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1 B () Ver 2014 0 2014/10 2015/1 http://www-cr.scphys.kyoto-u.ac.jp/member/tsuru/lecture/... 1. ( ) 2. 3.

3 1 7 1.1..................................................... 7 1.2............................................. 7 1.2.1................................................... 7 1.2.2 1g.......................................... 8 1.2.3 1 cgs........................................... 8 1.3..................................... 8 1.3.1................................................. 8 1.3.2 2........................ 9 1.3.3 2.................................. 9 1.3.4 n....................... 9 1.4 ( )................................................ 10 1.4.1................................................... 10 1.4.2................................................. 10 1.4.3............................................. 10 1.4.4............................................ 11 1.5 ( )................................................ 11 1.5.1................................................... 11 1.5.2............................................. 12 1.5.3 1:............................... 12 1.5.4 2:............................. 12 2 13 2.1 ( )..................................... 13 2.1.1............................................... 13 2.1.2......................................... 13 2.1.3................................... 14 2.2............................................. 14 2.2.1 :........................ 14 2.3................................................. 15 2.4 : 1 ( ).............................. 16 2.4.1.................................... 16 2.4.2................................... 17 2.4.3.................................... 18 2.5.......................................... 18 2.5.1 :............................ 18 2.5.2 :......................... 19 2.5.3 :.................... 20 2.6 (?): 2 ( ).................... 20 2.6.1................................... 20

4 2.6.2 : 2 ( )................. 21 2.7............................................ 22 2.7.1.................................. 22 2.7.2............................................ 23 2.7.3............................. 24 2.7.4 : gradient ( )....................... 25 2.8................................................. 27 2.8.1................................... 27 2.8.2............................. 28 2.9................................................... 28 2.10........................................ 29 2.11........................................... 29 3 31 3.1............................................. 31 3.1.1................................................... 31 3.1.2 (div ).............................. 32 3.2 2 ( ).................................. 33 3.2.1 (rot )................................... 35 3.3 ( )................................... 36 3.3.1............................................... 36 3.3.2.................................................. 36 3.3.3 ( ).......................................... 37 3.3.4 ( )........................................ 37 3.4................................................ 38 3.4.1............................... 38 3.4.2 :........................ 38 3.4.3 1................................ 39 3.4.4 2.................................... 39 3.5........................ 40 4 43 4.1....................................................... 43 4.1.1.............................................. 43 4.1.2............................................. 43 4.1.3......................................... 43 4.1.4................................................. 45 4.2.................................................. 45 4.2.1............................... 45 4.2.2......................................... 45 4.2.3.................................. 46 4.2.4................................ 46 4.2.5.......................................... 47 4.2.6..................................... 47 4.3 ( )........................................... 47 5 49

5 5.1............................................. 49 5.1.1............................................. 49 5.1.2 ( ).................................. 49 5.2.................................................. 49 5.3.......................................... 50 6 53 6.1 :............................................ 53 6.1.1 ( ).................................. 53 6.1.2 :............................. 53 6.1.3 :.................................. 53 6.2 :................................. 54 6.2.1.................................................. 54 6.2.2....................................... 54 6.3 : ( )........................ 55 6.3.1.................................. 55 6.3.2 : ( B i )................ 55 6.3.3 ( ) (?).................................. 56 6.3.4............................................ 56 6.4............................................. 57 6.4.1 A ( A B )............................. 57 6.4.2 ( A B i ).................. 58 6.4.3 ( i A )..................... 58 6.4.4 : - ( i A B )................ 59 6.4.5....................................... 60 6.5............................................ 62 6.6........................................... 62 6.6.1.............................. 62 7 65 7.1 :.................................... 65 7.1.1............................................. 65 7.1.2 :............................ 65 7.1.3.................................... 65 7.2 :................................. 67 7.2.1.............................................. 67 7.2.2............................................. 67 7.3.......................................... 67 8 : 69 8.1........................................... 69 8.2 :......................... 69 8.2.1.............................. 69 8.2.2................................... 70 8.3................................................... 71 8.4 -:................... 72 9 75

6 9.1............................................. 75 9.1.1.................................... 75 9.1.2............................... 75 9.2...................................................... 76 9.3...................................................... 77 9.4................................................ 78 9.5.................................................. 79 9.5.1 grad............................................... 79 9.5.2 div................................................ 79 9.5.3 rot................................................ 79 9.5.4................................................... 79 9.5.5.................................................. 79 9.6..................................................... 80

7 1 1.1 1 e C e = 1.60217733 10 19 [C] (1.1) 1g N A = 6.02 10 23 (1.2) e N A = 9.6 10 4 [C] (1.3) 1.2 1.2.1 R 12 2 q 1 q 2 F 12 F 12 F 12 = k q 1q 2 R 2 12 (1.4) 1[N] = 1[kg m s 2 ] (1.5) MK MK [m] [kg] [sec] [A] (1.6) 1A 1 1[C] 1[A] 1[sec] (1.7) MKA k k k k = 1 4πε 0 ε 0 (permittivity of vacuum) ε 0 = 8.854 10 12 [C 2 N 1 m 2 ] (1.9) 4π 4πR 2 1m 2 F 12 1 e 2 F 12 = 4πε 0 R12 2 = 1 ( 1.60 10 19 C ) 2 4πε 0 (1m) 2 = 2.3 10 28 [N] (1.10) (1.8)

8 1 1.2.2 1g (FY2013/1 : 2013/10/03 ) 1g 10 23 6.02 10 23 3 10 23 1g 1g 1g q = 6.02 10 23 1.60217733 10 19 = 9.6 10 4 [C] (1.11) 2 R = 1m F = 1 4πε 0 q 2 R 2 = 8.3 1019 [N] (1.12) 1[kg] 9.8[N] 10 19 10 19 [kg] 1 2000[km] 1[g] 1.2.3 1 cgs MKA cgs-gauss [cm] [g] [sec] (esu) e e = 4.803 10 10 [esu] (1.13) Gauss B MKA cgs-gauss 1m = 100cm 2 F 12 ( F 12 = e2 4.80 10 10 [esu] ) 2 R12 2 = (100[cm]) 2 = 2.3 10 23 [dyn] = 2.3 10 28 [N] (1.14) 1.3 1.3.1 3 A B A = (A x, A y, A z ), B = (Bx, B y, B z ) (1.15) A A = (A 2 x + A 2 y + A 2 z) 1/2 (1.16) A B A + B = (A x + B x, A y + B y, A z + B z ) (1.17) A + B 2 A B = (A x B x, A y B y, A z B z ) (1.18) B A

1.3. 9 1.3.2 2 (x, y, z) r r = (x, y, z) (1.19) 2 P 1 P 2 r 1 r 2 P 2 P 1 R 12 R 12 = r 1 r 2 (1.20) R 12 2 ( )R 12 R 12 = [ (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2] 1/2 (1.21) n 12 n 12 = R 12 R 12 n 12 1 (1.22) 1.3.3 2 P 1 P 2 () q 1 q 2 q 1 q 2 q 1 q 2 F 12 F 12 = 1 4πε 0 q 1 q 2 r 1 r 2 2 n 12 (1.23) F 12 = q 1q 2 4πε 0 n 12 r 1 r 2 2 n 12 = R 12 R 12 = r 1 r 2 r 1 r 2 (1.24) (1.25) F 12 = q 1q 2 4πε 0 r 1 r 2 r 1 r 2 3 ( F 12 = q 1q 2 x 1 x 2 4πε 0 [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 ], y 1 y 2, (1.27) 3/2 [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 3/2 ] z 1 z 2 [(x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 ] 3/2 ) (1.26) (1.28) 1.3.4 n n q 1, q 2,..., q n r 1, r 2,..., r n q, r F 1, F 2,..., Fn F n F = F i = 1 n qq i ( r r i ) 4πε 0 r r i 3 i=1 i=1 (1.29)

10 1 2 Coulomb i = 1 n n i F i n n q i q j r ij F i = F ij = 4πε 0 r ij 3 j i j i r ij = r i r j (1.30) 1.4 ( ) 1.4.1 2 A B θ ( ) A = A, B = B, A B AB cos θ (1.31) (1.32) θ A B cos θ A B A B = B A A ( B + C) = A B + A B (1.33) (1.34) 0 A A = A 2 = A 2 (1.35) 1.4.2 OP 1 P 2 O θ P 2 P 1 2 = OP1 2 + OP2 2 2OP1 OP 2 cos θ (1.36) OP 1 = r 1, OP 2 = r 2, P 2 P 1 = R 12 R 2 12 = R 12 2 = ( r 1 r 2 ) ( r 1 r 2 ) (1.37) = r 1 r 1 + r 2 r 2 2 r 1 r 2 (1.38) = r 2 1 + r 2 2 2 r 1 r 2 = r 2 1 + r 2 2 2r 1 r 2 cos θ (1.39) 1.4.3 A = (A x, A y, A z ), B = (B x, B y, B z ) x,y,z i, j, k A = (A x, A y, A z ) = A x i + A y j + A z k (1.40) B = (B x, B y, B z ) = B x i + B y j + B z k (1.41)

1.5. ( ) 11 A B = (A x i + A y j + A z k) (B x i + B y j + B z k) (1.42) = A x B x i i + A y B y j j + A z B z k k (1.43) +(A x B y + A y B x ) i j + (A y B z + A z B y ) j k + (A z B x + A x B z ) k i (1.44) = A x B x + A y B y + A z B z (1.45) 1.4.4 F s θ W W = F s cos θ (1.46) F s W = F s (1.47) 1.5 ( ) 1.5.1 2 A B θ ( ) C C = A B C = A B = AB sin θ (1.48) (1.49) C A B C A B A B A B C x,y,z x,y,z A, B, C A B = B A (1.50) A ( B + C) = A B + A C (1.51) θ = 0 A A = 0 (1.52) i, j, k i i = j j = k k = 0 i j = k, j k = i, k i = j (1.53)

12 1 1.5.2 A = (A x, A y, A z ) = A x i + A y j + A z k (1.54) B = (B x, B y, B z ) = B x i + B y j + B z k (1.55) A B = (A x i + A y j + A z k) (B x i + B y j + B z k) (1.56) = A x B x i i + A y B y j j + A z B z k k (1.57) +A x B y ( i j) + A y B x ( j i) + A y B z ( j k) (1.58) +A z B y ( k j) + A z B x ( k i) + A x B z ( i k) (1.59) = (A y B z A z B y ) i + (A z B x A x B z ) j + (A x B y A y B x ) k (1.60) ( A B) x = A y B z A z B y ( A B) y = A z B x A x B z ( A B) z = A x B y A y B x (1.61) (1.62) (1.63) (FY2009/1 : 2009/10/1 ) 1.5.3 1: R F θ N N = R F N = R F sin θ (1.64) (1.65) N R F N R F 1.1: 1.5.4 2: ( A B) C = ( B C) A = ( C A) B (1.66) 3

13 2 2.1 ( ) 2.1.1 q q 1 F = qq 1 4πε 0 r r 1 r r 1 3 (2.1) E( r) E( r) = q 1 4πε 0 r r 1 r r 1 3 (2.2) F = q E( r) (2.3) E( r) r = (x, y, z) E( r) E x (x, y, z) = E y (x, y, z) = E z (x, y, z) = q 1 x x 1 4πε 0 [(x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 ] 3/2 (2.4) q 1 y y 1 4πε 0 [(x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 ] 3/2 (2.5) q 1 z z 1 4πε 0 [(x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 ] 3/2 (2.6) E ( ) [N C 1 ] [C] = [A sec] [N] = [m kg sec 2 ] [m kg sec 3 A 1 ] [V] [V m 1 ] [N C 1 ] = [m kg sec 3 A 1 ] = [V m 1 ] (2.7) (FY2013/1 : 2013/10/03 ) 2.1.2 n q i r r E( r) n i = 4πε 0 r r i 3 = E i ( r) (2.8) E i ( r) = i=1 q i 4πε 0 r r i r r i 3 i=1 E i ( r) r i i r ( E i ( r, r i ) ) (2.9)

14 2 2.1.3 r ρ( r )[C m 3 ] r V [m 3 ] ρ( r ) V r ρ( r ) V 4πε 0 r r r r 3 r i, V i E( r) = i ρ( r i ) V i 4πε 0 r r i r r i 3 (2.10) (2.11) V V dv E( r) = 1 r r 4πε 0 r r 3 ρ( r )dv (2.12) V 3 xyz V = x y z (2.13) dv = dx dy dz (2.14) x E x (x, y, z) = 1 (x x )ρ(x, y, z ) 4πε 0 [(x x ) 2 + (y y ) 2 + (z z ) 2 ] 3/2 dx dy dz (2.15) (x 0, y 0, z 0 ) (x 1, y 1, z 1 ) E x (x, y, z) = 1 4πε 0 z1 y1 x1 z 0 y 0 (x x )ρ(x, y, z ) x 0 [(x x ) 2 + (y y ) 2 + (z z ) 2 ] 3/2 dx dy dz (2.16) 2.2 2.2.1 : (FY2013/2 : 2013/10/17 ) r ( ) λ s s r θ 0 s r λ s cos θ 4πε 0 (r 2 + s 2 ) = λ s 4πε 0 r (r 2 + s 2 ) 3/2 (2.17) s = E(r) = λ r 4πε 0 (r 2 + s 2 ) 3/2 ds (2.18)

2.3. 15 s = r tan θ r 2 + s 2 = r 2 (1 + tan 2 θ) = r2 cos 2 θ = r2 sec 2 θ (2.19) 2s ds = r 2 ( 2) sin θ cos 3 θ dθ (2.20) s ds = r 2 sin θ cos 3 θ dθ = 1 s r2 cos 2 tan θ dθ = r θ cos 2 dθ (2.21) θ ds = r sec 2 θdθ (2.22) s = θ = π/2 π/2 E(r) = = π/2 π/2 λ 1 4πε 0 r λ 4πε 0 π/2 π/2 r r 3 sec 3 θ r sec2 θdθ (2.23) cos θdθ = λ 2πε 0 r (2.24) Δs A s 0 r θ E P 2.1: 2.3 (electric line of force) q r E(r) = q ε 0 1 4πr 2 (2.25) E(r) 4πr 2 = q ε 0 (2.26) q q ( ) 2.2( ) q N N q ε 0 (2.27) q N q/ε 0 1 2.2 1 1

16 2 2.2: 1 ( ) ( ) 2.4 : 1 ( ) 2.4.1 2.3( ) q ( ) N = q/ε 0 2.3( ) 2.3( ) N = q/ε 0 2.3( ) 0 ( ) = ()/ε 0 (2.28) 2.3:

2.4. : 1 ( ) 17 2.4.2 2.4 2.4 N (= ) N ( ) 2.4( ) = cos θ (2.29) θ ( E ) n 2.4: N/ N/ E = N = N cos θ n (2.30) N = E = E cos θ = ( E n) = E n (2.31) ( E n E n ) N N = N = ( E n) (2.32) q q/ε 0 N N = N = ( E n) = q ε 0 (2.33) N = dn = E { n}d = q (2.34) ε 0 r N = dn = { E( r) n( r)}d = q (2.35) ε 0

18 2 f( r)d (2.36) f( r) () r 3 (x 0, y 0 ) (x 1, y 1 ) y1 x1 f( r)d = f(x, y)dxdy (2.37) y 0 x 0 1 q V N = dn = { E( r) n( r)}d = q = 1 q i = 1 ρ( r)dv (2.38) ε 0 ε 0 ε 0 i V r 2 r V 3 0 0 { E( r) n( r)}d = 0 (2.39) (FY2013/2 : 2013/10/17 ) 2.4.3 D(electric flux density) D( r) ε 0 E( r) (2.40) D( r) n( r)d = ρ( r)dv V (2.41) E D 2.5 2.5.1 : λ[c m 1 ] r E(r) N = λl ε 0 l[m] λl N r l r (2.42) (r) = 2πr l (2.43)

2.5. 19 r E(r) E = N = λl/ε 0 2πr l = λ 2πε 0 r (2.44) { E( r) n( r)}d = 1 ρ( r)dv ε 0 V (2.45) l r (r) { E( r) n( r)}d = E = E 2πrl 1 ρ( r)dv = λl ε 0 ε 0 V E 2πrl = λl E(r) = ε 0 λ 2πε 0 r (2.46) (2.47) (2.48) (2.49) 2.5.2 : R Q 4π 4π r E( r) n( r)d = E(r) 4πr 2 = 1 ρ( r)dv (2.50) ε 0 r V r < R r E( r) n( r) = E(r) 4πr 2 = 1 ρ( r)dv = 0 (2.51) ε 0 V E(r) = 0 (r < R) (2.52) r > R r Q 1 ρ( r)dv = Q ε 0 ε 0 V E( r) n( r) = E(r) 4πr 2 = Q (2.54) ε 0 E(r) = (2.53) Q 4πϵ 0 r 2 (r > R) (2.55) Q

20 2 2.5.3 : R ρ r r ρ E(r) = 1 ε 0 ρ 4πr3 3 1 4πr 2 = ρ 3ε 0 r (r < R) (2.56) R E(r) = 1 ε 0 ρ 4πR3 3 1 4πr 2 = ρr3 3ε 0 r 2 (r > R) (2.57) 2.6 (?): 2 ( ) 2.6.1 2.5 (a) Q q 1 q O P A B W = F cos θ s, W = qe cos θ s F = qe (2.58) (2.59) QO QP QA QB P A B q A B W s = s cos θ W = F s, F = qe (2.60) W = qe s = qe s cos θ = W (2.61) A B A B OP OP q O P W O P W O P 2.5(a) (b) RR R P O P OP r t( r) E( r) OP W W = q ( E t) s = q ( E( r) t( r))ds (2.62) OP OP q ( E( r) t( r))ds (2.63) OP OP OP

2.6. (?): 2 ( ) 21 q 1 E(r) = q 1 4πε 0 1 r 2 q 1 r Q O r 0 Q P P r E t r [ ( E( r) t( r))ds = ( E( r) q 1 1 t( r))ds = OP OP r 0 4πε 0 s 2 ds = q ] r0 1 1 = q ( 1 1 4πε 0 s 4πε 0 r 1 ) (2.65) r 0 W q W = qq ( 1 1 4πε 0 r 1 ) r 0 r (2.64) (2.66) s θ t E F 2.5: Q q 1 q O P (a) O P (b) Q O P 2.6.2 : 2 ( ) 2.6 C OA P OAP O O C 0 ( E( r) t( r)) ds = 0 (2.67) C Faraday (FY2009/3 : 2009/10/29 )

22 2 O A A' C P 2.6: C 2.7 2.7.1 O q P U( r) = q ( E( r ) t( r ))ds (2.68) OP q ( ) φ( r) = ( E( r ) t( r ))ds (2.69) OP (electostatic potential) V ev J (FY2013/3 : 2013/10/24 ) r E( r) = q 1 4πε 0 r 2 r = r φ(r) = ( E( r ) t( r ))ds = E(s)ds = OP r [ ] r q 1 q1 4πε 0 s 2 ds = = q 1 4πε 0 s 4πε 0 r : ds r r dr ds ds = dr (2.70) (2.71) r 1 q 1 O r φ( r) = q 1 4πε 0 r r 1 (2.72)

2.7. 23 r 1, r 2,... r n, n φ( r) = n i=1 q i 4πε 0 r r i ρ( r ) φ( r) = 4πε 0 r r dv V (2.73) (2.74) 2.7.2 2 +q q 2.7 0 2.7: 2 +q q 2.2( ) (FY2010/3 : 2010/10/21 )

24 2 2.7.3 ( ) 2.8 s P P PP t E P P φ P φ P q ( qφ P qφ P = qe ) t s (2.75) φ P φ P = ( E ) t s (2.76) φ E t = φ s = dφ(s) ds (2.77) t t P P P x P x E x = E i = φ x φ(x, y, z) E x (x, y, z) = x E x (x, y, z) (x, y, z) x φ P P' (2.78) (2.79) φ P E t P Δs (FY2009/4 : 2009/11/5 ) 2.8: ( E(x, y, z) = E( r) φ = (E x, E y, E z ) = x, φ y, φ ) = φ( r) = grad φ( r) (2.80) z

2.7. 25 Gradient nabla x ( ) φ x, y, z φ(x, y, z) x φ φ(x, y, z) φ(x + x, y, z) φ(x, y, z) = lim x x x 0 x 3 2 y z x φ (2.81) φ(x, y, z) = x 2 y 5 z 3 φ(x, y, z) x = 2xy 5 z 3, φ(x, y, z) y = 5x 2 y 4 z 3, φ(x, y, z) z (2.82) = 3x 2 y 5 z 2 (2.83) 1 f( r) f( r 0 + r) = f(x 0 + x, y 0 + y, z 0 + z) (2.84) [ ] [ ] [ ] f(x, y, z) f(x, y, z) f(x, y, z) f(x 0, y 0, z 0 ) + x + y + z (2.85) x r= r 0 y r= r 0 z r= r 0 r x, y, z 3 r(x, y, z) f r f(r) f(r) r r x, y, z 3 f(r) x, y, z f(x, y, z) f(r) x = df(r) r(x, y, z) dr x (2.86) r 2 x 2 + y 2 + z 2 r x = 1 ( x 2 + y 2 + z 2) 1/2 x 2x = 2 r (FY2013/4 : 2013/10/31 ) (2.87) (2.88) : nabla ( x, y, ) z ( φ x, φ y, φ ) = φ = grad φ z grad gradient φ (2.89) (2.90) (FY2011/3 : 2011/10/27 ) 2.7.4 : gradient ( ) ( 2.9) ( ) ( ) = ( ) (2.91)

26 2 (x, y) h(x, y) x h/ x y h/ y ( h = x, h ) y h(x, y) (2.92) h(x, y) φ( r) E( r) 2 3 ( φ E = x, φ y, φ ) = φ z (2.93) h(x, y) h(x, y) (FY2013/5 : 2013/11/7 ) 2.9: : q 1 r = (x, y, z) E φ (x, y, z) r φ( r) = φ(r) = q 1 1 4πε 0 r (2.94) E = φ ( φ(x, y, z) (E x, E y, E z ) =, x r = r(x, y, z) = x 2 + y 2 + z 2 φ(x, y, z), y ) ( φ(x, y, z) r = z x dφ(r) dr, r y dφ(r) dr, r z ) dφ(r) dr (2.95) (2.96) (2.97) r x = x r, dφ(r) dr r y = y r, = q 1 4πε 0 1 r 2 r z = z r (2.98) (2.99) (2.100)

2.8. 27 ( ) q1 x E = (E x, E y, E z ) = 4πε 0 r 3, q 1 y 4πε 0 r 3, q 1 z 4πε 0 r 3 (2.101) (FY2010/4 : 2010/10/28 ) 2.8 2.8.1 2 2 q 1 q 2 r 12 q 1 q 2 r 12 q 2 U = q 1q 2 q 1 q 2 = 4πε 0 r 12 4πε 0 r 1 r 2 = q 1φ 2 ( r 1 ) = q 2 φ 1 ( r 2 ) (2.102) r 1 r 2 2 r 12 = r 1 r 2 q 2 q 1 3 q 1 q 2 q 3 q 1 q 2 U = U 12 + U 13 + U 23 = + q 1q 3 + q 2q 3 q 1 q 2 = 4πε 0 r 12 4πε 0 r 13 4πε 0 r 23 4πε 0 r 1 r 2 + q 1 q 3 4πε 0 r 1 r 3 + q 2 q 3 (2.103) 4πε 0 r 2 r 3 = q 1 φ 2 ( r 1 ) + q 1 φ 3 ( r 1 ) + q 2 φ 3 ( r 2 ) = q 2 φ 1 ( r 2 ) + q 3 φ 1 ( r 3 ) + q 3 φ 2 ( r 3 ) (2.104) n 2 2 U = 1 n q i q j 4πε 0 r i r j = 1 n q i q j 8πε 0 r i r j (i,j) i j (2.105) q i q i q i U = 1 n 1 q i 2 4πε 0 = 1 2 i=1 n i=1 n j( i) q i φ i, φ i = 1 4πε 0 q j r i r j n j( i) q j r i r j 1/2 (2.106) (2.107)

28 2 2.8.2 n U = 1 ρ( r)φ( r)dv, φ( r) = 1 ρ( r ) 2 4πε 0 r r dv (2.108) 2 U = 1 ρ( r)φ( r)dv = 1 2 2 ρ( r) ρ( r ) 4πε 0 r r dv dv = 1 2 ρ( r)ρ( r ) 4πε 0 r r dv dv (2.109) 2.9 z d 2 +q q (0, 0, +d/2) (0, 0, d/2) (x, y, z) { } φ(x, y, z) = q 1 4πε 0 [x 2 + y 2 + (z d/2) 2 ] 1 1/2 [x 2 + y 2 + (z + d/2) 2 ] 1/2 [ x 2 + y 2 + φ = qd 4πε 0 (2.110) (x, y, z) d ( z ± d ) ] 2 1/2 ( x 2 + y 2 + z 2 ± zd ) 1/2 ( x 2 + y 2 + z 2) ( 1/2 1 1 2 2 ) zd x 2 + y 2 + z 2 (2.111) z (x 2 + y 2 + z 2 ) = p z 3/2 4πε 0 r 3 (2.112) p = qd, r 2 x 2 + y 2 + z 2 (2.113) p = qd φ( r) = 1 p r 4πε r 3 (2.114) (FY2009/5 : 2009/11/12 ) (FY2011/4 : 2011/11/17 ) E( r) = φ (2.115) r 2 = x 2 + y 2 + z 2 r x = 1 ( x 2 + y 2 + z 2) 1/2 x 2x = 2 r r 2 = x 2 + y 2 + z 2 2r r x = 2x r x = x r (2.116) (2.117) (2.118) (2.119) (2.120)

2.10. 29 E x = φ x = p 3zx 4πε 0 r 5 E y = φ y = p 3zy 4πε 0 r 5 E z = φ z = p 3z 2 r 2 4πε 0 r 5 p qd (2.121) (2.122) (2.123) (2.124) (x, y, z ± d/2) ±q r = (x, y, z) { } E( r) = q (x, y, z) (0, 0, d/2) (x, y, z) (0, 0, d/2) (2.125) 4πε 0 [x 2 + y 2 + (z d/2) 2 3/2 ] [x 2 + y 2 + (z + d/2) 2 ] 3/2 [ x 2 + y 2 + (z d/2) 2] 3/2 ( x 2 + y 2 + z 2 zd ) 3/2 = r 3 E x (x, y, z) = p 3zx 4πε 0 r 5 E y (x, y, z) = p 3zy 4πε 0 r 5 E z (x, y, z) = p 3z 2 r 2 4πε 0 r 5, p qd ( 1 zd ) 3/2 ( r 2 r 3 1 ± 3 ) zd 2 r 2 (2.126) (2.127) (2.128) (2.129) (2.130) 2.10 2.11 q q 1 F = qq 1 4πε 0 r r 1 r r 1 3 (2.131) E( r) E( r) = q 1 4πε 0 r r 1 r r 1 3, F = q E( r) (2.132) q = 0, q 1 = 0 (2.133) q = 0, q 1 0 (2.134) F = 0 E( r) = 0 E( r) 0 r 0

30 2 0 2 () () 2 1 2 1 2 2 1

31 3 3.1 3.1.1 ρ( r)dv = V D( r) n( r)d, or V ρ( r) dv = E( r) n( r)d (3.1) ε 0 3.1 x,y,z x, y z ρ( r) dv = ρ( r) x y z = ρ( r) V (3.2) ε 0 ε 0 ε 0 V V = x y z (3.3) 6 E( r) n( r) x A A A n(x, y, z) = ( 1, 0, 0) (3.4) E n = E x n x + E y n y + E z n z = E x (x, y, z) (3.5) A E( r) n( r)d = E x (x, y, z) y z A A (3.6) n(x + x, y, z) = (1, 0, 0) (3.7) E n = E x n x + E y n y + E z n z = E x (x + x, y, z) (3.8) A E( r) n( r)d = Ex (x + x, y, z) y z (3.9) A A A+A E( r) n( r)d = {Ex (x + x, y, z) E x (x, y, z)} y z (3.10) = E x(x + x, y, z) E x (x, y, z) V (3.11) x y B B z C C E y (x, y + y, z) E y (x, y, z) E( r) n( r)d = V (3.12) B+B y E z (x, y, z + z) E y (x, y, z) E( r) n( r)d = V (3.13) C+C z

32 3 E( r) n( r)d = B+B C+C E( r) n( r)d = E( r) n( r)d = E( r) n( r)d (3.14) = E x(x + x, y, z) E x (x, y, z) V x (3.15) + E y(x, y + y, z) E y (x, y, z) V y (3.16) + E z(x, y, z + z) E y (x, y, z) V z (3.17) V ρ( r) dv = E( r) n( r)d ε 0 V ρ( r) = E x(x + x, y, z) E x (x, y, z) ε 0 x + E y(x, y + y, z) E y (x, y, z) y + E z(x, y, z + z) E y (x, y, z) z x 0, y 0, x 0 ρ( r) ε 0 = E x x + E y y + E z z ρ( r) = D( r), or (3.18) (3.19) (3.20) (3.21) (3.22) ρ( r) ε 0 = E( r) (3.23) div (Diversence ) ρ( r) = div D( r), or ρ( r) ε 0 = div E( r) (3.24) x,y,z x,y,z x y 0 0 3.1.2 (div ) 2 E( r) = E x x + E y y (3.25) 3.2 y E x x = 0, E y y = 0 (3.26) E = E x x + E y y = 0 (3.27)

3.2. 2 ( ) 33 z n = ( 1, 0, 0) A A (x, y, z) z O y n = (1, 0, 0) x y x 3.1: 3.2 E y y < 0 (3.28) E x x > 0 (3.29) E y E x E y y = E x x (3.30) E = E x x + E y y = 0 (3.31) 3.2 E x x > 0 E y y > 0 (3.32) E = E x x + E y y > 0 (3.33) (FY2009/6 : 2009/11/19 ) 3.2 2 ( ) (2 ) ( E( r) t( r)) ds = 0 (3.34) C

34 3 y E y y + 0 x 0 x 0 x x 3.2: : : : 3.3 x PQRP P Q s t P Q t = (0, 1, 0) (3.35) y 0 E(x, y + s, z) tds = y 0 E y (x, y + s, z)ds (3.36) R s t R t = (0, 1, 0) (3.37) y 0 E(x, y + y s, z + z) tds = y 0 E y (x, y + y s, z + z)ds (3.38) y + y s = y + s y s = s ds = ds s : 0 y, s : y 0 y 0 E y (x, y + s, z + z)ds s s P Q R y E tds = E y (x, y + s, z) E y (x, y + s, z + z)ds (3.40) PQ+R 0 y 0 s z y [ = E y (x, y, z) + E ] [ y(x, y, z) s E y (x, y, z) + E y(x, y, z) y y = 0 E y(x, y, z) zds z s + E y(x, y, z) z z (3.39) ] ds (3.41) (FY2013/6 : 2013/11/28 ) s y P Q R E tds = E y(x, y, z) z y (3.43) z PQ+R Q R P E tds = E z(x, y, z) z y y QR+P (FY2010/5 : 2010/11/11 ) x (3.42) (3.44) y z = x ( ( E( r) t( r)) Ez (x, y, z) ds = E ) y(x, y, z) x (3.45) y z

3.2. 2 ( ) 35 z (x, y, z + z) R(x, y + y, z + z) t O P(x, y, z) Q(x, y + y, z) y x 3.3: x,y,z ( E( r) t( r)) ds = y z ( E( r) t( r)) ds = x y z ( Ex (x, y, z) z ( Ey (x, y, z) x E ) z(x, y, z) y (3.46) x E x(x, y, z) y ) z (3.47) ( E( r) t( r)) ds = 0 (3.48) C x,y,z 0 ( Ez y E ) ( y Ex = 0, z z E ) ( z Ey = 0, x x E ) x = 0 (3.49) y 3 ( E = x, y, ) ( Ez (E x, E y, E z ) = z y E y z, E x z E z x, E y x E ) x = 0 (3.50) y rot E = curl E = 0 (3.51) 3.2.1 (rot ) rot 3.4 y E x E y x E x y > 0 (3.52) = 0 (3.53) E y x E x y > 0 (3.54)

36 3 3.4 y x y E y x E x y > 0 (3.55) > 0 (3.56) 2 E y x E x = 0 (3.57) y 0 E y x E x y z ( rote Ez = y E y z, E x z E z x, E y x E ) x y (3.58) (3.59) y y x rot E x rot E = 0 3.4: : y : x y 3.3 ( ) 3.3.1 b a df(x) dx dx = [f(x)]b a = f(b) f(a) (3.60) f(x) f(x) 3.3.2 F tds OP (3.61)

3.3. ( ) 37 3.3.3 ( ) ( ) ( ) D n d = ρdv ( ) ρ = D V (3.62) (3.63) ρ ( ) D n d = DdV V F V ( ) F n d = F dv V (3.64) (3.65) 3.3.4 ( ) ( ( E( r) t( r)) Ez (x, y, z) ds = x y ( ( E( r) t( r)) Ex (x, y, z) ds = y z ( ( E( r) t( r)) Ey (x, y, z) ds = x z E ) y(x, y, z) x (3.66) z E z(x, y, z) x E x(x, y, z) y ) y (3.67) ) z (3.68) x,y,z ( {( E( r) t( r)) ds = E ) } n (3.69) n d C ( {( E( r) t( r)) ds = E ) } n d (3.70) C 0 0 F C { {( F ( r) t( r)} ds = F ) ( r) C (FY2011/5 : 2011/12/1 ) } n( r) d (3.71)

38 3 3.4 3.4.1 (3.109) (3.108) E = φ (3.72) E = ( φ) (3.73) x { E( r) } x = E z( r) y E y( r) z = y 2 φ( r) y z = 2 φ( r) z y { φ( r) } { φ( r) } = 2 φ( r) z z y y z + 2 φ( r) z y { E( r) } = 0 (3.76) x y z 0 (3.74) (3.75) E = φ E = 0 φ = 0 (3.77) φ φ 3.4.2 : E ρ (3.104) φ E (3.109) ( ρ( r) = ε E( r) = ( φ( r)) = x, y, ) ( φ( r) z x, φ( r) y, φ( r) ) z = { φ( r) } + { φ( r) } + { φ( r) } x x y y z z { 2 } φ( r) = x 2 + 2 φ( r) y 2 + 2 φ( r) z 2 (Laplacian) 2 ( ) 2 2 x 2 + 2 y 2 + 2 z 2 = = (3.81) ρ( r) ε 0 = 2 φ( r) = φ( r) (3.82) (Poission s equation) (3.78) (3.79) (3.80) 2 φ( r) = 0 (3.83) (Laplace s equation)

3.4. 39 3.4.3 1 R ρ ρ( r) = ρ ( r < R), (3.84) = 0 ( r > R) (3.85) r 2 = x 2 + y 2 + z 2 2r r x = 2x r x = x r φ x = r dφ(r) = x x dr r 2 φ x 2 = r2 x 2 dφ(r) r 3 dr dφ(r) dr + x2 d 2 φ(r) r 2 dr 2 y, z 2 φ(r) = 2 φ x 2 + 2 φ y 2 + 2 φ z 2 (3.86) (3.87) (3.88) (3.89) (3.90) (3.91) = 3r2 (x 2 + y 2 + z 2 ) dφ(r) r 3 + d2 φ(r) dr dr 2 = 2 dφ(r) + d2 φ(r) r dr dr 2 = 1 d 2 [rφ(r)] (3.92) r dr2 d 2 2 φ(r) = 1 r dr 2 [rφ(r)] = ρ ε (r < R) (3.93) = 0 (r > R) (3.94) 4 r φ(r) 0 (3.95) r = 0 φ(r) r = R φ(r) r = R dφ(r) dr (3.96) (3.97) (3.98) φ(r) = ρr2 2ε 0 ρ 6ε 0 r 2 (r < R) (3.99) = ρr3 3ε 0 r (r > R) (3.100) 3.4.4 2 (pn TBW)

40 3 3.5 3 3 3.5 E( r) = 1 4πε 0 F = qe V r r r r 3 ρ( r )dv (3.101) (3.102) : ρdv = D nd V ρ = D D = ε 0E (3.103) (3.104) (3.105) ρ( r, t) = D( r, t) (3.106) C E tds = 0 (3.107) E = 0 (3.108) E = φ φ( r) = OP ( E( r ) t( r )) ds (3.109) (3.110) 3.110 O φ O = 0 ρ( r ) φ( r) = 4πε 0 r r dv V (3.111) ρ( r) ε 0 = 2 φ( r) () (3.112) U( r) = qφ( r) (3.113) (FY2009/7 : 2009/11/26 )

3.5. 41 ρ( r ) φ( r) = V 4πε 0 r r dv ρ ρ( r) ε 0 ポアソン = 2 φ( r) E( r) = 1 4πε 0 φ ガウス ρ ε 0 = E V r r E = φ r r 3 ρ( r )dv ( φ = E t) ds OP E 静電気力 F ( r) = qe( r) 位置エネルギー U = qφ 渦無し E tds = 0 C E = 0 ガウス ( D n ) d = V ρ ε 0 = E ρdv 3.5: ρ E φ 3

43 4 4.1 4.1.1 () = () (FY2010/6 : 2010/11/18 ) (FY2013/7 : 2013/12/05 ) 4.1.2 4.1 4.1 4.2 (1) (2) 0 () ( ) (3) 0 (4) 0 ( ) (5) 4.1.3 4.3 ( )

第4章 44 導体と静電場図 4.1: 一様静電場中に導体球を置いた場合の電気力線図 4.2: 一様電場中に導体球を置いた場合の静電誘導 (a) 外部から加わる電場 (b) 導体表面に誘起される電荷が作る電場 (c) 両者の合計の電場とそれが作る等電位面るガウスの法則は以下のように書くことができる { } r) n( r) d = 1 E( ρ( r)dv ε0 V 1 E = σ ε0 σ E = ε0 (4.1) (4.2) (4.3) ここで σ( r) は電荷の面密度であり単位は [C/m2 ] となるより一般には導体表面の位置 r に対して面の垂直な向きの単位ベクトルを n( r) とすると r) E( = 1 σ( r) n( r) ε0 図 4.3: 導体表面にガウスの法則を適用する場合に得られる電場と表面電荷の関係 (4.4)

4.2. 45 4.1.4 0 (FY2011/6 : 2011/12/8 ) 4.2 4.2.1 1 2 4.4 1V C = Q V (4.5) F( ) 1 1[F] [pf] [nf] [µf] + + + + + V - - - - - 4.4: 4.2.2 E( r) = 1 ε 0 σ( r) n( r) (4.6) E = σ ε 0 ( ) φ = E t ds (4.7) (4.8) d φ = d 0 Eds = Ed (4.9)

46 4 φ V Q = σ V = Ed = σd ε 0 C = Q V = ε 0 d = 1 ε 0 Qd (4.10) (4.11) 4.2.3 d = 0.1mm( ) C = 1F C = ε 0 d = Cd ε (4.12) 1 0.1 10 3 = 8.84 10 12 = (3km) 2 (4.13) 1F 1µF = 10 6 F (4.14) 1pF = 10 12 F (4.15) (4.16) google - - - - - V E σ + + + + + (FY2009/8 : 2009/12/3 ) 4.5: 4.2.4 q ( C) dq dw dw = V dq = q C dq (4.17) 0 Q Q Q q W = dw = V dq = C dq = 1 Q 2 2 C = 1 2 CV 2 (4.18) 0 0 U e U e = 1 Q 2 2 C = 1 2 CV 2 (4.19)

4.3. ( ) 47 4.2.5 2 (2 ) 2 E = Q (4.20) ε 0 C = ε 0 (4.21) d (FY2013/8 : 2013/12/12 ) Q C U e = 1 Q 2 2 C = 1 2 E2 ε 2 0 2 d ε 0 = ε 0 2 E2 d = ε 0 2 E2 v, (4.22) v d (4.23) E u e = 1 2 ε 0E 2 = 1 2ε 0 D 2 = 1 2 ED V U e = ε 0 E 2 ( r)dv 2 V (4.24) (4.25) 4.2.6 a b l λ ( +λ λ) r E(r) = λl ε 0 1 2πrl = λ ε 0 1 2πr 0 φ(r) = r b E tds = b r E(r)dr = b r λ 1 ε 0 2πr dr = λ [ln r] b r ε = 0 V = φ(a) φ(b) = λ ( ) b ln 2πε 0 a Q = λl C = Q V = 2πε 0l ln(b/a) λ 2πε 0 ln ( ) b r (4.26) (4.27) (4.28) (4.29) 4.3 ( ) (FY2010/7 : 2010/11/25 )

48 4 - - - - b + + + + + + a O + + r + - E(r) - - - - - 4.6:

49 5 5.1 5.1.1 I [A] = [C/sec] i n i[a/m 2 ] I I = i( r) n( r)d (5.1) i θ n 5.1: i I 5.1.2 ( ) { i( r) n( r)} d = 0 (5.2) i( r) = 0 (5.3) 5.2 I V R[Ω] R V I (5.4)

50 5 n i l 5.2: i I R = ρ l σ 1 ρ (5.5) (5.6) ρ[ω m] σ[1/ω m] I = V R = El σ l i = I = σe = σe (5.7) (5.8) i( r) = σe( r) (5.9) PN 5.3 m e E v m dv dt = ee 0 τ l l = 1 ee 2 m τ 2 v v = l τ = ee 2m τ (5.10) (5.11) (5.12)

5.3. 51 v n i i = en v = ne2 τ 2m E = σe σ = ne2 τ 2m τ l 1 W W = eel = e2 E 2 2m τ 2 P P = n W τ = ne2 E 2 2m τ = E i = σe 2 = 1 σ i2 (5.13) (5.14) (5.15) (5.16) (5.17) (FY2009/9 : 2009/12/10 ) (FY2011/7 : 2011/12/15 ) (FY2013/9 : 2013/12/19 )

53 6 6.1 : 6.1.1 ( ) F = qq 1 4πε 0 r r 1 r r 1 3 (6.1) q m q m1 F F = q mq m1 4πµ 0 r r 1 r r 1 3 (6.2) H H( r) = q m 1 r r 1 4πµ 0 r r 1 3 (6.3) F = q mh( r) (6.4) µ 0 µ 0 = 4π 10 7 [N/A 2 ] (6.5) q m H q m [Wb] ( ) (6.6) H[A/m] (6.7) E D H B B = µ 0 H (6.8) 6.1.2 : 6.1.3 : 2 ( 6.1) R 2 I I 1 l l : F = µ 0 II 1 2π R l (6.9)

54 6 I 1 F I l[m] R 6.1: D B l : F = µ 0 I 1 Il = BIl 2π R (6.10) B = µ 0 I 1 2π R (6.11) B B [T]( ) l = 1[m] I = 1[A] B = 1[T] F = 1[N] B = 1[T] = F/Il = 1[N/A m] (6.12) I 1 = 1[A] 1[m] B B = µ 0 I 1 2π R = 4π 10 7 2π 1 1 = 2 10 7 [T] (6.13) 1[gauβ] = 10 4 [T] (6.14) B H H = I 1 2πR 1A 1/2π[m] 1[A/m] (6.15) 0.24 0.66[gauβ]( 0.45[gauβ] ) A( ) 800[gauβ] ( 1984 ) 5000[gauβ] 8000[gauβ] 6.2 : 6.2.1 6.2.2 0

6.3. : ( ) 55 I B 6.2: { } B( r) n( r) d = 0 (6.16) B( r) = 0 (6.17) 6.3 : ( ) 6.3.1 C { B( r) t( r)} ds = 0 (6.18) C B( r) = 0 (6.19) 6.3.2 : ( B i ) B(R) = µ 0 I 2π R (6.20) 2πRB(R) = µ 0 I (6.21) 2πRH(R) = I (6.22) (6.23)

56 6 I R C { B( r) t( r)} ds = B(R) 2πR = µ 0 I 2π R 2πR = µ 0I (6.24) C ( 6.2) C C { B( r) t( r)} ds = µ 0 C ( 6.3) (FY2010/8 : 2010/12/2 ) { i( r) n( r)} d (6.25) { {( B( r) t( r)} ds = B ) } { n( r) d = µ 0 i( r) n( r)} d (6.26) C B( r) = µ 0 i( r) (6.27) H( r) = i( r) (6.28) I i B C 6.3: B 6.3.3 ( ) (?) ( ) ( &) 6.3.4 6.4 n[ /m] H H { B( r) t( r)} ds = µ 0 C C { H( r) t( r)} ds = { i( r) n( r)} d (6.29) { i( r) n( r)} d (6.30) C 6.4 ABCDA { ( ) = H( r) t( r)} ds = Hl (6.31) C

6.4. 57 ABCDA n[ /m] l[m] = nl[ ] I { ( ) = i( r) n( r)} d = nli (6.32) Hl = nli H = ni, B = µ 0 ni (6.33) (6.34) (FY2009/10 : 2009/12/17 ) (FY2011/8 : 2011/12/22 ) 6.4: 6.4 6.4.1 A ( A B ) E = 0 (6.35) E = φ (6.36) φ B = µ 0 i 0 (6.37) B = 0 (6.38) X ( X ) = 0 (6.39) ( ) B = A (6.40) A A A = 0 (6.41)

58 6 A 2 A = 0 A B = A A A( 0) = f F = f F = 0 F ( E = ρ/ε 0 E = 0 E ) ( A F ) = 0 ( A F ) = A A F A A 0 (AB ) AB ( ) 6.4.2 ( A B i ) () ρ ε 0 = 2 φ (6.42) φ E E ρ A i A(φ ) B( E ) B i(ρ ) 2 µ 0 i( r) = B( r) (6.43) B = A µ 0 i( r) = B( r) = ( A( r) ) (6.44) (6.45) X ( X) = ( X) 2 X (6.46) A = 0 µ 0 i = ( A) = ( A) 2 A = 2 A (6.47) (FY2013/10 : 2013/12/26 ) µ 0 i( r) = 2 A( r) (6.48) µ 0 i x = 2 A x ( r) x 2 + 2 A x ( r) y 2 + 2 A x ( r) z 2 x (6.49) 6.4.3 ( i A ) µ 0 i x ( r) = 2 A x ( r) x 2 + 2 A x ( r) y 2 + 2 A x ( r) z 2 (6.50)

6.4. 59 ρ( r) = 2 φ( r) = 2 φ( r) ε 0 x 2 + 2 φ( r) y 2 + 2 φ( r) z 2 (6.51) 1 ρ( r ) φ( r) = 4πε 0 r r dv A x ( r) = µ 0 4π A( r) = µ 0 4π V V V i x ( r ) r r dv i( r ) r r dv (6.52) (6.53) (6.54) ( 6.5) C I t A( r) = µ 0 4π = µ 0 4π V C i( r ) r r dv I t( r ) r r ds = µ 0I 4π C (6.55) t( r ) r r ds (6.56) A( r) = µ 0 i( r ) V 4π r r A( r) = µ 0 I t( r ) s 4π r r (6.57) (6.58) (6.59) V i( r ) A( r) t( r ) θ θ s V I C A( r) B( r) B( r) 6.5: - (FY2012/10 : 2012/12/20 ) 6.4.4 : - ( i A B ) 1: B A rotation r s I t r B B( r) = µ 0 I s t( r ) ( r r ) 4π r r 3 (6.60)

60 6 Appendix C C B( r) = µ 0 I t( r ) ( r r ) 4π r r 3 ds C B( r) = µ 0 i( r ) ( r r ) 4π r r 3 dv V - - E( r) = 1 4πε 0 V ρ( r )( r r ) r r 3 dv 2 (6.61) (6.62) (6.63) 2: - (6.60) r = 0 ( 6.6) I R B B(R) = µ 0 I 2π R ( λ) R E E(R) = 1 λ 2πε 0 R λ s 2 E E = 1 λ sin θ s 4πε 0 r 2 I s B = µ 0 I sin θ s 4π r 2 (6.64) (6.65) (6.66) (6.67) λ s r sin θ rotation rotation ( ) 6.6 I s t r I s t r t r sin θ I s t r B B B( r) = µ 0 I s t r 4π r 3 (FY2010/9 : 2010/12/9 ) (6.68) 6.4.5 6.7 a z z s B = µ 0I 4π s r 2 (6.69)

6.4. 61 R E A R B θ r θ r λ s t I s t 6.6: 6.7 z B cos α = µ 0I 4π cos α r 2 s C B(z) = µ 0I cos α 4π r 2 ds = µ 0I cos α 4π r 2 2πa = µ 0 Ia 2 (6.71) 2(z 2 + a 2 ) 3/2 B(z = 0) = µ 0I 2a, C I H(z = 0) = 2a (6.70) (6.72) B cos α α A B A B B sin α z r r C I O a α s I s t 6.7:

62 6 6.5 1 u e = D 2 = 1 2ε 0 2 ε 0E 2 = 1 2 DE U e = ε 0 E 2 ( r)dv 2 V u m = 1 2µ 0 B 2 = 1 2 µ 0H 2 = 1 2 BH V U m = µ 0 H 2 ( r)dv 2 V (6.73) (6.74) (6.75) (6.76) LC (FY2009/11 : 2010/1/14 ) 6.6 6.6.1 3 3 6.8 (FY2010/11 : 2010/1/14 ) (FY2011/9 : 2011/12/27 )

6.6. 63 A A( r) = µ 0 4π V i( r ) r r dv i µ 0 i( r) = 2 A( r) アンペールの法則 ( 渦の法則 ) B( r) = µ 0 4π µ 0 i( r) = B( r) ビオ-サバールの法則 V B( r) = A( r) i( r ) ( r r ) r r 3 dv B アンペールの力 F = I t s B F = i V B 磁場に関するガウスの法則 B( r) = 0 B( r) n( r) d = 0 A = 0 に注意アンペールの法則 ( 渦の法則 ) µ 0 i( r) = B( r) { µ 0 i n} d = C { B t} ds 6.8: i B A 3

65 7 7.1 : 7.1.1 B l I F = µ 0II 1 2πR l F = IBl, B = µ 0I 1 2πR (7.1) (7.2) (7.3) F = I t s B (7.4) F = i V B (7.5) (FY2013/11 : 2014/01/09 ) 7.1.2 : 7.1 a b a I b F = Ib B (7.6) N = F a sin θ 2 = IabB sin θ = IB sin θ (7.7) 2 = ab n N = I n B = µ 0 I n H = m H (7.8) m µ 0 I n (7.9) m 7.1.3 q m ( ) F = q m H = q m µ 0 B (7.10)

66 7 7.1: (a) (b) ±q m s m = q m s (7.11) N = m H (7.12) q m s µ 0 I n (7.13) µ 0 I N n 7.2:

7.2. : 67 7.2 : 7.2.1 q F = q v B (7.14) F = q( E + v B) (7.15) 7.2.2 B ( m q > 0) m d v dt = q v B F v B qvb r m v2 = qvb r r = mv qb v 0 r = mv 0 qb ω 0 = v 0 r = qb m ω 0 (7.16) (7.17) (7.18) (7.19) (7.20) 7.3 K E B q F = q( E + v B) (7.21) v B v? K K K K K E E E K E B F = q( E + v B) = qe (7.22) E = E + v B (7.23) K K

69 8 : 8.1 D( r) = ρ( r) (8.1) B( r) = 0 (8.2) E( r) = 0 (8.3) H( r) = i( r) (8.4) D( r) = ε 0 E( r), B( r) = µ0 H( r) (8.5) (FY2010/10 : 2010/12/16 ) 8.2 : 8.2.1 Φ φ em ( 8.1) 1 φ em = dφ dt B { } Φ = B n d φ em C( ) { φ em = E t} ds C 1 em induced electromotive force em ( ) (8.6) (8.7) (8.8)

70 8 : { E t} ds = d dt C { B n } d { {[ E t} ds = ] } E n d ( ) (8.10) C = d { } B n d (8.11) dt E( r, t) = B( r, t) t (8.9) (8.12) (8.3) (8.3) t = 0 B n C 8.1: φ em 8.2.2 8.2( ) a B v q F F = q v B (8.13) F = qvb F = qe = qvb E = vb

8.3. 71 φ em = ae φ em = ae = avb (8.14) B 8.2( ) AB CD φ em φ em ABCDA φ em = (B 1 B 2 )av (8.15) t v t Φ Φ = B 1 av t B 2 av t = (B 1 B 2 )av t (8.16) Φ t = (B 1 B 2 )av (8.17) dφ dt = (B 1 B 2 )av (8.18) (8.19) φ em = dφ dt (8.20) { E t} ds = d { } B n d (8.21) dt C 8.2.1 8.2.2 (FY2011/10 : 2012/1/5 ) 8.3 I(t) Q(t) ( ) = () (8.22) dq(t) = I(t) (8.23) dt ρ( r, t)dv = i( r, t) n( r)d (8.24) t V

72 8 : A F a B q B v D F 2 B 2 v a C φ em A B 1 F 1 a v B 8.2: : ( ) ( i n) d = i dv ρ( r, t) t V (8.25) = i( r, t) (8.26) 8.4 -: E( r) = 0 (8.27) E( r, t) = B( r, t) t (8.28) (8.4) H( r, t) = i( r, t) (8.29) (8.26) div ( H( r, ) t) = i( r, t) (8.30) ( H ) 0 ρ t 0

8.4. -: 73 ρ (8.1) t D( r, t) = ρ( r, t) (8.31) ( t D( r, ) t) = t ρ( r, t) = i( r, t) (8.32) D( r, t) = i( r, t) (8.33) t ( (8.30) (8.30) (8.33) D( r, ) t)/ t ( H( r, ) t) D( r, t) t = i( r, t) (8.34) H( r, t) = i( r, t) + D( r, t) t (8.35) 2 D( r, t)/ t - E( r, t) = B( r, t) t (8.36)

75 9 9.1 9.1.1 D( r, t) = ρ( r, t) (9.1) B( r, t) = 0 (9.2) : E( r, t) = B( r, t) t (9.3) - : H( r, t) = i( r, t) + D( r, t) t (9.4) D( r, t) = ε 0 E( r, t), B( r, t) = µ0 H( r, t) (9.5) m d v dt = q ( E + v B ) (FY2010/11 : 2011/01/13 ) (9.6) 9.1.2 ρ = 0 i = 0 D( r, t) = 0 (9.7) B( r, t) = 0 (9.8) E( r, t) = B( r, t) t H( r, t) = D( r, t) t (9.9) (9.10)

76 9 2 2 9.2 E B E + B t B E ε 0 µ 0 t E = 0 (9.11) B = 0 (9.12) = 0 (9.13) = 0 (9.14) (9.13) ( E ) + t B = 0 (9.15) 1 (9.11) ( E ) ( = E ) 2 E = 2 E (9.16) 2 (9.14) t B 2 E = ε 0 µ 0 t 2 ( 2 2 ) ε 0 µ 0 E t 2 = 0 (9.18) ( 2 2 ) ε 0 µ 0 B t 2 = 0 (9.19) E z (9.18) ( 2 z 2 ε 2 ) 0µ 0 E t 2 = 0 (9.20) ( 2 z 2 1 2 ) F c 2 t 2 = 0 (9.21) c c = (9.17) 1 1 = ε0 µ 0 8.854 10 12 4π 10 = 3.00 7 108 [m/s] (9.22) (FY2009/12 : 2010/1/21 )

77 9.3 (Partial Derivative Differentiation) ( ) f( r) f( r 0 + r) = f(x 0 + x, y 0 + y, z 0 + z) (9.23) [ ] [ ] [ ] f(x, y, z) f(x, y, z) f(x, y, z) f(x 0, y 0, z 0 ) + x + y + z (9.24) x r= r 0 y r= r 0 z r= r 0 f = f f f x + y + x y z z df = f f f dx + dy + x y z dz (9.25) (9.26) x = x(t), y = y(t), z = z(t) (9.27) f = f(x, y, z) = f (x(t), y(t), z(t)) = f(t) (9.28) df dt = f dx x dt + f dy y dt + f dz z dt (9.29) ( 1) f = f(r), r = r(x, y, z) (9.30) f r r x,y,z f/ x r = r(x, y, z) dr = r r r dx + dy + x y z dz df = df(r) dr dr (9.31) (9.32)

78 9 df = df(r) [ r r r dx + dy + dr x y ( ) ( df r df r = dx + dr x dr y df = f f f dx + dy + x y z dz f x = df r dr x ] z dz ) dy + ( df dr (9.33) ) r dz (9.34) z (9.35) (9.36) ( 2) x = x(u, v), y = y(u, v) (9.37) g = g(x, y) = g (x(u, v), y(u, v)) (9.38) x y u v g u v x u, g x, x v, g y, y u, g u, g v, g u g v = y v, x u x v y u y v g x g y g = g(x) g/ y = 0 g u = x g u x + y g x g (= 0) = u y u x = x g u x + y g u y x g v x + y g v y (9.39) (9.40) (9.41) (9.42) g x x u g u = x dg u dx (9.43) (9.44) 9.4 : 1 x r r n = [ (x x ) 2 + (y y ) 2 + (z z ) 2] n/2 x x = n x r r n+2 (9.45)

9.5. 79 9.5 9.5.1 grad 2.7.4 9.5.2 div 3.1.2 9.5.3 rot 3.2.1 9.5.4 ( ) X =0 ( X ) = ( X x ) + ( X x y ) + ( X y z ) (9.46) z = ( Xz x y X ) y + ( Xx z y z X ) z + ( Xy x z x X ) x (9.47) y ( ) ( ) ( ) 2 = y z 2 2 X x + z y z x 2 2 X y + x z x y 2 X z (9.48) y x = 0 (9.49) ( X ) ( = X ) 2 X ( ) 9.5.5 : (x, y, z) (r, θ, φ) x = r sin θ cos φ y = r sin θ sin φ z = r cos θ grad : V (r, θ, φ) grad ( = r, 1 ) r θ, 1 r sin θ φ ( V grad V = V = r, 1 ) V r θ, 1 V r sin θ φ (9.50) (9.51) (9.52) (9.53) (9.54) div : A = (A r, A θ, A φ ) div div A = A = 1 ( r 2 ) 1 r 2 A r + r r sin θ θ (sin θa θ) + 1 r sin θ φ (A φ) (9.55) 2 : V (r, θ, φ) 2 = 2 V = 1 ( r 2 r 2 V ) ( 1 + r r r 2 sin θ V ) 1 2 V + sin θ θ r r 2 sin 2 θ 2 φ (9.56)

80 9 rot : A = (A r, A θ, A φ ) rot ( ( rot A = A = rot A ) (, rot ) ( A, rot ) ) A (9.57) r θ φ ( rot A ) { 1 = r r sin θ θ (A φ sin θ) A } θ (9.58) φ ( rot A ) 1 = θ r sin θ φ A r 1 r r (ra φ) (9.59) ( rot A ) = 1 { φ r r (ra θ) } θ (A r) (9.60) 9.6 : (x, y, z) (ρ, φ, z) x = ρ cos φ y = ρ sin φ z = z (9.61) (9.62) (9.63) grad : V (r, θ, φ) grad ( = ρ, 1 ρ φ, ) z ( V grad V = V = ρ, 1 V ρ φ, V ) z (9.64) (9.65) div : A = (A ρ, A φ, A z ) div diva = A = 1 ρ ρ (ρa ρ) + 1 ρ φ (A φ) + z (A z) (9.66) 2 : V (r, φ, z) 2 = 2 V = 1 { ( ρ V ) + 1 2 } V ρ ρ ρ ρ φ 2 + V ρ 2 z 2 rot : A = (A ρ, A φ, A z ) rot ( ( rota = A = rot A ) ) ( rota ) ( rota ) ρ φ ( rota ) z = 1 A z ρ φ A φ z = A ρ z A z ρ = 1 { ρ ρ (ρa φ) A } ρ φ ρ (, rot A ) (, rot ) A φ z (9.67) (9.68) (9.69) (9.70) (9.71)

81 ( ) I, II,,, IBN4-00-007643-3, IBN4-00-007644-2 ( ),,, IBN4-00-007892-5 ( ),,, IBN4-00-007744-9 B (),,, IBN4-8375-0533-3,,, IBN4-7853-2223-3,,, IBN4-87361-662-X E H, D B,,, IBN4-320-03155-5,,,, IBN4-320-03308-6 div, grad, rot,,,, IBN4-320-03306-X,,,,, IBN4-87576-566-5 ( ),, IBN4-87576-566-5,,, ( / ),,, IBN4-00-007792-9,