r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2) permittivity F 12 1 2 F 21 1 F 12 F21 2 O O 2.1: Coulomb (1) F 12 = F 21 (2) 1 2 > 0 F 12 r 1 r 2 1 2 < 0 17
18 2 F 12 r 2 r 1 (3) Coulomb 2.1 10 km Coulomb M =1.988 10 30 kg F G =6.6742 10 11 (1.988 1030 ) (9.109 10 31 ) (10 10 3 ) 2 =1.21 10 18 [N]. Coulomb F C = 1 4π 8.854 10 12 (1.602 10 19 ) 2 (10 10 3 ) 2 =2.31 10 36 [N] (1.21 10 18 )/(2.31 10 36 ) 5 10 17 10 57
2.2 19 2.2 Coulomb (2.1) F 21 = 2 E 1 (2.3) E 1 = 1 1 r 2 r 1 4πε 0 r 2 r 1 2 r 2 r 1 (2.4) (2.4) E 1 2 1 r 2 r 1 1 electric field (2.3) 1 E 1 2 F 21 1 2 1 E 1 2 Coulomb 2.2: (2.4) (2.3) Coulomb (2.1) Coulomb (2.1) 1 2 (2.4) 2 1 E 1 (2.3) E 1 2 E 1 F 21 2
20 2 k r k ( k =1, 2,,n) k r E k (r) E(r) = n E k (r), E k (r) = k=1 k 4πε 0 r r k 2 r r k r r k. (2.5) ρ(r) v k r k ρ(r k ) v k E(r) = 1 n r r k 4πε 0 r r k 3 ρ(r k ) v k (2.6) k=1 v k 0 n ρ(r ) r E(r) = 1 r r 4πε 0 r r 3 ρ(r )dv. (2.7) dv dx dy dz ε 0 C 2 N 1 m 2 : N m 2 C C 2 m 2 = N C (2.8) dr r r de r r O 2.3:
2.2 21 Dirac δ function k r k ( k =1, 2,,n) n ρ(r )= k δ(r r k ) (2.9) δ(r r) = k=1 r = r 0 r r (2.10) δ(r r)dx dy dz =1, f(r )δ(r r)dx dy dz = f(r) (2.11) (2.9) (2.7) (2.11) (2.5) 2.2 a Q z z P z θ θ +dθ 2πa a sin θ dθ 4πa 2 dq = Q Q sin θ 2πa a sin θ dθ = dθ 4πa2 2 P de = 1 dq 4πε 0 R 2 R P z x y 0 2.4 ϕ z de z = 1 4πε 0 dq R 2 cos ϕ ψ R 2 + r 2 2Rr cos ϕ = a 2 θ a 2 + r 2 2ar cos θ = R 2 cos ϕ R 2 = 1 ( 1 2 Rr + r ) R 2 a2 R 2, sin θ dθ = R r ar dr
22 2 dθ R O a θ r ϕ P de z 2.4: R P P ( Q r+a 1 E z = 16πε 0 a r a r 2 + r ) R 2 a2 Q R 2 r 2 dr = 4πε 0 r 2 r>a ( Q r+a 1 E z = 16πε 0 a r 2 + r ) R 2 a2 R 2 r 2 dr =0 r<a a r
2.3 Gauss 23 2.3 Gauss 2.3.1 E(r) r =(x, y, z) E(x, y, z) = E x (x, y, z) E y (x, y, z) E z (x, y, z) flux, P d d P P n d = n d P E d d = E d = E n d flux of electric force (2.8) N m 2 C 1. n E d 2.5: E E d d ε 0 dψ = ε 0 E d = ε 0 E n d. (2.12)
24 2 d electric flux ε 0 C 2 N 1 m 2 N C 1 ψ : C (2.13) D = ε 0 E (2.14) electric flux density : C m 2 (2.15) dψ = D d = D n d (2.16) 2.3.2 ψ d dψ (2.12) (2.16) ψ = ε 0 E d = D d. (2.17) E (D) d r (2.17) r E(r) =E n, n = r r E E = 4πε 0 r 2 n r r n = r/r
2.3 Gauss 25 d = n d d dψ dψ = ε 0 E d = d (2.18) 4πr2 n n =1 ψ = dψ = 4πr 2 d = 4πr 2 4πr2 = (2.19) r 2 r 2 r (2.18) d dψ dψ = 4π d r 2. d/r 2 2.6 d dω (2.18) dψ = dω. (2.20) 4π d de d θ de d cosθ dω dω 2.6: (2.20) d 2.6 d d E θ d dω dω = d cos θ r 2 d E = /(4πε 0 r 2 ) dψ = ε 0 E d = ε 0 E cos θ d = 4π d cos θ r 2
26 2 (2.20) (2.20) ψ = E d = 4π 4π ψ = dω = 4π = (2.21) 4π 4π (2.19) dω dω d 1 d 2 d 3 dψ 1 = 4π dω, dω dψ 2 = 4π dω, dψ 3 = dω, 4π (2k 1) (2k) dψ 1 +dψ 2 =0, ψ = dψ = 4π (dω dω +dω dω + +dω)= dω = (2.22) 4π dω (2k 1) (2k) ψ = dψ = (dω dω +dω dω + +dω dω ) = 0 (2.23) 4π 0 2.3.3 Gauss n 1, 2,, n E k E k ψ
ψ k ψ = ε 0 = n k=1 E d ψ k = ε 0 n k=1 2.3 Gauss 27 E k d (2.24) ψ k k 0 (2.24) V dv dv ρ ρ dv V Gauss D = ε 0 E ψ = ε 0 E d = k (2.25) ψ = ε 0 E d = ρ(r)dv V ρ V Coulomb Maxwell Coulomb 2.3 a Q Q >0 Gauss Q <0
28 2 2.7 r Gauss a O r P E 2.7: r >a P E r d = n d ε 0 E d = ε 0 E 4πr 2 E r Q Gauss ε 0 E 4πr 2 = Q E = Q 4πε 0 r 2 r <a P Gauss ε 0 E 4πr 2 =0 E =0
2.4 Gauss 29 2.4 Gauss 2.4.1 Gauss Gauss D(r) n(r)d = ρ(r)dv (2.26) n(r) r V 2.8 x, y, z x y z V = x y z V D z y D y z D x x x+ x 2.8: Gauss (2.26) x y z x x + x n x x D n = D x y D y z D z x x n x x D n = D x x D n d = D x (x + x, y, z) y z D x (x, y, z) y z x = D x(x + x, y, z) D x (x, y, z) V x x/ x =1 x y z = V y z [ Dx (x + x, y, z) D D n d = x (x, y, z) + D y(x, y + y, z) D y (x, y, z) x y + D ] z(x, y, z + z) D z (x, y, z) V. z
30 2 V x, y, z 0 V 0 D lim x (x + x, y, z) D x (x, y, z) = D x x 0 x x D n d lim V 0 V = D x x + D y y + D z z (2.27) A V r A n d div A = lim (2.28) V 0 V V A div A A div A = A = A x x + A y y + A z z (2.29) (2.27) (2.29) (2.26) lim V 0 D n d V = div D (2.30) (2.26) (2.26) V V V 0 x, y, z 0 r =(x, y, z ) V 0 ρ(r )dv lim V 0 r V V = ρ(r) (2.31) (2.26) (2.30) (2.31) D = ε 0 E Gauss
2.4 Gauss 31 Gauss div E = ρ ε 0 (2.32) 2.4.2 Gauss Gauss A(r) div A V A n d = V div A dv (2.33) Gauss (2.28) Gauss V 1, V 2 1, 2 V = V 1 + V 2 (2.28) A n d 1 (div A) 1 V 1, A n d 2 (div A) 2 V 2 1 2 V 1 V 2 V A n d (div A) 1 V 1 + (div A) 2 V 2 V V n V 1, V 2,..., V n Gauss n n A n d = lim A n i n d i = lim (div A) i n i V i = div A dv. (2.34) V i=1 i=1