electromagnetism i i
DESCRIPTION
shmeivseis hlektromagnhtismou eapTRANSCRIPT
-
Maxwell
, () .
dS ,dl ,
J ( /m2) : F=q( E+uB)
: E (x , y , z )= x y z x y zEx E y E z Gauss and Stokes:
d S=
dV ,
Ad l=
Ad S
1 0
-
2 :(a) E= x a x2 , (b) E=a( x y+ y x ) , (c) E=a( x y2+ y x2+ z x y) :
1) , 0
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2) V= Edl (x0,y0,z0). (x0,y0,z0). . , (0,0,0) (x0,0,0), (x0,0,0) (x0,y0,0) (x0,y0,0) (x0,y0,z0):
V ( x0, y0, z0)=0
x0
E (x ,0,0)( x dx )0
y0
E(x0, y ,0)( y dy )0
z0
E(x0, y0, z)( z dz )=
(a) =0
x0
(a x2 x)( x dx)00=a0
x0
x2dx=a3x0
3
(b) =0
x0
(a x y )( x dx)0
y0
(a y x+a x0 y )( y dy )0=a x0 y0
.
(0,0,0) (x0,y0,z0). d l d l=r0dl , r0 r0=x0 x+ y0 y+z0 z . l 0 r0 .
3 =/0: Gauss: Feynman ( 5.11 5.8)
4 R1 R2. q, Q.() . () , r, .: . .:
() , ( Gauss). -q. Q+q, Q,
qQ
R1
R2
-qQ+q
-
. . ( ) ( Faraday). . () Q+q , . r>R1 :
V (r>R1)=1
4 0Q+qr
. (1)
( ) . :
V (R2
-
:
( r )= 14 0
qr2r
6 b, R, . qa qb. () , b, R.() () () ?() , , qc, .
: : () , qa , qb . , qa+qb . :
a=qa4 a2
b=qb4 b2
R=qa+qb4 R2
.
() , qa+qb :
( r )= 14 0
qa+qbr 2
r .
() , . :
Ea( r )=1
4 0
qarra
3 ( r ra)
Eb( r )=1
4 0
qbrrb
3 ( r rb),
-
ra rb .() , .() , Faraday, () R , ().
7 R1 R2 . Q. . .: , .: :
V 1=1
4 0
Q1R1
, V 2=1
4 0
Q2R2
.
:Q1R1=Q2R2
. (1)
Q1+Q2=Q . (2)
(1) (2) :
Q1=R1
R1+R2Q , Q2=
R2R1+R2
Q .
:
1=Q1
4 R12=
Q4
1R1(R1+R2)
2=Q2
4 R22=
Q4
1R2(R1+R2)
.
, (3) .
() 8 .: Ampere.: Ampere :
Bdl=0
JdS ,
J . , 0 . .
-
2 B . :
2 =0 B=0
2 .
U=
02 all space
E2dV = 02 all space
2dV = 12 allspace
dV
9 R1, R2 (>R1) q. .: , , .: Gauss, . , Gauss . Gauss . , :
E(r>R2)=0
E(R1
-
U=12(qV (R1)q V (R2))=
q2 ( q4 0 ( 1R1 1R2 )0)=12 q
2
4 0( 1R1 1R2) .
C=Q
V
: C=0d
, A , d .
: U=12CV 2=1
2Q2
C
10 R1, R2 (>R1): . .: :
V= q4 0
( 1R1 1R2)
C= qV
=4 0R1R2R2R1
.
, :
U=12q2
CC=1
2q2
U=1
2q2
12
q2
4 0( 1R1 1R2)=4 0
R1 R2R2R1
.
11 ( ) F. q , x (
-
. . .() , . . :
dU=FdxF= q2
2 0 .
/ .. ( )
12 R R/2 . Q. .: ( Qs > Q), ( Qs-Q Q-Qs) .:
:
= Q43 R
343 (R/2)
3= Q
76 R
3.
, :
Qs=43 R3= Q
76 R
3
43 R3=8
7Q .
Qs Q/7. . , . Gauss.
R
R/2
Q
A B
-
r :
EdS= q04 r2 E=
043 r3E=
3 0r , (r
-
13 -2q d . +q 3d .() +q.() .: .: . - (). .
() , :
F= 14 0
2q2
(2d)2+ 1
4 02q2
(4d)2+ 1
4 0q2
(6d)2=29
721
4 0q2
d2.
() . (+q,-q) :
=2 14 0
q(3d )2+r2
3d((3d )2+r 2)1/2
= 64 0
qd((3d )2+r 2)3 /2
.
:
=2 14 0
qd2+r2
d(d2+r2)1 /2
= 44 0
qd(d2+r2)3 /2
.
(3) :
=0 total=dq2 ( 2(d2+r2)3 /2 3((3d)2+r2)3 /2 ) .
-2q
2q
r
-q
V=0d
3d
0
+q
-
14 q d . . : .:
() - . . , :
F=x 14 0
q2
(2d )2 12( x+ y ) 1
4 0q2
8d2 y 1
4 0q2
(2d )2.
LaplaceIn Cartesian coordinates
In cylindrical coordinates,
In spherical coordinates,
f2 f
q-q
-qq
d
d
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F()=F (x+iy)=U (x , y)+iV (x , y ) U V Laplace . .
15 F()=ln .: F()=F (x+iy)=U (x , y)+iV (x , y ) . Feynman U V Laplace .: =x+iy=r ei (http://en.wikipedia.org/wiki/Complex_logarithm ):
ln()= ln (r )+ i=ln(x2+ y2)+i atan2( y , x ) .:U (x , y)=ln ( x2+ y2)=ln(r )V (x , y )=atan2( y , x)=
.
Gauss ( 5.2 Feynman). . U(x,y) Laplace .