tsimpoukis vasiki thewria
DESCRIPTION
electromagneticTRANSCRIPT
-
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(