Жданов С.К Цветков И.В - Основы физических процессов в плазме и в плазменных установках (2000)
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ȿ > 0.1Eɤɪ.
§ 11. Ⱦɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɩɥɚɡɦɵ
ɉɨɦɟɫɬɢɦ ɩɥɚɡɦɭ ɜɨ ɜɧɟɲɧɟɟ ɩɟɪɟɦɟɧɧɨɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɢ ɩɨɩɵɬɚɟɦɫɹ ɩɪɨɫɥɟɞɢɬɶ ɢɡɦɟɧɟɧɢɟ ɟɟ ɫɜɨɣɫɬɜ, ɩɨɫɬɟɩɟɧɧɨ ɭɜɟɥɢɱɢɜɚɹ ɟɝɨ ɱɚɫɬɨɬɭ. ɋɬɚɬɢɱɟɫɤɨɟ ɜɧɟɲɧɟɟ ɩɨɥɟ, ɤɚɤ ɦɵ ɭɠɟ ɡɧɚɟɦ, ɩɪɨɧɢɤɚɟɬ ɜ ɩɥɚɡɦɭ ɧɚ ɬɨɥɳɢɧɭ ɦɚɫɲɬɚɛɚ ɞɟɛɚɟɜɫɤɨɝɨ ɫɥɨɹ. ɗɬɨ ɜɵɬɟɤɚɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ ɷɤɪɚɧɢɪɨɜɤɢ (ɫɦ. §3), ɤɨɬɨɪɨɟ, ɧɚɩɪɢɦɟɪ, ɞɥɹ
ɨɞɧɨɦɟɪɧɨɝɨ ɫɥɭɱɚɹ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ: |
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d 2ϕ |
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ϕ |
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= |
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(1.74) |
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dx2 |
r2 |
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De |
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Ɂɞɟɫɶ rDe – ɷɥɟɤɬɪɨɧɧɵɣ ɞɟɛɚɟɜɫɤɢɣ ɪɚɞɢɭɫ. ɍɞɨɛɧɨ ɩɟɪɟɩɢɫɚɬɶ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɢɫɩɨɥɶɡɭɹ |
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ɩɪɟɞɫɬɚɜɥɟɧɢɟ |
Ɏɭɪɶɟϕ ~ ϕk exp(ikx − iωt) . Ɋɟɡɭɥɶɬɚɬ ɞɥɹ ɚɦɩɥɢɬɭɞɵ |
Ɏɭɪɶɟ-ɝɚɪɦɨɧɢɤɢ |
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ɩɨɬɟɧɰɢɚɥɚ ϕɤ, ɤɚɤ ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɛɭɞɟɬ ɫɥɟɞɭɸɳɢɦ: |
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k 2 (1 + |
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(1.75) |
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k 2r2 |
k |
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ɋɪɚɜɧɢɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɫ ɭɪɚɜɧɟɧɢɟɦ ɞɥɹ ɢɧɞɭɤɰɢɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ |
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dD |
= 0 . |
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(1.76) |
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dx |
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Ⱦɥɹ Ɏɭɪɶɟ-ɝɚɪɦɨɧɢɤɢ ɟɝɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ: |
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k2εk ϕk = 0 , |
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(1.77) |
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ɝɞɟ εk - ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɩɥɚɡɦɵ, ɨɩɢɫɵɜɚɸɳɚɹ ɨɬɤɥɢɤ ɩɥɚɡɦɵ ɧɚ ɫɬɚɬɢɱɟɫɤɨɟ ɜɨɡɞɟɣɫɬɜɢɟ. Ʉɚɤ ɜɢɞɢɦ ɢɡ ɫɪɚɜɧɟɧɢɹ (1.75) ɢ (1.77), ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ
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εk = 1 + |
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(1.78) |
k 2r2 |
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Ɉɬɤɥɢɤ ɩɥɚɡɦɵ ɧɚ ɫɬɚɬɢɱɟɫɤɨɟ ɜɨɡɞɟɣɫɬɜɢɟ ɨɤɚɡɵɜɚɟɬɫɹ ɜɩɨɥɧɟ ɷɤɜɢɜɚɥɟɧɬɟɧ ɷɮɮɟɤɬɭ ɩɨɥɹɪɢɡɚɰɢɢ ɨɛɵɱɧɨɝɨ ɞɢɷɥɟɤɬɪɢɤɚ, ɩɨɦɟɳɟɧɧɨɝɨ ɜɨ ɜɧɟɲɧɟɟ ɩɨɥɟ, ɯɨɬɹ, ɤɨɧɟɱɧɨ, ɦɟɯɚɧɢɡɦ ɩɨɥɹɪɢɡɚɰɢɢ ɢɧɨɣ: ɟɫɥɢ ɞɟɛɚɟɜɫɤɢɣ ɪɚɞɢɭɫ ɦɚɥ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɯɚɪɚɤɬɟɪɧɨɣ ɞɥɢɧɨɣ ɜɨɥɧɵ, ɬɨ ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɨɤɚɡɵɜɚɟɬɫɹ ɨɱɟɧɶ ɛɨɥɶɲɨɣ ɢ ɩɨɥɟ ɜ ɩɥɚɡɦɭ, ɮɚɤɬɢɱɟɫɤɢ, ɧɟ ɩɪɨɧɢɤɚɟɬ.
ɉɪɢ ɧɢɡɤɢɯ, ɧɨ ɧɟɧɭɥɟɜɵɯ, ɱɚɫɬɨɬɚɯ ω ɤɚɱɟɫɬɜɟɧɧɨ ɤɚɪɬɢɧɚ ɧɟ ɢɡɦɟɧɢɬɫɹ − ɡɚɪɹɞɵ ɛɭɞɭɬ ɷɤɪɚɧɢɪɨɜɚɬɶ ɜɧɟɲɧɟɟ ɩɨɥɟ ɜ ɫɥɨɹɯ ɦɚɫɲɬɚɛɚ ɞɟɛɚɟɜɫɤɢɯ. ɉɥɚɡɦɚ ɛɭɞɟɬ ɜɟɫɬɢ ɫɟɛɹ ɤɚɤ ɩɪɨɜɨɞɧɢɤ − ɜɧɟɲɧɟɟ ɩɨɥɟ ɜ ɧɟɟ ɧɟ ɛɭɞɟɬ ɩɪɨɧɢɤɚɬɶ.
ɇɨ ɟɫɥɢ ɱɚɫɬɨɬɚ ɩɨɥɹ ɛɭɞɟɬ ɜɟɥɢɤɚ, ɢ ɛɭɞɟɬ ɩɪɟɜɵɲɚɬɶ ɩɥɚɡɦɟɧɧɭɸ ɱɚɫɬɨɬɭ, ɬɨ ɤɚɪɬɢɧɚ ɤɚɱɟɫɬɜɟɧɧɨ ɢɡɦɟɧɢɬɫɹ: ɷɥɟɤɬɪɨɧɵ ɢɡ-ɡɚ ɢɧɟɪɰɢɢ ɧɟ ɛɭɞɭɬ ɭɫɩɟɜɚɬɶ ɩɨɞɫɬɪɚɢɜɚɬɶɫɹ ɩɨɞ ɤɨɥɟɛɚɧɢɹ ɩɨɥɹ, ɨɧɢ ɛɭɞɭɬ ɨɫɰɢɥɥɢɪɨɜɚɬɶ ɨɤɨɥɨ ɧɟɤɨɬɨɪɨɝɨ ɫɪɟɞɧɟɝɨ ɩɨɥɨɠɟɧɢɹ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɟ ɫɦɨɠɟɬ ɩɪɨɧɢɤɧɭɬɶ ɜ ɩɥɚɡɦɭ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ, ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɫɬɶ ɫɥɭɱɚɸ ɧɢɡɤɢɯ ɱɚɫɬɨɬ, ɩɪɨɧɢɤɚɸɳɟɟ ɜ ɩɥɚɡɦɭ ɩɨɥɟ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ. ɑɬɨɛɵ ɧɚɣɬɢ ɞɢɷɥɟɤɬɪɢɱɟɫɤɭɸ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɩɥɚɡɦɵ ɜ ɷɬɨɦ ɩɪɟɞɟɥɟ ɜɵɫɨɤɢɯ ɱɚɫɬɨɬ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɩɟɪɟɦɟɧɧɨɟ ɩɨɥɟ ɝɚɪɦɨɧɢɱɟɫɤɢɦ:
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= E0e |
iω t |
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(1.79) |
E |
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ɋɦɟɳɟɧɢɟ ∆ɯ ɷɥɟɤɬɪɨɧɚ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɬɚɤɨɝɨ ɩɨɥɹ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ
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= eE0e |
iωt |
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me x = eE |
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(1.80) |
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∆x = − |
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E. |
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mω 2 |
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ɉɨɞɫɱɢɬɚɟɦ ɬɟɩɟɪɶ ɢɧɞɭɤɰɢɸ ɩɨɥɹ ɜ ɩɥɚɡɦɟ |
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+ 4π P, |
(1.81) |
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D = εω E |
= E |
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ɝɞɟ Ɋ = ɩɟ∆x - ɞɢɩɨɥɶɧɵɣ ɦɨɦɟɧɬ ɟɞɢɧɢɰɵ ɨɛɴɟɦɚ ɩɥɚɡɦɵ, ɨɛɭɫɥɨɜɥɟɧɧɵɣ ɫɦɟɳɟɧɢɟɦ ɷɥɟɤɬɪɨɧɨɜ. ɉɪɨɢɡɜɟɞɹ ɩɨɞɫɬɚɧɨɜɤɭ, ɩɨɥɭɱɢɦ ɜɟɥɢɱɢɧɭ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɢ:
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§ω p · |
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ω 2 |
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4πne2 |
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ε = 1 − ¨ |
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¸ |
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= |
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(1.82) |
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ω |
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ω ¹ |
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p |
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me |
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ɨɩɢɫɵɜɚɸɳɟɣ ɨɬɤɥɢɤ ɩɥɚɡɦɵ ɧɚ ɜɵɫɨɤɨɱɚɫɬɨɬɧɨɟ ɢ ɨɞɧɨɪɨɞɧɨɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ.
ɉɨɥɭɱɟɧɧɭɸ ɮɨɪɦɭɥɭ ɞɥɹ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɢ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɜɨɡɞɟɣɫɬɜɢɹ ɧɚ ɩɥɚɡɦɭ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ, ɡɚɤɨɧ ɞɢɫɩɟɪɫɢɢ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɹɟɬ
ɭɪɚɜɧɟɧɢɟ (ɩɨɞɪɨɛɧɟɟ ɫɦ. Ƚɥɚɜɭ 3): |
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N 2 = ε , |
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ɝɞɟ N=ɤɫ/ω – ɩɨɤɚɡɚɬɟɥɶ ɩɪɟɥɨɦɥɟɧɢɹ. ɂɡ ɮɨɪɦɭɥɵ (1.82) ɨɱɟɜɢɞɧɨ, ɱɬɨ ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɟ ɜɨɥɧɵ ɫ ɱɚɫɬɨɬɨɣ ω > ωp ɦɨɝɭɬ ɩɪɨɧɢɤɚɬɶ ɜ ɩɥɚɡɦɭ ɢ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɜ ɧɟɣ, ɬɚɤ ɤɚɤ ɞɥɹ ɧɢɯ ɛɭɞɟɬ N2>0. ɇɚɩɪɨɬɢɜ, ɜ ɨɛɥɚɫɬɢ ɱɚɫɬɨɬ ω<ωp ɫɨɝɥɚɫɧɨ (1.82) ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɨɬɪɢɰɚɬɟɥɶɧɚɹ, ɬɚɤ ɱɬɨ ɞɥɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɫ ɬɚɤɨɣ ɱɚɫɬɨɬɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɟɥɨɦɥɟɧɢɹ ɨɤɚɡɵɜɚɟɬɫɹ ɱɢɫɬɨ ɦɧɢɦɵɦ, ɢ ɩɨɩɟɪɟɱɧɚɹ ɜɨɥɧɚ ɧɟ ɦɨɠɟɬ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɜ ɩɥɚɡɦɟ. Ɉɞɧɚɤɨ ɜ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɜɨɡɦɨɠɧɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɜ ɩɥɚɡɦɟ ɩɪɨɞɨɥɶɧɵɯ - ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ, ɥɟɧɝɦɸɪɨɜɫɤɢɯ ɜɨɥɧ.
Ʉɚɤ ɦɵ ɜɢɞɢɦ, ɜ ɫɬɚɬɢɱɟɫɤɨɦ ɩɪɟɞɟɥɟ ɢ ɜ ɜɵɫɨɤɨɱɚɫɬɨɬɧɨɦ ɩɪɟɞɟɥɟ ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɩɥɚɡɦɵ ɨɩɢɫɵɜɚɟɬɫɹ ɪɚɡɥɢɱɧɵɦɢ ɮɨɪɦɭɥɚɦɢ. ȼ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ, ɨɛɟ ɨɧɢ ɹɜɥɹɸɬɫɹ ɩɪɟɞɟɥɶɧɵɦɢ ɫɥɭɱɚɹɦɢ ɨɞɧɨɣ ɛɨɥɟɟ ɨɛɳɟɣ ɮɨɪɦɭɥɵ:
εk ,ω = 1 − |
ω p2 |
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(1.84) |
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ω 2 − k 2r2 |
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ɨɛɴɟɞɢɧɹɸɳɟɣ, ɤɚɤ ɥɟɝɤɨ ɜɢɞɟɬɶ, ɨɛɚ ɩɪɟɞɟɥɶɧɵɯ ɫɥɭɱɚɹ.
ȽɅȺȼȺ 2
ɉɅȺɁɆȺ ȼ ɆȺȽɇɂɌɇɈɆ ɉɈɅȿ
§ 12. Ɉɞɧɨɱɚɫɬɢɱɧɨɟ ɪɚɫɫɦɨɬɪɟɧɢɟ
ɋɭɬɶ ɨɞɧɨɱɚɫɬɢɱɧɨɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ, ɢɥɢ ɩɪɢɛɥɢɠɟɧɢɹ, ɜ ɨɩɢɫɚɧɢɢ ɩɥɚɡɦɟɧɧɵɯ ɩɪɨɰɟɫɫɨɜ ɫɜɨɞɢɬɫɹ ɤ ɢɡɭɱɟɧɢɸ ɞɜɢɠɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɢɰ ɩɥɚɡɦɵ ɜ ɩɨɥɹɯ, ɤɨɬɨɪɵɟ ɫɱɢɬɚɸɬɫɹ ɡɚɞɚɧɧɵɦɢ ɢɡɧɚɱɚɥɶɧɨ. Ɍɟɦ ɫɚɦɵɦ ɩɪɟɧɟɛɪɟɝɚɟɬɫɹ ɜɥɢɹɧɢɟɦ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɩɥɚɡɦɵ ɧɚ ɩɨɥɹ «ɭɩɪɚɜɥɹɸɳɢɟ» ɢɯ ɞɜɢɠɟɧɢɟɦ, ɬ.ɟ. ɩɪɟɧɟɛɪɟɝɚɟɬɫɹ ɷɮɮɟɤɬɚɦɢ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɢ ɜɵɡɵɜɚɟɦɨɝɨ ɢɦ ɢɡɦɟɧɟɧɢɹ ɩɨɥɹ. ȼ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɲɚɝɚ ɬɚɤɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɜɩɨɥɧɟ ɨɩɪɚɜɞɚɧɨ ɢ ɩɨɜɫɟɦɟɫɬɧɨ ɢɫɩɨɥɶɡɭɟɬɫɹ, ɨɫɨɛɟɧɧɨ, ɟɫɥɢ ɟɫɬɶ ɜɨɡɦɨɠɧɨɫɬɶ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɷɮɮɟɤɬɢɜɧɨ ɭɱɟɫɬɶ ɜɥɢɹɧɢɟ ɩɨɥɟɣ ɨɫɬɚɥɶɧɵɯ ɱɚɫɬɢɰ ɩɥɚɡɦɵ ɧɚ ɞɜɢɠɟɧɢɟ ɤɚɤɨɣ-ɥɢɛɨ ɨɞɧɨɣ ɜɵɞɟɥɟɧɧɨɣ ɱɚɫɬɢɰɵ. ɉɨɞɱɟɪɤɧɟɦ, ɜɦɟɫɬɟ ɫ ɬɟɦ, ɱɬɨ ɧɟɞɨɭɱɟɬ ɷɮɮɟɤɬɨɜ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɢɹ ɫɨɞɟɪɠɢɬ ɜ ɫɟɛɟ ɦɧɨɠɟɫɬɜɨ ɧɟɩɪɢɹɬɧɵɯ «ɫɸɪɩɪɢɡɨɜ», ɢ ɜ ɢɫɬɨɪɢɢ ɪɚɡɜɢɬɢɹ ɢɫɫɥɟɞɨɜɚɧɢɣ ɜ ɨɛɥɚɫɬɢ ɭɩɪɚɜɥɹɟɦɨɝɨ ɬɟɪɦɨɹɞɟɪɧɨɝɨ ɫɢɧɬɟɡɚ ɬɚɤɢɯ ɩɪɢɦɟɪɨɜ ɩɪɟɞɨɫɬɚɬɨɱɧɨ. ɇɚɩɪɢɦɟɪ, ɜ ɤɥɚɫɫɢɱɟɫɤɢɯ ɚɤɫɢɚɥɶɧɨ-ɫɢɦɦɟɬɪɢɱɧɵɯ ɚɞɢɚɛɚɬɢɱɟɫɤɢɯ ɥɨɜɭɲɤɚɯ (ɫɦ. §18), ɩɪɟɤɪɚɫɧɨ ɭɞɟɪɠɢɜɚɸɳɢɯ ɨɬɞɟɥɶɧɵɟ ɱɚɫɬɢɰɵ ɩɥɚɡɦɵ ɨɱɟɧɶ ɧɢɡɤɨɣ ɩɥɨɬɧɨɫɬɢ, ɤɨɝɞɚ ɨɞɧɨɱɚɫɬɢɱɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɜɩɨɥɧɟ ɨɩɪɚɜɞɚɧɨ, ɫ ɩɨɜɵɲɟɧɢɟɦ ɩɥɨɬɧɨɫɬɢ ɩɥɚɡɦɵ, ɤɨɝɞɚ ɜɫɟ ɛɨɥɶɲɭɸ ɪɨɥɶ ɧɚɱɢɧɚɸɬ ɢɝɪɚɬɶ ɫɨɛɫɬɜɟɧɧɵɟ ɩɨɥɹ ɩɥɚɡɦɵ, ɩɨɹɜɥɹɟɬɫɹ ɧɟɭɫɬɨɣɱɢɜɨɫɬɶ ɢ ɭɞɟɪɠɚɧɢɟ ɩɥɨɬɧɨɣ ɩɥɚɡɦɵ ɛɟɡ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɢɥɢɣ ɩɪɨɫɬɨ ɧɟɜɨɡɦɨɠɧɨ.
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɡɚɞɚɧɧɵɯ ɜɧɟɲɧɢɯ
ɩɨɥɹɯ ɢɦɟɟɬ ɜɢɞ: |
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× B + F , |
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ɝɞɟ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɫɩɪɚɜɚ ɨɬɜɟɱɚɟɬ ɫɢɥɟ ɫɨ ɫɬɨɪɨɧɵ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ, ɜɬɨɪɨɟ ɨɩɢɫɵɜɚɟɬ ɫɢɥɭ ɫɨ ɫɬɨɪɨɧɵ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ (ɫɢɥɚ Ʌɨɪɟɧɰɚ), ɚ ɩɨɫɥɟɞɧɟɟ ɨɛɨɡɧɚɱɚɟɬ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɭɸ ɩɪɨɱɢɯ ɜɧɟɲɧɢɯ ɫɢɥ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɞɟɣɫɬɜɨɜɚɬɶ ɧɚ ɱɚɫɬɢɰɭ.
ɍɪɚɜɧɟɧɢɟ (2.1) ɦɨɠɟɬ ɛɵɬɶ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɧɨ ɬɨɥɶɤɨ ɜ ɨɬɞɟɥɶɧɵɯ, ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɵɯ ɫɥɭɱɚɹɯ. ɗɬɨ ɩɨɧɹɬɧɨ, ɟɫɥɢ ɭɱɟɫɬɶ, ɱɬɨ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɱɚɫɬɢɰɭ ɩɨɥɹ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɤɨɨɪɞɢɧɚɬ ɢ ɜɪɟɦɟɧɢ ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɱɬɨ ɨɧɢ ɫɚɦɢ ɫɜɹɡɚɧɵ ɭɪɚɜɧɟɧɢɹɦɢ Ɇɚɤɫɜɟɥɥɚ. ɉɨɷɬɨɦɭ ɪɚɫɫɦɨɬɪɢɦ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɵɟ, ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɳɢɟɫɹ ɫɥɭɱɚɢ. Ɍɚɤ, ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɢɯ, ɧɟɷɥɟɤɬɪɢɱɟɫɤɢɯ ɫɢɥ, ɨɛɵɱɧɨ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɚ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɱɚɫɬɨ ɨɬɫɭɬɫɬɜɭɟɬ. ɗɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɜɢɠɟɧɢɸ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɩɨɫɬɨɹɧɧɨɦ ɜɨ ɜɪɟɦɟɧɢ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ. Ɋɚɫɫɦɨɬɪɢɦ ɫɧɚɱɚɥɚ ɞɜɢɠɟɧɢɟ ɜ ɨɞɧɨɪɨɞɧɨɦ ɩɨɥɟ, ɬ.ɟ. ɧɟ ɦɟɧɹɸɳɟɦɫɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.
§13. Ⱦɜɢɠɟɧɢɟ ɜ ɩɨɫɬɨɹɧɧɨɦ ɢ ɨɞɧɨɪɨɞɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ
ȼɫɥɭɱɚɟ ɟɫɥɢ ɞɪɭɝɢɯ ɫɢɥ, ɤɪɨɦɟ ɫɢɥɵ Ʌɨɪɟɧɰɚ ɧɟɬ, ɬɨ ɭɪɚɜɧɟɧɢɟ (2.1) ɢɦɟɟɬ ɜɢɞ
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ɍɦɧɨɠɚɹ ɫɤɚɥɹɪɧɨ ɩɪɚɜɭɸ ɢ ɥɟɜɭɸ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (2.2) ɧɚ ɫɤɨɪɨɫɬɶ, ɭɱɢɬɵɜɚɹ, ɱɬɨ ɫɢɥɚ Ʌɨɪɟɧɰɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɤ ɨɛɨɢɦ ɜɟɤɬɨɪɚɦ, ɜɯɨɞɹɳɢɦ ɜ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ (2.2), ɩɨɥɭɱɢɦ ɫɨɨɬɧɨɲɟɧɢɟ
&& ≡ d mv2 = mvv dt 2 0 .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ:
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ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɨɫɬɨɹɧɧɨɟ ɜɨ ɜɪɟɦɟɧɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɧɟ ɩɪɨɢɡɜɨɞɢɬ ɪɚɛɨɬɵ ɧɚɞ ɱɚɫɬɢɰɟɣ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ, ɡɚɞɚɧɧɵɣ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɫɨɯɪɚɧɹɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ.
Ɋɚɡɥɨɠɢɜ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɧɚ ɤɨɦɩɨɧɟɧɬɵ: ɩɚɪɚɥɥɟɥɶɧɭɸ v|| = ( vB& ) / B ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ v& = v& − v||B / B ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ, ɩɨɥɭɱɢɦ ɞɥɹ ɩɪɨɞɨɥɶɧɨɝɨ
ɭɫɤɨɪɟɧɢɹ: &
mv|| = ce BB ( v& × B&) ≡ 0 ,
- ɩɨɫɬɨɹɧɧɨɟ ɢ ɨɞɧɨɪɨɞɧɨɟ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɧɟ ɦɟɧɹɟɬ ɩɪɨɞɨɥɶɧɭɸ ɤɨɦɩɨɧɟɧɬɭ ɫɤɨɪɨɫɬɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
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Ⱦɥɹ ɩɨɩɟɪɟɱɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɩɨɥɭɱɚɟɦ |
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ɢ, ɤɚɤ ɦɵ ɜɢɞɢɦ, ɩɨɩɟɪɟɱɧɚɹ ɤɨɦɩɨɧɟɧɬɚ ɦɟɧɹɟɬɫɹ ɬɨɥɶɤɨ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɚɫɬɢɰɚ ɪɚɜɧɨɦɟɪɧɨ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɫɢɥɨɜɨɣ ɥɢɧɢɢ ɩɨɥɹ, ɜɪɚɳɚɹɫɶ ɩɪɢ ɷɬɨɦ ɜɨɤɪɭɝ ɫɢɥɨɜɨɣ ɥɢɧɢɢ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ. Ɉɩɪɟɞɟɥɢɦ ɪɚɞɢɭɫ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ ρ ɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ω.
Ɍɚɤ ɤɚɤ v& ɹɜɥɹɟɬɫɹ ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶɧɵɦ ɭɫɤɨɪɟɧɢɟɦ, ɬɨ, ɜɫɩɨɦɧɢɜ ɨɩɪɟɞɟɥɟɧɢɟ
ɜɟɥɢɱɢɧɵ ɦɨɞɭɥɹ ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶɧɨɝɨ ɭɫɤɨɪɟɧɢɹ − ɨɬɧɨɲɟɧɢɟ ɤɜɚɞɪɚɬɚ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɤ ɪɚɞɢɭɫɭ ɨɤɪɭɠɧɨɫɬɢ, ɢɥɢ ɩɪɨɢɡɜɟɞɟɧɢɟ ɤɜɚɞɪɚɬɚ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɧɚ ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ, − ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
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Ɉɬɤɭɞɚ ɢ ɩɨɥɭɱɚɟɦ ɬɪɟɛɭɟɦɵɟ ɧɚɦ ɜɟɥɢɱɢɧɵ |
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ρ = ρɥɚ ɪɦ ≡ |
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Ɋɚɞɢɭɫ ρɥɚɪɦ ɧɚɡɵɜɚɸɬ ɥɚɪɦɨɪɨɜɫɤɢɦ ɪɚɞɢɭɫɨɦ ɱɚɫɬɢɰɵ, ɚ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ωɥɚɪɦ -
ɥɚɪɦɨɪɨɜɫɤɨɣ ɢɥɢ ɰɢɤɥɨɬɪɨɧɧɨɣ ɱɚɫɬɨɬɨɣ ɜɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ (ɞɥɹ ɷɬɢɯ ɜɟɥɢɱɢɧ ɭɩɨɬɪɟɛɢɬɟɥɶɧɵ ɢ ɞɪɭɝɢɟ ɨɛɨɡɧɚɱɟɧɢɹ, ɧɚɩɪɢɦɟɪ, ɫ ɢɧɞɟɤɫɨɦ “B”: ρȼ ɢ ωȼ, ɩɨɞɱɟɪɤɢɜɚɸɳɢɟ ɜ ɹɜɧɨɦ ɜɢɞɟ, ɱɬɨ ɨɧɢ ɨɬɧɨɫɹɬɫɹ ɤ ɫɥɭɱɚɸ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ). ɂɡ ɫɨɨɬɧɨɲɟɧɢɣ (2.6) ɢ (2.7) ɜɢɞɧɨ, ɱɬɨ ɰɢɤɥɨɬɪɨɧɧɚɹ ɱɚɫɬɨɬɚ ɡɚɜɢɫɢɬ ɨɬ ɦɚɫɫɵ ɢ ɡɚɪɹɞɚ ɱɚɫɬɢɰɵ, ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɧɨ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɟɥɢɱɢɧɵ
ɫɤɨɪɨɫɬɢ ɜɪɚɳɚɸɳɟɣɫɹ ɜ ɩɨɥɟ ɱɚɫɬɢɰɵ, ɚ ɥɚɪɦɨɪɨɜɫɤɢɣ ɪɚɞɢɭɫ ɡɚɜɢɫɢɬ ɨɬ ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɤɨɪɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ, ɩɪɢɱɟɦ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɟɟ ɨɧ ɜɨɡɪɚɫɬɚɟɬ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ. ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɜɟɤɬɨɪ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɩɨɥɨɠɢɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɚɧɬɢɩɚɪɚɥɥɟɥɟɧ, ɚ ɨɬɪɢɰɚɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ (ɧɚɩɪɢɦɟɪ, ɷɥɟɤɬɪɨɧɚ) - ɩɚɪɚɥɥɟɥɟɧ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ. Ɍɚɤ ɤɚɤ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɦɨɠɧɨ ɭɩɨɞɨɛɢɬɶ ɤɪɭɝɨɜɨɦɭ ɬɨɤɭ j = qω/2π, ɬɨ ɜɪɚɳɟɧɢɸ ɱɚɫɬɢɰɵ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɦɨɠɧɨ ɫɨɩɨɫɬɚɜɢɬɶ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ, ɪɚɜɧɵɣ ɦɚɝɧɢɬɧɨɦɭ ɦɨɦɟɧɬɭ ɷɬɨɝɨ ɤɪɭɝɨɜɨɝɨ ɬɨɤɚ:
& |
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j |
& |
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µ = |
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S . |
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(2.8) |
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c |
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Ɂɞɟɫɶ S |
- ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɪɢɟɧɬɢɪɨɜɚɧɧɚɹ ɩɥɨɳɚɞɶ ɤɪɭɝɚ, ɨɯɜɚɬɵɜɚɟɦɚɹ ɥɚɪɦɨɪɨɜɫɤɨɣ |
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ɨɤɪɭɠɧɨɫɬɶɸ, ɪɚɜɧɚɹ S = πρ2. ɉɨɞɫɬɚɜɥɹɹ ɡɧɚɱɟɧɢɹ j ɢ S ɜ ɭɪɚɜɧɟɧɢɟ (2.8), ɩɨɥɭɱɚɟɦ |
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& |
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B& |
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mv2 / 2 |
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µ = −µ |
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, |
µ = |
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(2.9) |
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B |
B |
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Ɇɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ ɜɪɚɳɚɸɳɟɣɫɹ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɱɚɫɬɢɰɵ ɜɫɟɝɞɚ ɧɚɩɪɚɜɥɟɧ ɩɪɨɬɢɜ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɨɠɢɬɟɥɶɧɨ ɢ ɨɬɪɢɰɚɬɟɥɶɧɨ ɡɚɪɹɠɟɧɧɵɟ ɱɚɫɬɢɰɵ ɜɪɚɳɚɸɬɫɹ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɧɚɩɪɚɜɥɟɧɢɹɯ (ɪɢɫ. 2.1). ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɬɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɫɬɨɹɧɧɨɦ ɢ ɨɞɧɨɪɨɞɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɩɢɪɚɥɶ.
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Ʉɨɧɤɪɟɬɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɜɪɚɳɟɧɢɹ |
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ɱɚɫɬɢɰɵ ɜ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ |
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ɢɡɜɟɫɬɧɨɦɭ «ɩɪɚɜɢɥɭ Ʌɟɧɰɚ», ɫɨɝɥɚɫɧɨ |
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ɤɨɬɨɪɨɦɭ, ɥɸɛɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ |
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ɫɬɪɟɦɢɬɫɹ |
ɞɜɢɝɚɬɶɫɹ |
ɬɚɤ, |
ɱɬɨɛɵ |
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ɩɪɨɬɢɜɨɞɟɣɫɬɜɨɜɚɬɶ |
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ɩɪɢɱɢɧɟ, |
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ɜɵɡɵɜɚɸɳɟɣ ɟɟ ɞɜɢɠɟɧɢɟ. ȼɪɚɳɚɸɳɚɹɫɹ |
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ɩɨ ɥɚɪɦɨɪɨɜɫɤɨɣ |
ɨɤɪɭɠɧɨɫɬɢ |
ɱɚɫɬɢɰɚ |
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Ɋɢɫ.2.1. Ⱦɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɵɯ ɱɚɫɬɢɰ ɜ ɩɨɫɬɨɹɧɧɨɦ |
ɜɟɞɟɬ ɫɟɛɹ |
ɤɚɤ |
ɞɢɚɦɚɝɧɟɬɢɤ |
− |
ɨɧɚ |
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ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ (ɩɪɨɞɨɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɧɚɩɪɚɜɥɟɧɚ |
ɫɬɪɟɦɢɬɫɹ |
ɨɫɥɚɛɢɬɶ ɨɯɜɚɬɵɜɚɟɦɵɣ |
ɟɺ |
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ɜɞɨɥɶ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɜ ɨɛɨɢɯ ɫɥɭɱɚɹɯ) |
ɥɚɪɦɨɪɨɜɫɤɨɣ |
ɨɤɪɭɠɧɨɫɬɶɸ |
ɩɨɬɨɤ |
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ɜɧɟɲɧɟɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. ɗɬɨ ɜɵɪɚɠɚɟɬ ɫɭɬɶ ɬɚɤ ɧɚɡɵɜɚɟɦɨɝɨ ɩɪɢɧɰɢɩɚ ɞɢɚɦɚɝɧɟɬɢɡɦɚ ɫɜɨɛɨɞɧɵɯ ɱɚɫɬɢɰ.
ȼɨɡɧɢɤɚɟɬ «ɡɚɤɨɧɧɵɣ» ɜɨɩɪɨɫ: ɟɫɥɢ ɛɵ ɜɪɚɳɟɧɢɟ ɱɚɫɬɢɰɵ ɩɨ ɥɚɪɦɨɪɨɜɫɤɨɣ ɨɤɪɭɠɧɨɫɬɢ ɜɵɡɵɜɚɥɨ ɢɡɦɟɧɟɧɢɟ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɜ ɤɨɬɨɪɨɦ ɨɧɚ ɜɪɚɳɚɟɬɫɹ, ɬɨ ɷɬɨ ɞɨɥɠɧɨ ɛɵɥɨ ɛɵ ɫɤɚɡɵɜɚɬɶɫɹ ɧɚ ɬɪɚɟɤɬɨɪɢɢ ɟɺ ɞɜɢɠɟɧɢɹ, ɚ ɢɡɦɟɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ ɩɪɢɜɟɥɨ ɛɵ ɤ ɞɨɩɨɥɧɢɬɟɥɶɧɨɦɭ ɢɫɤɚɠɟɧɢɸ ɩɨɥɹ ɢ ɬ.ɞ. ȼ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɷɬɨ ɥɨɠɧɵɣ ɩɭɬɶ. ȼ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɟ ɩɨɫɬɭɥɢɪɭɟɬɫɹ, ɱɬɨ ɬɨɱɟɱɧɵɟ ɡɚɪɹɠɟɧɧɵɟ ɱɚɫɬɢɰɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɱɟɪɟɡ ɩɨɫɪɟɞɫɬɜɨ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɫɨɡɞɚɜɚɟɦɨɝɨ ɢɦɢ ɜ ɨɤɪɭɠɚɸɳɟɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬɟɦ ɫɚɦɵɦ «ɫɚɦɨɞɟɣɫɬɜɢɟ» ɱɚɫɬɢɰ ɢɫɤɥɸɱɚɟɬɫɹ. ɇɚɩɪɢɦɟɪ, ɜ ɪɚɫɫɦɨɬɪɟɧɧɨɦ ɫɥɭɱɚɟ ɞɜɢɠɟɧɢɹ ɜɵɞɟɥɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɡɚɞɚɧɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɷɬɨ ɩɨɥɟ ɫɨɡɞɚɟɬɫɹ ɞɪɭɝɢɦɢ, ɞɜɢɠɭɳɢɦɢɫɹ ɡɚɞɚɧɧɵɦ ɨɛɪɚɡɨɦ ɡɚɪɹɠɟɧɧɵɦɢ ɱɚɫɬɢɰɚɦɢ, ɡɞɟɫɶ ɭɧɟɫɟɧɧɵɦɢ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɶ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɜɥɢɹɧɢɟɦ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɱɚɫɬɢɰɵ ɧɚ ɞɜɢɠɟɧɢɟ ɢɫɬɨɱɧɢɤɨɜ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, − ɚ ɬɨɥɶɤɨ ɬɚɤ, ɜɥɢɹɹ ɧɚ ɬɪɚɟɤɬɨɪɢɢ ɷɬɢɯ ɢɫɬɨɱɧɢɤɨɜ, ɢ ɡɚɬɪɚɱɢɜɚɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɷɧɟɪɝɢɸ, ɱɚɫɬɢɰɚ ɦɨɝɥɚ ɛɵ «ɭɩɪɚɜɥɹɬɶ» ɫɜɨɟɣ ɬɪɚɟɤɬɨɪɢɟɣ.
§ 14. Ⱦɜɢɠɟɧɢɟ ɜ ɫɢɥɶɧɨɦ ɦɟɞɥɟɧɧɨ ɦɟɧɹɸɳɟɦɫɹ ɩɨɥɟ. Ⱦɪɟɣɮɨɜɨɟ ɩɪɢɛɥɢɠɟɧɢɟ
ȿɫɥɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɦɟɞɥɟɧɧɨ ɦɟɧɹɟɬɫɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢ ɜɨ ɜɪɟɦɟɧɢ, ɬɨ, ɞɜɢɠɭɳɚɹɫɹ ɜ ɧɟɦ ɱɚɫɬɢɰɚ, ɩɪɟɠɞɟ ɱɟɦ ɩɨɱɭɜɫɬɫɬɜɭɟɬ ɜɥɢɹɧɢɟ ɢɡɦɟɧɟɧɢɹ ɩɨɥɹ, ɫɨɜɟɪɲɢɬ ɜ ɧɟɦ ɦɧɨɠɟɫɬɜɨ ɥɚɪɦɨɪɨɜɫɤɢɯ ɨɛɨɪɨɬɨɜ, ɧɚɜɢɜɚɹɫɶ ɧɚ ɫɢɥɨɜɭɸ ɥɢɧɢɸ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. ɗɬɨ ɞɜɢɠɟɧɢɟ, ɮɚɤɬɢɱɟɫɤɢ ɜ ɩɨɫɬɨɹɧɧɨɦ ɩɨɥɟ, ɦɵ ɭɠɟ ɢɡɭɱɢɥɢ. ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɜɢɠɟɧɢɟ ɧɟ ɫɨɛɫɬɜɟɧɧɨ ɱɚɫɬɢɰɵ, ɚ ɟɺ ɦɝɧɨɜɟɧɧɨɝɨ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ, ɬɚɤ
ɧɚɡɵɜɚɟɦɨɝɨ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ (ɜ ɡɚɪɭɛɟɠɧɨɣ ɥɢɬɟɪɚɬɭɪɟ ɬɚɤɨɣ ɩɨɞɯɨɞ ɢɡɜɟɫɬɟɧ ɤɚɤ
ɩɪɢɛɥɢɠɟɧɢɟ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ - guiding center approximation). ɋɥɟɞɭɟɬ, ɨɞɧɚɤɨ, ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɦɟɧɹɟɬɫɹ ɜɨ ɜɪɟɦɟɧɢ, ɬɨ ɧɭɠɧɨ ɭɱɢɬɵɜɚɬɶ ɢ
ɫɥɚɛɵɟ ɷɥɟɤɬɪɢɱɟɫɤɢɟ ɩɨɥɹ. ɋɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɹɦ Ɇɚɤɫɜɟɥɥɚ, ɜ ɫɢɥɭ
1 ∂ B& rotE = − c ∂ t ,
ɩɟɪɟɦɟɧɧɨɟ ɜɨ ɜɪɟɦɟɧɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɜɵɡɵɜɚɟɬ ɩɨɹɜɥɟɧɢɟ ɜɢɯɪɟɜɨɝɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ.
ɇɚ ɦɟɞɥɟɧɧɨɟ ɞɜɢɠɟɧɢɟ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ, ɨɩɪɟɞɟɥɹɟɦɨɝɨ ɤɚɤ ɩɪɨɞɨɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ ɱɚɫɬɢɰɵ, ɬɚɤ ɢ ɜɥɢɹɧɢɟɦ ɫɥɚɛɨɝɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɢ ɫɥɚɛɵɯ ɧɟɨɞɧɨɪɨɞɧɨɫɬɟɣ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɧɚɥɨɠɢɬɫɹ ɛɵɫɬɪɨɟ ɜɪɚɳɟɧɢɟ ɱɚɫɬɢɰɵ ɜɨɤɪɭɝ ɫɢɥɨɜɵɯ ɥɢɧɢɣ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. Ɍɚɤɨɟ ɪɚɡɞɟɥɶɧɨɟ ɪɚɫɫɦɨɬɪɟɧɢɟ ɛɵɫɬɪɨɝɨ ɜɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ ɩɨ ɥɚɪɦɨɪɨɜɫɤɨɣ ɨɤɪɭɠɧɨɫɬɢ ɢ ɦɟɞɥɟɧɧɨɝɨ «ɞɪɟɣɮɚ» ɰɟɧɬɪɚ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜɨ, ɟɫɥɢ ɢɡɦɟɧɟɧɢɟ ɥɚɪɦɨɪɨɜɫɤɨɝɨ ɪɚɞɢɭɫɚ ɧɚ ɨɞɧɨɦ ɨɛɨɪɨɬɟ ɛɭɞɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɧɶɲɟ ɫɚɦɨɝɨ ɥɚɪɦɨɪɨɜɫɤɨɝɨ ɪɚɞɢɭɫɚ. ɗɬɨ ɭɫɥɨɜɢɟ, ɨɱɟɜɢɞɧɨ, ɛɭɞɟɬ ɜɵɩɨɥɧɟɧɨ, ɟɫɥɢ ɯɚɪɚɤɬɟɪɧɨɟ ɜɪɟɦɹ ɢɡɦɟɧɟɧɢɹ ɩɨɥɟɣ ɛɭɞɟɬ ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟ ɜɪɟɦɟɧɢ ɥɚɪɦɨɪɨɜɫɤɨɝɨ ɨɛɨɪɨɬɚ, ɢ ɟɫɥɢ ɯɚɪɚɤɬɟɪɧɵɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɦɚɫɲɬɚɛ ɢɡɦɟɧɟɧɢɹ ɩɨɥɟɣ ɛɭɞɟɬ ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɵɲɚɬɶ ɥɚɪɦɨɪɨɜɫɤɢɣ ɪɚɞɢɭɫ:
∆t ɯɚ ɪ |
>> T |
, ∆l ɯɚ ɪ |
>> ρ |
ɥɚ ɪɦ |
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ɩɨɥɹ |
ɥɚ ɪɦ |
ɩɨɥɹ |
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Ʉɨɥɢɱɟɫɬɜɟɧɧɨ ɷɬɢ ɤɪɢɬɟɪɢɢ ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
∂ B ∂ t |
<< 1 , |
∂ E ∂ t |
<< 1 , |
ωɥɚɪɦ B |
ωɥɚ ɪɦ E |
B |
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ρɥɚɪɦ B |
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ρɥɚ ɪɦ E |
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(2.10) |
Ɉɱɟɜɢɞɧɨ, ɷɬɢ ɭɫɥɨɜɢɹ ɜɵɩɨɥɧɟɧɵ ɬɟɦ ɥɭɱɲɟ, ɱɟɦ ɛɨɥɶɲɟ ɜɟɥɢɱɢɧɚ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɬɚɤ ɤɚɤ ɥɚɪɦɨɪɨɜɫɤɚɹ ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ ɜɨɡɪɚɫɬɚɟɬ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɜɟɥɢɱɢɧɟ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɚ ɥɚɪɦɨɪɨɜɫɤɢɣ ɪɚɞɢɭɫ ɭɛɵɜɚɟɬ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɜɟɥɢɱɢɧɟ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. ȼɦɟɫɬɟ ɫ ɬɟɦ, ɜɟɥɢɱɢɧɚ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ E ɧɟ ɞɨɥɠɧɚ ɛɵɬɶ ɫɥɢɲɤɨɦ ɛɨɥɶɲɨɣ. ɇɚ ɨɞɧɨɦ ɨɛɨɪɨɬɟ ɱɚɫɬɢɰɵ ɟɟ ɫɤɨɪɨɫɬɶ ɞɨɥɠɧɚ ɦɟɧɹɬɶɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɨ, ɧɚɫɬɨɥɶɤɨ ɧɟɡɧɚɱɢɬɟɥɶɧɨ, ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ ɭɫɥɨɜɢɟ
δv ~ |
qE |
T |
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qE |
≡ c |
E |
<< v . |
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mωɥɚ ɪɦ |
B |
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m ɥɚ ɪɦ |
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Ɉɬɤɭɞɚ ɢ ɩɨɥɭɱɚɟɦ ɬɪɟɛɭɟɦɨɟ ɨɝɪɚɧɢɱɟɧɢɟ ɧɚ ɜɟɥɢɱɢɧɭ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ
v |
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B . |
(2.11) |
ɉɪɢ ɜɵɩɨɥɧɟɧɢɢ ɭɫɥɨɜɢɣ (2.10) ɢ (2.11) ɢɫɬɢɧɧɚɹ ɬɪɚɟɤɬɨɪɢɹ ɱɚɫɬɢɰɵ ɨɛɵɱɧɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɟɞɥɟɧɧɨ ɢɡɝɢɛɚɸɳɭɸɫɹ ɫɩɢɪɚɥɶ ɫ ɦɟɞɥɟɧɧɨ ɦɟɧɹɸɳɢɦɫɹ ɪɚɞɢɭɫɨɦ ɢ ɲɚɝɨɦ. ɉɪɨɟɤɰɢɹ ɬɪɚɟɤɬɨɪɢɢ ɱɚɫɬɢɰɵ ɧɚ ɩɥɨɫɤɨɫɬɶ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɤ ɫɢɥɨɜɵɦ ɥɢɧɢɹɦ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɬɪɨɯɨɢɞɭ.
Ⱦɜɢɠɟɧɢɟ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ ɨɛɵɱɧɨ ɧɚɡɵɜɚɸɬ ɞɪɟɣɮɨɜɵɦ ɞɜɢɠɟɧɢɟɦ, ɚ ɩɪɢɛɥɢɠɟɧɧɨɟ ɨɩɢɫɚɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɤɚɤ ɞɜɢɠɟɧɢɟ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ - ɞɪɟɣɮɨɜɵɦ ɩɪɢɛɥɢɠɟɧɢɟɦ. Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɫɥɭɱɚɢ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɟ ɨɛɳɢɣ ɢɧɬɟɪɟɫ, ɬɚɤ ɤɚɤ ɤ ɧɢɦ ɦɨɠɧɨ ɫɜɟɫɬɢ ɦɧɨɝɢɟ ɜɢɞɵ ɞɜɢɠɟɧɢɹ ɡɚɪɹɠɟɧɧɵɯ ɱɚɫɬɢɰ ɜ ɧɟɨɞɧɨɪɨɞɧɵɯ ɦɚɝɧɢɬɧɵɯ ɩɨɥɹɯ ɢ ɜ ɩɨɥɹɯ ɞɪɭɝɢɯ ɜɧɟɲɧɢɯ ɫɢɥ.
§ 15. Ⱦɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɜ ɨɞɧɨɪɨɞɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ
ȿɫɥɢ ɧɚ ɱɚɫɬɢɰɭ, ɩɨɦɢɦɨ ɫɢɥɵ Ʌɨɪɟɧɰɚ, ɞɟɣɫɬɜɭɟɬ ɩɨɫɬɨɹɧɧɚɹ ɫɢɥɚ F& , ɬɨ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɢɦɟɟɬ ɜɢɞ:
& |
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× B + F . |
(2.12) |
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Ɋɚɡɥɨɠɢɜ ɜɟɤɬɨɪɵ v& ɢ F& ɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɢ ɩɚɪɚɥɥɟɥɶɧɭɸ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɤɨɦɩɨɧɟɧɬɵ, ɤɚɤ ɷɬɨ ɭɠɟ ɞɟɥɚɥɨɫɶ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɚɪɚɝɪɚɮɟ, ɨɩɪɟɞɟɥɢɦ, ɱɬɨ
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mv = F + |
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(2.13) |
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Ɇɵ ɜɢɞɢɦ, ɱɬɨ ɩɪɨɞɨɥɶɧɨɟ ɢ ɩɨɩɟɪɟɱɧɨɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɨɤɚɡɵɜɚɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ, ɢ ɢɯ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɪɚɡɞɟɥɶɧɨ. ɉɪɢ ɷɬɨɦ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɢɥɵ, ɩɚɪɚɥɥɟɥɶɧɚɹ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ, ɨɛɭɫɥɚɜɥɢɜɚɟɬ ɧɟɧɭɥɟɜɨɟ ɩɪɨɞɨɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɱɚɫɬɢɰɵ, ɤɨɬɨɪɨɟ ɢɡɦɟɧɹɟɬ ɩɪɨɞɨɥɶɧɭɸ ɤɨɦɩɨɧɟɧɬɭ ɫɤɨɪɨɫɬɢ. ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɢɥɵ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ, ɱɚɫɬɢɰɚ ɫɨɜɟɪɲɚɟɬ ɫɥɨɠɧɨɟ ɞɜɢɠɟɧɢɟ, ɹɜɥɹɸɳɟɟɫɹ ɫɭɩɟɪɩɨɡɢɰɢɟɣ ɜɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ ɢ ɫɢɫɬɟɦɚɬɢɱɟɫɤɨɝɨ ɫɧɨɫɚ (ɞɪɟɣɮɚ) ɫ ɧɟɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ. ɑɬɨɛɵ ɭɛɟɞɢɬɶɫɹ ɜ ɷɬɨɦ, ɩɪɟɞɫɬɚɜɢɦ ɫɤɨɪɨɫɬɶ ɩɨɩɟɪɟɱɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜ ɜɢɞɟ ɫɭɦɦɵ
v& = v&d + v&r , |
(2.14) |
ɝɞɟ v&d — ɩɨɫɬɨɹɧɧɚɹ ɫɤɨɪɨɫɬɶ, ɚv&r — ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɜɨɤɪɭɝ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ. Ɍɚɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɢ ɤɚɤ ɩɟɪɟɯɨɞ ɜ ɞɜɢɠɭɳɭɸɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ, ɧɨ
ɡɚɪɚɧɟɟ ɧɟ ɢɡɜɟɫɬɧɨɣ (!), ɫɤɨɪɨɫɬɶɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ. ɉɨɞɫɬɚɧɨɜɤɚ ɜɵɪɚɠɟɧɢɹ (2.14) ɜɨ ɜɬɨɪɭɸ ɢɡ ɮɨɪɦɭɥ (2.13) ɞɚɟɬ
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Ɍɚɤ ɤɚɤ ɩɟɪɜɵɟ ɞɜɚ ɫɥɚɝɚɟɦɵɯ ɫɩɪɚɜɚ − ɩɨɫɬɨɹɧɧɵɟ ɜɟɤɬɨɪɵ, ɬɨ ɨɧɢ ɦɨɝɭɬ ɤɨɦɩɟɧɫɢɪɨɜɚɬɶ ɞɪɭɝ ɞɪɭɝɚ. ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɩɨɬɪɟɛɨɜɚɬɶ ɜɵɩɨɥɧɟɧɢɟ ɪɚɜɟɧɫɬɜɚ
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ɗɬɨ ɬɪɟɛɨɜɚɧɢɟ ɢ ɨɩɪɟɞɟɥɹɟɬ ɜɟɥɢɱɢɧɭ ɫɤɨɪɨɫɬɢ ɞɪɟɣɮɚ ɱɚɫɬɢɰɵ: |
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Ɉɫɬɚɜɲɚɹɫɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (2.15) |
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ɨɩɢɫɵɜɚɟɬ ɜɪɚɳɟɧɢɟ ɜɨɤɪɭɝ ɜɟɞɭɳɟɝɨ ɰɟɧɬɪɚ, ɤɚɤ ɷɬɨ ɨɱɟɜɢɞɧɨ ɢɡ ɫɪɚɜɧɟɧɢɹ ɫ ɭɪɚɜɧɟɧɢɟɦ (2.5). ɋɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɜ ɬɚɤɨɦ ɞɜɢɠɟɧɢɢ, ɤɚɤ ɦɵ ɭɠɟ ɨɛɫɭɠɞɚɥɢ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɚɪɚɝɪɚɮɟ, ɦɨɠɟɬ ɬɨɥɶɤɨ ɩɨɜɨɪɚɱɢɜɚɬɶɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ, ɧɟ ɦɟɧɹɹɫɶ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ. ȼɟɥɢɱɢɧɚ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɚ ɩɨ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ, ɤɨɬɨɪɚɹ, ɤɨɧɟɱɧɨ, ɞɨɥɠɧɚ ɛɵɬɶ ɢɡɧɚɱɚɥɶɧɨ ɡɚɞɚɧɚ:
vr =|v& 0 − v&d | , |
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ɝɞɟv& 0 - ɩɨɩɟɪɟɱɧɚɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɤɨɦɩɨɧɟɧɬɚ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ. ɇɚɩɪɢɦɟɪ, ɜ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ v& 0 = 0 ɩɨɥɭɱɚɟɦ
vr = vd = |
cF |
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qB |
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Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬ ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ:
ρ = |
mcvr |
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(2.20) |
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qB |
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ɩɨ ɤɨɬɨɪɨɣ ɜɪɚɳɚɟɬɫɹ ɱɚɫɬɢɰɚ ɜ ɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɞɜɢɠɭɳɟɣɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ
(2.16).
ɇɟ ɫɨɫɬɚɜɥɹɟɬ ɛɨɥɶɲɨɝɨ ɬɪɭɞɚ ɧɚɣɬɢ ɢ ɬɨɱɧɵɣ ɡɚɤɨɧ ɩɨɩɟɪɟɱɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɱɚɫɬɢɰɵ: |
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( v& |
− v& |
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v& |
− v& |
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r |
= r |
+ v |
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t + |
0 |
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(1 − cosω t ) + |
0 |
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sinω t . |
(2.21) |
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ω B |
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0 |
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ɂɧɞɟɤɫɨɦ ɧɨɥɶ ɡɞɟɫɶ ɩɨɦɟɱɟɧɵ ɧɚɱɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɥɢɱɢɧ. ɋɨɫɬɚɜɢɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɪɚɡɧɨɨɛɪɚɡɢɢ ɜɨɡɦɨɠɧɵɯ ɬɪɚɟɤɬɨɪɢɣ ɩɨɩɟɪɟɱɧɨɝɨ
ɞɜɢɠɟɧɢɹ ɦɨɠɧɨ ɩɨ ɧɢɠɟɫɥɟɞɭɸɳɟɦɭ ɪɢɫ. 2.2:
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Ʉɪɨɦɟ ɬɨɝɨ, ɫɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɟɫɥɢ |
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ɩɨɩɟɪɟɱɧɚɹ ɤɨɦɩɨɧɟɧɬɚ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɜ |
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ɬɨɱɧɨɫɬɢ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɤɨɪɨɫɬɶɸ ɞɪɟɣɮɚ: |
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v& 0 = v&d , |
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(2.22) |
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ɬɨ, ɤɚɤ ɷɬɨ ɫɥɟɞɭɟɬ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ, ɱɚɫɬɢɰɚ |
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ɞɜɢɠɟɬɫɹ ɪɚɜɧɨɦɟɪɧɨ ɩɨɩɟɪɟɤ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. |
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Ɍɚɤɚɹ |
ɜɨɡɦɨɠɧɨɫɬɶ |
ɩɪɟɞɫɬɚɜɥɹɟɬ |
ɛɨɥɶɲɨɣ |
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ɢɧɬɟɪɟɫ ɜ ɩɪɨɛɥɟɦɟ ɬɪɚɧɫɩɨɪɬɚ ɱɚɫɬɢɰ ɩɨɩɟɪɟɤ |
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Ɋɢɫ.2.2. Ɍɢɩɵ ɜɨɡɦɨɠɧɵɯ ɬɪɚɟɤɬɨɪɢɣ ɞɥɹ |
ɫɢɥɶɧɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ. |
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ɇɚɦɢ |
ɧɟ |
ɜɵɫɤɚɡɵɜɚɥɨɫɶ& |
ɧɢɤɚɤɢɯ |
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ɧɟɤɨɬɨɪɵɯ ɱɚɫɬɧɵɯ ɫɥɭɱɚɟɜ ɧɚɩɪɚɜɥɟɧɢɹ |
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ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɩɨɩɟɪɟɱɧɨɝɨ ɞɜɢɠɟɧɢɹ |
ɩɪɟɞɩɨɥɨɠɟɧɢɣ |
ɨ ɩɪɢɪɨɞɟ |
ɫɢɥɵ F , |
ɩɨɷɬɨɦɭ |
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ɱɚɫɬɢɰɵ ɜ ɫɤɪɟɳɟɧɧɵɯ ɩɨɥɹɯ (ɲɬɪɢɯɨɜɚɹ ɜɵɜɨɞ |
ɨ ɯɚɪɚɤɬɟɪɟ ɞɜɢɠɟɧɢɹ ɫɩɪɚɜɟɞɥɢɜ ɞɥɹ |
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ɥɢɧɢɹ − ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɥɚɪɦɨɪɨɜɫɤɨɣ ɥɸɛɨɣ |
ɩɨɫɬɨɹɧɧɨɣ |
ɫɢɥɵ |
F& |
: ɩɪɨɞɨɥɶɧɨɟ (ɜ |
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ɨɛɳɟɦ |
ɫɥɭɱɚɟ) |
ɭɫɤɨɪɟɧɢɟ |
ɢ |
ɫɢɫɬɟɦɚɬɢɱɟɫɤɢɣ |
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ɫɧɨɫ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɜ ɫɨɜɨɤɭɩɧɨɫɬɢ ɫ ɪɚɜɧɨɦɟɪɧɵɦ ɜɪɚɳɟɧɢɟɦ ɜ ɩɨɩɟɪɟɱɧɨɣ ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɩɥɨɫɤɨɫɬɢ.
ɂɡ ɮɨɪɦɭɥɵ (2.16) ɫɥɟɞɭɟɬ, ɱɬɨ ɫɤɨɪɨɫɬɶ ɞɪɟɣɮɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɦɚɫɫɵ ɱɚɫɬɢɰɵ, ɧɨ ɡɚɜɢɫɢɬ ɨɬ ɡɧɚɤɚ ɟɟ ɡɚɪɹɞɚ, ɟɫɥɢ ɫɢɥɚ ɨɬ ɡɧɚɤɚ ɡɚɪɹɞɚ ɧɟ ɡɚɜɢɫɢɬ. ɉɨɥɟɡɧɨ ɨɬɦɟɬɢɬɶ ɬɚɤɠɟ, ɱɬɨ ɞɟɣɫɬɜɢɟ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ, ɩɪɢɜɨɞɢɬ (ɜ ɫɪɟɞɧɟɦ ɩɨ ɨɫɰɢɥɥɹɰɢɹɦ!) ɧɟ ɤ ɭɜɟɥɢɱɟɧɢɸ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ, ɚ ɤ ɞɜɢɠɟɧɢɸ ɟɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ. ɋ ɩɨɯɨɠɢɦ ɹɜɥɟɧɢɟɦ ɦɵ ɜɫɬɪɟɱɚɟɦɫɹ ɜ ɦɟɯɚɧɢɤɟ ɩɪɢ ɢɡɭɱɟɧɢɢ ɩɪɟɰɟɫɫɢɢ ɨɫɢ ɝɢɪɨɫɤɨɩɚ, ɤɨɝɞɚ ɩɪɢɥɨɠɟɧɧɵɣ ɤ ɝɢɪɨɫɤɨɩɭ ɦɨɦɟɧɬ ɜɵɡɵɜɚɟɬ ɜɪɚɳɟɧɢɟ ɟɝɨ ɨɫɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɤɚɤ ɤ ɦɨɦɟɧɬɭ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɝɢɪɨɫɤɨɩɚ, ɬɚɤ ɢ ɤ ɦɨɦɟɧɬɭ ɫɢɥɵ.