚɜɧɟɧɢɣ (6.38) ɢ (6.40), ɤɚɠɞɨɟ ɢɡ ɤɨɬɨɪɵɯ ɫɨɞɟɪɠɢɬ ɢ
, ɢ H.
Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɪɚɧɟɟ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɦɟɯɚɧɢɱɟɫɤɢɯ ɜɨɥɧ, ɞɢɮɮɟɪɟɧ-
ɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ
222 2
www w
22222
www w
yzvt
1ȟȟȟ ȟ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ. ȼɫɹɤɚɹ ɮɭɧɤɰɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ
ɬɚɤɨɦɭ ɭɪɚɜɧɟɧɢɸ, ɨɩɢɫɵɜɚɟɬ ɧɟɤɨɬɨɪɭɸ ɜɨɥɧɭ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɪɚɜɧɟɧɢɹ
(6.47) ɢ (6.48) ɭɤɚɡɵɜɚɸɬ ɧɚ ɬɨ, ɱɬɨ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɩɨɥɹ ɦɨɝɭɬ ɫɭɳɟ-
ɫɬɜɨɜɚɬɶ ɜ ɜɢɞɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ, ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɤɨɬɨɪɵɯ ɪɚɜɧɚ
111
İİȝȝ İȝ İȝ
00 00
(6.49)
.v
136
Ⱦɥɹ ɜɚɤɭɭɦɚ ɩɨ ɮɨɪɦɭɥɟ (6.49) ɩɨɥɭɱɚɟɬɫɹ
E
E
x
x
11
|
v
HP
00
4910
410
S
S
7
9
8
310 ɦ/ ɫ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ
c
v
(6.50)
HP
ɢ ɜ ɜɚɤɭɭɦɟ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɤɨɪɨɫɬɶɸ ɫɜɟɬɚ.
6.6. ǛǷǺǽǶǬȋ ȉǷDZǶǾǼǺǸǬǯǹǴǾǹǬȋ ǮǺǷǹǬ
ɉɟɪɟɞ ɢɫɫɥɟɞɨɜɚɧɢɟɦ ɩɥɨɫɤɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɡɚɩɢɲɟɦ ɩɪɨɟɤɰɢɢ
ɭɪɚɜɧɟɧɢɣ (6.29) ɢ (6.31) ɧɚ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ. Ɍɨɝɞɚ, ɩɪɢɧɹɜ ɜɨ ɜɧɢɦɚɧɢɟ
ɮɨɪɦɭɥɵ (6.36), ɩɨɥɭɱɢɦ ɜɦɟɫɬɨ ɤɚɠɞɨɝɨ ɢɡ ɜɟɤɬɨɪɧɵɯ ɭɪɚɜɧɟɧɢɣ ɬɪɢ
ɫɤɚɥɹɪɧɵɯ:
E
w
ww
ww w
ww
EE
ww w
w
E
ww w
HD
ww
zx
yz t
ww w
ww
HH
xz
ww w
zx t
w
H
y
ww w
y
zx
yz t
xz
zx t
ww
y
xz
yt
w
H
y
ww
HD
xz
yt
½
B
,
°
°
°
w
B
°
y
(6.51)
,
¾
°
°
B
.
°
°
¿
j
x
½
,
°
°
°
w
D
°
y
j
y
(6.52)
,
¾
°
j
z
°
.
°
°
¿
ɍɪɚɜɧɟɧɢɹ (6.30) ɢ (6.32) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɫɤɚɥɹɪɧɨɦ ɜɢɞɟ, ɢɫɩɨɥɶɡɨɜɚɜ
ɫɨɨɬɧɨɲɟɧɢɟ (6.37):
w
w
B
w
D
w
w
B
B
w
x
x
x
x
y
z
w
y
D
w
y
y
w
0
; (6.53)
z
w
D
w
z
.ȡ
z
w
(6.54)
ɉɭɫɬɶ ɩɥɨɫɤɚɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɚɹ ɜɨɥɧɚ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜ ɨɞɧɨɪɨɞɧɨɣ
00 j ,ȡ
ɧɟɩɪɨɜɨɞɹɳɟɣ ɫɪɟɞɟ (
). ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɜɨɥɧɨ-
137
ɜɵɦ ɩɨɜɟɪɯɧɨɫɬɹɦ. Ɍɨɝɞɚ E ɢ H, ɚ ɡɧɚɱɢɬ ɢ ɢɯ ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɟ ɛɭɞɭɬ
x
x
E
x
x
x
ɡɚɜɢɫɟɬɶ ɨɬ ɤɨɨɪɞɢɧɚɬ ɭ ɢ z.
ɉɨɷɬɨɦɭ ɭɪɚɜɧɟɧɢɹ (6.51) – (6.54) ɭɩɪɨɳɚɸɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
H
w
x
0,
t
w
E
w
z
ȝȝ
0
ww
xt
E
w
y
ȝȝ
ww
w
E
w
H
w
ww
H
w
ww
0
x
0,
t
z
İİ
0
y
İİ
0
xt
Hxw
x
w
Exw
x
w
½
°
°
H
w
°
y
,
(6.55)
¾
°
°
w
H
z
.
°
t
¿
½
°
°
E
w
°
y
,
(6.56)
¾
t
°
°
E
w
z
.
°
¿
(6.57)
0
(6.58)
0
ɉɟɪɜɨɟ ɢɡ ɭɪɚɜɧɟɧɢɣ (6.56) ɢ ɭɪɚɜɧɟɧɢɟ (6.58) ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɫɨɫɬɚɜɥɹɸɳɚɹ
ɢ ɭɪɚɜɧɟɧɢɟ (6.57) ɞɚɸɬ ɚɧɚɥɨɝɢɱɧɵɣ ɪɟɡɭɥɶɬɚɬ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɟɣ
ɧɟ ɦɨɠɟɬ ɡɚɜɢɫɟɬɶ ɧɢ ɨɬ t, ɧɢ ɨɬ ɯ. ɉɟɪɜɨɟ ɢɡ ɭɪɚɜɧɟɧɢɣ (6.55)
E
x
H
x
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɟ ɜɨɥɧɵ ɧɟ ɢɦɟɟɬ ɫɨɫɬɚɜɥɹɸɳɢɯ ɜɞɨɥɶ ɨɫɢ ɯ, ɬɨ ɟɫɬɶ
ɜɟɤɬɨɪɵ
ɢ H ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɤ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ –
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɩɨɩɟɪɟɱɧɵ.
ȼɬɨɪɵɟ ɢ ɬɪɟɬɶɢ ɭɪɚɜɧɟɧɢɹ (6.55) ɢ (6.56) ɦɨɠɧɨ ɨɛɴɟɞɢɧɢɬɶ ɜ ɞɜɟ ɝɪɭɩɩɵ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
E
w
y
ȝȝ
ww
H
w
ww
w
ww
w
0
z
İİ
0
xt
E
z
ȝȝ
0
H
y
İİ
0
ww
½
w
H
z
,
°
°
t
(6.59)
¾
w
E
y
°
.
°
¿
H
w
½
y
,
°
°
t
(6.60)
¾
w
E
°
z
.
°
t
¿
.
138
ɉɟɪɜɚɹ ɝɪɭɩɩɚ ɭɪɚɜɧɟɧɢɣ ɫɜɹɡɵɜɚɟɬ ɫɨɫɬɚɜɥɹɸɳɢɟ yE ɢ
H
x
x
E
x
H
x
ɫɬɚɜɥɹɸɳɢɟ
ɦɟɧɧɨɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ
ɢ yH . ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɛɵɥɨ ɫɨɡɞɚɧɨ ɩɟɪɟ-
E
z
E , ɧɚɩɪɚɜɥɟɧɧɨɟ ɜɞɨɥɶ ɨɫɢ ɭ. ɋɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ
y
ɭɪɚɜɧɟɧɢɸ (6.59) ɷɬɨ ɩɨɥɟ ɩɨɪɨɠɞɚɟɬ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ
ɜɞɨɥɶ ɨɫɢ z. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɟɪɜɵɦ ɭɪɚɜɧɟɧɢɟɦ (6.59) ɩɨɥɟ
ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ
E ɢ ɬ.ɞ. ɇɢ ɩɨɥɟ
y
, ɧɢ ɩɨɥɟ yH ɩɪɢ ɷɬɨɦ ɧɟ ɜɨɡɧɢɤɚ-
E
z
ɸɬ. Ⱥɧɚɥɨɝɢɱɧɨ, ɟɫɥɢ ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɛɵɥɨ ɫɨɡɞɚɧɨ ɩɨɥɟ
ɭɪɚɜɧɟɧɢɹɦ (6.60) ɩɨɹɜɢɬɫɹ ɩɨɥɟ, ɤɨɬɨɪɨɟ ɩɨɪɨɞɢɬ ɩɨɥɟ
ɫɥɭɱɚɟ ɧɟ ɜɨɡɧɢɤɚɸɬ ɩɨɥɹ
E ɢ
y
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɨɩɢɫɚɧɢɹ ɩɥɨɫɤɨɣ
H
z
, ɜɬɨɪɚɹ – ɫɨ-
H
z
, ɧɚɩɪɚɜɥɟɧɧɨɟ
H
z
ɫɨɡɞɚɫɬ
H
z
, ɬɨ ɫɨɝɥɚɫɧɨ
E
z
ɢ ɬ. ɞ. ȼ ɷɬɨɦ
E
z
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɞɨɫɬɚɬɨɱɧɨ ɜɡɹɬɶ ɨɞɧɭ ɢɡ ɫɢɫɬɟɦ ɭɪɚɜɧɟɧɢɣ (6.59) ɢ
(6.60), ɩɪɢɧɹɜ ɫɨɫɬɚɜɥɹɸɳɢɟ, ɭɱɚɫɬɜɭɸɳɢɟ ɜ ɞɪɭɝɨɣ ɫɢɫɬɟɦɟ, ɪɚɜɧɵɦɢ ɧɭɥɸ.
ȼɨɡɶɦɟɦ ɞɥɹ ɨɩɢɫɚɧɢɹ ɜɨɥɧɵ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.59), ɩɨɥɨɠɢɜ
=yH = 0. ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ ɯ ɢ ɩɪɨɢɡɜɟɞɟɦ ɡɚɦɟɧɭ
E
z
ɉɨɞɫɬɚɜɢɜ ɡɚɬɟɦ
ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɟɣ
ww ww
zz
ttx
ww ww
H
w
z
ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɩɨɥɭɱɢɦ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ
w
E :
y
22
ww
yy
İİȝȝ
22
ww
00
H
.
E
.
(6.61)
t
ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɜ ɩɨ ɯ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ (6.59), ɩɨɫɥɟ ɚɧɚɥɨɝɢɱɧɵɯ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɢɦ ɜɨɥɧɨɜɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɟɣ
22
ww
zz
İİȝȝ
00
22
ww
H
.
(6.62)
t
:
H
z
ɍɪɚɜɧɟɧɢɹ (6.61) ɢ (6.62) ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɭɪɚɜɧɟɧɢɣ (6.47) ɢ (6.48). Ɋɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (6.61) ɹɜɥɹɟɬɫɹ ɝɚɪɦɨɧɢɱɟɫɤɚɹ
ɮɭɧɤɰɢɹ
EE Ȧtkxij (6.63)
cos .
ym
1
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.62) ɢɦɟɟɬ ɚɧɚɥɨɝɢɱɧɵɣ ɜɢɞ
HH Ȧtkxij (6.64)
cos .
zm
2
ɉɨɞɫɬɚɜɢɦ ɮɭɧɤɰɢɢ (6.63) ɢ (6.64) ɜ ɫɢɫɬɟɦɭ (6.59)
sin sin ,
kE ȦtkxijȝȝȦH Ȧtkxij
mm
kH ȦtkxijȦE Ȧtkxij
mm
sin sin .
10 2
HH
20 1
139
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɷɬɢ ɪɚɜɟɧɫɬɜɚ ɜɵɩɨɥɧɹɥɢɫɶ, ɧɟɨɛɯɨɞɢɦɵ ɫɥɟɞɭɸɳɢɟ
ȝ
E
E
S
S
E
H
ɭɫɥɨɜɢɹ:
,
ijij
12
ȝȦH
kE
mm
0
ȦEkH
HH
mm
0
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɟ ɤɨɥɟɛɚɧɢɹ ɜɟɤɬɨɪɨɜ
,
.
ɢ H
ɩɪɨɢɫɯɨɞɹɬ ɫ ɨɞɢɧɚɤɨɜɨɣ ɮɚɡɨɣ, ɚ ɚɦɩɥɢɬɭɞɵ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ
EHȝȝHH (6.65)
00
mm
.
ɍɦɧɨɠɢɜ ɭɪɚɜɧɟɧɢɟ (6.63) ɧɚ ɨɪɬ ɨɫɢ ɭ, ɚ ɭɪɚɜɧɟɧɢɟ (6.64) ɧɚ ɨɪɬ ɨɫɢ z,
ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ ɩɥɨɫɤɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɜ ɜɟɤɬɨɪɧɨɣ ɮɨɪɦɟ:
cos ,
EE Ȧtkxij
HH Ȧtkxij
m
cos .
m
½
°
(6.66)
¾
°
¿
ɇɚ ɪɢɫ. 6.21 ɩɨɤɚɡɚɧ «ɦɨɦɟɧɬɚɥɶɧɵɣ ɫɧɢɦɨɤ» ɩɥɨɫɤɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ
ɜɨɥɧɵ. ɂɡ ɧɟɝɨ ɜɢɞɧɨ, ɱɬɨ ɜɟɤɬɨɪɵ
ɢ H ɨɛɪɚɡɭɸɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ
ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɩɪɚɜɨɜɢɧɬɨɜɭɸ ɫɢɫɬɟɦɭ. ȼ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɬɨɱɤɟ
ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɟɤɬɨɪɵ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɟɣ
ɢɡɦɟɧɹɸɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ.
y
v
Ɋɢɫ. 6.21
6.7. ǩǹDZǼǯǴȋ Ǵ ǴǸǻǿǷȈǽ ȉǷDZǶǾǼǺǸǬǯǹǴǾǹǺǵ ǮǺǷǹȇ
Ⱦɥɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɟɪɟɧɨɫɚ ɷɧɟɪɝɢɢ ɥɸɛɨɣ ɜɨɥɧɨɣ ɜ ɮɢɡɢɤɟ ɜɜɟɞɟɧɚ
ɜɟɤɬɨɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɧɚɡɵɜɚɟɦɚɹ ɩɥɨɬɧɨɫɬɶɸ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ
. Ɉɧɚ ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɤɨɥɢɱɟɫɬɜɭ ɷɧɟɪɝɢɢ, ɩɟɪɟɧɨɫɢɦɨɣ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ
ɟɞɢɧɢɱɧɭɸ ɩɥɨɳɚɞɤɭ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɤ ɧɚɩɪɚɜɥɟɧɢɸ, ɜ ɤɨɬɨɪɨɦ
ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜɨɥɧɚ. ɇɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ
140
ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ
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