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Уравнения Максвелла (55

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1 F

wFkl

1 F

wFml

wFlk

1 F

wFml

)

1 G

) .

(10.3)

4 wx

ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ:

§F

1 G

·.

(10.4)

mn mn ¸

wxl

4S wxl ©

ɇɚ ɨɫɧɨɜɚɧɢɢ ɩɨɫɥɟɞɧɟɝɨ ɪɚɜɟɧɫɬɜɚ ɩɨɥɨɠɢɦ

§F

1 G

·.

(10.5)

kl mn

mn ¸

4S©

ɉɨɥɭɱɟɧɧɵɣ ɧɚɦɢ ɬɟɧɡɨɪ ɷɧɟɪɝɢɢ-ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬ-

ɧɨɝɨ ɩɨɥɹ ɫɢɦɦɟɬɪɢɱɟɧ:
Tkl	Tlk .	(10.6)
ȿɝɨ ɫɥɟɞ (ɫɭɦɦɚ ɞɢɚɝɨɧɚɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ) ɪɚɜɟɧ ɧɭɥɸ:
Tll	0 .	(10.7)
ɂɧɬɟɪɩɪɟɬɚɰɢɹ ɤɨɦɩɨɧɟɧɬ	Tkl ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ	ɪɚɜɟɧɫɬɜɟ

(10.1) ɢ ɧɚ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɤɨɦɩɨɧɟɧɬ Tkl(m) . ȼɪɟɦɟɧɧɚɹ ɤɨɦɩɨ-

ɧɟɧɬɚ T44 ɢɧɬɟɪɩɪɟɬɢɪɭɟɬɫɹ ɤɚɤ ɜɡɹɬɚɹ ɫ ɨɛɪɚɬɧɵɦ ɡɧɚɤɨɦ ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ; ɫɦɟɲɚɧɧɚɹ ɤɨɦɩɨɧɟɧɬɚ TD4 – ɤɚɤ D-ɹ ɩɪɨɟɤɰɢɹ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɩɨɥɹ,

ɭɦɧɨɠɟɧɧɚɹ ɧɚ ic , ɢɥɢ ɠɟ ɤɚɤ D-ɹ ɩɪɨɟɤɰɢɹ ɩɥɨɬɧɨɫɬɢ ɢɦ-

ɩɭɥɶɫɚ, ɭɦɧɨɠɟɧɧɚɹ ɧɚ ic ; ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɚɹ ɤɨɦɩɨɧɟɧɬɚ TDE –

ɤɚɤ E-ɹ ɩɪɨɟɤɰɢɹ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ D-ɣ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ.

11. ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɢ ɦɚɝɧɢɬɧɨɟ ɩɨɥɹ

Ⱥɩɩɚɪɚɬ 4-ɬɟɧɡɨɪɨɜ ɨɫɨɛɟɧɧɨ ɭɞɨɛɟɧ ɢ ɷɮɮɟɤɬɢɜɟɧ ɩɪɢ ɨɛɫɭɠɞɟɧɢɢ ɨɛɳɢɯ ɜɨɩɪɨɫɨɜ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɢ. ɉɪɢ ɚɧɚɥɢɡɟ ɤɨɧɤɪɟɬɧɵɯ ɫɢɬɭɚɰɢɣ ɛɵɜɚɟɬ ɭɞɨɛɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɚɩɩɚɪɚɬ 3-

M iA4 .

ɬɟɧɡɨɪɨɜ. ȼ ɱɚɫɬɧɨɫɬɢ, ɲɟɫɬɶ ɧɟɡɚɜɢɫɢɦɵɯ ɤɨɦɩɨɧɟɧɬ ɚɧɬɢ-

ɫɢɦɦɟɬɪɢɱɧɨɝɨ 4-ɬɟɧɡɨɪɚ Fkl			wAl wxk wAk wxl		ɦɨɠɧɨ ɜɵɪɚ-
ɡɢɬɶ ɱɟɪɟɡ ɤɨɦɩɨɧɟɧɬɵ ɞɜɭɯ 3-ɜɟɤɬɨɪɨɜ E& ɢ B&, ɤɨɬɨɪɵɟ ɨɩɪɟ-
ɞɟɥɹɸɬɫɹ ɪɚɜɟɧɫɬɜɚɦɢ:		1 wA&
&				M,	(11.1)
E		c	wt

	B&		[ A&] .		(11.2)

Ɂɞɟɫɶ ɜɜɟɞɟɧɨ ɨɛɨɡɧɚɱɟɧɢɟ

(11.3)

ɉɨɥɹɪɧɵɣ ɜɟɤɬɨɪ E& ɧɚɡɵɜɚɸɬ ɧɚɩɪɹɠɟɧɧɨɫɬɶɸ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ (ɢɥɢ ɩɪɨɫɬɨ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɩɨɥɟɦ), ɚɤɫɢɚɥɶɧɵɣ ɜɟɤ-

ɬɨɪ B& – ɢɧɞɭɤɰɢɟɣ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ (ɢɥɢ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ),

ɩɨɥɹɪɧɵɣ ɜɟɤɬɨɪ A& – ɜɟɤɬɨɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ, ɚ ɢɫɬɢɧɧɵɣ ɫɤɚɥɹɪ M – ɫɤɚɥɹɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ. ɉɪɟɞɫɬɚɜɥɹɹ Fkl ɜ ɜɢɞɟ ɬɚɛ-

ɥɢɰɵ (ɩɟɪɜɨɦɭ ɢɧɞɟɤɫɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɬɪɨɤɚ, ɚ ɜɬɨɪɨɦɭ – ɫɬɨɥɛɟɰ ɬɚɛɥɢɰɵ), ɩɨɥɭɱɢɦ:

	§	0
	¨	Bz
F	¨	Bz
F	¨
kl	¨	B
	¨	B	y
	¨	iE	x
	©		x

B	z	B	y		iE	x	·
	z		y			x	¸
0		Bx			iEy ¸
B		0			iE		¸.	(11.4)
	x					z ¸
iE	y	iE		z	0		¸
	y			z			¹

Ⱦɥɹ ɬɟɧɡɨɪɨɜ ɬɚɤɨɝɨ ɬɢɩɚ ɩɪɢɦɟɧɹɟɬɫɹ ɬɚɤɠɟ ɤɪɚɬɤɚɹ ɫɢɦɜɨɥɢɱɟɫɤɚɹ ɡɚɩɢɫɶ:

Fkl B&, iE& .	(11.5)
ɂɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɹ (11.4) ɞɥɹ Fkl	ɢ (10.5) ɞɥɹ ɬɟɧɡɨɪɚ

ɷɧɟɪɝɢɢ-ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ Tkl , ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɤɨɦɩɨɧɟɧɬɵ ɩɨɫɥɟɞɧɟɝɨ ɱɟɪɟɡ ɤɨɦɩɨɧɟɧɬɵ ɩɨɥɟɣ E& ɢ B&. ȼ ɪɟɡɭɥɶɬɚɬɟ ɧɚɣɞɟɦ, ɱɬɨ ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ( T44 ) ɪɚɜɧɚ

			E2 B2			,	(11.6)
				8S		,	(11.6)
				8S
ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ( ice&DTD4 ) ɪɚɜɧɚ
			c		&&
				[EB],			(11.7)
			4S	[EB],			(11.7)
			4S
ɩɥɨɬɧɨɫɬɶ ɢɦɩɭɥɶɫɚ (	i	e&DTD4 ) ɪɚɜɧɚ
ɩɥɨɬɧɨɫɬɶ ɢɦɩɭɥɶɫɚ (	c	e&DTD4 ) ɪɚɜɧɚ
1					&&
					[EB].		(11.8)
			4Sc		[EB].		(11.8)
			4Sc

Ɂɞɟɫɶ e&D ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜ ɧɚɩɪɚɜɥɟɧɢɢ D-ɣ ɞɟɤɚɪɬɨɜɨɣ

ɨɫɢ, ɚ ɩɨ ɩɨɜɬɨɪɹɸɳɢɦɫɹ ɝɪɟɱɟɫɤɢɦ ɢɧɞɟɤɫɚɦ ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ ɫɭɦɦɢɪɨɜɚɧɢɟ ɨɬ 1 ɞɨ 3.

3-ɬɟɧɡɨɪ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɢɦɟɟɬ ɜɢɞ

TDE	E2 B2	GDE	EDEE BDBE	.	(11.9)
	8S		4S

ɉɭɫɬɶ ɜɧɟ ɬɟɥɚ, ɨɝɪɚɧɢɱɟɧɧɨɝɨ ɧɟɤɨɬɨɪɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ,

ɢɦɟɟɬɫɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ ɩɨɥɟ. ɋɢɥɚ F&(S ) , ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɟɞɢɧɢɱɧɭɸ ɩɥɨɳɚɞɤɭ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɫɨ ɫɬɨɪɨɧɵ ɷɬɨɝɨ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɪɚɜɧɚ ɢɦɩɭɥɶɫɭ, ɜɬɟɤɚɸɳɟɦɭ ɜ ɬɟɥɨ ɱɟɪɟɡ ɷɬɭ ɩɥɨɳɚɞɤɭ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɬɨ ɟɫɬɶ

			E2 B2 &		& & &	& & &
&	(S )	&	E2 B2 &		E(EN ) B(BN )
F		eDTDENE		N			, (11.10)
F		eDTDENE	8S	N	4S		, (11.10)

ɝɞɟ N& – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜɧɭɬɪɟɧɧɟɣ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ.

Ⱦɥɹ ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɫɨ ɫɬɨɪɨɧɵ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɧɚ ɞɜɢɠɭɳɢɟɫɹ ɜ ɧɟɦ ɡɚɪɹɞɵ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (8.1) ɢɦɟɟɦ:

&	(V )	&		(V )	1	&	j	&	1	&&	(11.11)
F		e F			c	e F	j	UE	c	[ jB] ,	(11.11)
		D	D		c	D Dl	l		c

ɚ ɞɥɹ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɨɱɟɱɧɵɣ ɡɚɪɹɞ qa , ɩɨɥɭɱɢɦ:


&	& &	q	a ª		&	& &	º
Fa	qa E(ra ,t)		a ª				º	(11.12)
Fa	qa E(ra ,t)	c		¬va (t)B(ra ,t)¼.				(11.12)

12. ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɩɨɥɟɣ

ɇɚɣɞɟɦ ɡɚɤɨɧ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɥɟɣ E& ɢ B& ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ K ɤ ɞɪɭɝɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ K c. ɉɪɢ ɬɚɤɨɦ ɩɟɪɟɯɨɞɟ ɤɨɦɩɨɧɟɧɬɵ 4- ɬɟɧɡɨɪɚ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɤɚɤ ɩɪɨɢɡɜɟɞɟɧɢɹ ɤɨɦɩɨɧɟɧɬ 4- ɜɟɤɬɨɪɨɜ. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɬɟɧɡɨɪɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɢɦɟɟɦ

			Fklc /kp/lq Fpq .		(12.1)
ɉɭɫɬɶ ɫɢɫɬɟɦɚ			Kc ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ		K ɫɨ
ɫɤɨɪɨɫɬɶɸ	&		c	ɧɚɩɪɚɜɢɬɶ ɩɚɪɚɥɥɟɥɶ-
	u	. ȿɫɥɢ ɞɟɤɚɪɬɨɜɵ ɨɫɢ x ɢ x

ɧɨ ɷɬɨɣ ɫɤɨɪɨɫɬɢ, ɬɨ ɭ ɦɚɬɪɢɰɵ / ɨɤɚɠɭɬɫɹ ɨɬɥɢɱɧɵɦɢ ɨɬ ɧɭɥɹ ɥɢɲɶ ɫɥɟɞɭɸɳɢɟ ɷɥɟɦɟɧɬɵ:

/	/	44	*		1		, /	22	/	33	1,
/	/		*				, /		/		1,
11				1 u2		c2
				1 u2		c2
			/	/	41	i*u .					(12.2)
			14		41		c
							c

ɂɫɩɨɥɶɡɭɹ (11.4), (12.1) ɢ (12.2), ɩɨɥɭɱɢɦ ɮɨɪɦɭɥɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɦɩɨɧɟɧɬ ɩɨɥɟɣ:

Exc iF14c i/1 p/4q Fpq					i/11/44 F14 i/14/41F41
i*	2	§	u2		·	iF14	Ex ,
		¨1	c	2	¸F14
		©			¹

c	c		/2 p/3q Fpq		/22/33F23				F23		Bx ,
Bx	F23		/2 p/3q Fpq		/22/33F23				F23		Bx ,
c	c	i/2 p		/4q Fpq	i/22 /41F21 /44 F24
Ey	iF24	i/2 p		/4q Fpq	i/22 /41F21 /44 F24
	*u F			i*F	*	§	u B E			·	,
		c	21	24		¨	c	z		y ¸
		c				©	c			¹
					24

Ezc	§u	·	, Bcy
Ezc	*¨	By Ez ¸	, Bcy
	© c	¹

*§¨By u Ez ·¸, Bzc © c ¹

*§¨Bz u Ey ·¸. © c ¹

ɉɨɥɭɱɟɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ, ɧɟ ɫɜɹɡɚɧɧɨɦ ɫ ɜɵɛɨɪɨɦ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ:


&	&	&	§ &		1		&&		·	,
E\|\|c	E\|\| ; EAc		*¨EA		c		[uBA]		¸	,
			©		c				¹
&	&	&	§ &	1			&&	·	.	(12.3)
B\|\|c	B\|\|	; BAc	*¨BA	c		[uEA]¸			.	(12.3)
			©	c				¹

ɋɨɫɬɚɜɥɹɸɳɢɟ ɩɨɥɟɣ E&|| ɢ B&|| ɩɚɪɚɥɥɟɥɶɧɵ, ɚ ɫɨɫɬɚɜɥɹɸɳɢɟ

E&A ɢ B&A ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɫɤɨɪɨɫɬɢ u&.

13. ɂɧɜɚɪɢɚɧɬɵ ɩɨɥɹ

ɂɡ ɜɟɤɬɨɪɨɜ E& ɢ B& ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ 4-ɫɤɚɥɹɪɵ, ɬɨ ɟɫɬɶ ɢɧɜɚɪɢɚɧɬɧɵɟ ɜɟɥɢɱɢɧɵ, ɧɟ ɦɟɧɹɸɳɢɟɫɹ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɤ ɞɪɭɝɨɣ. ɉɪɟɠɞɟ ɜɫɟɝɨ ɡɚɦɟɬɢɦ, ɱɬɨ 4-ɫɤɚɥɹɪ ɨɛɹɡɚɬɟɥɶɧɨ ɞɨɥɠɟɧ ɛɵɬɶ 3-ɫɤɚɥɹɪɨɦ, ɩɨɷɬɨ-

ɦɭ ɜɧɚɱɚɥɟ ɡɚɣɦɟɦɫɹ 3-ɫɤɚɥɹɪɚɦɢ. ɂɡ ɞɜɭɯ ɜɟɤɬɨɪɨɜ E& ɢ B&

ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɬɨɥɶɤɨ ɬɪɢ ɧɟɡɚɜɢɫɢɦɵɯ 3-ɫɤɚɥɹɪɚ: E2 , B2 ɢ

(EB) . (Ɍɨɱɧɟɟ, (EB) – ɩɫɟɜɞɨɫɤɚɥɹɪ.) ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɹ

(12.3), ɩɨɥɭɱɢɦ:

E||c

])

2 ·

(13.1)

EAc

E||

¨EA

(EA[uBA

BA ¸

B||c

(13.2)

BAc

B||

¨BA

(BA[uEA])

&c&c

c c

& &

(E B )

E||B||

(EABA) E||B|| *

(EABA)

([uBA][uEA]) ¸

E||B||

& &

(EABA)

(EB) .

(13.3)

&&
ɂɡ (13.3) ɫɥɟɞɭɟɬ, ɱɬɨ 3-ɫɤɚɥɹɪ (EB) ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɢ 4-
ɫɤɚɥɹɪɨɦ. ɗɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɩɪɢɧɹɬɨ ɡɚɩɢɫɵɜɚɬɶ ɜ ɜɢɞɟ
&&	(13.4)
(EB) inv.	(13.4)

ɂɫɩɨɥɶɡɭɹ (13.1) ɢ (13.2), ɧɟɬɪɭɞɧɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ 4-

ɫɤɚɥɹɪɨɦ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɪɚɡɧɨɫɬɶ ɤɜɚɞɪɚɬɨɜ ɩɨɥɟɣ:
E2 B2 inv.	(13.5)

ɋɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɩɨɥɟɣ 4-ɫɤɚɥɹɪɨɦ ɧɟ ɹɜɥɹɟɬɫɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɭɳɟɫɬɜɭɟɬ ɥɢɲɶ ɞɜɚ ɧɟɡɚɜɢɫɢɦɵɯ ɢɧɜɚɪɢɚɧɬɚ.

14.ɍɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ

ȼɤɥɚɫɫɢɱɟɫɤɨɣ ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɟ ɩɪɢ ɨɩɢɫɚɧɢɢ ɥɸɛɵɯ ɩɪɨɰɟɫɫɨɜ ɜ ɩɪɢɧɰɢɩɟ ɦɨɠɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ ɬɨɥɶɤɨ ɬɟɧɡɨɪɨɦ ɩɨɥɹ Fkl , ɧɟ ɩɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ ɟɝɨ ɫɜɹɡɶ (7.2) ɫ 4-ɩɨɬɟɧɰɢɚɥɨɦ

Ak ɢ ɜɨɨɛɳɟ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɹ ɩɨɫɥɟɞɧɢɣ. ɉɪɢ ɬɚɤɨɦ ɩɨɞɯɨɞɟ ɪɚɜɟɧɫɬɜɨ (7.3)

wFml wFkm wFlk			0	(14.1)
wx	wx	wx
k	l	m

ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɧɟ ɤɚɤ ɬɨɠɞɟɫɬɜɨ, ɚ ɤɚɤ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɦɭ ɞɨɥɠɟɧ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɬɟɧɡɨɪ Fkl . Ʉɪɨɦɟ ɬɨɝɨ, ɬɟɧɡɨɪ Fkl

ɞɨɥɠɟɧ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɪɚɜɧɟɧɢɸ (9.1)

wFkl	4S	jk .	(14.2)
wxl	c

Ɉɛ ɷɬɢɯ ɭɪɚɜɧɟɧɢɹɯ ɝɨɜɨɪɹɬ ɤɚɤ ɨɛ ɭɪɚɜɧɟɧɢɹɯ Ɇɚɤɫɜɟɥɥɚ, ɡɚɩɢɫɚɧɧɵɯ ɜ ɪɟɥɹɬɢɜɢɫɬɫɤɢ ɤɨɜɚɪɢɚɧɬɧɨɦ ɜɢɞɟ.

ȼɵɪɚɡɢɜ ɫɨɝɥɚɫɧɨ (11.4) ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɚ Fkl ɱɟɪɟɡ ɤɨɦ-

ɩɨɧɟɧɬɵ ɜɟɤɬɨɪɨɜ E& ɢ B&, ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ ɜ ɬɪɟɯɦɟɪɧɨɦ ɜɢɞɟ:

	&	&
	&	1 wB	0,
°rot E		1 wB	0,			(14.3)
®	&	c wt				(14.3)
°	&	0.
¯div B		0.
	&	1 wE&	4S &		,
°rot B		c wt	c	j	,	(14.4)
®	&	c wt	c			(14.4)
°	&	4SU.
¯div E		4SU.

ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ ɮɨɪɦɚ ɭɪɚɜɧɟɧɢɣ Ɇɚɤɫɜɟɥɥɚ. ɂɯ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɢ ɜ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɨɪɦɟ:

°³(E&

°L ® & °°³(B ¯SV

&	1 d	³	&	&
dl )			(B dS ) 0,
	c dt
		SL		(14.5)
dS&)	0.

°³(B&

°L ® & °°³(E ¯SV

&	1 d	³	&	&
dl )			(E dS)
dl )	c dt		(E dS)
	c dt	SL
dS&)		SL
dS&)	4S³UdV.

4S	³	&	&
c		( j dS),
	SL		(14.6)

Ɂɞɟɫɶ L – ɡɚɦɤɧɭɬɵɣ ɤɨɧɬɭɪ, ɧɚ ɤɨɬɨɪɵɣ ɧɚɬɹɧɭɬɚ ɩɨɜɟɪɯɧɨɫɬɶ SL , ɚ SV – ɡɚɦɤɧɭɬɚɹ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɢɜɚɸɳɚɹ ɨɛɴ-

ɟɦ V .

15. ȼɟɤɬɨɪɵ ɩɨɥɹɪɢɡɨɜɚɧɧɨɫɬɢ

ɢ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ

ȼ ɫɥɭɱɚɟ ɩɪɨɢɡɜɨɥɶɧɨɣ ɷɥɟɤɬɪɨɧɟɣɬɪɚɥɶɧɨɣ ɜ ɰɟɥɨɦ ɫɢɫɬɟ-

ɦɵ ɩɥɨɬɧɨɫɬɶ ɡɚɪɹɞɚ U(r&,t) ɢ ɩɥɨɬɧɨɫɬɶ ɬɨɤɚ &j(r&,t) ɦɨɠɧɨ ɜɵ-

ɪɚɡɢɬɶ ɱɟɪɟɡ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɩɨɥɹɪɢɡɨɜɚɧɧɨɫɬɶ P&(r&,t) ɢ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ M& (r&,t) :

			&			& &			(15.1)
		U(r ,t)			div P(r ,t) ,
&	&		&		&	&	&
				wP(r,t)
j	(r	,t)				c rotM (r		,t) ,	(15.2)
					wt

ɢɥɢ ɠɟ ɜ ɱɟɬɵɪɟɯɦɟɪɧɨɣ ɡɚɩɢɫɢ:

j	c wmkl ,	(15.3)
k	wxl
	wxl
ɝɞɟ mkl – ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɵɣ ɬɟɧɡɨɪ (ɫɪɚɜɧɢɬɟ ɫ (11.5)):
mkl	M&,iP& .	(15.4)

ɇɢɠɟ ɦɵ ɞɨɤɚɠɟɦ ɪɚɜɟɧɫɬɜɚ (15.1) ɢ (15.2) ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ

ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɮɭɧɤɰɢɣ P&(r&,t) ɢ M& (r&,t) ɱɟɪɟɡ ɮɭɧɤ-

ɰɢɢ U(r&,t) ɢ &j(r&,t) .

Ɋɚɜɟɧɫɬɜɨ (15.1) ɜɵɬɟɤɚɟɬ ɢɡ ɫɥɟɞɭɸɳɟɣ ɰɟɩɨɱɤɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɪɨɜɨɞɢɬɫɹ ɩɨ ɜɫɟɦɭ ɩɪɨɫɬɪɚɧɫɬɜɭ):

&	c	&c	&	&c
U(r,t)	³dV U(r ,t)G(r			r )

³dV cU(r&c,t)G(r& r&c) G(r&)³dV cU(r&c,t)

³dV cU(r&c,t)[G(r& r&c) G(r&)]

³dV cU(r&c,t)[1 exp(r&c )]G(r& r&c)

c	&c		&c	&c			1	&	&c
c	&c		&c	exp(r		)	1	&	&c
³dV U(r ,t)(r )				&				G(r	r )
				c
				(r	)
	c	&c		&c				&	&c
	c	&c	&cexp(r			) 1		&	&c	(15.5)
div ³dV U(r ,t) r				&				G(r	r ) .	(15.5)
				(r	c
				(r		)

ɉɪɢ ɷɬɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ ɭɱɬɟɧɚ ɷɥɟɤɬɪɨɧɟɣɬɪɚɥɶɧɨɫɬɶ ɫɢɫɬɟɦɵ:

	c	&c	0 ,		(15.6)
³dV U(r ,t)			0 ,		(15.6)
				&c
ɚ ɬɚɤɠɟ ɢɫɩɨɥɶɡɨɜɚɧ ɨɩɟɪɚɬɨɪ ɫɞɜɢɝɚ exp(r ) :
&		&c	&	&c	(15.7)
G(r )	exp(r )G(r			r ) .	(15.7)

ɂɡ ɫɪɚɜɧɟɧɢɹ (15.1) ɢ (15.5) ɧɚɯɨɞɢɦ ɫɨɨɬɧɨɲɟɧɢɟ, ɜɵɪɚ-

&	&		&
ɠɚɸɳɟɟ P(r ,t) ɱɟɪɟɡ U(r ,t) :
					&
	&				c
	&	&	c &c	&cexp(r ) 1		&	&c	(15.8)
	P(r ,t)		³dV U(r ,t) r		&	G(r	r ) .	(15.8)
					c
					(r )
				28

Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɨɛɟ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (15.8) ɩɨ ɜɪɟɦɟɧɢ ɢ

ɭɱɢɬɵɜɚɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɡɚɪɹɞɚ
	&
	c			c		&c
	wU(r ,t)			c	&	&c				(15.9)
	wt		j(r ,t) 0 ,							(15.9)
	wt
ɩɨɥɭɱɚɟɦ
&			&c						&c
wP	³dV cr&cexp(&r			) 1G(r& r&c) wU(r ,t)
wt			c					wt
wt		(r )						wt
	&
		c						&
³dV cr&cexp(&r ) 1G(r&								&
³dV cr&cexp(&r ) 1G(r&							r&c)( cj(r&c,t))
	c
	(r	)
						&
	&					c
	&							) 1G(r&	r&c)
³dV c( cj(r&c,t)) r&cexp(&r								) 1G(r&	r&c)
						c
						(r	)
						&
	&						c		r&c) .	(15.10)
³dV c( j(r&c,t) c) r&cexp(&r ) 1G(r&									r&c) .	(15.10)
						c
						(r		)

ɉɟɪɜɵɣ ɢɡ ɩɨɥɭɱɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɢɧɬɟɝɪɚɥ ɩɨ ɛɟɫɤɨɧɟɱɧɨ ɭɞɚɥɟɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɤɨɬɨɪɵɣ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɜɫɥɟɞɫɬɜɢɟ ɨɬɫɭɬɫɬɜɢɹ ɧɚ ɷɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɨɤɨɜ:

			&
	&		c		r&c)
³dV c( cj(r&c,t)) r&cexp(&r ) 1G(r&					r&c)
			c
	(r )
	&
& &		c
& &			) 1G(r&	r&c) 0 .		(15.11)
³(dScj(r&c,t)) r&cexp(&r			) 1G(r&	r&c) 0 .		(15.11)
	c
	(r	)

ȼ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɦ ɜɵɪɚɠɟɧɢɢ ɜɬɨɪɨɝɨ ɢɧɬɟɝɪɚɥɚ ɜɵɩɨɥɧɹɟɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɩɨ ɲɬɪɢɯɨɜɚɧɧɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɭɱɢɬɵ-

& c

ɜɚɹ ɫɨɨɬɧɨɲɟɧɢɟ ( j )G(r

r )

( j )G(r

r ) :

1G(r& r&c)

( j c) r&cexp(&r )

j exp(&r ) 1G(r& r&c)

)

(r )

exp(r ) 1

exp(r

) 1

r ( j )

G(r r )

G(r

r )

(r )

)

(r )exp(r ) exp(r

) 1

r ( j )

G(r

r )

(r )

exp(r ) 1

r ( j )

G(r r )

)

exp(r ) 1 (r )

jG(r r ) j(r ) r ( j )

G(r

r )

(r )

exp(r ) 1 (r )

jG(r r ) [ [r j(r ,t)]]

G(r

r ) .

)

ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ:

exp(r

) 1 (r

)

³dV ® j(r ,t) [ [r j(r ,t)]]

¾G(r r )

(r )

exp(r ) 1 (r )

(r,t)

rot ³dV [r

(r ,t)]

G(r

r ) . (15.12)

(r )

Ɉɬɫɸɞɚ

ɜɵɬɟɤɚɟɬ

ɪɚɜɟɧɫɬɜɨ

(15.2) ɢ

ɫɨɨɬɧɨɲɟɧɢɟ,

ɜɵɪɚ-

ɠɚɸɳɟɟ M (r

,t) ɱɟɪɟɡ j(r ,t) :

c &c

exp(r

) 1

(r )

(15.13)

M (r,t)

³dV [r

j(r

,t)]

G(r

r ) .

)

ɜ ɜɢɞɟ ɪɹɞɚ ɢ ɜɵɩɨɥɧɢɜ

Ɂɚɩɢɫɚɜ ɨɩɟɪɚɬɨɪ ɫɞɜɢɝɚ exp(r )

ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɲɬɪɢɯɨɜɚɧɧɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɢɡ (15.8) ɢ (15.13) ɩɨɥɭɱɢɦ:

Dn 1 xD1 xD2

xDn 1Ur&

n 1 n!

( r )Ur ... ,

(15.14)

D1 D2 Dn 1 xD1 xD2 xDn 1

[r j

]

1)!

c n 1

& &&

[r j]

( r )[r j]

... .

(15.15)

ɉɟɪɜɵɟ ɱɥɟɧɵ ɜ ɷɬɢɯ ɪɚɡɥɨɠɟɧɢɹɯ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɥɨɬɧɨɫɬɢ ɞɢɩɨɥɶɧɵɯ ɦɨɦɟɧɬɨɜ: ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɜ (15.14) ɢ ɦɚɝɧɢɬɧɨɝɨ ɜ (15.15). Ɍɨɥɶɤɨ ɷɬɢ ɱɥɟɧɵ ɢ ɦɨɝɭɬ ɞɚɜɚɬɶ ɧɟɧɭɥɟɜɵɟ ɜɤɥɚɞɵ ɜ ɨɛɴɟɦɧɵɟ ɢɧɬɟɝɪɚɥɵ ɩɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɩɨ ɜɫɟ-

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