2.5 (Gauss) 2.5.1 (flux) v(r)( ) n v n v n (1) v n = v n = v, n. n n v v I(2012), ec. 2. 5 p. 1/30
i (2) lim v(r i ) i = v(r) d. i 0 i (flux) I(2012), ec. 2. 5 p. 2/30
2.5.2 ( ) ( ) q 1 r 2 E 2 q r 1 E 1 1 2 r 1 r 2 ( ) (3) = 1 + 2 + (4) E(r) d = E(r) d + E(r) d 1 2 I(2012), ec. 2. 5 p. 3/30
E(r) d = q 1 (5) d 1 4πε 0 r1 2 1 E(r) d = q 1 (6) d 2 4πε 0 r2 2 2 1 d 2 d = r2 1 (7) r2 2 (8) 1 E(r) d = 2 E(r) d (9) E(r) d = 0 I(2012), ec. 2. 5 p. 4/30
1 2 q r 1 r 2 1 n θ E 2 i 1/cosθ i E E E n = E n = Ecosθ (10) E 1 = E 2 (9) I(2012), ec. 2. 5 p. 5/30
q q (11) E(r) d = 0, q I(2012), ec. 2. 5 p. 6/30
q 1 2 q E 1 1 2 E 2 q n e n i q n I(2012), ec. 2. 5 p. 7/30
(q ) (12) E d = 0, + ( n i ) n e (13) E d = E d. ( ) r E d = 1 q 4πε 0 r 2 4πr2 = q (14) (= N ). ε 0 ( r ) I(2012), ec. 2. 5 p. 8/30
(13) E d = q (15), q. ε 0 { 0, q (16) E d = q/ε 0, q ( ) I(2012), ec. 2. 5 p. 9/30
(17) E = i E i (18) E d = i Q int. i E i d = q i i = Q int., ε 0 ε 0 q i =. I(2012), ec. 2. 5 p. 10/30
( ) E d = Q int. (19), Q int. ρ(r) dv ε 0 V (20) E(r) d = Q int. ε 0 V Q int. 1/r 2 I(2012), ec. 2. 5 p. 11/30
1: O a r Q r a ( ) ρ (21) Q = 4π 3 a3 ρ. O r(> a) (22) E(r) d = Q ε 0 O a I(2012), ec. 2. 5 p. 12/30
E(r) E(r) (23) E(r) d = E(r) d = 4πr 2 E(r) (24) E(r) = Q 4πε 0 1 r 2, r > a. ( Q ) r < a E(r) d = 1 4 (25) ε 0 3 πr3 ρ. I(2012), ec. 2. 5 p. 13/30
(26) (27) (28) E(r) = E r E(r) d = 4πr 2 E(r). E(r) = ρ 3ε 0 r = 1 4πε 0 Q a 3r. Q r 4πε 0 a 3, r < a Q 1 4πε 0 r 2, r > a r I(2012), ec. 2. 5 p. 14/30
a E(r) =, r < a ( Q ), r > a Q a I(2012), ec. 2. 5 p. 15/30
2: (cf. 2. 3. 3 2) λ R L E(r) R R E(r) d = λl (29), ε 0 = E(R) (30) E(R) = λ 2πε 0 1 R. d = 2πRLE(R). ( 2. 3. 3 2 ) L E(R) I(2012), ec. 2. 5 p. 16/30
3: ( σ) E ( A) A E E(r) d = Aσ (31). ε 0 E E A( )+E A( ) = Aσ (32). ε 0 I(2012), ec. 2. 5 p. 17/30
(33) E = σ 2ε 0. I(2012), ec. 2. 5 p. 18/30
P 0 -q (test charge ) P 0 P 0 P 0 -q -q P 0 ( ) E (34) E d < 0. P 0 E P 0 I(2012), ec. 2. 5 p. 19/30
E = φ P 0 E = φ = 0 φ φ(p) = P P 0 E dr P E 0 E P 0 φ E dr < 0 φ(p) > 0. (φ(p 0 ) = 0 ) dr P φ P 0 E I(2012), ec. 2. 5 p. 20/30
2.5.3 ( ) A(r) (35) A(r)(= diva) = A x x + A y y + A z z ( ) A (divergence) ( x 0, y 0, z 0 + z) V 0 n ( x 0, y 0, z 0 ) 0 A (, + y, x 0 z 0 + z y 0 z 0 ) ( x 0 + x, n y 0, z 0 ) z 0 I(2012), ec. 2. 5 p. 21/30
z z0 ( ) z0 + z( ) z0 ( ) (36) A z (x,y,z 0 )dxdy. z0 z0 + z (37) A z (x,y,z 0 + z)dxdy. z0 + z ( V 0 ) (38) {A z (x,y,z 0 + z) A z (x,y,z 0 )}dxdy Az (x,y,z 0 ) z zdxdy A z(x 0,y 0,z 0 ) z z x y = A z z V 0. I(2012), ec. 2. 5 p. 22/30
x y ( Ax A(r) d 0 x + A y y + A ) z (39) V 0 z = ( A) r=(x 0,y 0,z 0 ) V 0. A ( ) : 2 V 1,V 2 V 1(2) 1(2) (40) A d = A V 1, 1 n 2 (41) A d = A V 2. 2 V1 V2 n 1 I(2012), ec. 2. 5 p. 23/30
V 1 V 2 V 1+2 1+2 V 1 V 2 (n ) (42) A d + A d = A d. 1 2 1+2 1+2 A d = A V 1 + A V 2. ( ) V V ( V i i ) = A d (43) A V i = A dv ( ) i V I(2012), ec. 2. 5 p. 24/30
A(r) V ( V ) (44) A(r) d = V A(r) dv I(2012), ec. 2. 5 p. 25/30
E(r) d = Q int. = 1 (45) ρ(r)dv. ε 0 ε 0 ( 2. 5. 2 ) (46) E(r) d = E(r)dV. V V (45) E(r)dV = 1 (47) ε 0 V V ρ(r)dv. I(2012), ec. 2. 5 p. 26/30
V (48) E(r) = ρ(r) ε 0. r E x (E y,e z = const.) (49) E x (x+ x,y,z) E x (x,y,z) E x x x = E(x,y,z) x = ρ(x,y,z) ε 0 x. (50) E x (x+ x,y,z) E x (x,y,z)+ ρ(x,y,z) ε 0 x. I(2012), ec. 2. 5 p. 27/30
1: 2. 5. 2 1 (51) (52) E(r) = ρ(r) = { ρ, r < a 0, r > a ρ 3ε 0 r, r < a Q 4πε 0 r r 3, r > a, Q = 4πa3 ρ/3 r < a (53) E = ρ 3ε 0 r = ρ 3ε 0 ( x x + y y + z ) z = ρ ε 0 I(2012), ec. 2. 5 p. 28/30
r > a (54) E = Q 4πε 0 r r 3 = Q 4πε 0 ( ) 3 r 3(x2 +y 2 +z 2 ) 3 r 5 = 0 x ( x r 3 ) = I(2012), ec. 2. 5 p. 29/30
: (55) (56) E(r) = ρ(r) ε 0. E(r) = 0. (55) ρ,e ( 5 ) E(r,t) = ρ(r,t) ε 0 I(2012), ec. 2. 5 p. 30/30