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

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

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 .