6

6.1

- . , , ( ) 16 .

, , - . , ( ), . ( ) , ( ) .

18 - -, Coulomb - . - , . ,, , - .

, , . , , - , , . - , , .

464 6

Oersted 1820, - , , - Ampre . , Faraday - , , - . Faraday Maxwell Ampre .

, , , - , - .

, - Oersted, - ( ) ( ). - . , .

, -, . - .

. - . , , .

6.2

, , -

6.2 465

, , , . - , - - .

, , .

, , , , , Coulomb, Ampre,Ohm, Faraday .., - , - .

-, . , - Maxwell, - . , , , Maxwell, .

(.. - ,, , -), , ,, :

) , - . , , , Coulomb Ohm, . , , , , - , .., . -, , , .

, , , , - .

466 6

) , , , , , - .

) Maxwell, , , - . , - , , - .

) , , , , .

6.3

, , - .

- , , .

, , , - .

we , wm , pt P - , , .

6.3.1

, - E .

6.3 467

E .

, , - , we. , , we - . ,

we = E2; (6.1)

- . , , . , - MKSA = =2, (6.1)

we =12E2: (6.2)

E MKSA , , Volt/meter(V=m) Farad/meter (F=m).

6.3.2

, - : H .

(magnetic field intensity)H - . H .

H , , wm.

, , , wm

468 6

. ,

wm = H2; (6.3)

- . (permeability).

MKSA = 2, (6.3)

wm =12H2: (6.4)

H MKSA , , Ampre/meter(A=m) Henry/meter (H=m). r

r =

0(6.5)

0 = 4 107H=m . , E H

, w -

w =12E2 + H2

: (6.6)

6.3.3 Joule

, pt ( Joule) ,

pt = E2; (6.7)

, MKSA mho/meter (f=m) Siemens/meter (S=m).

-, .

6.4 MAXWELL 469

(6.7)

pt =

TE2; (6.8)

T ( T ) (5.38). T , - .

6.3.4

- . - , - . - . , , - .

- , P E H

P = E H (6.9) P Poynting. - ( Poynting) MKSA Watt/meter2 (W=m2).

6.4 Maxwell

6.4.1

, , V S,

470 6

6.1. - , , . WeWm V - t. Joule - , , pt P .

t t+ dt - , , ,We+dWe Wm+dWm, , dWe+ dWm - .

, , Joule dWt [t; t+dt] dWr S .

W W Pe m t, ,

S

V

P

XHMA 6.1: .

, , ,

dWe + dWm + dWt + dWr = 0: (6.10)

, , (6.10).

We V , (6.2),

We =V

wedV =12

V

E2dV =12

V

E EdV: (6.11)

, dWe

dWe =@We@t

dt =12

V

@(E E)

@tdV

dt

6.4 MAXWELL 471

=

V

E @E@t

dV

dt: (6.12)

, Wm dWm, (6.4),

Wm =V

wmdV =12

V

H2dV =12

V

H HdV; (6.13)

dWm =@Wm@t

dt =12

V

@(H H)

@tdV

dt

=

V

H @H@t

dV

dt: (6.14)

Joule dWt, (6.7),

dWt =

V

ptdV

dt =

V

E2dV

dt =

V

E EdVdt: (6.15)

, (6.9),

dWr =

S

P dSdt =

S

(E H) dSdt (6.16)

, Gauss (6.16),

dWr =

V

r (E H)dVdt: (6.17)

(6.12), (6.14), (6.15) (6.17) (6.10) V

E @E@t

dV +V

H @H@t

dV +V

E EdV

+V

r (E H)dVdt = 0 (6.18)

V

E @E

@t+ H @H

@t+ E E +r (E H)

dV = 0: (6.19)

(6.19) V , , ,

E @E@t

+ H @H@t

+ E E +r (E H) = 0: (6.20)

472 6

(6.20),

r (E H) =H r E E r H; (6.21)

@E@t

+ E rHE +

@H

@t+rE

H = 0: (6.22)

(6.22) . , (6.22) , E0 H 0

E0 = E; (6.23)

H 0 =H +H0; (6.24)

H0 ( ), - (6.22),

@E0

@t+ E0 rH 0

E0 +

@H 0

@t+rE0

H 0 = 0: (6.25)

(6.25), (6.23), (6.24) rH0 = @H0=@t = 0,

@E@t

+ E rHE +

@H

@t+rE

(H +H0) = 0: (6.26)

(6.26) (6.22)

H0 @H

@t+rE

= 0 (6.27)

, H0 , , ,

rE = @H@t

: (6.28)

, (6.28) (6.22), @E

@t+ E rH

E = 0 (6.29)

6.4 MAXWELL 473

E 6= 0,

rH = E + @E@t

: (6.30)

( - ) D, J (magnetic induction) ( ) (magnetic flux density)B, ( -)

D = E (6.31)

J = E (6.32)

B = H (6.33)

(6.28) (6.30)

rE = @B@t

(6.34)

rH = J + @D@t

(6.35)

B MKSA Tesla (T) Webermeter2 (1 Wbm2 = 1 Vsm2 = 104 Gauss).

(6.34) (6.35) Maxwell -, ; ., (6.34) (6.35) , .

- , . ,, , - (6.2), (6.4), (6.7) ( )

we =

D

0E dD (6.36)

474 6

wm =

B

0H dB (6.37)

pt = J E (6.38)

- ; ; , E H .

6.1

B H - ( ) ( Frchlich)

B =H

a+ bH;

a; b . - , H = Ha = 9a=b. ( = 0) H = Ha = 9a=b;

(6.37),

B =H

a+ bH:

, , (6.37)

wm = B(H)0

HdB = H0

Hd

H

a+ bH

= a

Ha0

H

(a+ bH)2dH

= a

a

b2(a+ bH)+

1b2

ln(a+ bH)HaH=0

=a2

b2(a+ bHa) ab2

+a

b2lna+ bHa

a

:

, Ha = 9a=b,

wm =a

10b2 ab2

+a

b2ln 10 = 1;4

a

b2:

6.4 MAXWELL 475

, - , (6.4),

wm =120H

2a =

8120a2

b2:

6.4.2

(6.34), ,

r r E = r @B@t

(6.39)

r @=@t , , ,

@

@tr B = 0 (6.40)

r B = const:; (6.41)

, - .

, , - rB = 0, (6.41), , ,

r B = 0 (6.42)

(6.42) Mawxell, - ( ) . -, (6.42) , , , Maxwell (6.34).

B (magnetic flux). - , d, - dS,

d = B dS (6.43)

476 6

, S,

=S

B dS (6.44)

MKSA Weber Voltsec (Wb Vs, 1 Wb = 108 Maxwell).

, B, - . H . , , B H . , , - .

(6.42) V S,

V

r BdV = 0: (6.45) (6.45) , -

Gauss, , S

B dS = 0 (6.46)

(6.46) ( (6.42) (6.35) - , , ) - , .

, (6.46) . , ( ) - ( ), ( ) ( ).

6.4.3

(6.35),

r r H = r J +r @D@t

: (6.47)

6.4 MAXWELL 477

(6.47), r @=@t ,

r J + @@tr D = 0: (6.48)

(6.48) V S

V

r JdV +V

@

@tr DdV = 0 (6.49)

, Gauss ,

S

J dS + @@t

V

r DdV = 0: (6.50)

, , (6.50), , Q - S,

@Q@t

+@

@t

V

r DdV = 0: (6.51)

(6.51), - ,

V

r DdV = Q: (6.52)

V , (6.52)

V

r DdV =V

dV ; (6.53)

r D = (6.54)

Maxwell. (6.54), , , .

478 6

(6.54) (6.48)

r J + @@t

= 0 (6.55)

, ,

r J = 0: (6.56)

, (6.54) , Maxwell (6.35), (6.55).

, (6.52) -, Gauss

S

D dS = Q (6.57)

. (.. ), (6.57), Q , .

6.4.4

, , - S 1 2.

( 6.2()) , - h, (S = Sn^, n^ 1 2). , 6.2() , , `, , , -h, . 6.2() - 6.2() . , h 6.2() , , , .

6.4 MAXWELL 479

Dh

S

DS n

t

t

Sc

Dh

Dl

Sk

1

2

1 1

2

() ()

k

XHMA 6.2: S.

, s, K. , s, K . K s -, , . K ( ) MKSA , , Ampre/meter (A=m).

(6.46) St 6.2(), S1; S2 S - , ,

St

B dS =S1

B dS +S2

B dS +S

B dS = 0: (6.58)

, h - (h ! 0), . , S , B1 B2 , , , , , (6.58)

Bn1S Bn2S = 0; (6.59) Bn1 ; Bn2 .

(6.59)

Bn1 = Bn2 (6.60)

480 6

n^ (B2 B1) = 0 (6.61)

( ).

, (6.57) 6.2(),

St

D dS =S1

D dS +S2

D dS +S

D dS = Q; (6.62)

Q V . h (h ! 0), , , S, (6.62)

Dn2S Dn1S = Q; (6.63) Dn2 ; Dn1 D .

, h! 0, - , , s - , Q

Q = sS: (6.64)

, (6.64) (6.63)

Dn2 Dn1 = s (6.65)

n^ (D2 D1) = s (6.66)

- ( -).

, (6.34) S 6.2(),

S

rE dS = S

@B

@t dS (6.67)

6.4 MAXWELL 481

, Stokes (6.67), cE d` =

S

@B

@t dS; (6.68)

c . h (h! 0),

(6.68), ,

Et1` Et2` = @B1@t

^h1` @B2@t

^h2` (6.69)

Et1 Et2 = @B1@t

h1 +@B2@t

h2

^; (6.70)

Et1 ; Et2 - , ^ = t^ n^ - h1;h2 (h = h1+h2) .

(6.70), h! 0,

Et1 Et2 = 0 (6.71)

n^ (E1 E2) = 0 (6.72)

- .

, , (6.35) S 6.2(),

S

rH dS =S

J dS +S

@D

@t dS (6.73)

, Stokes ,cH d` =

S

J dS +S

@D

@t dS: (6.74)

(6.74), h1;h2,

Ht1`Ht2` = J ^`h+@D

@t ^`h: (6.75)

482 6

I = J ^`h; (6.76) , K, (6.76)

I = J ^`h =K ^`: (6.77) (6.77) (6.75)

Ht1`Ht2` =K ^`+@D

@t ^`h (6.78)

Ht1 Ht2 =K ^+

@D

@t ^h: (6.79)

(6.79) h! 0,

Ht1 Ht2 =K ^ (6.80)

K - t^ ( - ).

^ - (6.80), (6.80)

(H1 H2) t^ =K ^: (6.81) , (6.81), t^

t^ = ^ n^; (6.82)

(H2 H1) (^ n^) =K ^: (6.83)

(6.83),

A (B C) = (C A) B (6.84)

[n^ (H2 H1)] ^ =K ^ (6.85)

6.4 MAXWELL 483

(6.80)

n^ (H2 H1) =K (6.86)

Ht1 = Ht2 (6.87)

n^ (H2 H1) = 0 (6.88)

- H .

6.2

E H ( = 0;J = 0) ( !1) y = 0 y = b 6.3:

E =b

E0 cos

yb

sin(!tz)y^

+ E0 sinyb

cos(!tz)z^ (0 6 y 6 b); (i)

H = !0b

E0 cosyb

sin(!tz)x^ (0 6 y 6 b); (ii)

!; ;E0 , !

=

r!200

b

2: (iii)

m0 0,

s

s z x

y

b

XHMA 6.3: O .

:) Maxwell.

484 6

) y = 0 y = b. ( = 0; =0; = 0), (E =H = 0, y < 0 y > b).

) E;H; , , E H:

rE =@Ez@y

@Ey@z

x^+

@Ex@z

@Ez@x

y^ +

@Ey@x

@Ex@y

z^

=@Ez@y

@Ey@z

x^ @Ez

@xy^ +

@Ey@x

z^

=hbE0 cos

yb

cos(!tz)

+2b

E0 cos

yb

cos(!tz)

x^ 0y^ + 0z^

=

b+2b

E0 cos

yb

cos(!tz)x^

=

b+

b

!200

b

2E0 cos

yb

cos(!tz)x^;

rE = b!200E0

cosyb

cos(!tz)x^; (iv)

rH =@Hz@y

@Hy@z

x^+

@Hx@z

@Hz@x

y^ +

@Hy@x

@Hx@y

z^

=@Hx@z

y^ @Hx@y

z^;

rH = !0bE0

cosyb

cos(!t z)y^

!0E0 sinyb

sin(!tz)z^; (v)

r E = @Ex@x

+@Ey@y

+@Ez@z

=@Ey@y

+@Ez@z

= E0 sinyb

sin(!tz) E0 sin

yb

sin(!tz);

6.4 MAXWELL 485

r E = 0; (vi)

r H = @Hx@x

+@Hy@y

+@Hz@z

=@Hx@x

;

r H = 0; (vii)

@E

@t=

@Ey@t

y^ +@Ez@t

z^;

@E

@t=

!bE0

cosyb

cos(!tz)y^

!E0 sinyb

sin(!tz)z^; (viii)

@H

@t=

@Hx@t

x^;

@H

@t= !

20bE0

cosyb

cos(!tz)x^: (ix)

, (vi)-(ix),

D = E = 0E; B = H = 0H; (x)

r D = 0r E = 0; (xi)r B = 0r H = 0; (xii)

@D

@t= 0

@E

@t=

!0bE0

cosyb

cos(!t z)y^

!0E0 sinyb

sin(!t z)z^; (xiii)

@B

@t=0

@H

@t=!

200bE0

cosyb

cos(!tz)x^: (xiv)

(iv), (v), (xi)-(xiv), J = 0; = 0, Maxwell:

rE = @B@t

; (xv)

rH = J + @D@t

; (xvi)

486 6

r B = 0; (xvii)r D = : (xviii)

) s K -

s = n^ (D2 D1); K = n^ (H2 H1); (xix) - n^ 1 2. , , y = 0 (n^ = y^, D2 = 0E2 = 0E,D1 = 0E1 = 0, H2 = H , H1 = 0) y = b (n^ = y^, D2 = 0E2 = 0,D1 = 0E1 = 0E,H2 = 0,H1 =H) ,

sjy=0 = y^ (0E)jy=0 = 0y^ (Eyy^ + Ezz^)jy=0 = 0Eyjy=0

=0bE0

sin(!tz); (xx)

sjy=b = y^ (0E)jy=b = 0y^ (Eyy^ + Ezz^)jy=b = 0Eyjy=b

=0bE0

sin(!tz); (xxi)

Kjy=0 = y^ Hjy=0 = y^ Hxx^jy=0 = Hxz^jy=0

=!0bE0

sin(!tz)z^; (xxii)

Kjy=b = y^ (H)jy=b = y^ Hxx^jy=b = Hxz^jy=b

=!0bE0

sin(!tz)z^: (xxiii)

6.4.5

- , - Maxwell

rH = J + @D@t

rE = @B@t

r B = 0r D =

(6.89)

487

, (6.89)

f = (E + v B) (6.90)

Lorentz. (6.90) , - ,

V

fdV =V

(E + v B)dV (6.91)

f , - (E;B), - v.

(6.89) Maxwell, H E Ampre () Faraday ( ), . -, B D, , , - ( ) Gauss ( ) .

Maxwell, -, (.. - ) . - Maxwell , - . , , - , , .

, , , - , - , Maxwell.

6.1 ,

. S

488 6

c, - ( H E) Maxwell. , -, , , - Ampre - Faraday. ; , , , , - E;H; ; .

6.2 E;H , (J = 0; = 0), Maxwell.) Maxwell (-

)

E0 = E + H; H 0 = E + H;

; = (=)1=2.) , 2 + 2 = 1, -

, Poynting.

) = 0 = 1;)

; ;

E0 = E + H; H 0 = E + H;

E;H E0;H 0 Maxwell (J 6= 0; 6= 0). ; ; ;

6.3 a I . H - P (; '; z) , ,

H =I

2a2'^;

z -. B - H ( MKSA) B = 0;3H0;2, .

489

6.4 (J = 0; = 0) ( = 0) , E

E = E0 cos(!t z)x^;

E0; !; .) ; ;E0; ! -

;H0,

H = H0 cos(!t z)y^

.)

.

) ;

6.5 1 2 - 1 = 20; 1 = 0 2 = 0; 2 = 5000, , - x+ y = 5. 1 (x+ y 5 < 0)

E1 = 120(x^+ 2y^) V=m; H1 = 2x^+ y^ KA=m;

, E2;H2;D2 B2 - . s (s = 0) K K = p2z^KA=m.

: - , .

7

7.1

, , - . () .

, @D=@t = 0 @B=@t =0, Maxwell (6.89), - ,

rE = 0 r D = Dn2 Dn1 = s Et1 Et2 = 0

(7.1)

rH = J r B = 0Bn2 Bn1 = 0 (H1 H2) n^ =K

(7.2)

(7.1) (7.2) ( ) ; s;J K. -, , .

(7.1), D = E, , -, - .

(7.2), B =H , .

( -) - (7.2), . ,

492 7

(7.2) rH = 0; (7.3)

, , , - -. -, , .

, , - .

, - , :

) -.

) - .

) -.

(7.2). ,

J =rH;K = (H1 H2) n^:

(7.4)

- .

7.2 Ampre

7.2.1

, , Maxwell (6.89)

rH = J : (7.5)

7.2 AMPRE 493

S

dS

H

c

XHMA 7.1: S.

S 7.1, c, (7.5) S,

S

rH dS =S

J dS: (7.6)

Stokes (7.6) I , S,

I =S

J dS; (7.7) (7.6)

cH d` = I (7.8)

(7.8) H ( - ) c.

, I , , c. I , c, .

, (7.8) , H - c I .

Ampre - - . ( ) , , (7.5).

494 7

I1

I2 I

3

I4

c

XHMA 7.2: I .

c , , I1; I2; I3 I4 7.2, Ampre

cH d` = I = I1 I2 + I3 I4; (7.9)

S c.

7.2.2 Ampre

7.2.2.1

7.3, I . z .

Ampre.

c

O r

I

z

H=Hj$

dl

XHMA 7.3: .

, - ,

7.2 AMPRE 495

. , , , A Biot-Savart. , .

Ampre , (7.8), H d` -,

cH d` =

cH'^ d`'^ =

cHd` = I; (7.10)

'^ . (7.10),

H , , , ,

H

cd` = I

H2 = I; (7.11)

H =I

2'^; (7.12)

B =0I

2'^: (7.13)

, H (- ) , - , .

7.2.2.2

, , 7.4, I . I -

J =I

a2z^: (7.14)

496 7

, , . , 7.4() , P , ( ) P1 P2 .

, P .

, P , Ampre c1 . ,

H =I

2'^ ( > a): (7.15)

P , c2, Ampre

c2

H d` = H2 = I(); (7.16)

z

I

a

r

r

H

H

S1

c1

S2

c2

()

P1

P2

P

d dHH= j$

a

r

()

XHMA 7.4: .

7.2 AMPRE 497

I() c2. S2 c2, I(), (7.14),

I() =S2

J dS =S2

JdS =I

a2

S2

dS =I

a22;

I() = I2

a2: (7.17)

(7.17) (7.16)

H =I

2a2'^ ( 6 a): (7.18)

H

Hmax

HI

amax=

2p

0 a r

H H a= maxr

HI

=2pr

XHMA 7.5: H .

7.5 - , .

7.2.2.3

Ampre 7.6, I () , , (-).

Ampre I II , - ,

H =I

2a2'^ ( 6 a); (7.19)

H =I

2'^ (a 6 6 b): (7.20)

498 7

I

I

I

z

a

b

c

II a b

c(I)

(II)

(III)

(IV)

A A

A-A

H

r0 a b c

(I)(II)

(III)

(IV)

( )

( )

( )

XHMA 7.6: .

III (b 6 6 c), J

J =I

(c2 b2) ; (7.21)

I(), ,

I() = I (2 b2)J = I 2 b2c2 b2 I;

,

I() =c2 2c2 b2 I: (7.22)

, , Ampre

H2 =c2 2c2 b2 I (7.23)

H =I

2c2 2c2 b2 '^ (b 6 6 c): (7.24)

, IV I() ,

H = 0 ( > c): (7.25)

7.3 499

H , , 7.6().

7.3

7.3.1

, , V , , - .

Maxwell, - (J = 0),

rH = 0: (7.26)

(7.26),

rE = 0 (7.27)

, -H - m,

H = rm (7.28)

m, - , (scalarmagnetic potential) ( MKSA) Ampre (A) Ampre- (AE).

AMB ANB , V , A B ( 7.7), Ampre AMBNA,

AMCNA

H d` =AMB

H d`+BNA

H d` = 0 (7.29)

c1

H d`c2

H d` = 0; (7.30)

c1

H d` =c2

H d` =cH d` = Um;AB (7.31)

500 7

c A B. (7.31) Um;AB,

V , .

J=0

A

B

cc

1

c2

V

M

N

XHMA 7.7: V .

Um;AB, A B , , V , (magnetomotive for-ce) ( MMF) .

(7.31), (7.28),

Um;AB =cH d` =

c(rm) d` = m(A) m(B); (7.32)

Um;AB A B m(A) m(B) - m A B, .

, , . , , - .

, ( 7.8 V 0 V ).

, - . , , V 7.8, N I .

7.3 501

A

B

N

V

V

V

c

c1

c2

e

e

J=0

M

XHMA 7.8: .

, , Ampre AMBNA, NI ,

cH d` = NI: (7.33)

(7.33), (7.28), - A A,

cH d` =

c(rm) d` = m(A) 0m(A) = NI; (7.34)

m(A) 0m(A) m , .

(7.34) A NI . c ,

m(A) 0m(A) = 2NI: (7.35), c n ,

m(A) 0m(A) = nNI: (7.36)

, m .

, , .

502 7

, , - A.

, , , - m, ( 7.8 ee0) . , - .

7.3.2

, , - m.

, , Maxwell

r B = r (H) = r H = 0 (7.37), (7.28), Laplace

r2m = @2m@x2

+@2m@y2

+@2m@z2

= 0; (7.38)

, , - - .

H x=H0$

x

y

P z( , , )r j

j

r

zm

0mm

0

a

b(I)

(II)

(III)

XHMA 7.9: .

, , ,

7.3 503

, - H = H0x^ ( 7.9).

, - m, I, II III,, ,

m1 =A1+

B1

cos'+ C1; (7.39)

m2 =A2+

B2

cos'+ C2; (7.40)

m3 =A3+

B3

cos'+ C3; (7.41)

Ai; Bi; Ci (i = 1; 2; 3) .

(7.39), (7.40), (7.41) sin', '.

:) m

. , , () ( = 0),

B1 = 0: (7.42)

) III ( b), H0x^.

H3j(b) = rm3 j(b) = r(A3 cos'+ C3)

= r(A3x+ C3) = @(A3x)@x

x^ = A3x^ = H0x^;

A3 = H0: (7.43)

) - = a = b, H - ,

m1 j=a = m2 j=a ; (7.44)m2 j=b = m3 j=b : (7.45)

504 7

(7.44) (7.45), (7.39)-(7.41), A1a+

B1a

cos'+ C1 =

A2a+

B2a

cos'+ C2; (7.46)

A2b+

B2b

cos'+ C2 =

A3b+

B3b

cos'+ C3: (7.47)

(7.46) (7.47), ', -

A1a+B1a

= A2a+B2a; (7.48)

A2b+B2b

= A3b+B3b; (7.49)

C1 = C2 = C3: (7.50)

C1; C2 C3 , ' = =2, (7.50)

C1 = C2 = C3 = 0: (7.51)

) B B = a = b,

B1 j=a = B2 j=a ; (7.52)

B2 j=b = B3 j=b (7.53), B = H H = @m=@,

0

@m1@

=a

= @m2@

=a

; (7.54)

@m2@

=b

= 0

@m3@

=b

: (7.55)

(7.39), (7.40) (7.41) (7.54) (7.55)

A1 B1a2

= r

A2 B2

a2

; (7.56)

r

A2 B2

b2

=A3 B3

b2

; (7.57)

7.3 505

r . (7.42), (7.43), (7.48), (7.49),

(7.51), (7.56) (7.57) Ai; Bi; Ci(i = 1; 2; 3):

A1 =4r

a2

b2(r 1)2 (r + 1)2

H0; B1 = 0; C1 = 0; (7.58)

A2 =(r + 1)2r

A1; B2 =(r 1)a2

2rA1; C2 = 0; (7.59)

A3 = H0; B3 = (2r 1)(a2 b2)

4rA1; C3 = 0: (7.60)

( < a), - m1 , (7.39) (7.58),

m1 =4rH0

a2

b2(r 1)2 (r + 1)2

cos'

=4rH0

a2

b2(r 1)2 (r + 1)2

x; (7.61)

H1

H1 = rm1 = @m1@x

x^ =4rH0

(r + 1)2 a2

b2(r 1)2

x^: (7.62)

(7.62) -, , , H0. , , (7.62), H1 H0 .

(magnetic shielding), ,

s =H1H0

(7.63)

506 7

7.1:

a = 5 cm; r = 500; d = b ad (mm) s = H1=H0

1 0;1714

2 0;0962

5 0;0442

10 0;0256

, (7.62),

s =4r

(r + 1)2 a2

b2(r 1)2

: (7.64)

(r 1), (7.64)

s ' 4r

1 a

2

b2

; (7.65) r ! 1 .

s, - = 5000 a 5 cm, 7.1 d = b a. , , - , . , d = 1 cm, 2;6% .

, , - .

, - (.. , , , , ...) , m. -, , .

7.4 A 507

7.4 A

7.4.1

7.3, m - .

, , - Maxwell

rH = J ; (7.66) , , , .

Maxwell

r B = 0 (7.67) . (7.67), - , - , .

(7.67), -, - , - A, - B

B = rA (7.68)

, (7.68) (7.67),

r B = r r A = 0: (7.69) A, (7.68),

(magnetic vector potential). ,, B. , A B,

A0 = A+r ; (7.70)

508 7

, . :

B0 A0, (7.68) (7.70)

B0 = rA0 = r (A+r ) = rA+rr : (7.71)

(7.71), , - ,

B0 = rA = B: (7.72)

, , A A0, - , . , A (7.68).

A, , . , Helmholtz, - , - A. , - A,

r A = 0 (7.73) Coulomb.

, A , , - . A MKSA Volt sec/meter (Vs=m).

7.4.2 Poisson

(7.66), , , (7.38),

rB = J : (7.74), B (7.68),

rrA = J (7.75)

7.4 A 509

(7.75),

rrA = r(r A)r2A (7.76)

(7.76), (7.73), ,

r2A = J (7.77)

Ax; Ay; Az Jx; Jy; Jz A J , , - Poisson (7.77) Poisson

r2Ax = Jx; (7.78)r2Ay = Jy; (7.79)r2Az = Jz: (7.80)

(7.78)-(7.80)

=14

V 0

dV 0

R(7.81)

Poissonr2 =

(7.82)

, ,

Ax =

4

V 0

JxdV0

R; (7.83)

Ay =

4

V 0

JydV0

R; (7.84)

Az =

4

V 0

JzdV0

R; (7.85)

R dV 0 P , A, Jx; Jy; Jz J dV 0.

, , (7.81), (7.83)-(7.85) - , (.. ).

510 7

(7.83), (7.84) (7.85) -

A = Axx^+Ayy^ +Azz^ =

4

V 0

JdV 0

R(7.86)

, , A, (7.77), (7.86) V 0.

, , - r2A r2Ax;r2Ay;r2Az r2A -. , - , r2A (7.76), .

I S0, d`0 , dV 0 = S0d`0, JdV 0

JdV 0 = JS0d`0 = Id`0 = Id`0; (7.87) (7.86)

A =

4

Id`0

R(7.88)

A, , .

, - c S, (7.68) (6.44),

=S

B dS =S

rA dS: (7.89)

(7.89), Stokes , -, ,

=cA d` (7.90)

7.4 A 511

, A.

7.4.3

, , AB 7.10, , z. ` I .

AB - P , , - ' ( ).

A, (7.88), I , z,

A(; z) = Az(; z)z^ =0Iz^

4

z2z1

dz0

R

=0Iz^

4

z2z1

dz0p2 + (z z0)2 ; (7.91)

z

x

y

O

j r

A

B

R

r

z

R1

R2

j$

q1

q2

q

z

dz

I

z1

z2

l

P z( , , )r j

B=Bjj$

A z=Az$P

XHMA 7.10: .

512 7

R =p2 + (z z0)2 dz0

P . (7.91)

A = Azz^ =0Iz^

4ln

"z z1 +

p2 + (z z1)2

z z2 +p2 + (z z2)2

#

A =0Iz^

4lnz z1 +R1z z2 +R2

(7.92)

R1; R2 P A B , .

7.10, - B (7.68), ,

A = A' =@Az@'

= 0; (7.93)

B = rA = @Az

@'^: (7.94)

, -B B'.

(7.92) (7.94)

B = B''^ = 0I4

2664p

2 + (z z1)2z z1 +

p2 + (z z1)2

p2 + (z z2)2

z z2 +p2 + (z z2)2

3775 '^

B = 0I4

R1(z z1 +R1)

R2(z z2 +R2)'^ (7.95)

PP 0 P z, - PP 0A PP 0B (7.95)

7.4 A 513

B = B''^ = 0I41(1 cos 1) 1

(1 cos 2)

'^;

B = B''^ =0I

4(cos 1 cos 2)'^ (7.96)

1 2 z PA PB, .

7.4.4

, (7.96) 1 = 0 2 = ,

B = B''^ =0I

2'^: (7.97)

, (7.97) (7.13) Ampre.

, (7.92), , - . , , , - , .

, , ( '; z, ) , z (A = A()z^), (7.97) ,

B =0I

2'^ = rA =

@A@z

@Az@

'^ = dA

d'^;

A = 0I2

ln + C; (7.98)

= , C = 0I2 ln .

514 7

, -

A = 0I2

ln

z^: (7.99)

(7.98), - .

7.4.5

A, , I . P , , r a(r a). , P x = 0. , 7.11, = =2 '^ = x^.

I

q

wj

j

Oa

dl

M

N

P r,( ,2

)qp-

y

x

z

rR

K

dl

c

XHMA 7.11: .

A - A', y - ( y) Id` Id`0, ,.

7.4 A 515

, (7.88),

A = A''^ =0I

4

c0

d`0

R=

0I

4

20

a'^0

Rd'0;

'^0 d`0 ,

A' =0Ia

4

20

sin'0

Rd'0: (7.100)

PK P y PN;KN - P K OM , OPM , ,

R2 = r2 + a2 2ar cos!; ! PM OPM .

, , OKN OPK,

r cos! = (ON) = (OK) sin'0 = r sin sin'0; (7.101)

R =r2 + a2 2ar sin sin'01=2

= r1 +

a2

r2 2a

rsin sin'0

1=2: (7.102)

(7.102), r a ,

1R' 1

r

1 2a

rsin sin'0

1=2=

1r

1 +

a

rsin sin'0 +O

a2

r2

; (7.103)

O(a2=r2) - a=r .

, (7.103) O(a2=r2),

1R' 1

r

1 +

a

rsin sin'0

; (7.104)

516 7

(7.100)

A' =0Ia

4

20

1r

1 +

a

rsin sin'0

sin'0d'0

=0Ia

4r

20

sin'0d'0 +a

rsin

20

sin2 '0d'0

=0Ia

4r

0 +

a

r sin

A = A''^ =0Ia

2

4sin r2

'^ (7.105)

B, (7.105) (7.68), Ar; A - A 7.11 ,

B = rA = 1r sin

@ (sin A')@

r^ 1r

@ (rA')@r

^

B = Brr^ +B^ =0Ia

2

4r32 cos r^ + sin ^

(7.106)

B , (7.106), E ( (1.183), (1.184)) . I - (magnetic dipole). (magneticdipole moment)M - , . ,

M = a2Iz^ (7.107)

, - , (1.185) , ( (7.148)).

7.4 A 517

x

z

d

O a

rq

b

I

c

S

S

XHMA 7.12: .

. , , , , b, d ( 7.12).

(7.105) (7.90),

=S

B dS =cA d` =

cA' d` = A'

cd` = A'2b; (7.108)

S c . (7.108), A' (7.105),

=0Ia

2

2sin2 r

=0Ia

2b2

2 (b2 + d2)3=2: (7.109)

. , S0 - r O , (7.106),

=S0B dS0 =

S0

Brr^ +B^

dS0r^ =

S0BrdS

0

= 0

20

0Ia2

4r32 cos r2 sin d d' =

0Ia2

2sin2 r

; (7.110)

, , (7.109).

518 7

7.4.6 , ,

V 0, r P .

V

dV

O

r

R

r

P

JdV

XHMA 7.13: .

A P , (7.86),

A =

4

V 0

JdV 0

R; (7.111)

R P dV 0. O

V 0. r r0 O

P dV 0, , 7.13, P dV 0

R = r r0: (7.112) (7.112)

R2 = r2 2r r0 + r02

1R

=1r

1 2r r

0

r2+r02

r2

1=2: (7.113)

(7.113) r0=r (, , r0=r 1 r0=R 1), (7.113)

1R' 1

r+r r0r3

(7.114)

7.4 A 519

(7.111)

A =

4

V 0

1r+r r0r3

JdV 0: (7.115)

(7.115), r - ,

A =

41r

V 0JdV 0 +

41r3

V 0(r r0)JdV 0: (7.116)

(7.116), J - V 0, . V 0 . , Ii i- V 0i ,

V 0JdV 0 =

Xi

V 0iJdV 0 =

Xi

Ii

c0id`0i = 0; (7.117)

c0i ., (7.116)

A =

41r3

V 0(r r0)JdV 0: (7.118)

(7.118),

A (B C) = (A C)B (A B)C; (7.119)

(r r0)J = (r0 J) r + (r J)r0

(r r0)J = 1

2(r0 J) r + 1

2(r r0)J + (r J)r0 : (7.120)

(7.120) (7.118)

A =

8r3

V 0(r0 J) rdV 0+

V 0

(r r0)J + (r J)r0dV 0 (7.121)

Q =V 0

(r r0)J + (r J)r0 dV 0: (7.122)

520 7

, (7.122) u,

u Q =V 0

(r r0)(u J) + (r J)(u r0) dV 0 =

V 0UdV 0: (7.123)

U (7.123), r0 x0; y0; z0 V 0, r0(u r0) = u r0(r r0) = r, ,

U = (r r0)(u J) + (r J)(u r0) = J r0 (r r0)(u r0)= r0 (u r0)(r r0)J (u r0)(r r0)r0 J (7.124)

, r0 J = 0; (7.125)

U = r0 (u r0)(r r0)J : (7.126) (7.126) (7.123)

u Q =V 0r0 (u r0)(r r0)J dV 0 (7.127)

, Gauss,

u Q =S0(u r0)(r r0)J dS0; (7.128)

S0 V 0., , V 0,

J S0 , (7.128)

u Q = 0 (7.129) , u , -

Q = 0: (7.130)

, (7.121), (7.122) (7.130),

A =

8r3

V 0(r0 J) rdV 0: (7.131)

7.4 A 521

(7.131), r , , ,

A =

4r3

12

V 0

r0 J dV 0 r: (7.132)

M =12

V 0

r0 J dV 0 (7.133)

. (7.131) (7.133) -

A, -, M r

A =

4M rr3

(7.134)

B, (7.134) (7.68)

B =

4r

M rr3

: (7.135)

, , (7.135), -

r (fA) = frA+rf A; (7.136)

rM rr3

=

1r3r (M r) +r

1r3

(M r): (7.137)

(7.137),

r1r3

= 3r

r5; (7.138)

r1r3

(M r) = 3r

r5 (M r) (7.139)

522 7

,

A (B C) = (A C)B (A B)C; (7.140)

r1r3

(M r) = 3

r5[(r r)M (r M)r] ; (7.141)

r1r3

(M r) = 3

r5(M r)r r2M : (7.142)

, , x; y; z Px; Py; Pz - (7.137),

r (M r) = Pxx^+ Pyy^ + Pzz^; (7.143) x Px

Px = [r (M r)]x

=@

@y(Mxy Myx) @

@z(MzxMxz) = 2Mx; (7.144)

Mx;My;Mz M , - x; y; z P .

Py = [r (M r)]y = 2My; (7.145)Pz = [r (M r)]z = 2Mz: (7.146)

(7.144), (7.145) (7.146)

r (M r) = 2M : (7.147), (7.142) (7.147) (7.137),

(7.135)

B =

4

3(M r)r

r5M

r3

(7.148)

(7.148) (1.185) - E M =qa.

7.4 A 523

O

dS

r

I

S

dl

XHMA 7.14: .

( 7.14) I , (7.133),

JdV 0 = Id`0; (7.149)

M =

I

2

c0r0 d`0 = I

S

dS; (7.150)

dS , , d`0 O.

O , (7.150)

M = IS (7.151)

S - , - I .

, (7.151) (7.134)

A =I

4S rr3

(7.152)

, (7.105) (7.107) 7.4.4 - (7.152) (7.151), ,

S = a2z^ (7.153)

S r = a2r sin '^: (7.154)

524 7

7.5 Biot-Savart

(7.86) A , - V 0 ( 7.15), B H .

O

R

P x y z( , , )

( , , )x y z

dV

V

XHMA 7.15: x; y; z x0; y0; z0.

, (7.86) (7.68)

B = rA = 4r

V 0

JdV 0

R(7.155)

(7.155), ,

B =

4

V 0r

J

R

dV 0: (7.156)

, - V 0 x; y; z P , .

r ('A) = r'A+ 'rA (7.157)

(7.156),

rJ

R

= r

1R

J + 1

Rr J : (7.158)

, , (7.158) J - x0; y0; z0 ,

7.5 BIOT-SAVART 525

x; y; z .

, (7.158)

rJ

R

= r

1R

J = R J

R3: (7.159)

(7.159) (7.156), Biot-Savart

B =

4

V 0

J RR3

dV 0 (7.160)

H , (7.160) B = H ,

H =14

V 0

J RR3

dV 0 (7.161)

(7.161) H - .

- E D -, ,

E =14

V 0

R

R3dV 0; (7.162)

D =14

V 0

R

R3dV 0 (7.163)

(7.160), (7.161), (7.162) (7.163), B - () , E . - , B , () H , D . , , - 1= .

526 7

qRI

P x y z( , , )

dB

dl

c

XHMA 7.16: .

, , - I ( 7.16).

, , , JdV 0 - Id`0, (7.160)

B =I

4

c0

d`0 RR3

(7.164)

c0 . H , -

(7.161),

H =I

4

c0

d`0 RR3

(7.165)

(7.164) (7.165), dB dH B H , Id`0 , , ,

dB =I

4d`0 RR3

(7.166)

dH =I

4d`0 RR3

(7.167)

Biot-Savart. dB dH P

7.5 BIOT-SAVART 527

d`0, d`0 R .

d`0 R, dB dH , ,

dB =I

4d`0

R2sin ; (7.168)

dH =I

4d`0

R2sin : (7.169)

, - Biot-Savart.

7.5.1

AB 7.10 I P ; '; z.

dB dz0 , (7.168),

dB =0I

4dz0

R2sin '^: (7.170)

7.10, ,

z0 = z tan

; (7.171)

R =

sin : (7.172)

(7.171), P z ,

dz0 = d

1tan

=

tan2 1

cos2 d =

sin2 d: (7.173)

(7.172) (7.173) (7.170)

dB =0I

4sin d'^: (7.174)

AB (7.174)

B =0I

4

21

sin d'^

528 7

B =0I

4(cos 1 cos 2)'^ (7.175)

, (7.96) A. 1 !0 2 ! , (7.175) (7.13), .

7.5.2

x = 0, K = Is = Isz^ z ( 7.17).

I zs s=I $

x

z y

l l

y

dyR

P x y( , ,0)

d dBB= j$

L

c

XHMA 7.17: .

, 2`, , z. , - B, Biot-Savart, z.

, , dy0, -, , (7.13),

dB =0Isdy

0

2R'^ =

0Isdy0

2z^ RR2

; (7.176)

Isdy0 dy0 , '^

7.5 BIOT-SAVART 529

P z (0; y0; 0) R P 1.

(7.176),

z^ R = z^ xx^+ (y y0)y^ = xy^ (y y0)x^; (7.177)

dB =0Is2

(y0 y)x^+ xy^x2 + (y y0)2 dy

0: (7.178)

B (7.178)

B =0Is2

``

(y0 y)x^+ xy^x2 + (y y0)2 dy

0

B =0Is2

12lnx2 + (` y)2x2 + (`+ y)2

x^

tan1

y `x

tan1

y + `x

y^

(7.179)

x = 0, (7.179) `!1,

B =0Is2

x

jxj y^ (7.180)

(7.180) - Ampre c 7.17, Bx .

7.5.3

, , - B a I ( 7.18). dB d`0 , -, , , dBz , .

1, , P z = 0.

530 7

I

z

P z(0,0, )

R

dB

dBz

a

x

y

z

w

w

dj

j

=adjj$dl

XHMA 7.18: .

, (7.168), = =2 ( d`0 R ),

dB =0I

4d`0

R2;

dBz

dBz =0I

4d`0

R2sin!z^ =

0I

4ad'0

R2a

Rz^

dBz =0Ia

2

4(a2 + z2)3=2d'0z^: (7.181)

B, P , (7.181)

B = Bzz^ = 20

dBz =0Ia

2z^

4(a2 + z2)3=2

20

d'0 (7.182)

B =0Ia

2

2(a2 + z2)3=2z^ (7.183)

7.5 BIOT-SAVART 531

, z = 0,

B =0I

2az^ (7.184)

7.5.4

7.19, I a `. N , - NI ( 7.19()), -

K = Is =NI

`: (7.185)

B P , dB - dz0 .

I

z

N

a

l

z

B

E

M

O

P

q1

q2

z

dz

z

I NIs=

l

() ()

XHMA 7.19: ) N . ) -.

532 7

, dz0 - a

Isdz0 =

NI

`dz0: (7.186)

-B P , (7.183),

dB =NIdz0a2

2` [a2 + (z z0)2]3=2z^: (7.187)

(7.187) -

B =NIa2

2`

`0

dz0

[a2 + (z z0)2]3=2z^

B =NI

2`

"zp

z2 + a2+

` zp(` z)2 + a2

#z^ (7.188)

, ,

B =NI

2`(cos 1 + cos 2) z^ (7.189)

(7.188) z = `=2,

BM =NI

(4a2 + `2)1=2z^: (7.190)

, z = 0 z = `, , (7.188),

BE =NI

2(a2 + `2)1=2z^: (7.191)

(` a), (7.190) (7.191)

BM ' NI`z^ = Isz^; (7.192)

7.6 533

BE ' NI2` z^ =12Isz^: (7.193)

, - - . , , (` a), (7.190), (7.191) (7.184) ( N , I - NI).

7.6

, , , , .

B

I

S c

XHMA 7.20: A.

, , 7.20, - I .

S c , (7.89) (7.90),

=S

B dS =cA d`: (7.194)

- c I , I (flux linkage), -

= =S

B dS: (7.195)

534 7

I

B

I

XHMA 7.21: B.

c - I , , ,

= 2: (7.196), N ,

,

= N (7.197)

, 7.21, -, 1;2; : : : ;m N1; N2; : : : ; Nm , ,

=mXi=1

Nii =cA d` (7.198)

, , .

MKSA Weber(1Wb = 1Vs) Weber- (Wb-).

7.7 535

7.7

7.21 I . I - ,

L =I

(7.199)

I , (self-inductance) . , - , - .

MKSA Henry (H), ,, mH H.

L C . - , - L I . : Q, - U , I .

, .

7.7.1

() - 7.22. , .

, - , , H . , N , Ampre,

cH d` = NI; (7.200)

536 7

r

dr

O

A

A

c

r

dr

ab

d

A-A

( ) ( )

XHMA 7.22: .

H2 = NI; (7.201)

H =NI

2(7.202)

B =

NI

2: (7.203)

, (7.203) 7.22(),

=S

B dS = ba

NId

2d =

NId

2

ba

d

=NId

2lnb

a

: (7.204)

(7.204), , , ,

= N =N2Id

2lnb

a

: (7.205)

7.7 537

, , , (7.205) (7.199),

L =N2d

2lnb

a

(7.206)

, - S, -

L =N2S

2m(7.207)

m O -.

7.7.2

7.5.4, B ` a

B =NI

`(7.208)

., ,

(, ) ,

= N = NBS = NBa2

, (7.208),

=N2Ia2

`: (7.209)

(7.209) (7.199)

L =N2a2

`(7.210)

538 7

I-I

O r

dr

r

dr

B B

cb

a

XHMA 7.23: .

7.7.3

7.23. - () a () b. - ( = 0). , , , :

) B - .

B (a 6 6b), (7.20),

Be() = B'() =0I

2; (7.211)

e . , , , -

, (a 6 6 b) d,

de = de = Be()d =0I

2d

(a 6 6 b): (7.212)

(7.212)

e = ba

0I

2d

=

0I

2lnb

a

; (7.213)

7.7 539

Le =eI

=02

lnb

a

(7.214)

) ,, B .

di(), , +d, ( 6 a), , (7.19),

Bi() = B'() =0I

2a2 ( 6 a);

di() = Bi()d =

0I

2a2d ( 6 a); (7.215)

i . I , J -

J =I

a2

Ii(), ,

Ii() = 0

20

J d d' = J2 = I2

a2: (7.216)

(7.215) , Ii()=I ,

di() = N()di() =Ii()I

0I

2a2d

, (7.216),di() =

0I

2a43 d: (7.217)

(7.217)

i =0I

2a4

a03d =

0I

8; (7.218)

540 7

Li =iI

=08

(7.219)

= b0d =

a0di +

bade = i +e

, (7.213) (7.218),

=0I

8+0I

2lnb

a

: (7.220)

(7.220) , ,

L =I= Li + Le =

08

+02

lnb

a

(7.221)

, (7.221) . - , , ., d0i d 0i , -

, (b 6 6 c) d, (7.22) (7.24),

d0i = B()d = 0H()d =0I

2c2 2c2 b2 d (b 6 6 c);

d 0i =I()I

d0i =0I

2(c2 b2)2(c2 2)2

d (b 6 6 c);

0i = cb

d0i =0I

2

c4

(c2 b2)2 lncb

+

b2 3c24(c2 b2)

: (7.222)

t, , 0i (7.221). ,

7.7 541

t = + 0i

=0I

2

14+ ln

b

a

+

c4

(c2 b2)2 lncb

+

b2 3c24(c2 b2)

: (7.223)

Lt

Lt =tI

=02

14+ ln

b

a

+

c4

(c2 b2)2 lncb

+

b2 3c24(c2 b2)

: (7.224)

(7.224) , , - (b=c! 1) (7.221).

7.7.4

7.24, - . a d . I , , .

I

I

x

y

a

(1) (2)

B1

B2

xdx

d

a

1m

XHMA 7.24: .

, , - dx 7.24, B , B1 B2 ,

B = B1 +B2 =0I

2

1x+

1d x

y^; (7.225)

542 7

= = x=dax=a

d =0I

2

daa

1x+

1d x

dx;

=0I

lnd aa

: (7.226)

(7.226)

L =I=

0lnd aa

(7.227)

(7.227), d a (d a),

L ' 0lnd

a

(7.228)

, - .

, - 2

L =04

+0lnd aa

(7.229)

L ' 04

+0lnd

a

: (7.230)

2 8.11 - .

7.8 543

7.8

(6.90) Lorentz, - F q v - (E;B),

F = q(E + v B) (7.231)

, ,

E0 =F

q= E +Em = E + v B (7.232)

(7.232) E Em = v B, - .

, , v B.

, (6.90), - f v B, J = v,

f = (v B) = J B: (7.233), F V

F =V

(J B)dV (7.234)

(7.234), JdV = Id`,

F = Ic(d`B) (7.235)

F - ( ) I (7.25). (7.235)

dF = I(d`B) (7.236)

544 7

dl

I

c

B

d I dF B= ( )l

XHMA 7.25: .

Laplace, - dF d` , I , B.

dF (7.236) .

7.1

I1. 2a;

p2a;

p2a I2

, 2a c , 7.26. .

I1

I2

y

z

x

A

B

G

a

a

c a

x

dx

dlAG

dlBG

(1)

(2)

XHMA 7.26: .

7.8 545

1 2, - B1 x, , (7.13), = x '^ = y^,

B1 =0I12x

y^: (i)

F12, , FAB;FB ;FA -, ,

F12 = FAB + FB + FA: (ii)

Laplace ((7.235))

FAB = I2A`B

d A`B B1 = I2 z=az=a

(dzz^)0I12c

y^

= 0I1I22c

aa

dzx^ = 0I1I2ac

x^;

FB = I2B`

d B` B1 = I2 c+ax=c

(dxx^+ dzz^)0I12x

y^

, dz = dx,

FB =0I1I22

c+ac

dx

xz^ +

c+ac

dx

xx^

=0I1I22

lnc+ ac

x^+

0I1I22

lnc+ ac

z^;

FA = I2`A

d `A B1 = I2 c+ac

(dxx^ dxz^)0I12x

y^

=0I1I22

c+ac

dx

xz^ +

c+ac

dx

xx^

=0I1I22

lnc+ ac

x^ 0I1I2

2lnc+ ac

z^:

FAB;FB ;FA (ii) -

F12 = 0I1I2

a

c ln

a+ cc

x^: (iii)

, , F12 -

546 7

. , , , F12:

F21 = F12 = 0I1I2

a

c ln

a+ cc

x^: (iv)

, , FB FA , , , z , x (FB+FA =2(FB)xx^).

7.8.1

, I , B = B0 ( 7.27), F , (7.235),

F = Icd`B

, ,

F = IB0 cd` = 0; (7.237)

, , cd` = 0: (7.238)

(7.237) F , I , B, .

c

I

B0

F=0

T0

XHMA 7.27: c B0.

7.8 547

a/2

a/2

xy

z

b/2 b/2

a a

b

b

u1

u2

u3

u4

T13

T24(1) (2)

(3)

(4)

B x y z= + +B B Bx y z$ $ $

( )

y

zB

F1

F3

aa

b/2 b/2

bb

( )

XHMA 7.28: ) xy. ) .

, , F , .

, , c 7.28, z x y. - I , B Bx; By; Bz x; y z, .

F1 (1) , (7.235),

F1 = Ic1

d`B = IB c1

d` = I(Bxx^+Byy^ +Bzz^) (ax^)

F1 = Ia(Byz^ Bzy^): (7.239)

548 7

F3, (3) , F1,

F3 = F1 = Ia(Bzy^ Byz^): (7.240) T13 F1 F3, (7.239)

(7.240),

T13 = F1 u1 + F3 u3 = 2F1 u1

= 2Ia(Byz^ Bzy^)b

2y^

= IabByx^: (7.241)

, F2 F4, (2) (4) , ,

F2 = IB c2

d` = I(Bxx^+Byy^ +Bzz^) (by^)

F2 = Ib(Bzx^Bxz^) (7.242)

F4 = F2 = Ib(Bxz^ Bzx^): (7.243)

, T24 F2 F4

T24 = 2F2 u2 = 2Ib(Bzx^Bxz^)a2x^

T24 = IabBxy^: (7.244)

(7.241) (7.244)

T = T13 + T24 = Iab(Byx^+Bxy^): (7.245)

S = abz^ (7.246)

, - (7.245)

T = IS(z^ B) = IS B (7.247)

T =M B (7.248)

7.8 549

M . (7.248) - T M B.

, , (7.248) ( ) M , B.

7.8.2

, , (), , , - (1) (2), I1 I2, , a. - z x = 0, 7.29.

x

y

z

O

aI

1I

2

B2

dF21

d dzl $= z

(1) (2)

XHMA 7.29: .

B2 (2), (1), (7.13),

B2 =0I22a

x^: (7.249)

dF21 d` -, , (7.236),

dF21 = I1(d`B2) = I1dz z^ 0I2

2ax^

=

0I1I2dz

2ay^ (7.250)

550 7

(7.250) ` (1)

F21 =0I1I2y^

2a

z+`z

dz:

F21 =0I1I2`

2ay^ (7.251)

(7.251) ` -. , -, F12 ` - F21. , , F12 ( F21) - 7.29, I1 I2 .

, , , - .

I1 = I2 =I , , , - (7.251),

F =0I

2

2a: (7.252)

(7.252) .

, , , - , - .

7.8.3 Hall

, , , d, I ( 7.30). B - .

( ) - I , Lorentz,

Fm = ev B; (7.253)

7.8 551

d

l

I

Fm

v

B

I

XHMA 7.30: Hall.

e () v . - , - () ( ). - , , - E Fm. , , , Fe Fm ( ). Fe -

eE = ev B: (7.254)

Hall. VH ( Hall) -

,

VH = E` = vB`; (7.255)

v B.

(7.255), N S ,

VH =IB`

NeS=

IB`

Ne`d=

IB

Ned: (7.256)

552 7

, VH , , - B (7.256).

7.1 H = H0z^ (H0 )

, R r. , - , ( )

m =

8>>>:H0r cos r 6 R"

1 + R

r

3#H0r cos r > R;

; , :) H

B .)

.

7.2 -

A(x; y) = I(x2 + y2 + 2xy)z^; I .) H(x; y) -

(x; y) .) B .) ,

B A = (x; y)z^, (x; y) .

7.3 r = a - (r 6 a) (r > a) :

A =

8>:03K0r sin '^ r 6 a

03K0

a3

r2sin '^ r > a:

553

- ( ).

z

a

K

m0

Om0

(1)

(2)

XHMA 7.31: .(. 7.3)

z

r1m

Om1

(1)(2)

m2

(3)

r2

q

P r( , , )q j

H0

XHMA 7.32: . (. 7.4)

7.4 , H0 = H0z^.) , -

r1 1, , r2 2.

) 1 2, , .

) , - H0, 1, .

7.5 1 2, . I z = 0

554 7

h. - . ( .)

m1

y x

z

PI

h

m2

(1)

(2)

XHMA 7.33: . (. 7.5)

7.6 , , , , , a b = 2a, I J , ,

J = J0a

2z^;

J0 z . :

) H B .

) .

7.7 , , a b, 0. b a, c. I , - , -

B =0cI

2(a2 b2) . -, A A0 . ( , - .)

555

O O

cx

y

z

a

b

AA

XHMA 7.34: , , - . (. 7.7)

x

y

z

I

A(0,0,0) B a(2 ,0,0)

C a a( , ,0)

K

XHMA 7.35: . (. 7.8)

7.8 ABC 7.35 I . A(0; 0; 0), B(2a; 0; 0) C(a; a; 0), K ABC P (0; 0; 5a) z.

7.9 a t, I J ( ). O z . J

J = J0

a'^;

J0 '^ . :) B P

z .) B0 O ;

BK

556 7

K a I .

) ; '; z (

p2 + z2 a) ,

.

P z(0,0, )

z

O

r

a

tJ

XHMA 7.36: .(. 7.9)

x

y

z

I

M

A

B

I

q

l/2

l/2

d

x

y

z

I

M

d

I

() ()

XHMA 7.37: ` ) z = 0 ) y = 0. (.7.10)

7.10 I y 7.37. I 0 AB - `, M (d; 0; 0) x. F AB , :) AB z = 0 ( 7.37()) -

x ( : 0; =3 =2)

557

) AB z ( 7.37()).

7.11 , z - Oxyz, I , - , . x = 0 w y = 0. F ( - ).

x

y

z

I

d

w/2

w/2

XHMA 7.38: . (. 7.11)

x

y

I

a b b

cI

(1)

(2)

(3)

XHMA 7.39: T . (. 7.12)

7.12 I 0 I , 7.39. F1;F2;F3 , .

7.13 1 M1 = 5x^Am2 x = 0 O - Oxyz. , , - 2 M2 = 3y^Am2 y = 3 P P (4;3; 10). T2 -. ( .)

558 7

Oy

x

z

M1

M2

(1)

(2)

XHMA 7.40: . (. 7.13)

7.14 a . H

H() = H0

12

a 13

a

2'^;

H0 , '^ . , :) J I .) H ( > a).) .)

.) .

z

ab

z

l

XHMA 7.41: .(. 7.15)

559

7.15 ` a N b. I , (z =`=2).

7.16 , , I . c d 7.42. :) -

.) -

.) ` a; b; c

.

x

y

z

x

z

I

a

I

I

l c

I

cb

d

XHMA 7.42: , . (. 7.16)

7.17 7.43 BKA a AB I . K, I 0 - O. K

TK =0aII 0(sin cos ):

, .

560 7

x

y

a

C

K

q

I

I

O

q

B

A

XHMA 7.43: . (. 7.17)

7.18 , , , , a, I z = 0 z = a z. N ,:) B dB=dz,

d2B=dz2 z, 0 6 z 6 a.) H z0

Bmax.) BmaxB(z)Bmax 100%

0 6 z 6 a.( Helmholtz)

7.19 ( 0) (x=a)2 + (y=b)2 = 1. , z Oxyz, J = J z^. :)

A(x; y) = Ja2b2

2(a2 + b2)

x2

a2+y2

b2 1z^:

) ( ).

) `

Li =`

4ab

a2 + b2:

8

8.1 Faraday

, 1831 Faraday, .

- E c , () . , , , - .

, , E, Maxwell

rE = @B@t

; (8.1)

, , - B.

(8.1), Maxwell-Faraday, - (electromagnetic induction) Faraday.

, ( ) c, - ( 8.1). (8.1) S - c,

S

rE dS = S

@B

@t dS = @

@t

S

B dS: (8.2)

562 8

S

dS

cdl

XHMA 8.1: H S.

(8.2), Stokes S,

cE d` = @

@t: (8.3)

(8.3) - E ( ) c. c - N (.. N ), (8.3)

cE d` = N d

dt(8.4)

, ,

cE d` = d

dt(8.5)

(8.5) - Faraday , () E - . c , .

Faraday :

) (8.5), ,

8.2 563

. , - , () , , , , .

) c , () , .

) (8.5) Lentz -, , , , - . , Lenz -, ,, , - .

, , - . , ., , , - (.. , -, -, ...). , , -

f , , = c=f(c ) , (quasi-static) . , .., f < 105Hz > 3 Km, - .

8.2

`, ( 8.2), v

564 8

B. Em - , (7.232),

Em = v B: (8.6)

-e

Fm

Em

v

B

N

M

l

XHMA 8.2: .

,

Fm = ev B (8.7) . () . , , E Em,

E = Em = B v: (8.8) E MN , ,

E =

N

M

(v B) d` = vB `0d`;

E = vB` (8.9)

(8.9) - .

8.3 565

8.3

8.1 Faraday(8.5), .

O

r

S S

B

c t( ) c t dt( + )

dSdS

dSp

v

dr

dl

XHMA 8.3: .

, (, , rE = @B=@t = 0), 8.3. - , (, ).

, v d`

dE = E0 d` = (E +Em) d` = [E + (v B)] d`: (8.10) ,

Stokes,

E =cE0 d` =

cE d`+

c(v B) d`

=S

rE dS +c(v B) d`;

566 8

E =

c(v B) d`: (8.11)

, , c(t) c0(t + dt) t t+dt, -. , , S S0 c c0, . r ( ) d`, dt - v,

dr = vdt: (8.12)

(8.11), (8.12),

E =c

dr

dtB

d` = 1

dt

c(dr B) d`; (8.13)

,

(AB) C = (C A) B; (8.14)

E = 1

dt

c(d` dr) B; (8.15)

dt dr t r t! 0.

(8.15), d`dr dS S d` dt,

E = 1dt

S

B dS: (8.16)

, , (6.46) St S; S0 S,

St

B dS = 0 (8.17)

S

B dS S

B dS +S0B dS0 = 0; (8.18)

8.4 567

S S0 , , - .

(8.18), - (t) (t+ dt) t t+ dt, ,

S

B dS = (t) (t+ dt) = d: (8.19)

(8.19) (8.16)

E = ddt

; (8.20)

(8.5). , , (8.5) (8.20) , - , .

8.4

8.3 - , , , .

(7.232), c

E =cE0 d` =

c(E + v B) d` (8.21)

, Stokes,

E =S

rE dS +c(v B) d`: (8.22)

(8.22), (8.1) , ,

E = S

@B

@t dS +

c(v B) d`: (8.23)

, , (8.23), ( (8.16) (8.18))

568 8

t + dt -

c(v B) d` = 1

dt

S

B(t+ dt) dS (8.24)

c(v B) d` = 1

dt

S

B(t+ dt) dS S0B(t+ dt) dS0

: (8.25)

B S t+ dt Taylor

B(t+ dt) = B(t) +@B

@tdt+ h(dt2); (8.26)

h(dt2) (8.25)

c(v B) d` = 1

dt

S

B(t) dS S0B(t+ dt) dS0

+S

@B

@t dS: (8.27)

(8.27) (8.23)

E = 1dt

S

B(t) dS S0B(t+ dt) dS0

= d

dt

S

B dS = ddt

; (8.28)

(8.4)., , (8.23) (8.28),

-

E = ddt

= ddt

S

B dS = S

@B

@t dS +

c(v B) d` (8.29)

Faraday.

8.5 E A 569

(8.29) - , . , - ( ) ( ).

:) (8.29)

rE = @B@t

(8.30)

, , , .

) E,

E = E0 v B (8.31) E0 Em =v B, .

, (8.30) (8.31) - .

8.5 E A

A.

(8.1) Faraday, - (7.68),

rE = @@trA (8.32)

, -,

rE +

@A

@t

= 0: (8.33)

(8.33) E + @A=@t, - , .

570 8

, r E = 0, E (E = r) . E + @A=@t, ,

E +@A

@t= r: (8.34)

(8.34) - , , @A=@t , , E .

E , (8.34), A

E = r @A@t

(8.35)

, , (8.35) - , , .

8.6

, , S, - ! x - B = B0eatz^ ( 8.4).

- - .

, , (8.29),

E = S

@B

@t dS +

c(v B) d`: (8.36)

(8.36)

8.6 571

q w= t

I

y

z

x

q w= t

v1

v3

d dSS n= $

B z=B e0

-at$

w/2

w/2

l/2

l/2

y

x

z

B

c1

c2

c3

c4

() ()

XHMA 8.4: ) . ) t.

S

@B

@t dS =

S

aB0eatz^ (dSn^)= aB0eat

S

(z^ n^)dS

= aB0eat cos!tS

dS

= aB0Seat cos!t: (8.37)

O , c1; c2; c3; c4 , -

c(v B) d` =

c1

(v B) d`+c2

(v B) d`

+c3

(v B) d`+c4

(v B) d`

=c1

(v B) d`+c3

(v B) d`; (8.38)

v B d` c2 c4. v = !w=2 v1 v3 c1 c3,

v1 = v3 = v(sin!t y^ + cos!t z^): (8.39)

572 8

(8.39) (8.38) c(v B) d` =

`0

v(sin!t y^ + cos!t z^) B0eatz^ (dx x^)

+ `0

v(sin!t y^ + cos!t z^) B0eatz^ (dx x^)= 2

`0vB0e

at sin!t [(y^ z^) x^] dx

= 2B0v`eat sin!t = B0`w!eat sin!t

= B0S!eat sin!t: (8.40)

, , (8.36), (8.37) (8.40),

E = aB0Seat cos!t+B0S!eat sin!t: (8.41) N , , ,

E = NB0Seat(a cos!t+ ! sin!t): (8.42) (8.42),

(a = 0)

E = NB0S! sin!t; (8.43), .

(8.42)

E = ddt

= N ddt

= N ddt

S

B dS: (8.44)

, (8.44)

E = N ddt

S

B0e

atz^ (dSn^) = N d

dt

B0e

at cos!tS

dS

= N d

dt

B0Se

at cos!t= NB0Seat(a cos!t+ ! sin!t):

8.7

, 7.7 - .

8.7 573

Y11

Y12

I1

I2

R12

dl1

dl2

c1

c2

(I) (II)

XHMA 8.5: I1.

, , (I) (II) 8.5, I1 I2, .

12 - I1 (I) (II), (7.198),

12 =c2

A1 d`2 (8.45)

A1 I1 d`2 c2 (II).

(mutual inductance) M12 ( , , , , -) (I) (II)

M12 = 12=I1 (8.46)

, 21 (I) I2 (II),

M21 = 21=I2: (8.47)

, M12 M21 L12 L21, .

574 8

, , M12 M21 , M =M12 =M21 (I) (II).

, (8.45) -A1 (7.88)

12 =c2

I14

c1

d`1R12

d`2 = I14

c1

c2

d`1 d`2R12

; (8.48)

R12 d`1 d`2 c1 c2, .

, (8.48), , (8.46),

M12 =12I1

=

4

c1

c2

d`1 d`2R12

: (8.49)

, I2 , (8.49)

M21 =21I2

=

4

c2

c1

d`2 d`1R21

: (8.50)

(8.49) (8.50), d`1 d`2 = d`2 d`1 R12 = R21,

M =M12 =M21 =

4

c1

c2

d`1 d`2R12

(8.51)

Neumann. (8.51),

. MKSA Henry (H).

I1 I2, L1 (L11) L2 (L22) , 1 2 , ,

1 = 11 +21 = L1I1 +MI2; (8.52)

2 = 21 +22 = L2I2 +MI1; (8.53)

8.7 575

11 (I) I1, I2 = 0, 22 (II) I2, I1 = 0.

, , , c1 c2, , .

(8.52) (8.53) (n) Lij (i; j = 1; 2; : : : ; n).

8.1

- ( ) - ( ) 8.6. , z, .

dr

B=Bjj$r

a b

I -I

I

-I

g

d

rag

rad

rbg

rbd

(1)

(2)

xz

y

XHMA 8.6: .

, , - 12, (2), - (1). ., ,

, , , , (2),

576 8

. d

d , B ( < < )

B = B''^ =0I

2'^; (i)

d = d = B''^ d'^ = 0I2

d

:

=

0I

2d

=

0I

2ln

: (ii)

(2)

=

0I

2d

=

0I

2ln

: (iii)

, 12 (2) (1)

12 = = 0I2 ln

: (iv)

, (8.46),

M =M12 =M21 =02

ln

: (v)

, , - , , = ; = , = ; =, .

8.8

(I) (II) 8.7, N1 N2 , . - (I) (II) , ,M , L1 L2.

, (I) I1 (I2 =

8.8 577

0), . , 11 , 12, I1 , 11 .

, , 11 , 12 - 11. 11 12 .

I1

(I)

I2

(II)

N1

N

2

XHMA 8.7: .

, ,

12 = k1211 (jk12j 6 1) ; (8.54) k12 , (8.54) (7.197) , ,

k12 =12=N211=N1

: (8.55)

(8.55), (7.199), (8.46), (8.51),

k12 =N1N2

M

L1: (8.56)

, k12, , M .

, I2 ( I1 = 0),

21 = k2122 (jk21j 6 1) ; (8.57)

578 8

k21 =N2N1

M

L2: (8.58)

(8.56) (8.58)

M = kpL1L2 (8.59)

k

k =pk21k12 (8.60)

(coefficient of coupling) -.

, k12; k21 k (8.59)

M =pL1L2: (8.61)

(8.61) ., k

, k .

, s

s = 1 k = 1 MpL1L2

(8.62)

(leakage).

8.9

, , - 8.8. cj j- Sj ( ), dVj - d`j , ,

dVj = Sj d`j ; (8.63) Sj d`j .

8.9 579

H B,

I t( )

cj

Sj

dVjdlj

E

R

XHMA 8.8: .

, , . (6.57), dwm

dwm =H dB: (8.64)

d2Wm;j dVj , ,

d2Wm;j = dwmdVj =H dBdVj = (H d`j) (Sj dB): (8.65)

, j- ,

dWm;j =cj

(Sj dB) (H d`j) (8.66)

, Sj dB dj - j- , , ,

dWm;j = djcj

H d`j : (8.67)

(8.67), Nj cj , Ampre,

dWm;j = djNjI = I(Njdj) = Idj ; (8.68)

580 8

j = Njj Nj , I .

, , dWm ,

dWm =nX

j=1

Idj = InX

j=1

dj ;

dWm = Id ; (8.69)

d - , n .

I = = 0, Wm , ( , , I(t)),

Wm = 0

Id (8.70)

(8.70), (.. ), = (I).

, -, (7.199),

= LI; (8.71)

L . , , (8.70), (8.71),

Wm = 0

Id = I0ILdI = L

I0IdI;

Wm =12LI2 =

12I (8.72)

I .

8.9 581

, , (8.72)

We =12CU2 =

12QU (8.73)

. 7.7, L; I; - C;U;Q .

(8.72)

L =2WmI2

(8.74)

- L .

, , (8.69) , , ., E R , , ,

E ddt

= IR; (8.75)

d=dt , Faraday, .

(8.75) Idt

EIdt = I2Rdt+ Id : (8.76)

(8.76) dt, ( Joule) R , (8.69) - .

, - A.

, d , - (7.198),

d = d

cA d`

=

cdA d`; (8.77)

582 8

c - . , (8.77) (8.69) -

dWm =cIdA d`: (8.78)

(8.78), , Id` JdV ,

dWm =V

J dA dV ; (8.79)

V . , J = 0.

, , Wm , A,

Wm =V

A0J dA

dV (8.80)

, (7.198) (8.72),

Wm =I

2

cA d` = 1

2

cA Id` (8.81)

Wm =12

V

J A dV (8.82)

, (8.82)

We =12

V

dV (8.83)

.

(8.82), , (6.4). , Wm

(6.4) (7.68)

8.9 583

Wm =12

V

H2dV =12

V

H B dV = 12

V

H r A dV : (8.84)

, , Maxwell (6.35) @D=@t

rH = J : (8.85) (8.84),

r (AH) = (rA) H (rH) A; (8.86) (8.85) Gauss,

Wm =12

V

J A dV + 12

V

r (AH) dV

=12

V

J A dV + 12

S

(AH) dS: (8.87)

, , S V - - r (r !1), (7.134) (7.148), A B ( H) r2 r3, . , , S r2, (8.87) r3, ., , r ! 1, (8.87) -

Wm =

12

V

J A dV ; (8.88)

., (8.88) (7.86)

Wm =

8

V

V

J J 0R

dV dV 0; (8.89)

R dV dV 0.

8.2

a I , - J = J z^, z^ z . , :

584 8

) H B .) i.) A = J4 (a

2 2) z^, , .) , ` ,

.

z

J z=J$

a

r

l

ar

dr

H=Hjj$

() ()

XHMA 8.9: ) . ) - .

) H -B Ampre 8.9., ,

I() = 0

20

Jdd' = 2J 0

d = J2; (i)

Ampre , - ,

I() =c()

H() d` = 20

H'()d' = 2H';

H() = H'()'^ =I()2

'^ =J

2'^ =

I

2a2'^: (ii)

8.9 585

, , (6.33) ,

B() = B'()'^ = H() =J

2'^ =

I

2a2'^: (iii)

) - , d, ` ( 8.9()), -, ,

d() = B() dS = B'()'^ `d'^ = B'()`d = J`2 d:

d(), d() - I()=I , , ,

d() =I()I

d() =2J

a2J

J`

2d =

J`

2a23d:

, , -

i = a0

d() =J`

2a2

a0

3d =`Ja2

8=

`I

8: (iv)

) - A B = r A, ,

rA =1

@Az@'

@A'@z

^+

@A@z

@Az@

'^+

1

@(A')@

@A@'

z^

rA = @Az@

'^ = @@

J

4(a2 2)

'^ =

J

2'^:

, , (iii) - , , , B = rA. A

Poisson (7.77). -, , ,

r2A =r2A 2

2@A'@'

A2

^+

r2A' + 2

2@A@'

A'2

'^

+ (r2Az)z^ = r2Azz^ =1

@

@

@Az@

+

12

@2Az@'2

+@2Az@z2

z^

586 8

=1

@

@

@Az@

z^ =

1

@

@

@

@

J

4(a2 2)

z^

= J41

@

@(22)z^ = J z^ = J ;

(7.77).) , (ii),

Wm =

wmdV =

12H2dV =

`0

a0

20

12H2() d d' dz

=

2

`0

a0

20

I

2a2

2 d d' dz =

2

I

2a2

22`

a0

3d

=`

16I2: (v)

, , (8.72) (8.82). , i (iv) (8.72)

Wm =12Ii =

`

16I2;

A (8.82) , ,

Wm =12

J AdV = 1

2

JAdV =

J

2

AdV

=J

2

`0

a0

20

J

4(a2 2) d d' dz

=2J2`

8

a0

(a2 2) d = J2`a4

16=

`

16I2;

.,

(7.199) (8.74) (iv), (v)

Li =iI

=2WmI2

=`

8: (vi)

8.10

(1) (2), , 8.10, I1 I2

8.10 587

( ) J1(r1) J2(r2).

I1

I2

dV1

dV2

r1

r2

R r r=2 1-

J r1 1( )

J r2 2( )

(1) (2)

V1 V2

XHMA 8.10: .

A1;B1;H1 , , , A2;B2;H2 , , (8.82),

Wm =12

V

J A dV = 12

V

J (A1 +A2) dV

=12

V

J1(r1) [A1(r1) +A2(r1)] dV1

+12

V

J2(r2) [A1(r2) +A2(r2)] dV2

=12

V

J1(r1) A1(r1) dV1 + 12V

J2(r2) A2(r2) dV2

+12

V

J1(r1) A2(r1) dV1 + 12V

J2(r2) A1(r2) dV2; (8.90)

J1(r1) J2(r2) V1 V2, .

To (8.90) - W12

588 8

W12 =V

J1(r1) A2(r1) dV1 =V

J2(r2) A1(r2) dV2

=

4

V

V

J1(r1) J2(r2)R

dV1dV2

(8.91)

(8.91), (7.66), (7.68), (8.86) Gauss

W12 =V

J1 A2 dV =V

(rH1) A2 dV

=V

H1 (rA2) dV V

r (A2 H1) dV

=V

H1 B2 dV S

(A2 H1) dS: (8.92)

, , S . , (8.92)

W12 =V

H1 B2 dV : (8.93)

(8.91), -

W12 =V

H1 B2 dV =V

H2 B1 dV (8.94)

8.11

, , n ( ) I1; I2; : : : ; In, . .

Ii; Vi ci , i- , , , (8.88) JdV ! Id`,

Wm =12

V

J A dV = 12

nXi=1

Vi

J A dV

8.11 589

=12

nXi=1

ci

A Iid` = 12nXi=1

Ii

ci

A d`: (8.95)

(8.95), (7.198),

Wm =12

nXi=1

Iii (8.96)

i i- . , (n = 1), (8.96)

(8.72). ji i-

Ij j- , Lij = Lji , , ,

ji = LjiIj ; (8.97)

i =nX

j=1

ji =nX

j=1

LjiIj ; (8.98)

Lii Li i- -. (8.96) i (8.98)

Wm =12

nXi=1

nXj=1

LijIiIj (8.99)

(n = 2), L11; L22 L12 L1; L2 M , , (8.99) Wm -

Wm =12L1I

21 +

12L2I

22 +MI1I2 (8.100)

(8.100) 1 ( I2 = 0) 2 ( I1 = 0), . (8.100) W12 -

W12 =MI1I2 = I121 = I212 (8.101)

590 8

(8.101) M (8.51) Neumann

W12 =I1I24

c1

c2

d`1 d`2R12

(8.102)

, (7.88),

W12 = I1c1

A2 d`1 = I2c1

A1 d`2 (8.103)

A1 A2 I1 I2, . (8.103) (8.91) .

B1 B2 , Wm B = B1+B2, - (8.84)

Wm =12

V

H B dV =V

B B dV2

=V

(B1 +B2) (B1 +B2) dV2

Wm =V

B212

dV +V

B222

dV +V

B1 B2

dV (8.104)

(8.104)

W12 =V

B1 B2

dV (8.105)

(8.101), (8.102), (8.103) (8.105) - M

M =12I1

=21I2

=W12I1I2

=

4

c1

c2

d`1 d`2R12

=1I2

c1

A2 d`1 = 1I1

c2

A1 d`2 = 1I1I2

V

B1 B2

dV

(8.106)

8.11 591

L, (8.71),(8.72), (8.81), (8.82) (8.84)

L =I=

2WmI2

=

V

H B dVI2

=

cA d`I

=

V

J A dVI2

(8.107)

8.3

, 7.6.

Wm1;Wm2;Wm3 Wm4 I; II; III IV, , (8.84), (7.19), (7.20), (7.24) (7.25),

Wm1 =02

V1

H21dV1 =02

a0

I

2a2

22d =

0I2

16; (i)

Wm2 =02

V2

H22dV2 =02

ba

I

2

22d =

0I2

4lnb

a

; (ii)

Wm3 =02

V2

H23dV3 =02

cb

I

2c2 2c2 b2

22d

=0I

2

4

c4

(c2 b2)2 lncb

+

b2 3c24(c2 b2)

; (iii)

Wm4 = 0: (iv)

, , Wm

Wm =Wm1 +Wm2 +Wm3 +Wm4

=0I

2

4

14+ ln

b

a

+

c4

(c2 b2)2 lncb

+

b2 3c24(c2 b2)

: (v)

(v) (8.74) - L

L =2WmI2

=02

14+ ln

b

a

+

c4

(c2 b2)2 lncb

+

b2 3c24(c2 b2)

: (vi)

(vi) , , (7.224).

592 8

8.12

n . I - , R E - , Kirchhoff Faraday,

E = IR + ddt

; (8.108)

- .

(8.108), Idt,

EIdt = I2Rdt+ Id: (8.109)

, , ,

nX=1

EIdt =nX

=1

I2Rdt+nX

=1

Id: (8.110)

, (8.110) - dt ( Joule (8.110)).

, , , dt, dx -F , F Fxdx, Fx F x^ . dt , , dWm -.

, , , Joule, dWm . , ,

nX=1

EIdt =nX

=1

I2Rdt+ Fxdx+ dWm: (8.111)

8.12 593

(8.110) (8.111)

nX=1

Id = Fxdx+ dWm (8.112)

M , , - , , .

8.12.1

, - (I(t) = I).

, (8.96), - ,

dWm =12d

nX=1

I =12

nX=1

Id: (8.113)

(8.112) (8.113)

Fxdx = dWm =12

nX=1

Id (8.114)

Fx =@Wm@x

( ) (8.115)

(8.114), . , (8.115) - F , .

(8.115), Wm (8.99),

Fx =12

nXi=1

nXj=1

IiIj@Lij@x

(8.116)

(8.116) , Lii (i = 1; 2; : : : ; n)

594 8

, .

, , (8.116) - .

(8.116), F;x x F - - @Lij=@x = 0 i 6= j 6= ,

F;x = InX

j=1

Ij@Lj@x

(8.117)

, (8.98),

F;x = I@@x

(8.118)

- . , (8.116) L12 =

L21 =M , Fx = I1I2

@M

@x(8.119)

, , (8.119) - Fy Fz F ,

F = I1I2rM (8.120)

(8.120), M (8.51) Neumann

F =I1I24

c1

c2

r

1R12

(d`1 d`2) (8.121)

,

r1r

= r

r3; (8.122)

F = I1I24

c1

c2

R12

d`1 d`2R312

(8.123)

8.12 595

, (8.115) (8.119) T ', ,

T =@Wm@'

( ) (8.124)

T = I1I2@M

@'(8.125)

(8.123) Laplace (7.236) : F , (7.164) (7.235),

F = I2c2

d`2 B1 = I1I24c2

d`2 c1

d`1 R12R312

=I1I24

c1

c2

d`2 (d`1 R12)R312

: (8.126)

(8.126),

A (B C) = (A C)B (A B)C;

F =I1I24

c1

c2

(d`2 R12)R312

d`1

I1I24

c1

c2

(d`1 d`2)R312

R12: (8.127)

(8.127), (8.122) - ,

c1

c2

(d`2 R12)R312

d`1 =c1

d`1

c2

r

1R12

d`2 = 0: (8.128)

, , (8.127) (8.128) (8.123).

596 8

8.12.2

(d = 0; = 1; 2; : : : ; n), (8.112)

Fxdx+ dWm = 0; (8.129)

Fx = @Wm@x

( ) (8.130)

. , F .

, Wm , (8.96),

Wm =12I11 +

12I2;2 (8.131)

1 2

1 = L1I1 +MI2 (8.132)

2 = L2I2 +MI1: (8.133)

1;2 L1; L2 -, (8.132) (8.133)

@1@x

= L1@I1@x

+M@I2@x

+ I2@M

@x= 0 (8.134)

@2@x

= L2@I2@x

+M@I1@x

+ I1@M

@x= 0: (8.135)

@I1=@x; @I2=@x (8.134) (8.135)

@I1@x

=MI1 L2I2L1L2 M2

@M

@x; (8.136)

@I2@x

=MI2 L1I1L1L2 M2

@M

@x: (8.137)

8.12 597

(8.130), (8.131), (8.132), (8.133), (8.136) (8.137),

Fx = 121

@I1@x

+2@I2@x

= 1

2

MI1 L2I2L1L2 M2

(L1I1 +MI2)

@M

@x

+MI2 L1I1L1L2 M2

(L2I2 +MI1)

@M

@x

(8.138)

, ,Fx = I1I2

@M

@x(8.139)

, (8.119), .

, , (8.124) (8.125) - T

T = @Wm@'

( ) (8.140)

T = I1I2@M

@'(8.141)

8.4

- 7.1. , , 7.1 .

, 7.1, - 1 2 ( 8.11), 2

12 = 12 =SAB

B1 dS2

=SAB

0I12x

y^

(dxdyy^) = 0I1

2

SAB

1xdxdy:

598 8

I1

I2

y

z

x

A

B

G

a

a

c a

xdx

(1)

(2)

XHMA 8.11: - .

, B A ,, y = x+ c+ a y = x c a,

12 =0I12

c+ax=c

1x

x+c+axca

dy

dx =

0I12

c+ac

2a+ c x

xdx

=0I1

(a+ c)

c+ac

dx

x c+ac

dx

;

12 =0I1

(a+ c) ln

a+ cc

a: (i)

, , M12

M12 =12I1

=0

(a+ c) ln

a+ cc

a: (ii)

, (8.120) (ii) ( c x)

F12 = I1I2rM12 = I1I2 @M12@x

x=c

x^

=0I1I2

@

@x

(a+ x) ln

a+ xx

ax=c

x^;

8.12 599

F12 =0I1I2

lna+ cc

ac

x^ (iii)

, , 7.1.

8.5

a1 a2 d (d a1; a2), 8.12. N1; N2 I1; I2 , . ( a1 a2 d.)

O2

O1

j2

j2

j1

q

P1

P2

P

P2

R12

dl1

dl2

x

y

z

x

y

I1

I2

z d=

a1

a2

XHMA 8.12: .

z - O1xyz O2x0y0z0, O1 O2 , O1x;O2x0 O1y;O2y0. P 02 P2

, P1 , R12 , P1P2P 02

R12 =q(P2P 02)2 + (P1P

02)2

, O1P1P 02,

600 8

R12 =qd2 + a21 + a

22 2a1a2 cos('1 '2):

M12 , (8.51) Neumann d`1; d`2 ('1 '2),

M12 =04

c1

c2

d`1 d`2R12

=0N1N2

4

a1a2 cos('1'2) d'1d'2[d2 + a21 + a

22 2a1a2 cos('1'2)]1=2

=0N1N2

4

d'2

'2'2

a1a2 cos'd'(d2 + a21 + a

22 2a1a2 cos')1=2

; (i)

' = '1'2. (i), 2, - ('2), ('2) ; , , '2,

M12 =0N1N2a1a2

2

cos'd'(d2 + a21 + a

22 2a1a2 cos')1=2

:

, , d a1; a2, (a21 + a

22 2a1a2 cos')=d2

1, , ,

M12 =0N1N2a1a2

2

(d2 + a21 + a22 2a1a2 cos')1=2 cos'd'

=0N1N2a1a2

2d

1 +

a21 + a22 2a1a2 cos'

d2

1=2cos'd'

' 0N1N2a1a22d

1 1

2a21 + a

22 2a1a2 cos'

d2

cos'd'

=0N1N2a1a2

2d

1 a

21 + a

22

2d2

cos'd'+

a1a2d2

cos2 'd'

=0N1N2a

21a

22

2d3

cos2 'd' =0N1N2a

21a

22

2d3

1 + cos 2'

2

d'

M12 ' 0N1N2a21a

22

2d3: (ii)

(8.120) (ii) z. , (ii) z d,

8.12 601

1 2

F = Fzz^ = I1I2dM12dz

z=d

z^ ' I1I2 ddz

0N1N2a

21a

22

2z3

z=d

z^;

F ' 30I1I2N1N2a21a

22

2d4z^ = 30M1M2

2d4z^;

M1 = N1I1a21; M2 = N2I2a

22

1 2, . -, . -, .

- Neumann . - , , - (8.106) . , A1 -

I1 , (7.105), - P O1 R,

A1 =0N1I1a

21

4R2sin '^ =

0N1I1a21a2

4R3'^; (iii)

sin = a2=R. , ,

M12 =12I1

=1I1

c2

A1 d`2 = 1I1

20

0N1N2I1a21a2

4R3('^ a2d''^)

=0N1N2a

21a

22

4R3

20

d' =0N1N2a

21a

22

2R3=

0N1N2a21a

22

2(d2 + a22)3=2

=0N1N2a

21a

22

2d31 +

a2d

23=2 ' 0N1N2a21a222d3 (ii).

602 8

8.13

, , 8.13. () - , , - , , , . - x .

I

B

Fx

x

XHMA 8.13: .

S B x ,

Wm = wmV =B2

20Sx: (8.142)

F , ( ) , , (8.130),

F = dWmdx

x^ = B2

20Sx^: (8.143)

(8.143), F - . , ()

p =F

S=

B2

20: (8.144)

- .

8.14 603

8.14 Maxwell

3.6.3.2 - , Maxwell -, ( -), . , -, , J 0 0. -, , F V Lorentz , (6.91) (5.19),

F =V

fdV =V

(E + v B)dV =V

(E + J B)dV: (8.145)

f = E + J B (8.146) E H , , , J Maxwell (6.89) (6.31)-(6.33)

f = (r D)E +rH @D

@t

B

= 0(r E)E + 10(rB)B 0@E

@tB

= 0(r E)E + 10(rB)B 0

@

@t(E B)E @B

@t

= 0(r E)E + 1

0(rB)B 0 @

@t(E B) 0E (rE)

f = 0 [(r E)E E (rE)]+ 0 [(rH)H] 00 @

@t(E H): (8.147)

604 8

(8.147),

r F 2 = 2(F r)F + 2F (r F ); (8.148) F = E F = H r B = 0r H = 0,

f = 0 [(r E)E + (E r)E)] + 0 [(r H)H + (H r)H]

12r 0E2 + 0H2 00 @

@t(E H): (8.149)

, , (8.149), , , x :

0 [E(r E) + (E r)E] x^

= 0

@

@x

E2x+

@

@y(ExEy) +

@

@z(ExEz)

; (8.150)

0 [H(r H) + (H r)H] x^

= 0

@

@x

H2x+

@

@y(HxHy) +

@

@z(