2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30
(2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i i i+1 n 3 2 r 2 F i n i 2 i n 1 r 1 1 ( ) A ) dr = n(r)dr n(r) r Γ Γ i B I(2011), Sec. 2. 4 p. 2/30
(4) w W q = B A F B q dr = A E dr. ( ) w Γ w Γ 1 ( ) q q B (5) E(r) = q 4πε 0 ˆr r 2. (i) A B ( A A B) q O r A A r A r B A I(2011), Sec. 2. 4 p. 3/30
w = B A E(r) dr = A A E(r) dr B A E(r) dr 1 A A (ˆr) (dr) E dr = 0. A B dr = ˆrdr (6) w = rb r A q 4πε 0 dr r 2 = q 4πε 0 ( 1 1 ). r A r B (ii) A A A B B q O A A A I(2011), Sec. 2. 4 p. 4/30
(7) w = ra r A = q 4πε 0 q dr 4πε 0 r 2 rb r A ra r A dr r 2 = q 4πε 0 q dr 4πε 0 r 2 ( 1 1 ) r A r B. rb q dr 4πε 0 r 2 = ( ) w (iii) q O B b x a y r θ c E A I(2011), Sec. 2. 4 p. 5/30
abc E (E bc abc ) (8) E dr = E y = E rcosθ. (9) a c a b c E dr = E rcosθ. ( E cosθ E ac ) E(r) w = B A E(r) dr I(2011), Sec. 2. 4 p. 6/30
2.4.2 P 0 φ( ) P 0 A ( A r A ) (10) φ(r A ) w(p 0 A) = A P 0 E(r) dr, B ( : ) A (11) φ(r B ) = B P 0 E(r) dr. A B w(a B) = w(a P 0 )+w(p 0 B) (12) = w(p 0 A)+w(P 0 B) = φ(r B ) φ(r A ). P 0 I(2011), Sec. 2. 4 p. 7/30
(13) (14) w(a B) = B A B A E(r) dr. E(r) dr = φ(r B ) φ(r A ). φ: ( ) (15) r φ(r) = E(r ) dr. P 0 P 0 q (6) r A, r B = r (16) φ(r) = q 4πε 0 1 r. I(2011), Sec. 2. 4 p. 8/30
( (2. 3. 4)) (17) E(r) = i E i (r), (E i i ) (18) r r φ(r) = E(r ) dr = E i (r ) dr P 0 P 0 i = r E i (r ) dr = φ i (r) i P 0 i (φ i i ) = 1 q i 4πε i 0 r r i. I(2011), Sec. 2. 4 p. 9/30
( ρ(r)) (19) φ(r) = 1 4πε 0 ρ(r )dv r r. cf. (2. 3. 5) (15) (N/C)m = J/C 1J/C 1V ( Volt) E V/m 2.4.3 2 (x,y,z) (x+dx,y,z) (x,y,z) (x+dx,y,z) x x + dx (20) w = E dr = E x dx. I(2011), Sec. 2. 4 p. 10/30
(21) w = φ(x+dx,y,z) φ(x,y,z) = φ x dx. E x = φ x. y z (22) E = (E x,e y,e z ) = ( φ x, φ y, φ z ). = ( x, y, z ) ( ) I(2011), Sec. 2. 4 p. 11/30
(23) E = φ(r) (= gradφ). (gradient: ) (15) ) f f f f gradient f 2.4.4 rotation( ) A(r) ( Az A(r) = y A y z, A x z A z x, A y x A ) x (24), y A rotation( ) ( A = rota = curla ) (23) I(2011), Sec. 2. 4 p. 12/30
(25) ( E) z = E y x E x y = ( ) φ + ( ) φ x y y x = 2 φ x y + 2 φ y x = 0. x,y (26) E(r) = 0. ) φ A(r) = f(r) (f(r) ) A(r) = 0 (26) (27) E(r) = q 4πε 0 r r 0 r r 0 3 I(2011), Sec. 2. 4 p. 13/30
(28) = = ( E) z = E y x E x y ( q y y 0 4πε 0 x r r 0 3 y [ q (y y 0 ) x 4πε 0 ) x x 0 r r 0 3 1 r r 0 3 (x x 0) y ] 1 r r 0 3 = 0. ) (29) E = 0. ( ) ( ) I(2011), Sec. 2. 4 p. 14/30
2.4.5 (Stokes) ( )Γ A(r) Γ Γ 1 dr ( ) A t (30) A t (r)dr = A(r) dr. Γ ( 1 A t A ) (circulation) (cf. (1)) Γ Γ A P 1 Γ 1 Γ 2 P 2 I(2011), Sec. 2. 4 p. 15/30
P 1, P 2 Γ 1 P1 P 2 A(r) dr = P2 P 1 A(r) dr, Γ 2 P2 A(r) dr, P 1 (31) A(r) dr = A(r) dr + A(r) dr. Γ Γ 1 Γ 2 Γ I(2011), Sec. 2. 4 p. 16/30
(32) =. Γ y 0 + y y ( ) xy y 0 z x 0 x 0 + x x (33) A(r) dr = x0 + x + x 0 x0 x 0 + x A x (x,y 0,z)dx+ y0 + y y 0 A x (x,y 0 + y,z)dx+ A y (x 0 + x,y,z)dy y0 y 0 + y A y (x 0,y,z)dy I(2011), Sec. 2. 4 p. 17/30
= x0 + x + x 0 y0 + y y 0 [A x (x,y 0,z) A x (x,y 0 + y,z)]dx [A y (x 0 + x,y,z) A y (x 0,y,z)]dy A x (x,y 0 + y,z) = A x (x,y 0,z)+ A x(x,y 0,z) y A x (x,y,z) y ( : 2 y +O(( y) 2 ) y=y0 ) x, y 2 x0 + x x 0 A x(x 0,y 0,z) y [ A ] x(x,y 0,z) y y y0 + y dx+ y 0 y x+ A y(x 0,y 0,z) x [ Ay (x 0,y,z) x y x ] x dy I(2011), Sec. 2. 4 p. 18/30
= ( Ay x A ) x x y = ( A) z x y = ( A) z ds y ( x y = = ds ) n( ) z ( ) = ( A) nds = ( A) ds. (ds nds ) ( ) (32) S Γ (34) = = ΓA(r) dr A(r) dr ( A) ds = ( A) ds. ( ) S I(2011), Sec. 2. 4 p. 19/30
(35) A(r) dr = Γ S ( A(r)) ds. : ω r = (x,y,0) v(r) = ( yω,xω,0) v(r) = rω (36) v(r) dr = 2πaaω = 2πa 2 ω. r=a y v r x (37) v(r) = (0,0,2ω) I(2011), Sec. 2. 4 p. 20/30
(38) r a v(r) ds = 2ωπa 2. ) A = 0 A 0 2 P 1, P 2 Γ 1 P 2 Γ (39) A(r) dr = A(r) dr. P 2 1 Γ 1 Γ 2 Γ = Γ 1 Γ 2 (40) A(r) dr = A(r) dr A(r) dr = 0. Γ Γ 1 Γ 2 I(2011), Sec. 2. 4 p. 21/30
( A(r) dr = 0 Γ 2 ) (35) (41) ( A) ds = 0. S S (42) A(r) = 0. ( rotation ) ( A = 0 A dr = 0) A(r) = 0 f(r) A(r) = f(r) 2 A(r) P 0 I(2011), Sec. 2. 4 p. 22/30
(43) P2 P 1 A(r) dr P0 = A(r) dr + P 1 = f(r 2 ) f(r 1 ). P2 P 0 A(r) dr P 1 r f(r) A(r ) dr P (44). 0 P 0 (45) f(r +dr) f(r) = A(r) dr. (dr = (dx,dy,dz)) r 1 P 2 r 2 P 0 r (46) f(r +dr) f(r) = f x dx+ f y dy + f z dz = f(r) dr. I(2011), Sec. 2. 4 p. 23/30
(47) A(r) = f(r). (f ) A = f A = 0 ( (25) (26) ) ( ) (5) (9) w = B A E(r) dr E(r) = g(r)ˆr E(r) r (ii) (48) = w = ra r A B A E(r) dr g(r)dr ra r A g(r)dr = rb r A g(r)dr. I(2011), Sec. 2. 4 p. 24/30
w φ E = φ E = 0 ( ) r (1/r 2 ) E = 0 ( ) (1/r 2 ) ( ) (= 2. 5) 2.4.6 gradient (49) f = ( f x, f y, f z ( f) x = f/ x f x f (cf. (46)) f f ). I(2011), Sec. 2. 4 p. 25/30
( ) (50) φ(r) = const. ( ) (51) E = φ E φ E E φ = const.? E I(2011), Sec. 2. 4 p. 26/30
: 1 q(> 0) +q 1: (cf. 2. 3. 3 ) z (18) d +q [ 2 φ(r) = q 1 (52) O 4πε 0 x2 +y 2 +(z d/2) 2 d q ] 2 1. x2 +y 2 +(z +d/2) 2 I(2011), Sec. 2. 4 p. 27/30
(r = x 2 +y 2 +z 2 d) (53) = = = [ ( φ(r) = q ) 1/2 ) ] 1/2 r 2 zd+ d2 (r 2 +zd+ d2 4πε 0 4 4 [ ( q 1 1 zd ) 1/2 ( 4πε 0 r r + d2 1+ zd ) ] 1/2 2 4r 2 r + d2 2 4r 2 [ q 1 1+ zd ( 4πε 0 r 2r 1 zd ) ] +O((d/r) 2 ) 2 2r 2 q zd 4πε 0 r, 3 p = (0,0,qd) 1 p r. 4πε 0 r 3 I(2011), Sec. 2. 4 p. 28/30
E(r) = φ(r) ( r n / x = nx/r n+2 ) E x 1 [ ] px (54) 4πε 0 r p r3x, 3 r 5 E y 1 [ ] py 4πε 0 r p r3y, 3 r 5 E z 1 [ ] pz 4πε 0 r p r3z. 3 r 5 p ] (55) E(r) = 1 4πε 0 [ 3(p r) r 5 r p r 3 p = (0,0,qd) (p r = qdz) = 1 4πε 0 3(p r)r r 2 p r 5. I(2011), Sec. 2. 4 p. 29/30
(56) E x 1 3qdzx 4πε 0 r 5 E z qd 3z 2 r 2 4πε 0 r 5 = qd 3xz 4πε 0 r, E 5 y qd 3yz 4πε 0 r, 5 = qd 4πε 0 ( 3z 2 r 5 1 r 3 ). z (r = (0,0,z)) (57) (2. 3. 17) xy (z = 0) (58) (xy ) I(2011), Sec. 2. 4 p. 30/30