Уравнения Максвелла (55
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ɇ. Ⱥ. ɉɨɩɨɜ
ɍɊȺȼɇȿɇɂə ɆȺɄɋȼȿɅɅȺ
Ɇɨɫɤɜɚ – 2012
ɍȾɄ 537.8(07) ȻȻɄ 22.313.2ɹ7 ɉ58
ɉɟɱɚɬɚɟɬɫɹ ɩɨ ɪɟɲɟɧɢɸ ɍɱɟɧɨɝɨ ɫɨɜɟɬɚ Ɇɨɫɤɨɜɫɤɨɝɨ ɩɟɞɚɝɨɝɢɱɟɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ
Ɋɟɰɟɧɡɟɧɬɵ:
ɩɪɨɮɟɫɫɨɪ ȼ. ɉ. Ʉɪɚɣɧɨɜ (ɆɎɌɂ), ɩɪɨɮɟɫɫɨɪ Ⱥ. ɇ. Ɇɚɧɫɭɪɨɜ (ɆɉȽɍ).
ɉ58 ɉɨɩɨɜ ɇ.Ⱥ. ɍɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ: ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ. – Ɇ.:
МПГУ, 2012. – 34 ɫ.
Ⱦɚɧɧɨɟ ɩɨɫɨɛɢɟ ɩɨɫɜɹɳɟɧɨ ɨɛɫɭɠɞɟɧɢɸ ɨɫɧɨɜɧɵɯ ɭɪɚɜɧɟɧɢɣ ɤɥɚɫɫɢɱɟɫɤɨɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ – ɭɪɚɜɧɟɧɢɣ Ɇɚɤɫɜɟɥɥɚ. ȼ ɫɨɝɥɚɫɢɢ ɫ ɫɨɜɪɟɦɟɧɧɵɦɢ ɬɟɧɞɟɧɰɢɹɦɢ ɜ ɪɚɡɜɢɬɢɢ ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɣ ɮɢɡɢɤɢ ɜ ɩɨɫɨɛɢɢ ɛɨɥɶɲɨɟ ɜɧɢɦɚɧɢɟ ɭɞɟɥɹɟɬɫɹ ɩɪɢɧɰɢɩɚɦ ɫɢɦɦɟɬɪɢɢ, ɜ ɫɜɹɡɢ ɫ ɱɟɦ ɜ ɩɨɫɨɛɢɢ ɫ ɫɚɦɨɝɨ ɧɚɱɚɥɚ ɢɫɩɨɥɶɡɭɸɬɫɹ ɪɟɥɹɬɢɜɢɫɬɫɤɢɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɢ ɪɟɥɹɬɢɜɢɫɬɫɤɢɣ ɮɨɪɦɚɥɢɡɦ, ɱɬɨ ɫɩɨɫɨɛɫɬɜɭɟɬ ɛɨɥɟɟ ɫɠɚɬɨɦɭ, ɚ ɜ ɢɞɟɣɧɨɦ ɫɦɵɫɥɟ – ɛɨɥɟɟ ɫɨɞɟɪɠɚɬɟɥɶɧɨɦɭ ɢɡɥɨɠɟɧɢɸ ɨɫɧɨɜɧɵɯ ɡɚɤɨɧɨɜ ɢ ɩɨɧɹɬɢɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ. ɉɨ ɷɬɢɦ ɠɟ ɩɪɢɱɢɧɚɦ ɩɪɢɧɰɢɩ ɤɚɥɢɛɪɨɜɨɱɧɨɣ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɨɞɧɨ ɢɡ ɢɫɯɨɞɧɵɯ ɬɪɟɛɨɜɚɧɢɣ, ɤɨɬɨɪɵɦ ɞɨɥɠɧɨ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɨɩɢɫɚɧɢɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. ɉɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɣ ɮɢɡɢɤɢ ɢ ɫɬɭɞɟɧɬɨɜ ɮɢɡɢɱɟɫɤɢɯ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ ɩɟɞɜɭɡɨɜ.
ISBN 978-5-4263-0105-4
©Н. А. Попов
©ɆɉȽɍ, 2012
©Оформление. ɂɡɞɚɬɟɥɶɫɬɜɨ «ɉɪɨɦɟɬɟɣ», 2012
ɋɨɞɟɪɠɚɧɢɟ
ɉɊȿȾɂɋɅɈȼɂȿ................................................................................................................. |
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ɗɅȿɄɌɊɂɑȿɋɄɂɃ ɁȺɊəȾ............................................................................................ |
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ɉɅɈɌɇɈɋɌɖ ɁȺɊəȾȺ ɂ ɉɅɈɌɇɈɋɌɖ ɌɈɄȺ............................................................ |
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4-ɌɈɄ .............................................................................................................................. |
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ɁȺɄɈɇ ɋɈɏɊȺɇȿɇɂə ɁȺɊəȾȺ................................................................................ |
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ɉɈɅȿȼȺə ɎɈɊɆȺ ɍɊȺȼɇȿɇɂə Ⱦȼɂɀȿɇɂə ɑȺɋɌɂɐ......................................... |
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ɌȿɇɁɈɊ ɗɇȿɊȽɂɂ-ɂɆɉɍɅɖɋȺ ɋɂɋɌȿɆɕ ɑȺɋɌɂɐ............................................. |
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ɉȿɊȿɆȿɇɇɕȿ ɋɈɋɌɈəɇɂə ɗɅȿɄɌɊɈɆȺȽɇɂɌɇɈȽɈ ɉɈɅə ɂ ɉɊɂɇɐɂɉ |
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ɄȺɅɂȻɊɈȼɈɑɇɈɃ ɂɇȼȺɊɂȺɇɌɇɈɋɌɂ..................................................................... |
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ɍɊȺȼɇȿɇɂȿ Ⱦȼɂɀȿɇɂə ɁȺɊəɀȿɇɇɕɏ ɑȺɋɌɂɐ ȼ ɗɅȿɄɌɊɈɆȺȽɇɂɌɇɈɆ |
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ɉɈɅȿ ................................................................................................................................ |
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ɍɊȺȼɇȿɇɂə Ⱦȼɂɀȿɇɂə ȾɅə ɗɅȿɄɌɊɈɆȺȽɇɂɌɇɈȽɈ ɉɈɅə........................... |
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ɌȿɇɁɈɊ ɗɇȿɊȽɂɂ-ɂɆɉɍɅɖɋȺ ɗɅȿɄɌɊɈɆȺȽɇɂɌɇɈȽɈ ɉɈɅə........................ |
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ɗɅȿɄɌɊɂɑȿɋɄɈȿ ɂ ɆȺȽɇɂɌɇɈȿ ɉɈɅə............................................................. |
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ɉɊȿɈȻɊȺɁɈȼȺɇɂȿ ɉɈɅȿɃ.................................................................................... |
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ɂɇȼȺɊɂȺɇɌɕ ɉɈɅə............................................................................................... |
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ɍɊȺȼɇȿɇɂə ɆȺɄɋȼȿɅɅȺ ...................................................................................... |
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ȼȿɄɌɈɊɕ ɉɈɅəɊɂɁɈȼȺɇɇɈɋɌɂ ɂ ɇȺɆȺȽɇɂɑȿɇɇɈɋɌɂ .............................. |
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ɉɪɟɞɢɫɥɨɜɢɟ
ɉɨɫɨɛɢɟ ɩɨɫɜɹɳɟɧɨ ɨɫɧɨɜɧɵɦ ɭɪɚɜɧɟɧɢɹɦ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ – ɭɪɚɜɧɟɧɢɹɦ Ɇɚɤɫɜɟɥɥɚ. ȼ ɩɨɫɨɛɢɢ ɫ ɫɚɦɨɝɨ ɧɚɱɚɥɚ ɢɫɩɨɥɶɡɭɸɬɫɹ ɪɟɥɹɬɢɜɢɫɬɫɤɢɟ ɢɞɟɢ ɢ ɪɟɥɹɬɢɜɢɫɬɫɤɢɣ ɮɨɪɦɚɥɢɡɦ. ȼ ɧɚɱɚɥɟ ɩɨɫɨɛɢɹ ɪɚɫɫɦɨɬɪɟɧɵ ɫɜɨɣɫɬɜɚ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɡɚɪɹɞɚ, ɜɜɟɞɟɧɵ ɩɨɧɹɬɢɹ ɩɥɨɬɧɨɫɬɢ ɡɚɪɹɞɚ ɢ ɩɥɨɬɧɨɫɬɢ ɬɨɤɚ, ɨɛɫɭɠɞɟɧ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɡɚɪɹɞɚ. Ɂɚɬɟɦ ɩɪɢɜɟɞɟɧɚ ɩɨɥɟɜɚɹ ɮɨɪɦɚ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ, ɜɜɟɞɟɧ ɬɟɧɡɨɪ ɷɧɟɪɝɢɢ-ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ. ȼ ɤɚɱɟɫɬɜɟ ɩɟɪɟɦɟɧɧɵɯ ɫɨɫɬɨɹɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɜɜɟɞɟɧɵ ɤɨɦɩɨɧɟɧɬɵ 4-ɩɨɬɟɧɰɢɚɥɚ ɢ ɢɯ ɩɟɪɜɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ ɢ ɜɪɟɦɟɧɢ. ɋɪɚɡɭ ɠɟ ɜɵɞɜɢɝɚɟɬɫɹ ɬɪɟɛɨɜɚɧɢɟ ɨɛ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɭɪɚɜɧɟɧɢɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɚɥɢɛɪɨɜɨɱɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ 4-ɩɨɬɟɧɰɢɚɥɚ. ɉɪɢ ɷɬɨɦ ɟɞɢɧɫɬɜɟɧɧɨɣ ɧɟɡɚɜɢɫɢɦɨɣ ɤɚɥɢɛɪɨɜɨɱɧɨ-ɢɧɜɚɪɢɚɧɬɧɨɣ ɤɨɧɫɬɪɭɤɰɢɟɣ ɢɡ ɩɟɪɟɦɟɧɧɵɯ ɫɨɫɬɨɹɧɢɹ ɨɤɚɡɵɜɚɟɬɫɹ 4-ɬɟɧɡɨɪ ɩɨɥɹ. ɉɨɷɬɨɦɭ ɢɦɟɧɧɨ ɷɬɨɬ ɬɟɧɡɨɪ ɜɵɫɬɭɩɚɟɬ ɜ ɤɚɱɟɫɬɜɟ ɩɪɟɞɫɬɚɜɢɬɟɥɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɜɨ ɜɫɟɯ ɭɪɚɜɧɟɧɢɹɯ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ ɢ ɜ ɜɵɪɚɠɟɧɢɹɯ ɞɥɹ ɬɚɤɢɯ ɜɟɥɢɱɢɧ, ɤɚɤ ɷɧɟɪɝɢɹ ɢɥɢ ɢɦɩɭɥɶɫ ɩɨɥɹ. ɋɨɩɨɫɬɚɜɥɟɧɢɟ ɲɟɫɬɢ ɧɟɡɚɜɢɫɢɦɵɦ ɤɨɦɩɨɧɟɧɬɚɦ ɬɟɧɡɨɪɚ ɩɨɥɹ ɤɨɦɩɨɧɟɧɬ ɞɜɭɯ 3-ɜɟɤɬɪɨɜ
– ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɢ ɢɧɞɭɤɰɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ – ɩɪɢɜɨɞɢɬ ɤ ɨɛɵɱɧɨɣ ɬɪɟɯɦɟɪɧɨɣ ɮɨɪɦɭɥɢɪɨɜɤɟ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ. ȼ ɩɨɫɥɟɞɧɟɦ ɪɚɡɞɟɥɟ ɩɨɫɨɛɢɹ ɪɚɫɫɦɨɬɪɟɧɵ ɜɟɤɬɨɪɵ ɩɨɥɹɪɢɡɨɜɚɧɧɨɫɬɢ ɢ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ ɢ ɨɛɫɭɠɞɟɧɨ ɢɯ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɞɥɹ ɨɩɢɫɚɧɢɹ ɪɚɫɩɨɥɨɠɟɧɢɹ ɢ ɞɜɢɠɟɧɢɹ ɡɚɪɹɞɨɜ.
1.ɗɥɟɤɬɪɢɱɟɫɤɢɣ ɡɚɪɹɞ
ȼɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɭɱɚɫɬɜɭɸɬ ɱɚɫɬɢɰɵ (ɬɟɥɚ), ɨɛɥɚɞɚɸɳɢɟ ɷɥɟɤɬɪɢɱɟɫɤɢɦɢ ɡɚɪɹɞɚɦɢ, ɷɥɟɤɬɪɢɱɟɫɤɢɦɢ ɞɢɩɨɥɶɧɵɦɢ ɦɨɦɟɧɬɚɦɢ, ɦɚɝɧɢɬɧɵɦɢ ɞɢɩɨɥɶɧɵɦɢ ɦɨɦɟɧɬɚɦɢ ɢɥɢ
4
ɞɪɭɝɢɦɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ. ɗɥɟɤɬɪɢɱɟɫɤɢɣ ɡɚɪɹɞ ɹɜɥɹɟɬɫɹ ɧɚɢɜɚɠɧɟɣɲɟɣ ɢɡ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɬɚɤɨɝɨ ɪɨɞɚ. ɂɡ ɚɧɚɥɢɡɚ ɨɩɵɬɧɵɯ ɞɚɧɧɵɯ ɜɵɬɟɤɚɸɬ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɡɚɪɹɞɚ:
1)Ɂɚɪɹɞ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ: ɜɟɥɢɱɢɧɟ ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɷɬɭ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɫɨɡɞɚɜɚɟɦɨɝɨ ɞɪɭɝɢɦɢ ɱɚɫɬɢɰɚɦɢ; ɜɟɥɢɱɢɧɟ ɡɚɪɹɞɚ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɢ ɩɨɥɟ, ɫɨɡɞɚɜɚɟɦɨɟ ɫɚɦɢɦ ɡɚɪɹɞɨɦ.
2)Ɂɚɪɹɞ ɹɜɥɹɟɬɫɹ ɜɧɭɬɪɟɧɧɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɱɚɫɬɢɰɵ, ɬɨ ɟɫɬɶ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɟɟ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ ɢ ɫɤɨɪɨɫɬɢ.
3)Ɂɚɪɹɞ – ɫɤɚɥɹɪɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɱɚɫɬɢɰɵ (3-ɫɤɚɥɹɪ). ɉɪɢ ɷɬɨɦ ɨɧ ɢɫɬɢɧɧɵɣ ɫɤɚɥɹɪ, ɬɨ ɟɫɬɶ ɢɧɜɚɪɢɚɧɬɟɧ ɩɨ ɨɬɧɨɲɟɧɢɸ ɧɟ ɬɨɥɶɤɨ ɤ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦ ɩɨɜɨɪɨɬɚɦ, ɧɨ ɢ ɤ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɢɧɜɟɪɫɢɢ.
4)Ɂɚɪɹɞ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɤ ɞɪɭɝɨɣ, ɬɨ ɟɫɬɶ ɹɜɥɹɟɬɫɹ 4-ɫɤɚɥɹɪɨɦ.
5)Ɂɚɪɹɞɵ ɛɵɜɚɸɬ ɞɜɭɯ ɡɧɚɤɨɜ: ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɢ ɨɬɪɢɰɚɬɟɥɶ-
ɧɵɟ.
6)Ɂɧɚɱɟɧɢɹ ɡɚɪɹɞɚ ɤɜɚɧɬɨɜɚɧɵ: ɡɚɪɹɞɵ ɬɟɥ ɤɪɚɬɧɵ ɷɥɟɦɟɧɬɚɪɧɨɦɭ ɡɚɪɹɞɭ.
7)ɗɥɟɤɬɪɢɱɟɫɤɢɣ ɡɚɪɹɞ ɚɞɞɢɬɢɜɟɧ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɡɚɪɹɞ ɥɸɛɨɝɨ ɬɟɥɚ ɪɚɜɟɧ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɡɚɪɹɞɨɜ ɱɚɫɬɟɣ ɷɬɨɝɨ ɬɟɥɚ.
8)ɉɪɢ ɥɸɛɵɯ ɩɪɨɰɟɫɫɚɯ ɜɵɩɨɥɧɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɡɚɪɹɞɚ.
2.ɉɥɨɬɧɨɫɬɶ ɡɚɪɹɞɚ ɢ ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ
ȼ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɟ ɛɵɜɚɟɬ ɭɞɨɛɧɨ ɞɚɠɟ ɞɥɹ ɫɢɫɬɟɦɵ ɬɨɱɟɱɧɵɯ ɱɚɫɬɢɰ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɡɚɪɹɞ ɢ ɦɚɫɫɭ ɤɚɤ ɪɚɫɩɪɟɞɟɥɟɧɧɵɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɧɟɩɪɟɪɵɜɧɵɦ ɨɛɪɚɡɨɦ. ɇɚɪɹɞɭ ɫ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɨ ɧɟ-
5
ɩɪɟɪɵɜɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɡɚɪɹɞɚ ɢ ɦɚɫɫɵ ɜɜɨɞɹɬ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɧɟɩɪɟɪɵɜɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɫɤɨɪɨɫɬɢ, ɨɩɢɫɵɜɚɟɦɨɦ ɮɭɧɤɰɢɟɣ v&(r&,t) (ɩɨɥɟ ɫɤɨɪɨɫɬɟɣ).
ɋɤɨɪɨɫɬɶ v&(r&,t) ɨɬɧɨɫɢɬɫɹ ɤ ɨɩɪɟɞɟɥɟɧɧɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɨɦ r&. Ɉɧɚ ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɬɨɣ ɱɚɫɬɢɰɵ, ɤɨɬɨɪɚɹ ɜ ɦɨɦɟɧɬ t ɧɚɯɨɞɢɬɫɹ ɜ ɷɬɨɣ ɬɨɱɤɟ. ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, ɭ ɱɚɫɬɢɰɵ ɫ
ɧɨɦɟɪɨɦ a ɜ ɦɨɦɟɧɬ t ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɪɚɜɟɧ r&a (t) |
ɢ ɫɤɨɪɨɫɬɶ ɪɚɜɧɚ |
v&a (t) , ɬɨ |
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v&(r&a ,t) v&a (t) . |
(2.1) |
Ⱦɥɹ ɭɫɤɨɪɟɧɢɹ ɱɚɫɬɢɰɵ ɩɪɢ ɷɬɨɦ ɩɨɥɭɱɚɟɦ:
& |
& & |
§ |
& |
· |
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dva (t) |
dv(ra ,t) |
¨ dra |
w& |
¸v&(r&a ,t) |
v&(r&a ,t) |
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wt |
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dt |
dt |
© |
dt |
wra ¹ |
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v&a a v&(r&a ,t) |
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v& |
(r&a ,t) . |
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Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɞɥɹ 4-ɭɫɤɨɪɟɧɢɹ ɢɦɟɟɦ:
dU (a) |
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dU (a) |
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& |
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& |
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& |
½ |
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k |
J |
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k |
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® v |
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U |
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(r ,t) |
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U |
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(r ,t)¾ |
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dt0(a) |
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dt |
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a ¯ |
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k |
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¿ |
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Ul (ra ,t) |
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Uk |
(ra ,t) , |
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(2.3) |
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wx(a) |
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ɝɞɟ ɩɨ ɞɜɚɠɞɵ ɩɨɜɬɨɪɹɸɳɟɦɭɫɹ (ɥɚɬɢɧɫɤɨɦɭ) ɢɧɞɟɤɫɭ l ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ ɫɭɦɦɢɪɨɜɚɧɢɟ ɨɬ 1 ɞɨ 4; ɩɨ a (ɧɨɦɟɪ ɱɚɫɬɢɰɵ) ɫɭɦɦɢɪɨ-
ɜɚɧɢɹ ɧɟɬ; dt0(a) – ɩɪɨɦɟɠɭɬɨɤ ɫɨɛɫɬɜɟɧɧɨɝɨ ɜɪɟɦɟɧɢ a -ɣ ɱɚɫɬɢ-
ɰɵ; dt – ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɩɨ ɱɚɫɚɦ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ v&a ;
U (a) |
(U&(a) ,U (a) ) |
(v& |
J |
a |
,icJ |
a |
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– 4-ɫɤɨɪɨɫɬɶ a -ɣ ɱɚɫɬɢɰɵ; |
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1 1 v2 |
c2 ; |
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) . Ʉɪɨɦɟ ɬɨɝɨ, ɡɞɟɫɶ ɜɜɟɞɟɧɨ ɩɨ- |
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wx(a) |
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icwt |
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ɥɟ 4-ɫɤɨɪɨɫɬɟɣ Uk (r&,t) , ɬɚɤ ɱɬɨ
6
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U (a) (t) |
U |
k |
(r& ,t) . |
(2.4) |
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Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢ ɞɜɢɠɟɧɢɹ |
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ɡɚɪɹɞɨɜ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɥɨɬɧɨɫɬɶ ɡɚɪɹɞɚ U(r&,t) |
ɢ ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ |
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& |
& |
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(r ,t) . ɉɥɨɬɧɨɫɬɶ ɡɚɪɹɞɚ – ɡɚɪɹɞ, ɩɪɢɯɨɞɹɳɢɣɫɹ ɧɚ ɟɞɢɧɢɰɭ ɨɛɴ- |
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ɟɦɚ. ɉɥɨɬɧɨɫɬɶ ɬɨɤɚ – ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɡɚɪɹɞɚ – ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɡɚɪɹɞɭ, ɩɟɪɟɧɨɫɢɦɨɦɭ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɟɞɢɧɢɱɧɭɸ ɩɥɨɳɚɞɤɭ, ɪɚɫɩɨɥɨɠɟɧɧɭɸ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɞɜɢɠɟɧɢɹ ɡɚɪɹɞɨɜ.
Ⱦɥɹ ɫɢɫɬɟɦɵ ɬɨɱɟɱɧɵɯ ɡɚɪɹɞɨɜ ɩɥɨɬɧɨɫɬɶ ɡɚɪɹɞɚ ɢ ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ ɡɚɩɢɫɵɜɚɸɬɫɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ:
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U(r&,t) |
¦qaG r& r&a (t) , |
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& |
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(2.6) |
j(r ,t) |
¦qava (t)G r |
ra (t) |
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a
ɝɞɟ qa – ɡɚɪɹɞ a -ɣ ɱɚɫɬɢɰɵ, r&a (t) – ɟɟ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ, ɚ v&a (t) – ɫɤɨɪɨɫɬɶ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t . ɉɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (2.5) ɩɨ ɧɟɤɨɬɨɪɨɦɭ ɨɛɴɟɦɭ ɩɨɥɭɱɚɟɦ, ɤɚɤ ɷɬɨ ɢ ɞɨɥɠɧɨ ɛɵɬɶ, ɫɭɦɦɭ ɡɚɪɹɞɨɜ ɱɚɫɬɢɰ, ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɦɨɦɟɧɬ t ɜɧɭɬɪɢ ɷɬɨɝɨ ɨɛɴɟɦɚ, ɚ ɩɪɢ ɚɧɚɥɨɝɢɱɧɨɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (2.6) – ɫɭɦɦɭ ɩɪɨɢɡɜɟɞɟɧɢɣ ɡɚɪɹɞɨɜ ɱɚɫɬɢɰ ɧɚ ɢɯ ɫɤɨɪɨɫɬɢ. ɂɫɩɨɥɶɡɭɹ ɪɚɜɟɧɫɬɜɚ (2.1) ɢ (2.5) ɢ ɨɞɧɨ ɢɡ ɫɜɨɣɫɬɜ ɞɟɥɶɬɚɮɭɧɤɰɢɢ –
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f (r&a )G(r& r&a ) |
f (r&)G(r& r&a ) , |
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(2.7) |
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ɩɪɟɨɛɪɚɡɭɟɦ ɜɵɪɚɠɟɧɢɟ (2.6) ɢ ɩɨɥɭɱɢɦ: |
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& & |
& |
& |
& |
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& & |
& |
& |
j(r ,t) |
¦qava (t)G r |
ra (t) |
¦qav(ra ,t)G |
r |
ra (t) |
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¦qav&(r&,t)G r& r&a (t) |
v&(r&,t)¦qa G r& r&a (t) , |
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ɢɥɢ ɠɟ |
& |
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& |
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j |
(r ,t) |
U(r ,t) v |
(r,t) . |
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ȿɫɥɢ ɧɟ ɭ ɜɫɟɯ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɡɧɚɤɢ ɡɚɪɹɞɨɜ ɨɞɢɧɚɤɨɜɵ, ɬɨ,
ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɧɟɨɛɯɨɞɢɦɨ ɜɜɨɞɢɬɶ ɞɜɚ ɩɨɥɹ ɫɤɨɪɨɫɬɟɣ: v&( ) (r&,t)
ɞɥɹ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɢ v&( ) (r&,t) ɞɥɹ ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɡɚɪɹɞɨɜ. ɉɪɢ ɷɬɨɦ ɛɭɞɟɦ ɢɦɟɬɶ
U |
U( ) U( ) , |
(2.9) |
&j &j ( ) &j ( ) |
U( ) v&( ) U( ) v&( ) . |
(2.10) |
Ɉɱɟɧɶ ɜɚɠɧɨ, ɱɬɨ ɩɥɨɬɧɨɫɬɶɸ ɡɚɪɹɞɚ ɢ ɩɥɨɬɧɨɫɬɶɸ ɬɨɤɚ ɦɨɠɧɨ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɢ ɱɚɫɬɢɰɵ ɫ ɧɭɥɟɜɵɦ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɡɚɪɹɞɨɦ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɬɨɱɟɱɧɨɣ ɱɚɫɬɢɰɵ ɫ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɞɢɩɨɥɶɧɵɦ
ɦɨɦɟɧɬɨɦ d&(t) : |
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U(r&,t) |
div^d&(t)G r& r&d (t) `, |
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j(r ,t) |
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d (t)G r |
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ɚ ɞɥɹ ɬɨɱɟɱɧɨɣ ɱɚɫɬɢɰɵ ɫ ɦɚɝɧɢɬɧɵɦ ɞɢɩɨɥɶɧɵɦ ɦɨɦɟɧɬɨɦ
P&(t) :
U(r&,t) 0, &j(r&,t) c rot^P&(t)G r& r&P(t) `. |
(2.12) |
ɉɨ ɚɧɚɥɨɝɢɢ ɫ ɩɥɨɬɧɨɫɬɶɸ ɡɚɪɹɞɚ U ɢ ɩɥɨɬɧɨɫɬɶɸ ɬɨɤɚ &j
ɞɥɹ ɫɢɫɬɟɦɵ ɬɨɱɟɱɧɵɯ ɱɚɫɬɢɰ ɦɨɠɧɨ ɜɜɟɫɬɢ «ɩɥɨɬɧɨɫɬɶ ɦɚɫɫɵ»
U(m) ɢ «ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ ɦɚɫɫ» &j (m) |
ɜɵɪɚɠɟɧɢɹɦɢ: |
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U(m) (r&,t) |
¦maG r& r&a (t) , |
(2.13) |
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&j (m) (r&,t) |
¦mav&a (t) G r& r&a (t) , |
(2.14) |
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ɝɞɟ |
m – ɦɚɫɫɚ a -ɣ ɱɚɫɬɢɰɵ. ȼɟɥɢɱɢɧɵ U(m) ɢ &j (m) |
ɫɜɹɡɚɧɵ ɫɨ- |
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ɨɬɧɨɲɟɧɢɟɦ |
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& & |
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(m) (r,t) |
U(m) (r,t) v(r ,t) , |
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ɚɧɚɥɨɝɢɱɧɵɦ ɫɨɨɬɧɨɲɟɧɢɸ (2.8). Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ «ɩɥɨɬɧɨɫɬɢ ɦɚɫɫɵ» U(m) ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ ɜɤɥɚɞ, ɨɛɭɫɥɨɜɥɟɧɧɵɣ ɨɬɧɨɫɢɬɟɥɶ-
ɧɵɦ ɞɜɢɠɟɧɢɟɦ ɱɚɫɬɢɰ ɢ ɢɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ.
3. 4-Ɍɨɤ
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ɉɭɫɬɶ ɜ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ K ɜ ɨɛɴɟɦɟ GV ɡɚ- |
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ɤɥɸɱɟɧ ɡɚɪɹɞ Gq |
UGV , ɞɜɢɠɭɳɢɣɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ K |
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ɫɨ |
ɫɤɨɪɨɫɬɶɸ v&. |
ɗɬɨɣ |
ɫɤɨɪɨɫɬɢ |
ɫɨɨɬɜɟɬɫɬɜɭɟɬ |
4-ɫɤɨɪɨɫɬɶ |
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Uk |
(v&J,icJ) . Ɉɛɴɟɦ GV |
ɫɜɹɡɚɧ ɫ ɨɛɴɟɦɨɦ |
GV0 , |
ɡɚɧɢɦɚɟɦɵɦ |
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ɬɟɦ ɠɟ ɡɚɪɹɞɨɦ Gq ɜ ɦɝɧɨɜɟɧɧɨ ɫɨɩɭɬɫɬɜɭɸɳɟɣ ɟɦɭ ɫɢɫɬɟɦɟ |
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ɨɬɫɱɟɬɚ |
K0 , |
ɫɨɨɬɧɨɲɟɧɢɟɦ |
JGV |
GV0 . |
ȼɟɥɢɱɢɧɚ |
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U0 |
Gq GV0 |
U J ɟɫɬɶ ɩɥɨɬɧɨɫɬɶ ɡɚɪɹɞɚ ɜ ɫɢɫɬɟɦɟ K0 . ȼɟɥɢɱɢ- |
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ɧɵ Gq ɢ GV0 , ɚ ɫ ɧɢɦɢ ɢ U0 |
ɹɜɥɹɸɬɫɹ 4-ɫɤɚɥɹɪɚɦɢ. ɉɨɷɬɨɦɭ ɱɟ- |
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ɬɵɪɟɯɤɨɦɩɨɧɟɧɬɧɚɹ ɜɟɥɢɱɢɧɚ |
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U |
& |
& |
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jk {U0Uk JUk |
(Uv,icU) |
( j,icU) |
(3.1) |
ɹɜɥɹɟɬɫɹ 4-ɜɟɤɬɨɪɨɦ. ɗɬɨɬ 4-ɜɟɤɬɨɪ ɧɚɡɵɜɚɸɬ 4-ɬɨɤɨɦ. ȿɫɥɢ
Gq Gq( ) Gq( ) , ɩɪɢɱɟɦ ɫɤɨɪɨɫɬɢ ɡɚɪɹɞɨɜ Gq( ) ɢ Gq( ) ɪɚɡɥɢɱ- |
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ɧɵ, ɬɨ j |
j( ) j( ) , ɝɞɟ |
j( ) ɢ |
j( ) – 4-ɬɨɤɢ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɢ |
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k |
k |
k |
k |
k |
ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɡɚɪɹɞɨɜ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. |
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ɉɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ K ɤ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ Kc, ɞɜɢɠɭɳɟɣɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ K ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ u&, ɤɨɦɩɨɧɟɧɬɵ 4-ɬɨɤɚ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɤɨɦɩɨɧɟɧɬɵ 4-
ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚ: |
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jkc /kl jl . |
(3.2) |
ɝɞɟ /kl – ɦɚɬɪɢɰɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɨɪɟɧɰɚ, ɚ ɩɨ ɩɨɜɬɨɪɹɸɳɟ-
ɦɭɫɹ (ɥɚɬɢɧɫɤɨɦɭ) ɢɧɞɟɤɫɭ l , ɤɚɤ ɨɛɵɱɧɨ, ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ
9
ɫɭɦɦɢɪɨɜɚɧɢɟ ɨɬ 1 ɞɨ 4. Ȼɨɥɟɟ ɩɨɞɪɨɛɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ (3.2)
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ: |
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& |
& |
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U u j|| |
c2 |
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& |
j|| uU |
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& |
& |
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j||c |
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jAc |
jA , Uc |
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(3.3) |
1 u2 c2 |
1 u2 |
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c2 |
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Ɂɞɟɫɶ ɜɟɤɬɨɪ |
ɩɥɨɬɧɨɫɬɢ |
3-ɬɨɤɚ |
ɩɪɟɞɫɬɚɜɥɟɧ |
ɜ ɜɢɞɟ |
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&j &j|| &jA , ɩɪɢɱɟɦ ɫɨɫɬɚɜɥɹɸɳɚɹ &j|| ɩɚɪɚɥɥɟɥɶɧɚ, ɚ |
&jA ɩɟɪɩɟɧ- |
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ɞɢɤɭɥɹɪɧɚ ɫɤɨɪɨɫɬɢ u&. |
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4. Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɡɚɪɹɞɚ
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɡɚɪɹɞɚ ɭɬɜɟɪɠɞɚɟɬ, ɱɬɨ ɜ ɡɚɦɤɧɭɬɨɣ ɮɢɡɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɩɪɢ ɩɪɨɬɟɤɚɧɢɢ ɜ ɧɟɣ ɥɸɛɵɯ ɩɪɨɰɟɫɫɨɜ ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɷɥɟɤɬɪɢɱɟɫɤɢɯ ɡɚɪɹɞɨɜ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ. Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɜ ɬɚɤɨɣ (ɤɚɤ ɝɨɜɨɪɹɬ, ɝɥɨɛɚɥɶɧɨɣ) ɮɨɪɦɭɥɢɪɨɜɤɟ ɧɟɨɛɯɨɞɢɦɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫɥɟɞɢɬɶ ɡɚ ɜɫɟɦɢ ɡɚɪɹɞɚɦɢ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ. ɉɨɷɬɨɦɭ ɡɚɱɚɫɬɭɸ ɛɨɥɟɟ ɭɞɨɛɧɵɦ ɨɤɚɡɵɜɚɟɬɫɹ ɥɨɤɚɥɶɧɵɣ ɩɨɞɯɨɞ, ɤɨɝɞɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɵɞɟɥɹɸɬ ɧɟɤɨɬɨɪɵɣ ɩɪɨɢɡɜɨɥɶɧɵɣ ɨɛɴɟɦ ɢ ɫɥɟɞɹɬ ɡɚ ɢɡɦɟɧɟɧɢɟɦ ɡɚɪɹɞɚ ɜɧɭɬɪɢ ɷɬɨɝɨ ɨɛɴɟɦɚ, ɩɨɥɚɝɚɹ, ɜ ɫɢɥɭ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɡɚɪɹɞɚ, ɱɬɨ ɷɬɨ ɢɡɦɟɧɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɨɛɭɫɥɨɜɥɟɧɨ ɥɢɲɶ ɩɟɪɟɦɟɳɟɧɢɟɦ ɡɚɪɹɞɚ ɱɟɪɟɡ (ɧɟɩɨɞɜɢɠɧɭɸ) ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɢɜɚɸɳɭɸ ɜɵɞɟɥɟɧɧɵɣ ɨɛɴɟɦ:
d |
& |
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³ |
& & & |
& |
³ |
& & |
& |
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³U(r |
,t)dV |
U(r,t)v(r ,t) dS |
j(r,t)dS . |
(4.1) |
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dt |
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V |
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SV |
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SV |
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ȼɧɟɫɹ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɜɪɟɦɟɧɢ ɩɨɞ ɡɧɚɤ ɢɧɬɟɝɪɚɥɚ ɢ ɩɪɟɨɛɪɚɡɨɜɚɜ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɟɦɨɣ Ƚɚɭɫɫɚ ɩɨɜɟɪɯɧɨɫɬɧɵɣ ɢɧɬɟɝɪɚɥ ɜ ɨɛɴɟɦɧɵɣ, ɩɨɥɭɱɢɦ:
³ |
§ wU |
&· |
(4.2) |
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¨ |
wt |
div j ¸dV 0 . |
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© |
¹ |
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V
Ɉɬɫɸɞɚ ɜɜɢɞɭ ɩɪɨɢɡɜɨɥɶɧɨɫɬɢ ɨɛɴɟɦɚ V ɫɥɟɞɭɟɬ:
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