1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1
1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2 > 0) (q 1 q 2 < 0) MKA k k = 1 ε 0 = 8.854 10 12 C 2 N 1 m 2 1.2 r 1, r 2 2
1.3 r (1,2) q 1, q 2 q 1 q 2 > 0 1.4 r (1,2) q 1, q 2 q 1 q 2 < 0 3
r 1 r 2 r 2 r 1 1.1 q 1 q 2 r 1 r 2 r 2 r 1 r 1 r 2 r 1 r 2 r 2 r 1 r 1 r 2 q 1 q 2 F 12 F 12 = q 1q 2 1 r 1 r 2 2 r 1 r 2 r 1 r 2 (1.2) = q 1q 2 r 1 r 2 r 1 r 2 3 (1.3) 2 1 F 21 F 21 = q 1q 2 1 r 1 r 2 2 r 2 r 1 r 1 r 2 (1.4) = q 1q 2 r 2 r 1 r 1 r 2 3 (1.5) q 1 q 2 3 r 1, r 2, r 3 q 1, q 2, q 3 q 1 q 2, q 3 F 1(23) F 1(2,3) = F 12 + F 13 = q 1q 2 r 1 r 2 r 1 r 2 3 + q 1q 3 r 1 r 3 r 1 r 3 3 (1.6) F 12, F 13 (??) q 1 q 2 q 1 q 3 r 1, r 2,, r N q 1, q 2,, q N q 1 q 2, q N F 1(2 N) 4
1.5 q 1, q 2, q3 1.6 N 5
F 1(2 N) = F 12 + F 13 + + F 1N = N F 1i (1.7) i=2 i 6
1. 1cm 2. r 0 = (1, 1, 1)[m] 1C r 1 = (3, 3, 3)[m] 2C 3. r 1 = (1, 1, 1)[m] 1C r 2 = ( 2, 2, 2)[m] 2C r = (x, y, z)[m] 2C 0 7
1.2 r q r 1 q 1 F F = qq 1 r r 1 r r 1 3 (1.8) r, r 1 q 1 r r 1 q q 1 r 1 r q 2 1C +q[c] 1 +q[c] q[c] +q[c] q[c] 1C +q[c] q[c] ±q[c] r q r 1 q 1 F 8
1.7 +q 1.8 ±q F F = qq 1 r r 1 r r 1 3 = E( r)q (1.9) E( r) = q 1 r r 1 r r 1 3 (1.10) 9
r 1 q 1 r q q E( r) r 1 q 1 r q 1.9 r 1 q 1 r E( r) r q F = E( r)q F (1.10) q = 1 1C r 1, r 2,, r N q 1, q 2,, q N r E( r) E( r) = = E i ( r) = N i=1 q i r r i r r i 3 N E i ( r) (1.11) i=1 q i r r i r r i 3 (1.12) (1.11) i 1 N q 1,, q N 10
E i ( r) r i q i 1 2 10 23 1m 3 ρ V Q V Q/V r ρ( r) ρ( r) x, y, z ρ( r) i r i x, y, z x i, y i, z i Q i Q i = ρ( r i ) x i y i z i r i Q i r E( r) = i ρ( r i ) r r i r r i 3 x i y i z i r E( r) x i, y i, z i 0 11
1.10 E( r) = = lim x i, y i, z i 0 ρ( r ) r r = 1 i ρ( r i ) r r i r r i 3 x i y i z i r r 3 dx dy dz ρ( r )( r r ) dv (1.13) r r 3 ρ( r ) 3 1m λ λ r λ( r) r C s i i r i Q i Q i = λ( r i ) s i 12
1.11 s i r E( r) = i λ( r i ) r r i r r i 3 s i C r E( r) s i 0 E( r) = = lim s i 0 i C λ( r i ) r r i r r i 3 s i λ( r ) r r r r 3 ds (1.14) 1m 2 σ σ( r) r i r i x i, y i σ( r i ) x i y i r E( r) = i σ( r i ) r r i r r i 3 x i y i 13
1.12 r i E( r) x i, y i 0 E( r) = = lim x i, y i 0 i σ( r i ) σ( r ) r r = 1 r r i r r i 3 x i y i r r 3 dx dy σ( r )( r r ) d (1.15) r r 3 14
λ 0 z z z + λ( r) = λ 0 x, y, z (1.14) E x ( r) = C = λ 0r x λ( r ) r x r x r r 3 ds 1 [r 2 x + r 2 y + (r z z ) 2 ] 3/2 dz (1.16) E y ( r) = C = λ 0r y λ( r ) r y r y r r 3 ds 1 [r 2 x + r 2 y + (r z z ) 2 ] 3/2 dz (1.17) E z ( r) = C = λ 0r y λ( r ) r z r z r r 3 ds 1 [r 2 x + r 2 y + (r z z ) 2 ] 3/2 dz (1.18) r x r = (R, 0, 0) (1.17) r y = 0 E y ( r) = 0 (1.18) E z ( r) z 1.13 z λ 0 15
E z ( r) = 0 E x ( r) (1.16) E x ( r) = λ 0R 1 (R 2 + z ) 3/2 dz (1.19) R > 0 z = R tan θ dz = R cos 2 θ dθ z (, + ) θ ( π 2, π ) (1.19) 2 E x ( r) = λ π 2 0R = λ 0R = λ 0 R π 2 π 2 π 2 π 2 1 R R 3 (1 + tan 2 θ) 3/2 cos 2 θ dθ π 2 1 R 2 1 cos3 θ cos 2 θ dθ cos θ dθ = λ 0 R = λ 0 2πε 0 R sin θ π 2 π 2 (1.20) E( r) x x z z z R E( r) = λ 0 2πε 0 R E( r) = λ 0(r x, r y, 0) 2πε 0 r 2 x + r 2 y 16
1.14 ρ( r) ρ( r) 0 1 x λ(x) a Q x 0 λ(x) x 0 a/2 x 0 + a/2 λ 0 Q λ 0 a = Q λ 0 = Q/a { Q/a, x0 a/2 < x < x λ(x) = 0 + a/2 0, a 0 λ(x) x 0 Q 17
δ(x) 1 δ(x x 0 ) = lim a 0 Q λ(x) Q = 1 λ(x) x 0 0 + infty δ(x x 0 )dx = 1 δ(x x 0 ) x 0 0 x 0 f(x) δ(x x 0 ) f(x) δ(x x 0 )f(x) + x 0 f(x) f(x 0 ) + δ(x x 0 )f(x)dx = f(x 0 ) r 1 Q ρ( r) ρ( r) = Qδ(r x r 1x )δ(r y r 1y )δ(r z r 1z ) r x, y, z r 1 x, y, z x, y, z 3 1.3???? Q > 0 R E( r) E( r) n( r)d = E( r) d (1.21) 18
1.15 R n( r) r n( r)d d 1.21 R n( r) = r/ r = r/r = r/r (1.10) E( r) n( r)d = = = Q r r 3 r rd d Q R 4 Q R 2 r r d = Q ε 0 (1.22) 1.16 n( r) r 19
d 4πR 2 E( r) n( r)d R 1.2 r σ L ( r) r e( r) e( r) = E( r)/ E( r) // e( r) : σ L ( r) = A E( r) = AE( r)) (1.23) A 1 (1.22) e( r) = n( r) E( r) n( r)d = E( r) e( r) n( r)d = σ L ( r)d (1.24) N L (1.22) 1.17 E( r) n( r) = r 20
(1.22) (1.22) σ L ( r)d σ L ( r)d σ L ( r)d i σ L n A n lim 0 r i i A i n i σ L ( r i ) i r i 1.18 21
1.19 A i θ i θ i = π 2 n i 0 n i i cos θ i n i = σ L ( r i ) i cos θ i cos θ i = e( r i ) n( r i ) n( r i ) A i A i n i n i = σ L ( r i ) e( r i ) n( r i ) i N L i 0 n i N L N L = lim i 0 i n i = lim σ L ( r i ) e( r i ) n( r i ) i i 0 i = σ L ( r) e( r) n( r)d (1.25) σ L ( r) e( r) n( r)d = σ L ( r) e( r) n( r)d (1.26) 22
σ L ( r) e( r) = E( r) E( r) n( r)d = E( r) n( r)d (1.27) N L (1.25) (1.25) 1.20 (1.25) (1.25) e( r) n( r) e( r) r n( r) e( r) n( r) > 0 e( r) n( r) < 0 (1.25) 1.20 23
σ L ( r) e( r) n( r)d = σ L ( r) e( r) n( r)d E( r) E( r) n( r)d = E( r) n( r)d (1.22) Q ε 0 N L = σ L ( r) e( r) n( r)d = E( r) n( r)d = E( r) n( r)d = Q (1.28) ε 0 ε 0 (1.12) N r E( r) E i ( r) E i ( r) q i E( r) = i E i ( r) N E( r) n( r)d = E i ( r) n( r)d = i i q i E i ( r) n( r)d = i ε 0 = Q ε 0 (1.29) 24
Q E( r) n( r)d ε 0 ρ( r) E( r) n( r)d = 1 ( ) ε 0 = 1 ρ( r)dv (1.30) ε 0 V V (1.30) *1 V a σ 0 r r r σ 0 > 0 r σ 0 < 0 r R E( r) n( r)d = E( r)d = E(R) d = E(R) 4πR 2 = 1 ( ) (1.31) ε 0 *1 25
E(R) R (1.30) R < a R > a 4πa 2 σ 0 0, r < a E( r) = a 2 r 2 σ 0, r > a 0, r < a E( r) = a 2 r 3 σ 0 r, r > a z λ 0 z z z z R t, b s E( r) n( r)d = E( r) n( r)d + E( r) n( r)d + E( r) n( r)d t b t, b t, b E( r) n( r) 0 s z s E( r) n( r)d = E( r)d s s = E(R) d s = E(R) 2πR (1.32) E(R) z R (1.30) λ 0 ε 0 E( r) = E(R) = λ 0 2πε 0 R 26 (1.33)
1. a > b σ a, σ b a 0 2. a ρ 0 27