50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq

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49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1

50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r r 2 r r r r (2.3 2.1.1 1.7 10 27 kg, 9.1 10 31 kg

2.2 51 R F C = e 2 /(4πε 0 R 2 F G = GMm/R 2 F G /F C = GMm/(e 2 /(4πε 0 = 4.5 10 40 2.1.2 m 2 l q 2θ q 2l sin θ F C = q 2 /(4πε 0 (2l sin θ 2 F C /(mg = tan(θ q = 4l πε 0 mg sin 3 θ/ cos θ 2.1.3 1 l q q 0 2.2

52 2 I A B B q r F E( r F = ( r q 2.2.1 r q r E( r = 1 4πε 0 E( r = q r r 2 n i=1 r r r r = 1 4πε 0 r r r r 3 q (2.4 E i ( r = 1 4πε 0 n i=1 E( r = 1 4πε 0 r r i r r i 3 q i (2.5 r r r r 3 ρ( rdv (2.6

2.2 53 2.2.1 d q, q (0, 0, d/2, (0, 0, d/2 1. r = (x, y, z 2. d E( r = 1 ( q( r (0, 0, d/2 q( r (0, 0, d/2 + 4πε 0 r (0, 0, d/2 3 r (0, 0, d/2 3 = q ( (x, y, z d 4πε 0 (x 2 + y 2 + (z d/2 2 3/2 (x, y, z + d (x 2 + y 2 + (z + d/2 2 3/2 r d E( r q 4πε 0 r 3 q 4πε 0 r 3 ( (x, y, z d/2 (1 dz r 2 3/2 (x, y, z + d/2 (1 + dz r 3/2 2 ( (x, y, z d/2(1 + 3dz/2 r 2 (x, y, z + d/2(1 3dz/2 r 2 ( q 3dz(x, y, z = 4πε 0 r 3 (0, 0, d + r 2 = qd ( 3z(x, y, z 4πε 0 r 3 (0, 0, 1 + r 2 = qd 4πε 0 r 5 (3xz, 3yz, 3z2 r 2

54 2 I 2.2.2 λ x z z z (0, 0, z (x, 0, 0 de(z = λdz 4πε 0 1 x 2 + z 2 x de (z = λdz 1 x 4πε 0 x 2 + z 2 x2 + z 2 E(x = λ 4πε 0 = λ 4πε 0 1 x = λ 4πε 0 1 x = λ 2πε 0 x π/2 π/2 π/2 x (x 2 + z 2 π/2 sec 2 θ sec 3 θ dθ cos θdθ 3/2 dz z = x tan θ 2.2.3 2 2q, q (0, 0, d/2, (0, 0, d/2 d 1

2.2 55 2.2.1 E( r = 1 ( 2q( r (0, 0, d/2 4πε 0 r (0, 0, d/2 ( 3 2(x, y, z d/2 (x 2 + y 2 + (z d/2 2 3/2 (x, y, z + d/2 (x 2 + y 2 + (z + d/2 2 3/2 = q 4πε 0 r d E( r = q 4πε 0 r 3 q 4πε 0 r 3 ( 2(x, y, z d/2 (1 dz r 2 3/2 q( r (0, 0, d/2 + r (0, 0, d/2 3 (x, y, z + d/2 (1 + dz r 3/2 2 ( 2(x, y, z d/2(1 + 3dz/2 r 2 r 2 (x, y, z + d/2(1 3dz/2 q 4πε 0 r 3 d 1 E( r q 4πε 0 r 3 ( (x, y, z 3d/2 + 9 (x, y, z d/6dz 2 r 2 ( (x, y, z 3d/2 + 9 (x, y, zdz 2 r 2 2.2.4 1. R λ 2. R σ

56 2 I 3. σ xy z 1. R(cos θ, sin θ, 0 ds = R dθ E(z = 2π 0 1 (0, 0, z R(cos θ, sin θ, 0 λr dθ 4πε 0 (R 2 + z 2 3/2 = 1 2ε 0 Rλ(0, 0, z (R 2 + z 2 3/2 2. r dr σdr E(z = = = = R 0 1 r(0, 0, z σ dr 2ε 0 (r 2 + z 2 3/2 (0, 0, zσ 2ε 0 (0, 0, zσ [ 2ε 0 (0, 0, zσ 2ε 0 3. R E(z = R 0 r dr (r 2 + z 2 3/2 (z 2 + r 2 1/2] R 0 ( 1 z 1 R2 + z 2 (0, 0, 1σ 2ε 0

2.2 57 2.2.2 E + + + + - 2.2 2.2.5 1. 1 A s ε 0 2. 1 A s ε 0 q r E = q 4πε 0r 2 r πr 2 q Nq E = q 4πε 0 r 2 = Nq 4πr 2 N = 1/ε 0 2.2.6 1.

58 2 I 2. 2.2.3 dω ds dω dω = d S r r 3 (2.7 d S ds r 0 4πr0 2 r2 0 4π dw PˆÊ r n q ds 2.3 q S E d S

2.2 59 S E ds = q r 4πε 0 r 3 d S = q r ds r 3 = q 4πε 0 S 4πε 0 S dω S }{{} 4π = q/ε 0 (2.8 S q dω = 0 S S E d S = 0 ε 0 S E d S = { q q S 0 q S (2.9 ε 0 S E ds = ρ( r dv (2.10 V V S 2.2.7 a ρ S E(r ε 0 E(rdS = ε 0 E(r ds = ε 0 E(r4πr 2 S S S q(r q(r = 4π 3 r3 ρ r < a 4π 3 a3 ρ r > a

60 2 I q dw PˆÊ r n q E ds n1 E 1 q q n 2 E 2 PˆÊ 2.4 ρ r 3ε E(r = 0 ρ a 3 3ε 0 r 2 r < a r > a 2.2.8 λ l λl r

2.2 61 r E(r = λ 2πε 0 r 2πrlE(r = λl ε 0 2.2.9 σ 2.2.4 ±z 1 L σl 2 /ε 0 = 2E(zL 2 E(z = σ/(2ε 0 2.2.10 R 1, R 2 σ 1, σ 2 r l

62 2 I 0 r < R 1 2πrlε 0 E(r = 2πR 1 lσ 1 R 1 < r < R 2 2π(R 1 σ 1 + R 2 σ 2 l R 2 < r E(r = 0 r < R 1 R 1σ 1 ε 0r R 1 < r < R 2 R 2 < r R 1σ 1+R 2σ 2 ε 0r 2.2.11 R ρ { πr 2πrlε 0 E(r = 2 ρl r < R πr 2 ρl r > R E(r = { ρ 2ε 0 r ρ R 2 2ε 0 r r < R r > R 2.2.4 r F ( r A P F ( r

2.2 63 P W W (P = P A ( F ( r d r (2.11 W (P P F ( r C 1, C 2 A P ( F ( r d r = C 1 ( F ( r d r C 2 (2.12 C 1 F ( r d r = F ( r d r + C 1 C 2 F ( r d r C 2 F ( r d r (2.13 C 1 C 2 A P A C F ( r d r = 0 (2.14 r ϕ( r C F ( r = ϕ( r (2.15 (ϕ( r + δ r ϕ( r = F δ r (2.16

64 2 I q q F ( r = 1 qq 4πε 0 r 2 r r ϕ( r = 1 4πε 0 q r (2.17 (2.18 F ( r = { ϕ( r}q (2.19 ϕ( r ϕ( r = 1 ρ( r 4πε 0 r r dv (2.20 V q q ϕ( r = r E( r d r (2.21 ϕ( r = r ϕ( r = δ r E δ r = 0 (2.22 J/C V

2.2 65 2.2.12 R σ E(r = { σr 2 ε 0 1 r 2 r > R 0 r < R ϕ(r = = = r E(r dr R { σr 2 1 ε 0 r σr 2 1 ε 0 R r σr 2 ε 0 1 r 2 dr σr 2 1 ε 0 r 2 dr r > R r < R r R r > R 0 dr r < R 2.2.13 a ρ 2.2.7 ρ a 3 E(r = 3ε 0 r 2 r > a ρ r r < a 3ε 0 E(r r

66 2 I E(r 0 a 2.5 r ϕ(r = = = r a E(rdr r ρ a 3 3ε 0 r 2 dr ρ a 3 3ε 0 r 2 dr + ρ a 3 3ε 0 r ρ 3a 2 r 2 3ε 0 2 a r r > a r < a ρ 3ε 0 rdr r > a r < a 2.2.14 d q, q (0, 0, d/2, (0, 0, d/2 ( ϕ( r = q 1 4πε 0 x2 + y 2 + (z d/2 2

2.2 67 φ(r 0 a r 2.6 1 x2 + y 2 + (z + d/2 2 q 4πε 0 x2 + y 2 + z 2 ( (1 + = qd 4πε 0 z r 3 zd/2 x 2 + y 2 + z 2 (1 zd/2 x 2 + y 2 + z 2 2.2.5 q 1, q 2 r 1, r 2 q 1 q 2 ϕ 1 ( r 2 = 1 q 1 4πε r 2 r 1 (2.23

68 2 I U = 1 q 1 q 2 4πε r 2 r 1 (2.24 U = 1 4πε n (i,j q i q j r j r i (2.25 n (i,j n U = 1 2 n ϕ iq i (2.26 i=1 ϕ i = 1 4πε 0 n j( i q j r i r j (2.27 ϕ i q i 1/2 2 U = 1 ϕ( rρ( rdv (2.28 2 ϕ( r = 1 ρ( r 4πε 0 r r dv (2.29 U = 1 2 ρ( rρ( r 4πε 0 r r dv dv (2.30 2.2.15 R ρ 0

2.2 69 r ρ dr 4πr 2 ρdr ρr2 3ε 0 2.2.13 W = R 0 = 4πρ2 3ε 0 ρr 2 4πr 2 ρdr 3ε 0 R 0 = 4πρ2 15ε 0 R 5 r 4 dr 2.2.16 1 V 2 1 ev 1 ev J 10 10 m ev 1 ev = 1.602 10 19 A s 1V = 1.602 10 19 J (1.602 10 19 2 4 3.14 8.854 10 12 10 10 J 1.602 10 19 = 4 3.14 8.854 10 12 10 = 14.4 ev 10 ev

70 2 I 2.2.17 R Q 0 Q Q 0 q 4πε 0 R dq = Q2 8πε 0 R 2.2.18 ±q a 1 1 1 ( 2q2 4πε 0 a + 2q2 2a 2q2 3a +... 1 q2 ( 1 i+1 4πε 0 a i i=1 = q2 4πε 0 a log 2 2.2.6 ±q d = r + r p = qd p p = lim qd ( r + r ( r + r = p lim d 0 d d 0 d (2.31

2.2 71 ϕ( r = 1 4πε 0 p r r 3 (2.32 n q i r i (i = 1 n i r i = 0 r r r i = ( r r i ( r r i = r 2 2 r r i + r i 2 r r i 1 r r i 1 ( 1 2 r i r r r 2 1 ( 1 + r i r r r 2 ϕ( r = 1 n q i 4πε 0 r r i 1 4πε 0 = 1 4πε 0 i=1 n i=1 n i=1 q i r ( 1 + r i r r 2 q i r + 1 4πε 0 1/2 n i=1 q i r i r r 3 = 1 4πε 0 Q r + 1 4πε 0 P r r 3 (2.33 Q = P = n q i (2.34 i=1 n q i r i (2.35 i=1

72 2 I Q P 2.2.19 1. 3 q, 2q, q d 2. 4 ±q 1 d 1. 2 p = q(0, 0, d (0, 0, d/2, (0, 0, d/2 ϕ( r = 1 4πε 0 = ( p ( r (0, 0, d/2 r (0, 0, d/2 3 p ( r (0, 0, d/2 + r (0, 0, d/2 3 1 4πε 0 r 3 d 1 ( p ( r (0, 0, d/2(1 + 3dz/2 r 2 + p ( r + (0, 0, d/2(1 3dz/2 r 2 ϕ( r = = 1 4πε 0 r 3 1 4πε 0 qr 3 ( p (0, 0, d p r 3dz ( p 2 3 ( p r2 r 2 2. 2 p = ±q(0, d, 0 (d/2, 0, 0, ( d/2, 0, 0 ϕ( r = 1 ( p ( r (d/2, 0, 0 4πε 0 r (d/2, 0, 0 3 p ( r ( d/2, 0, 0 r ( d/2, 0, 0 3 r 2

2.2 73 d 1 ϕ( r = 1 4πε 0 r 3 ( p ( r (d/2, 0, 0(1 + 3dx/2 r 2 p ( r + (d/2, 0, 0(1 3dx/2 r 2 ( 1 = 4πε 0 r 3 p (d, 0, 0 + p r 3dx r 2 = 3qd2 xy 4πε 0 r 5 2.2.7 v( r S v( r ds S f( r v( r ds = f( r dv S V V S E D = ε 0E (2.36

74 2 I S D ds = V ρ( rdv (2.37 ρ( rdv S V V v D f ρ D = 0 v 0 c v d s 0 c c D d s = 0 (2.38 c 2.3

2.3 75 2.3.1 S D( r ds = ρ( rdv V D( rdv = ρ( rdv V V V D( r = ρ( r (2.39 2.3.1 R ρ ρ( r = { ρ r R 0 r > R E( r = { ρ 3ε 0 r ρr 3 r 3ε 0 r 3 r R r > R E( r

76 2 I 2.3.2 R ρ { ρ x ρ( r = 2 + y 2 R 2 0 x 2 + y 2 > R 2 z L E( r = { ρ 2ε 0 (x, y, 0 x 2 + y 2 R 2 (x,y,0 x 2 +y 2 x 2 + y 2 > R 2 ρr 2 2ε 0 E( r 2.3.2 C E( r d s = 0 ( E( r ds = E( r d s S C

2.3 77 C E( r = 0 (2.40 2.3.3 1. F x = 2Axz, F y = 2Ayz, F z = A(x 2 + y 2 2z 2 2. F x = A(y 2 + z 2, F y = A(x 2 + z 2, F z = A(x 2 + y 2 3. F x = 2Axy, F y = A(x 2 y 2, F z = 0 1. F = 0 F = 0 ϕ( r = A ( (x 2 + y 2 z (2/3z 3 2. F 0 3. F = 0 F = 0 ϕ( r = A ( x 2 y y 3 /3 2.3.3 ϕ( r E( r E( r = ( ϕ( r = 0

78 2 I 2 ϕ( r = 1 ε 0 ρ( r (2.41 2 ϕ( r = 0 (2.42 = 2 2.3.4 q ϕ( r = q 4πε 0 r 2.3.5

2.3 79 2.3.6 ϕ( r = Ae κr r 1. r = r 2. 3. 1. ϕ( r = A( 1r 2 r r e κr + A r ( κe κr r r ( ϕ( r = ρ( r = ε 0 2 ϕ( r = A r 3 e κr (1 + κr r ( ( Ar 3 e κr (1 + κr r + ( A ( r 3 e κr (1 + κr r + ( Ar ( 3 e κr (1 + κr r ( + A ( r 3 e κr (1 + κr r = κ 2 A e κr r 2. E( r = ϕ( r ρ = ε 0 κ 2 e κr r E( r = A r 3 e κr (1 + κr r

80 2 I δ 4πε 0 δ 2 Ae κδ δ 4πε 2 0 A 3. 0 ρ(r4πr 2 dr = 4πε 0 Aκ 2 re κr dr = 4πε 0 A 0 2.3.4 ρ( r ϕ( r ψ( r (ϕ( r + φ( r = ϕ( r + φ( r = 1 ρ( r ε 0 ϕ( r + φ( r 1. 2 2.

2.4 81 2 ϕ 1 ( r, ϕ 2 ( r ϕ 1 ( r = ϕ 2 ( r = 1 ε 0 ρ( r ϕ 1 ( r = ϕ 2 ( r = ϕ 0 φ( r = ϕ 1 ( r ϕ 2 ( r φ( r = 0 ϕ 1 ( r = ϕ 2 ( r 2 2.4 2.4.1 ( 10 8 Ωm (10 5 Ωm 1.55 10 8 Ωm

82 2 I ε 0 E S = σ S (2.43 σ E = σ n (2.44 ε 0 n DS 2.7 2.4.1 100 V m 1 6400 km σ = ε 0 E

2.4 83 Q = 8.85 10 12 100 4π(6.4 10 6 2 = 4.6 10 5 A s 2.4.2 A q 1 q 2 q 1 q 1 2.4.3 d ρ 1. x E(x 2. f = d 0 ρe(xdx 1. x x < 0 E = ρd/ε 0 x > d 0 0 < x < d E(x = (ρ/ε 0 x + C C x = 0 0 < x < d E(x = ρ (x d ε

84 2 I 2. f = d 0 ρe(xdx = ρ2 ε d 0 (x ddx = ρ2 d 2 2ε 2.4.2 ρ( r 1 2 0 1 ϕ 1 2 0 ϕ 1 ( r 1 0 2 ϕ 2 ϕ 2 ( r 1 2 ϕ 1, ϕ 2 ϕ( r = ϕ 1 ( r + ϕ 2 ( r (2.45 ϕ( r 1 ϕ 1 ( r + ϕ 2 ( r = ϕ 1 + 0 ϕ( r 2 2.4.4 d q 1.

2.4 85 2. q q 3. 2 4. q 1. 2. 3. ϕ( r ϕ( r = q ( 1 4πε 0 r (0, 0, d 1 r (0, 0, d ϕ(x, y, 0 = 0 z = 0 E( r = ϕ( r = q ( (x, y, z d 4πε 0 (x 2 + y 2 + (z d 2 3/2 (x, y, z + d (x 2 + y 2 + (z + d 2 3/2 E(x, y, 0 = ϕ( r = q 2πε 0 (0, 0, d (x 2 + y 2 + d 2 3/2 σ = ε 0 E

86 2 I 4. (0, 0, z q 2 F = (0, 0, 1 4πϵ 0 (2z 2 W = d = q2 16πε 0 d q 2 4πϵ 0 (2z 2 dz 2.4.5 E 0 R 1. ϕ 0 ( r = E 0 r 2. ϕ 1 ( r = 1 p r 4πε 0 r 3 0 3. 4. 5. 1. 2. r = R n ϕ( r = E 0 R n + 1 p R n 4πε 0 R 3

2.4 87 = ( E p 0 + 4πε 0 R 3 R n 3. n = 1 p = 4πε 0 R 3 E0 ϕ( r = E 0 r + 1 4πε 0 R 3 E0 r 4πε 0 r 3 = (1 R3 E r 3 0 r 4. r R ϕ( r = 3R 3 r 4 r E r 0 r (1 R3 E r 3 0 r = R n ϕ( r = 3( E 0 n n ε 0 5. 0 E p = 4πε 0 R 3 E0 2.4.6 d λ

88 2 I x = 0 E r = (0, y, z 2πl r (0, 0, d ε 0 E l = λl E l ( r = λ r (0, 0, d 2πε 0 r (0, 0, d 2 = λ 2πε 0 (0, y, z d y 2 + (z d 2 E m E m ( r = λ r (0, 0, d 2πε 0 r (0, 0, d 2 = λ (0, y, z + d 2πε 0 y 2 + (z + d 2 z = 0 E( r = E l ( r + E m ( r = λ 2πε 0 λ π ( (0, y, z d (0, y, z + d y 2 + (z d 2 y 2 + (z + d 2 E( r = λ (0, 0, d πε 0 y 2 + d 2 d y 2 +d 2

2.4 89 2.4.3 q ϕ C Q = Cϕ (2.46 C/V F U = 1 2 qϕ = 1 2 Cϕ2 = 1 2C q2 (2.47 Q φ 2.8 1 ϕ 1 2 0 q 1, q 2 q 1 = C 11 ϕ 1, q 2 = C 21 ϕ 1 (2.48

90 2 I 1 0 2 ϕ 2 q 1, q 2 q 1 = C 12 ϕ 2, q 2 = C 22 ϕ 2 (2.49 1 2 ϕ 1, ϕ 2 ( ( ( q1 C11 C = 12 ϕ1 q 2 C 21 C 22 ϕ 2 C ij (2.50 C 12 = C 21 (2.51 2.4.7 a 2.2.12 2.2.12 a Q Q = 4πε 0 aϕ Q = 4 3 πa3 ρ 4πε 0 a 2.4.8 2 1. q 1, q 2 ϕ 1, ϕ 2 ( ( ( q1 C11 C = 12 ϕ1 q 2 C 21 C 22 ϕ 2

2.4 91 2. U = 1 2 (q 1ϕ 1 + q 2 ϕ 2 3. 1 δq 1 δu δu = U q 1 δq 1 4. δq δq 2 δu = ϕ 1 δq 1 1. ( ϕ1 ϕ 2 ( 1 ( C11 C = 12 q1 C 21 C 22 q 2 ( 1 C22 C = 12 C 11 C 22 C 12 C 21 C 21 C 11 2. ϕ 1 oϕ 2 ( q1 q 2 U = C 22q 2 1 (C 12 + C 21 q 1 q 2 + C 11 q 2 2 2(C 11 C 22 C 12 C 21 3. 4. δu = C 22q 1 C 12+C 21 2 q 2 C 11 C 22 C 12 C 21 δq δu = C 22q 1 C 12 q 2 C 11 C 22 C 12 C 21 δq 1 δu C 12 = C 21

92 2 I 2.4.9 R 2 r 1. Q 2. 3. 1. 1 U = (1/2CQ 2 = Q 2 /(8πε 0 R 2. 2 U = 2(1/2C(Q/2 2 = Q 2 /(16πε 0 R 3. 2.4.10 R 1 R 2 R 3 q 1, q 2 E(r = { q1 4πϵ 0r R 2 1 < r < R 2 R 3 < r q 1 +q 2 4πϵ 0 r 2

2.4 93 R 1, R 3 ϕ 1, ϕ 2 ϕ 2 = q 1 + q 2 R 3 4πϵ 0 r 2 dr = q 1 + q 2 4πϵ 0 R 3 R2 q 1 ϕ 1 = ϕ 2 + R 1 4πϵ 0 r 2 dr = q 1 + q 2 4πϵ 0 R 3 + q 1 4πϵ 0 ( 1 R 1 1 R 2 q 1, q 2 ( q1 q 2 ( R1 R 2 R = 4πϵ 2 R 1 R 1R 2 R 2 R 1 0 R1R2 R 2 R 1 R 3 + R1R2 R 2 R 1 ( ϕ1 ϕ 2 2.4.11 S d σ +Q A -Q B 2.9 E = σ ε 0 ϕ ϕ = B A Edx = σd ε 0 = d ε 0 S Q

94 2 I Q = σs C = ε 0S d 2.4.12 R 1 R 2 2.4.10 q 1 = q 2 = q ϕ 1 ϕ 2 = q ( 1 1 4πϵ 0 R 1 R 2 C = 4πϵ 0R 1 R 2 R 2 R 1 2.4.13 A 3 a, b, c d 1, d 2 a c b Q a c ab cb E ab d 1 = E cb d 2 b a b c σ ab = ε 0 E ab, σ cb = ε 0 E cb b Q = (σ ab + σ cb A d 2 d 1 σ ab = Q, σ cb = Q, A d 1 + d 2 A d 1 + d 2 E ab = Q d 2, E cb = Q d 1 ε 0 A d 1 + d 2 ε 0 A d 1 + d 2

2.4 95 2.4.4 ϕ q = Cϕ q q + q ϕ q 0 q W U = q 0 ϕdq = q 0 q C dq = 1 q 2 2 C (2.52 φ φ+ φ φ q q+dq q φ q Dq 2.10 U = 1 2 ε 0 ( 2 q Ad = 1 ε 0 A 2 ε 0E 2 Ad (2.53 A, d Ad u 0 u 0 = U Ad = 1 2 ε 0E 2 = 1 2 DE = 1 2ε 0 D 2 (2.54

96 2 I 2.4.14 q R E( r = q r 4πϵ 0 r 3 u( r = (1/2ε 0 E( r 2 U = R = q2 8πε 0 R q 2 32π 2 ε 0 r 4 4πr2 dr 2.4.15 2.4.14 R R 1 R 2 R2 U = 32π 2 ε 0 r 4 4πr2 dr ( = q2 1 1 8πε 0 R 1 R 2 R 1 q 2 2.4.16

2.4 97 2 ±q 2d q d 2 0 2.4.17 1. 2. 2 1. A d C = ε 0 A/d Q U(d = dq2 2ε 0 A F (d = d U(d = Q2 2ε 0 A d 2. E = σ/ε 0 = Q/(Aε 0 F = (1/2E(σA = Q 2 /(2ε 0 A 1/2 2

98 2 I 2.4.5 C 1, C 2,..., C n C C = C 1 + C 2 + + C n (2.55 Q V = Q + Q + + Q ( 1 = Q + 1 + + 1 (2.56 C 1 C 2 C n C 1 C 2 C n 1 C = 1 + 1 + + 1 (2.57 C 1 C 2 C n 2.5 2.5.1 2 1 H + 1 F 1 F = 1.602 10 19 6.02 10 23 = 9.64 10 4 C

2.5 99 2.5.2 S 0 S i( r d S = 0 (2.58 i( r = 0 (2.59 2.5.3 I ϕ I = ϕ R R (2.60 R l S R = ρ l S (2.61 ρ σ i( r = σ E( r (2.62

100 2 I 2.5.1 σ 1, σ 2 i E 1, E 2 σ 1 E 1 = σ 2 E 2 σ Sε(E 2 E 1 = Sσ ( 1 σ = εi 1 σ 2 σ 1 2.5.2 2 2 4 2.5.4 i( r = 0 (2.63

2.5 101 E( r = 0 (2.64 i( r = σ E( r (2.65 D( r = 0 E( r = 0 D( r = ε 0E( r D i, σ ε 0 *1 E 2.5.3 R 1, R 2 L σ I 1. r 2. 3. 4. 1. r i(r = I/(2πrL *1

102 2 I 2. E(r = i(r/σ 3. ϕ = R2 R 1 = I 2πLσ E(rdr = I 2πLσ log R2 1 R 1 ( R2 r dr R 1 4. R = ϕ/i = 1 2πLσ log ( R2 R 1 2.5.4 2 2.5.5 σ 2 a R a ϕ 2 ±q ±q/(4πϵ 0 a ϕ = 2q/(4πϵ 0 a

2.6 103 ε 0 σ, q I I = 2πσa ϕ R = 1/(2πσa 2.6 2.6.1 q m q m F = 1 q m q m r 4πµ 0 r 2 r, (2.66 µ 0 Wb=J/A H q m F = q m H 2.6.2 q m q m l p m p m p m = q m l S

104 2 I N H N = p m H (2.67 l q m N q S -q m H 2.11 2.6.3 B q v F = q v B (2.68 v E F = q( E + v B (2.69 v = 0

2.6 105 E = E + v B B I dl df t d l = tdl df = Id l B (2.70 Id l = i q i v i dl z y f v B x 2.12 2.6.1 z B = (0, 0, B v = (0, v, 0 q < 0

106 2 I m qvb 1 2 mv2 y v R F F v x 2.13 R ω v R a Rω 2 Rω 2 = q vb 2πR R T = m v R = mv q B, T = 2πm q B (2.71 2.6.2 a I B N

2.6 107 n ABCD z AB = a i, BC = a j, CD = a i, DA = a j AB CD BC DA N = r F 0 N = (0, a/2, 0 (ai i B + (a/2, 0, 0 (ai j B + (0, a/2, 0 ( ai i B + ( a/2, 0, 0 ( ai j B ( = Ia2 j (0, B z, B y + i (B z, 0, B x 2 j (0, B z, B y + i (B z, 0, B x = Ia 2 ( B y, B x, 0 = Ia 2 n B a ( b c ( a b c 2.6.3 5 cm 0.8 A 0.02 N 1 IB a F = IBa B = 0.02/(0.8 0.05 = 0.5 T

108 2 I 2.6.4 a I xy xy B( r a xy r 0 = (x 0, y 0, 0 B z ( r = B z ( r 0 + [ x B z ( r] r= r0 (x x 0 + [ y B z ( r] r= r0 (y y 0 y 2 x y0+a/2 y 0 a/2 y0 +a/2 y 0 a/2 IB z (x 0 + a/2, y, 0dy ( IB z (x 0 a/2, y, 0dy F x ( F x = I B z (x 0, y 0, 0 + x B z (x 0, y 0, 0 a a ( 2 I B z (x 0, y 0, 0 x B z (x 0, y 0, 0 a a 2 = I[ x B z ( r] r= r0 a 2 x 2 y F y F y = I[ y B z ( r] r= r0 a 2 2.6.5

2.6 109 E = i B nq n, q B i = nq v F = q v B E ( 0 = q E + v B E = v B = i B nq 2.6.4 - I d s r d B d B = µ 0I 4πr 2 d s r r (2.72 B( r = µ 0 i( r ( r r 4π r r 3 dv (2.73

110 2 I z y I ds r q x dh 2.14 H = 1 µ 0 B H( r = 1 i( r ( r r 4π r r 3 dv (2.74 2.6.6 (0, 0, (0, 0, I dz k (a, 0, 0 d H d H = I 4πr 2 (dz k r r k

2.6 111 r = (a, 0, z d H(a, 0, 0 = I (a, 0, z 4π (a 2 + z 2 (dz(0, 0, 1 a2 + z 2 (a, 0, 0 d H = H d H x z y H H = = I 4π Ia 4π(a 2 + z 2 π/2 = I 4πa π/2 π/2 π/2 3/2 dz a a 3 (1 + tan 2 θ 3/2 cos θdθ adθ cos 2 θ z = a tan θ H = I 2πa 2.6.7 a I O H xy H( r = I t( r ( r r 4π r r 3 ds r = (a cos θ, a sin θ, 0 t( r = ( sin θ, cos θ, 0 r = (0, 0, z ds = adθ

112 2 I θ 0 2π H( r = I ( sin θ, cos θ, 0 ( a cos θ, a sin θ, z 4π ( ( a cos θ2 + a 2 sin 2 θ + z 2 ds 3/2 = I 2π (z cos θ, z sin θ, a adθ 4π 0 (z 2 + a 2 3/2 I = 4π (z 2 + a 2 3/2 2π Ia 2 = (0, 0, 1 2 (z 2 + a 2 3/2 0 (z cos θ, z sin θ, aadθ 2.6.8 2.6.9 a xy y I z (0, 0, z (0, y, 0 x a/2 a/2 2.6.6 z (0, 0, z d H = (I/ady 2π y 2 + z 2

2.6 113 H(z d H = (I/ady 2π (1, 0, 0 (0, y, z = Idy (0, z, y y 2 + z 2 y2 + z 2 2πa (y 2 + z 2 H(z = d H = I a/2 (0, z, y 2πa a/2 (y 2 + z 2 dy z 0 y = z tan θ, z tan θ 0 = a/2 I H( r = (0, z, 0 2πaz = (0, 1, 0 Iθ 0 πa θ0 θ 0 dθ 2.6.5 a I n r(r a H( r = 1 4πr 3 ( m 3( m r r r 2 (2.75 m = IS n (S S = πa 2 *2 n = (0, 0, 1 r xz H( r = I t( r ( r r 4π r r 3 ds *2 m MKSA SI

114 2 I r = (a cos θ, a sin θ, 0 t( r = ( sin θ, cos θ, 0 r = (x, 0, z ds = adθ θ 0 2π H( r = I ( sin θ, cos θ, 0 (x a cos θ, a sin θ, z 4π ( (x a cos θ2 + a 2 sin 2 θ + z 2 ds 3/2 = I (z cos θ, z sin θ, x cos θ + a ds 4π (x 2 + z 2 + a 2 3/2 2ax cos θ ( 1 + I (z cos θ, z sin θ, x cos θ + a 4π (x 2 + z 2 + a 2 3/2 H x ( r = I 2π 0 z cos θ ( 1 + 3ax cos θ x 2 +z 2 +a 2 adθ 4π (x 2 + z 2 + a 2 3/2 3xz Iπa 2 x = 2 +z 2 +a 2 Iπa2 3xz 4π (x 2 + z 2 + a 2 3/2 r 3 r 2 I ( 2π 3ax cos θ z sin θ 1 + 0 x H y ( r = 2 +z 2 +a adθ 2 = 0 4π (x 2 + z 2 + a 2 3/2 H z ( r = 3ax cos θ x 2 +z 2 +a 2 ds I ( 2π 0 ( x cos θ + a 3ax cos θ 1 + x 2 +z 2 +a adθ 2 4π (x 2 + z 2 + a 2 3/2 = I 2π 0 (a 3ax2 cos 2 θ x 2 +z 2 +a 2 adθ 4π (x 2 + z 2 + a 2 3/2 = Iπa 2 x 2 + 2z 2 4π (x 2 + z 2 + a 2 3/2 x 2 + z 2 + a 2 Iπa2 x 2 + 2z 2 r 3 r 2

2.6 115 2.75 n = (0, 0, 1 z y 0 2.6.6 C H d r C H d r = 0 C I C I C (2.76 I C dh d r 2.15 H( r d r = i( r ds (2.77 C S

116 2 I H( r = i( r (2.78 2.6.10 R I H(r r { 0 r < R H(r = r > R I 2πr 2.6.11 H = 0 H 1. H = ( A(x 2 + y 2 y, A(x 2 + y 2 x, A 2. A (z > d H x = A z (d z d d A (z < d, H y = A, H z = 0 1. H = 2Axy + 2Axy + 0 = 0

2.6 117 2. H = i j k x y z A(x 2 + y 2 y A(x 2 + y 2 x A = (0, 0, 4A(x 2 + y 2 H = 0 H = 0 (z > d (0, A/d, 0 (d z d 0 (z < d, 2.6.12 j z x y yz y z 1. z 0 0 y 0 2. 1 a Ha ( Ha = ja H = j/2

118 2 I 2.6.13 n I 1 a 2 1 1 2 0 Ha = (nai H = ni 2.6.14 R I 1. i( r 2. - 3. 1. i( r = I πr 2 2. (x, y, z (x 0, y 0, z 0 dh = i 4πr 2 (dz (x 0 x, y 0 y, z 0 zdxdy k (x0 x 2 + (y 0 y 2 + (z 0 z 2

2.6 119 = i ( (y 0 y, x 0 x, 0dxdydz 4πr 2 (x0 x 2 + (y 0 y 2 + (z 0 z 2 z (x 0, y 0, z 0 (x 0, 0, 0 y d H = i (y, x 0 x, 0dxdydz 4πr 2 (x0 x 2 + y 2 + z 2 (x, y, z d H = i ( y, x 0 x, 0dxdydz 4πr 2 (x0 x 2 + y 2 + z 2 y y y 3. H r { πr H d r = 2πrH(r = 2 i r < R I r > R H(r = 1 2 ir = 1 I 2 πr 2 r I 2πr r < R r > R

120 2 I 2.7 2.7.1 E( r E( r = ϕ( r ϕ( r E( r = 0 *3 B( r = 0 (2.79 B( r = µ 0 i( r (2.80 B( r = A( r (2.81 2.79 *4 2.80 A( r 2.7.2 E( r ϕ( r c ϕ ( r = ϕ( r + c E( r *3 *4 ( a = 0

2.7 121 χ( r ( χ( r = 0 *5 A = A + χ( r (2.82 A 2.7.3 i( r A( r 2.80 B( r = A( r ( A( r = ( 2 A( r + A( r (2.83 A( r = 0 2 A( r = µ0 i( r (2.84 A( r = µ 0 4π i( r r r dv (2.85 A( r *6 *5 *6 2 ϕ( r = ρ( r ϕ( r = 1 4π ρ( r r r dv

122 2 I A( r A( r = 0 A( r = µ 0 4π = µ 0 4π ( i( r 1 dv r r ( i( r 1 r r dv (2.86 = ( x, y, z = ( x, y, z ( i( r r r = i( r r r ( + i( r 1 r r (2.87 A( r = µ 0 4π i( r r r dv µ 0 4π ( i( r r r dv (2.88 i( r = 0 2 A( r = 0 2.7.1 1. A1 = ( By, 0, 0, 2. A2 = (0, Bx, 0 3. A3 = ( 1 2 By, 1 2Bx, 0

2.7 123 A i = (0, 0, B 2.7.2 I r A( r (0, 0, l (0, 0, l l r d ( dx log a2 + x 2 1 + x = a2 + x 2 A( r = µ 0I 4π l l (0, 0, 1 r r dz r = (x, y, z, r = (0, 0, z A x ( r = A y ( r = 0 A z ( r x x y y l z A z ( r µ 0I 4π l l 1 x2 + y 2 + (z z 2 dz = µ 0I 4π [log( x 2 + y 2 + (z z 2 + (z z] l l ( µ 0I 4π log x2 + y 2 + l 2 + l x2 + y 2 + l 2 l ( µ 0I 4π log l 1 + 1 + (x 2 + y 2 /l 2 1 x 2 +y 2 2 l µ ( 0I 4l 2 4π log x 2 + y 2

124 2 I ( = µ 0I 2π log 2l x2 + y 2 z B x = y A z ( r = µ 0I 2πr yr = µ 0I 2πr B y = y A z ( r = µ 0I 2πr xr = µ 0I 2πr l x r y r 2.7.3 R σ 0 x v 0 xy 1. 2. r r R z > 0 3. 1. σ 0 x v 0 i 0 i 0 = σ 0 v 0 2. A( r = µ 0 i 0 4π (1, 0, 0 r r dx dy

2.7 125 r R r = (0, 0, z y,z x A x (z = µ 0 i 0 4π 1 x 2 + y 2 + z 2 dx dy dx dy rdθdr 3. A x (z = µ 0 i 0 4π = µ 0 i 0 2 = µ 0 i 0 2 2π 0 dθ R 0 [ r2 + z 2 ] R 1 rdr r2 + z 2 0 ( R2 + z 2 z B x (z = y A z z A y = 0 B y (z = z A x x A z = µ ( 0 i 0 z 2 R2 + z 1 2 µ 0 i 0 2 B z (z = x A y y A x = 0 2.7.4 R σ 0 v 0 1. 2.

126 2 I 1. R σ r E(r = 0 (r < R E(r = σr/(ε 0 r (r > R { σr ε ϕ(r = 0 log R (r < R σr ε 0 log r (r > R z ε 0 1/µ 0, σ I/(2πR A z (r = 2. B = A { µ 0 I 2π µ 0I 2π log R (r < R log r (r > R 2.7.5 - x y,z B x = y A z ( r z A y ( r ( i z ( r 1 y = µ 0 4π = µ 0 4π = µ 0 4π ( i z ( r (y y r r i y( r 1 z r r dv r r 3 i y( r (z z r r 3 dv { i( r ( r r } x r r 3 dv

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