2010 4

3 0 5 0.1......................................... 5 0.2...................................... 6 1 9 2 15 3 23 4 29 4.1............................................. 29 4.2.............................. 30 4.3............................................... 30 4.4 3................................... 31 4.5................................... 33 4.6................... 35 4.7..................................... 36 4.8......................................... 36 5 39 5.1............................................ 39 5.2......................................... 40 5.3.......................................... 41 5.4..................................... 41 5.5.................................... 42 5.6....................................... 42 5.7...................................... 43 5.8............................... 44 6 45 6.1.................................... 45 6.2....................................... 46 6.3..................................... 48 6.4........................................... 49 6.5........................................... 51 6.6................................... 52 6.7 ( )...................................... 53 7 55 7.1......................................... 55

4 7.2.......................................... 55 7.3 (Magnet)......................................... 57 7.4................................. 58 8 59 8.1...................................... 59 8.2..................................... 60 8.3..................................... 60 8.4........................................ 65 8.5....................................... 66 9 69 9.1....................................... 69 9.2....................................... 70 9.3................................... 72 9.4.......................... 72 10 (Displacement current) 73 11 (Electromagnetic Induction) 75 11.1 1831 8 29....................................... 75 11.2....................................... 75 11.3 Motional Induction...................................... 75 11.4 Transformer Induction.................................... 76 11.5 Motional induction Transformer induction.............. 79 11.6................................ 79 11.7................................... 80 12 1 81 A 87 A.1 1......................................... 87 A.2 2......................................... 90 A.3............................................ 94

5 0 0.1 1 2 20% 2 20% 30% 30% 60% 1, 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 III III 7

6 0 0.2 ( ) D., R., J. /., 2002 ( :3). 427 H-18.., 2008 ( : ; 3). 427 M-16. 21 [ / ], 2005, 427 T-11.., 1999 ( - ). 427 H-14.., 1997 ( ). 427 K-33. :., 1988. 427-25.., 1987 ( :4). 420.8 B-12 4.., 1987 ( :3). 427 K-27., Edward M.Purcell [ ]/., 1970-1971 ( :2). 420 B-9 2-1.,, /., 1969 ( :3). 420 F-5. :., 2004. 427 W-8. ( ), 2004 ( :2). 420.7 E-3 2.., 1998 ( One Point:2). 427.1 O-2. /., 2002 ( ). 427 K-41.., 2005 ( :B-1483). 408 B-2 1483. ( 2)., 1976. 427 H-2. ( 3)., 1980. 427 T-1 y. 427 T-7 b. :, 1, 2 V. D., M. G. /

0.2 7,., 1991-1992. 427 B-9 1., 1, 2., 2007. 427 O-15 1. 3., 1999. 427-11 3. ( ), J.D. /., 1994 (. 67.). 427.7 J-3 2,., 1970. 427 G-3.., 2002 ( :3). 427.7 A-4. :,., 1980. 427-1.., 1987 ( :5). 420.8 B-12 5. /,., 1981 ( :21). 420.8 K- 4 21-c.

9 1 7 *1 600 *2 1 p.17 360 *3 1600 *4 elektron electricity 1773 *5 1785 *6 1891 *7 1897 *8 1912 *9 1.592 10 19 *1 *2 624-546 *3 427 347 *4 1544-1603 *5 1731-1810 *6 1736-1806 *7 1826 1911 *8 1856-1940 J.J. 1906 *9 1868-1953

10 1 10 1879 100

11 r[m] q[c] Q[C] k qq ( k = 8.9876 10 9 Nm 2 C 2 r 2 [N] ε 0 k = 1 4πε 0 µ 0 = 4π 10 7 N/A 2 c = 1 ε0 µ 0 c = 299792458m/s k ε 0 = 1 c 2 µ 0 = 8.854 10 12 C 2 /N m 2 10 9 m (1 )

12 1 ~10-2 m ~10-10 m ~10-6 m + + ~10-9 m + + + + + + 10Å (= ) =,,,, + + m e + 1840me 1840me e +e 0 e = 1.602 10 19 C

13 1.1 1. 2. 3. 4. 1.2 1. 2. 3. 50mA 1.5V 1Ω 4. 5. 6. 1.3 1. 1cm ( ) ( ) ( ) ( ) ( ) 1.4 1 ( 10 4 m 3 ), ( 10 24 ).,. 10 18 m 3. ( ) 10

15 2 ( ) *1 (1) ( ) (2) ( ) (3) ( ) (4) ( ), r (x, y, z) r = (x, y, z) r t r(t) t,. r *1 material, object

16 2 (scalar),,,,,,,, 2 (x y ) a = (a x, a y ) = a x î + a y ĵ î = (1, 0), î = (0, 1) 2 ( r θ) a = (r cos θ, r sin θ) = r cos θî + r sin θĵ = rˆr ˆr = (cos θ, sin θ), ˆθ = ( sin θ, cos θ) a y y a a ^ ^ r j^ ^ i a x x 2 2 (source) (field) (particle) Q r q F = k qq r 3 r = q E E = k Q r 3 r r E qe (probe) q

17 *2 R 1 Q 1 R 2 Q 2 r Q E 1 = k r R 1 ( r R 1 ), Q E2 = k 3 r R 2 ( r R 2 ), 3 q Q 2 Q 1 R 1 r O R 2 *2 1791-1867

18 2 q r q E 1 ( r) q E 2 ( r) F = q E 1 + q E 2 r E = E 1 + E 2

19 4 ( Kane and ternheim, )

20 2 2.1 2 A( r) = (x, y) 2.2 2 A( r) = r r r = (x, y) r = x 2 + y 2 2.3 2 A( r) = ( y, x) 2.4 2.5 xy (0, a) +q (0, a) q r = (x, y) (1) y E x = 0, E y 2k P y 3 P = qd (2) x E x = 0, E y k P x 3 Mathematica

21 2.1 2 A( r) = (x, y) 2.2 2 A( r) = r r

22 2 2.3 2 A( r) = ( y, x)

23 3 (global) (local) t t + dt v x(t + dt) x(t) = vdt v = dx dt. ( ) 1 ( ) d

24 第3章 電場のフラックス 立体角 ガウスの法則 面積 d を取り出す 球の中心から微小領域までの距離を r とする このとき ドームの壁に映る 微小領域の面積 dω と 微小領域本来の面積 d との関係は d cos α = r2 dω となる ここで 板 の法線ベクトル 面に垂直な単位ベクトル を n として α は n と r のなす角 である ここに現れた dω を 微小領域d を見込む立体角という 立体角は参照球面の一部分の面積 なので これらを全部足し合わせれば必ず 4π になる このことを ZZ dω = 4π と表現する つまり 立体角の積分は 4π である 積分記号に を添えることで ガウス面 上を掃 きつくすように積分を実行することを明示する ガウス面 r 法線ベクトル 立体角 dω ᵼ n α d cosα フラックス 放射状に水を放出できるスプリンクラーを思い浮かべよう スプリンクラーを網で囲む 網面がガウス面 網上の単位面積を毎秒 v リットルの水流が貫くとする このとき 網全体を毎秒 通過する フラックスと呼ぶ は ZZ ZZ Φ= v cos αd = vr2 dω と書けるだろう α は水流が 網を貫く点で網の法線となす角である ここで は網の表面 ガウ ス面 を表す フラックのイメージと数学的な表現が理解できるだろうか 電場のフラックス 点電荷 q の作る電場は = E 1 q r 4πε0 r3

25 Φ = E cos αd α E cos αd = E d d = ˆnd d cos α = r 2 dω ( ) q 1 Φ = 4πε 0 r 2 r 2 dω 1 r 2 r2 r Φ = q 4πε 0 dω = q 4πε 0 4π = q ε 0 E d = q ε 0 q q 1, q 2,, q N q N j=1 q j E d = 1 ε 0 N q j j=1

26 3 N j=1 q j ( dv ) ρ ρdv V ρdv V E d = 1 ε 0 V ρdv ε 0

27 3.1 a σ 3.2 λ 3.3 a Q 3.4, 5, (3 10 6 N/C), λ = 1.0 10 3 C/m,, ) ε 0 = 8.85 10 12 C 2 /N m 2 3.5 a, (1) F = k qq r 2 F = k qq r 2.1 (2) 3.6 1904 J.J. ( ) a ( Q) e (1) (2) a = 0.53 10 10 m, e = 1.6 10 19 C, m = 9.1 10 31 kg

28 3 21 [ / ], p.38 21 [ / ],

29 4 4.1 Q x m q v = dx/dt m dv dt = k qq x 2 mv dv dt = k qq dx x 2 dt = d ( ) 1 dt 2 mv2 = d ( k qq ) dt x = d ( 1 dt 2 mv2 + k qq ) = 0 x = 1 2 mv2 + k qq x = E ( ) E E U (x) = k qq x ( ) F (x) x U (x) du (x) F (x) = dx mv dv dt = F (x)dx dt = d ( ) 1 dt 2 mv2 = d ( U (x)) dt = d ( ) 1 dt 2 mv2 + U (x) = 0 = 1 2 mv2 + U (x) = E ( U (x) F U x U(x) = F (x )dx x 0 x 0

30 4 4.1 (1) (2) 4.2 F (x) x 1 x 2., F W x1 x 2 = x2 x 1 F (x )dx du (x). F F (x) = dx x2 du (x ) W x1 x 2 = x 1 dx dx = U (x 1 ) U (x 2 ) 4.3 Q x E(x) = k Q x 2 U(x) = q x x 0 E(x )dx x U(x)/q = V (x) = E(x )dx x 0 dv (x) E(x) = dx V (x) x 0 x x 0 V (x) = x k Q x 2 dx = [ k Q ] x x = k Q x U(x) = qv (x) [V] 4.2 +Q A +Q B B v 0

4.4 3 31 4.4 3 V ( r) ( ) V ( r) V ( r) V ( r),, x y z V ( r) grad V ( r) ( r) ( V V = x, V y, V ). z V = 1 r r = x 2 + y 2 + z 2 V = r r 3 Q r E = k Q r 3 r q r Q V = r r 3 V = k Q r E = V 3 q m d v dt = q V v = d r dt m v d v dt d r = q dt V d ( ) 1 dt 2 m v2 = q dx V dt x q dy V dt y q dz V dt z d ( ) 1 dt 2 m v2 = q dv dt d ( ) 1 dt 2 m v2 + qv = 0 = 1 2 mv2 + qv = E (

32 4 qv V =?? r V ( r) = E( r ) d r r 0 r 0 r 4.3 r 4.4 r 0 Q r V ( r) V ( r) = 1 Q 4πε 0 r r 0 E( r) 4.5 r 1, r 2, r N, Q 1, Q 2, Q N, r V ( r) V ( r) = 1 4πε 0 N j=1 Q j r r j E( r) 4.6 p V ( r) = 1 p r 4πε 0 r 3 4.7 (1) (2) E Q (3) E P

4.5 33 4.5 r 0 r w = N 1 j=0 f( r j ) ( r j+1 r j ) r r0 r 0 r N w = r f( r) d r r 0 r C r0 C C 4.8 3 f = (x, y) f( r ) d r r 0 r 0 = (0, 0), r = (1, 1) 1: y = x 2: y = x 2 3: y = x y (1,1) r (0,0) x 1 f = (x, y) = (x, x) d r = (dx, dy) = (dx, dx) f d r = 2xdx 1 0 2xdx = 1

34 4 2 f = (x, y) = (x, x 2 ) d r = (dx, dy) = (dx, 2xdx) f d r = xdx + 2x 3 dx 1 0 (x + 2x 3 )dx = 1 3 f = (x, y) = (x, x) d r = (dx, dy) = (dx, 1 2 x 1/2 dx) f d r = xdx + 1 dx = (x + 1/2)dx 2 1 0 (x + 1/2)dx = 1 3 V ( r) = 1 2 (x2 + y 2 ) f x = V x, f y = V y, z ( V f = x, V ) V y r 0 = (0, 0), r = (1, 1) f d r = dv = V ( r) + V ( r 0 ) = 1 x y dv (x, y) dv = V V dx + x y dy V f = V 4.9 V ( r) = mgz m, g V

4.6 35 4.6 (x ) x v(x) v(x x/2) y z x v(x) x v(x+ x/2) y zn y zn 3 x, y, z ( ) (x, 0, 0) 1 2 (x x/2, 0, 0), (x+ x/2, 0, 0) 1 2 y z 1 2 v(x + x/2) y z } {{ } 2 v(x x/2) y z } {{ } 1 v(x) x x y z 1 2 ˆn, +ˆn 1 = y zˆn, 2 = + y zˆn,, 1 2 v 1, v 2, v 1 1 + v 2 2 = v x(x) x x y z 3, r = (x, y, z) v( r) 6 v 1 1 + v 2 2 + v 3 3 + v 4 4 + v 5 5 + v 6 6 [ vx (x) = + v y(x) + v ] z(x) x y z x y z = v( r) x y z

36 4 1 6 4 3 5 2 v( r) d = v( r)dv V, ( ),. 4.10 v( r) = (y 2, 2xy + z 2, 2yz) 1 1 6 4.7, V Q, ρ [C/m 3 ] Q = ρ( r)dv, E( r) d = 1 ρ( r)dv ε 0 V V, E( r) d = E( r)dv V E( r)dv = 1 ρ( r)dv ε 0, V V E( r) = 1 ε 0 ρ( r), 4.8, [ V ( r)] = 1 ρ( r) ε 0 ( ) 2 = x 2 + 2 y 2 + 2 z 2 V ( r) = 1 ρ( r) ε 0 = 2 V ( r) = 1 ε 0 ρ( r)

4.8 37 2 ( r) V = 1 ε 0 ρ 4.11, x V = V 0 (x/d) 4/3..

39 5 5.1 B At

40 5 n[ 10 28 /m 3 ] Na 2.65 Cu 8.47 Ag 5.86 Au 5.90 Fe 17.0 Mn 16.5 5.2 σ σ/ε 0 5.1 (1) σ/ε 0 (2) σ 2 /2ε 0 electric tension electric stress

5.3 41 5.3 (condenser) (capacitor) 5.4 21 [ / ],

42 5 5.5 Q Q V Q = CV C (capacitance) F( ) 1F 1µF= 10 6 F 1pF= 10 12 F 5.6 1. Q q A q B q A q B Q q C Q q D q C q D q A q B Q E Q V Ed Q q A q B q C q D q A q B q C q D q A q B q C q D C d q A q B q C q D 2. (a) Q Q Q Q b a Q Q r(a<r<b) Q E 4 r Q b V ae dr 4 a C 4 ab b a b 5.2 5.3 a :C = 4πε 0 a 5.4 a, ( 2a, 3a),, 1, 2 ε 0 [A] 1 ( ), Q [A-1] [A-2] [B] 1 2 ( ), Q [B-1]

5.7 43 Q [B-2] [B-3] 2a a 3a 1 2 5.7 d ±Q d U F E = E = E + E = σ ε 0 σ 2ε 0 E = σ 2ε 0 σ = Q/ F F = QE = σ2 2ε 0 = 1 2 QE U = F d = σ2 2ε 0 d = 1 2 QEd = 1 2 QV = 1 2 CV 2 = Q2 2C U = σ2 2ε 0 d

44 5 d u = U d = σ2 2ε 0 = 1 2 ε 0E 2 1 2 ε 0E 2 5.5 V (1) Q r (2) Q (3) Q V (4) C (5) ( ) 1 U = 2 ε 0E 2 dτ V dτ dv V (6) U = 1 2 CV 2 L b a 5.8 W = 1 2 N q i V ( r i ) i=1 W = ε 0 E 2 dτ 2 V 5.6

45 6 6.1 v (steady current) j A/m 2 n j = en v 6.1 j = en v 6.2 n 10 29 /m 3. A = 2mm 2 1A, [ v 3 10 5 m/s 1/300cm] ], 21 [ /

46 6 6.2 E J. J. Thomson Drude Drude τ τ 1 m a = e E a = e E/m τ v = ee m τ j = en v = ne2 τ m E j = σ E σ = ne2 τ m (electrical conductivity) (electrical resistivity) ρ=1/σ

6.2 47 21 [ / ], ρ 10 8 Ω m = ρ 10 4 10 7 Ω m = ρ 10 7 10 17 Ω m = (thermal conductivity) C V = 3 2 k B κ σ = nc V v 2 τ 3 κ = nc V v 2 τ 3 m ne 2 τ =k B m v 2 e 2 = k B 3 2 e 2 2 k BT = 3 2 ( ) 2 kb m v 2 = 3 2 2 k BT L = 3 ( kb 2 e ) 2 = 3 2 κ σ = 3 2 ( kb e ) 2 T ( 1.38 10 23 ) 2 [J/K] 1.6 10 19 1.1 10 8 [V 2 /K 2 ] [C] e

48 6 κ σt = L (Wiedemann-Franz) 6.3 ρ L R = ρ L (resistance) G = 1 R (conductance) E = σ 1 j L LE = Lσ 1 j = V = L j = V = RI }{{} σ }{{} =I =R 6.3 ε σ C R = ε σc 6.4 a ( l), b, σ. V,.

6.4 49 b a V 6.5 ( ) 6.4 R Q J = I 2 R 1840 J.P.Joule Joule 6.6 m F µ

50 6 6.7 t = 0 V C R (1) Q(t) (2) Q(t) (3) I(t) (4) R dq(t) dt + Q(t) C (5) (6) (7) (6) = V 6.8 ( ) 70mV 6.0 nm Benjamin Crowell, Electricity and Magnetism

6.5 51 6.5 1845 0 2 0 6.9 6.10 (Wheatstone bridge) R3 Rx V R1 R2 6.11 ρ R = ρ ( ) h π ab

52 6 6.6 V j d j. d dv t, j d } {{ } + V ρ t dv } {{ } j d = V = 0 jdv V jdv + V ( ρ dv = 0 = t V j+ ρ ) dv = 0 t, j+ ρ t = 0 (continuity equation) j d = I V ρ t dv = d ( ) ρdv = dq dt V dt

6.7 ( ) 53 I dq dt I + dq dt = 0 6.12 ( 1/e ). t = 0 ρ 0, t ρ. (1) j+ ρ t = 0 E = ρ ε j = σ E ρ t = σ ε ρ. (2) τ. (3) ( ), σ = 6.0 10 7 Ω 1 m 1. ε = 1.2 10 11 C 2 /N m 2. (4) 6.7 ( ) ( ) ( ) ( ) ( ) ( ) l x 1. de(x) = en(x) dx ε dv (x) 2. E(x) = dx 3. m dv dt = ee(x) 4. 1 2 mv2 = ev (x) 5. j = env

54 6 n(x) ( ) j x d2 V (x) dx 2 dv dx dv d 2 V (x) dx dx 2 = j ε = dv dx = = en(x) ε = j εv = j ( ) 1/2 m ε 2eV (x) { m dv 2eV (x) dx = d ( ) } 2 1 dv = j ( m ) 1/2 d dx 2 dx ε 2e dx ( ) 1/2 4j m V 1/4 = dv ( ) 1/2 4j m ε 2e V = dx 1/4 ε 2e ( ) 1/2 4j m x = dv V 1/4 = = V = ε ( 9j m 4ε 2e 2e ) 2 3 x 4 3 ) 1/2 dx = 4 3 V 3/4 = ( 4j ε m 2e (2V 1/2) x = l V = ( ) 9j m 2 3 l 4 3 4ε 2e j = 4ε 2e 9l 2 m V 3/2

55 7 7.1 2 a b ( ) a b, a b sin φ, a b a x a y a z a x b x b y b z b x a x b y a y b x a y b z a z b y a z b x a x b z a b z a b x a b y a b a b sinφ a b b a φ b a 7.2 B T T=N/A m q B v F L = q v B 7.1 B v 0 q

56 7 7.2 B v 0 q 7.3 7.4 Physics for cientists and Engineers 6th ed. (College Text), erway and Jewett 1777-1851 1820 1775-1836 2

7.3 (Magnet) 57 19 7.3 (Magnet) 1 N

58 7 7.4 N N N B d = 0, B = 0

59 8 8.1 K q E v K, K v c v c K B = 1 c 2 v E. *1 c = 1 ε0 µ 0. *1 11-3

60 8 8.1 (1)., λ[c/m]. r E. (2) v. r P B., P θ. 8.2 E = q r (q = e) v 4πε 0 r3 B = 1 c 2 v E = µ 0q v r 4π r 3 8.3 I d l r d B q = endl v v d l db = µ 0 ( endl) v r 4π r 3 = µ 0 ( en v) dl r 4π r 3 ( en v) dl = ( env) d l = Id l d B = µ 0I 4π d l r r 3 1820

8.3 61 db I r d` d l r θ d B db = µ 0I 4π sin θdl r 2 * 2 *2 11-3

62 8 8.2 (1) z (0, 0, z) B z z B db z = µ 0I 4π a (a 2 + z 2 ) 3/2 dl z db O I a Id` (2) a Benjamin Crowell, Electricity and Magnetism

8.3 63 8.3 ( a n ) I B z z z + dz ndz z B db = µ 0I 2 a 2 { a 2 + (z z ) 2} 3/2 ndz 8.4 (1) AB I. P. (2) P A 1 2 r B 8.5 1 2a I. (1) O. (2) d P. P d 2a 2a O

64 8 8.6 a 8.7 a b a b

8.4 65 8.4 M = µ 0 I r m q I = q T = qv 2πr (T = 2πr/v) ( qv ) M = µ 0 πr 2 ẑ 2πr ẑ l = mrvẑ ( qv ) M = µ 0 πr 2 1 2πr mrv l = ( µ0 q ) l = g l 2m g = µ 0q 2m ` M

66 8 8.5 H = 1 µ 0 B µ 0 = 4π 10 7 N/A 2 H d r = I C., C.. C I H dr,, j( r)., C, I I = j d j r d C,. C H d r = j d 8.8 8.9 I 1, I 2 ( r) F = µ 0I 1 I 2 2πr

8.5 67 8.10 r =1m I 1 = I 2 = I 1m 2 10 7 N/m I µ 0 8.11,. (1) (a) R i. xy.., x Q(x, 0) B 1 (x). 0 < x < R R < x. (2) (b) r i. xy. (a, 0)., a r < x < a + r, Q(x, 0) B 2 (x). (3) (c), R, a b. i. a r < x < a + r, Q(x, 0) B(x). (a) y (b) y (c) y R RR R O Q x O Q P r x O Q r P x a a

69 9 9.1 f = (f x, f y, f z ) ( f fz = y f y z, f x z f z x, f y x f ) x y f, rot f f = i j k x y z f x f y f z i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) 9.1 f (1) f = (x, y, 0) (2) f = ( y, x, 0) 9.2 (A) (F) y y y x x x y y y x x x

70 9 9.2 f = (f x, f y, f z ) xy A B C D A 1 f(x, y, z) W A B C D A. x y+ y D C y y y A x B x x+ x W A B C D A = W A B + W B C + W C D + W D A W A B = f x (x, y, z) x, [ W B C = f x (x + x, y, z) y f y (x, y, z) + f y(x, y, z) x [ W C D = W D C = f x (x, y + y, z) x W D A = W A D = f y (x, y, z) y. ] x y, f x (x, y, z) + f x(x, y, z) y y ] x,, [ W A B C D A = f x (x, y, z) x + [ f x (x, y, z) + f x(x, y, z) [ fy (x, y, z) = x f y (x, y, z) + f y(x, y, z) x ] y y f x(x, y, z) y ] x y x f y (x, y, z) y ] x y. W A B C D A = A B C D A B C D f d r. [ fy (x, y, z) f d r = f ] x(x, y, z) x y x y ABCD yz zx [ fz (x, y, z) f ] y(x, y, z) y z, y z [ fx (x, y, z) f ] z(x, y, z) z x z x. xy ABCDEF 1 2

9.2 71 D C F E A D B C F A E G B E G G E [A B C D E F A ] A G E F A ] + G B C D G ] G E E G A B C D E F A A G E F A G B C D G C xy, xy C C [ fy (x, y, z) f d r = f ] x(x, y, z) dxdy = [ C x y f] x dxdy = ( f) d d xy C xy C C C f d r = ( f) d

72 9 9.3 f = (0, 2xz + 3y 2, 4yz 2 ) 1 1 [ (0, 0, 0) (0, 1, 0), (0, 1, 1), (0, 0, 1)] 9.3 H d r = j d C ( H) d = j d, H = j 9.4 { E = ρ ε 0 E = 0 E = V ( ) V = 0 { B = 0 B = µ 0 j

73 10 (Displacement current) I(t)., ±Q(t).,., E(t) = Q(t) ε 0 I(t) = dq(t) dt,., I(t) ( ).,.,, I(t), I(t),, I D (t). E t + Q t Q t I t I t I t ID t I t } } }, I D = I I D (t) = ε 0 de(t) dt, ( d I D (t) = ε 0 dt ) E d.,,.,,.,,.

74 10 (Displacement current), C ( H d r = j d d + ε 0 dt ) E d., C H d r = ( j + ε ) E 0 d t.,. J = j + ε 0 E t

75 11 (Electromagnetic Induction) 11.1 1831 8 29 1791 1867 11.2 IH 11.3 Motional Induction l B v. V m = v B l = vbl sin θ, motional induction ( ). ( ) C V m = v B d r C

76 11 (Electromagnetic Induction) B ` v 10.1 Motional induction, V m = v B l = B l v = B d (t) dt.,, ( ). 11.4 Transformer Induction 2 C( ), B(t). Φ(t) = B(t) V t = dφ(t) dt, transformer induction. n B, E.. C E d r = dφ(t) dt. ( r B( r, t) ),. V t = C B( r, E d r = t) d t

11.4 Transformer Induction 77 10.2 z B(t) = (0, 0, B 0 sin(ωt)). xy r E(r).. ( z ).. 10.3 I. 1 a ABCD, AB CD. AB x. (1) x B(x). µ 0. (2) ABCD Φ(x). (3) x I, ABCD. di/dt. (4), I ABCD v. AB x ABCD. v = dx/dt. (5) I ABCD v. AB x ABCD. I A a a D O B x C

78 11 (Electromagnetic Induction) 10.4 IH N (1) B(t) = B 0 cos(ωt) AC [ V = 1 3 πb 0ωh 2 N 3 sin (ωt)] Φ = 1 2πN 2 B(t) {r(θ)} 2 dθ r(θ) = hθ/2π. (2) ρ Q J (3) = 0.1cm 2, ρ = 10 8 Ω m h = 0.1cm N = 100 Q J 0

11.5 Motional induction Transformer induction 79 11.5 Motional induction Transformer induction 10.5( ) l (θ =90) B t B(t) = B 0 cos(ωt) v., x. ` x v motional induction transformer induction V emf = V m + V t = d dt ( (t) B(t)), Φ Φ(t) = B(t) (t)., V emf = dφ(t) dt.. motional induction transformer induction.. 11.6 C., C. C. C, ( )., C., C B( r), Φ, Φ(t) = B d., Φ C V = dφ(t) dt., C ( ) E( r). V emf = E d r C

80 11 (Electromagnetic Induction) Br d = nd C Ein r. C E d r = d dt ( ) B d.. 10.6( ) B(r) = { B0 sin (ωt) (0 < r a) B 1 sin (ωt) (a < r) t = 0 r = 2a m e r = 2a (1) B 0 B 1 B 0 = 5B 1 (2) a = 6cm B 0 = 0.02T ω = 100s 1 28% 11.7 E d r = ( E) d, C ( E) d = d dt ( E = B t. ) B d

81 12 1 1 1. E ( r) = 1 4πε 0 Q r 3 r 2. 1. ρ ( r) Q 2. Q = ρdv E d = 1 ρdv ε 0 V V

82 12 1 3. E d = EdV V 4. E = 1 ε 0 ρ 2 1. 2. B d = 0 B = 0 3 1. B = µ 0 H 2. dh = I d l r 4π r 3 3.

83 1. j ( r) I 2. I = j d H d r = j d C 3. H d r = H d C 4. H = j

84 12 1 4 1. I 2. ( ) d I D = ε 0 E d dt E = ε 0 t d 3. H d r = I + I D C C H d r = ( j + ε ) E 0 d t 4. 5 H = j + ε 0 E t 1. Motional Induction 2. Transformer Induction 3. Φ = B d 4. 5. V = dφ(t) dt C E d r = d dt ( ) B d 6. 7. C E d r = E = B t E d E 0

85 6 1. E d = 1 ρdv ε 0 V B d = 0 2. E = 1 ε 0 ρ B = 0 C H d r = C E d r = d dt ( H = j + ε 0 E t E = B t ( j + ε ) E 0 d t ) B d

87 A A.1 1 1 (1) e = 1.60 10 n C n (2) 2 10 (3) 3 Q r E (4) λ r 2 xy ( a, 0) q (a, 0) +q r = (x, y) E( r) y r A Ο B x (1) = 2 d E( r) r d k (2) (a, a) x y (3) y (0, y) 1 y 3 3 (1) (5) (A) (F) (1) A( r) = (x, x) (2) A( r) = (x, 1)

88 A (3) A( r) = (x, y) (4) A( r) = (y, x) (5) A( r) = ( y, x) y y y x x x y y y x x x 4 a Q r E(r) r } (1) r > a E(r) (2) 0 < r < a E(r) (3) r (4) 1904 J.J. ( ) a ( e) e m T a ε 0 e m 5 6

A.1 1 89 a λ 7 ρ d

90 A A.2 2 A1 A3 B1 B3 1 A1 [9 ] (1) ρ( r) E( r) E( r) V ( r) E( r) = V ( r) ρ( r) (2) σ (3) e n v j = n = 8.47 10 28 /m 3 A = 2mm 2 1A, v = 3.69 10 n m/s n n = (4) ε σ ±Q σ ε Q I = σ ε C R = A2 [3 ] (1) p r = (x, y, z) ( p r) = ( ) 1 r 3 = r = x 2 + y 2 + z 2 1 p r r 2 (r n ) n r r (2) p V ( r) = 1 p r 4πε 0 r 3 E( r) = 1 4πε 0 ( f ( r) g ( r) (fg) = f g + g f A3 [4 ] v = (x + y, x y, z) 1 1 vdv = v d V )

A.2 2 91 1 6 4 3 5 2 vdv = V v d 1 2 0 1 2 B1 [4 ] [1] a b V b a V (1) C (2) ( ) 1 U = 2 ε 0E 2 dτ V U = 1 2 CV 2 dτ dv

92 A V r r + dr [2] ε σ b a V (3) r I(r) a < r < b r r (4) (3) A1 B2 [4 ] R V t = 0 V C R (1) t Q(t) Q(t) (2) (3) (4) (3) B3 [4 ] R Q (1) O x x (2) x (3) Q M x

A.2 2 93 O x

94 A A.3 1 [8 ] E d = 1 ρdv ε 0 V (A.1) a ρ (a) r 0 < r < a a < r (b) r V 0 < r < a a < r (c) (b) V V E V E = V (A.2) (d) E = ρ ε 0 (A.3) 2 [4 ] B B = 0 (A.4) B B 0 (A.5) (a) (A.4) (b) (A.4) (A.5) 3 [8 ] I dh = I d l r 4π r 3 (A.6) (a) (A.6) d l, r, d H (b) (b-1) a z H(z) (b-2) ( a n ) I H z z z + dz ndz z dh = I 2 a 2 ndz

A.3 95 (b-3) (b-2) H = ni z 4 [8 ] z a j z r 0 < r < a H = 1 2 j r r = (x, y, 0), r = x 2 + y 2 j = (0, 0, j) xy a/2 C (a) H H x H y H z j, x, y (b) H z (c) H d (d) H d r C C ( a r = 2 cos θ, a ) 2 sin θ, 0 θ 0 2π (c) (d) H d = H d r C 5 [6 ] a d V = V 0 sin (ωt)

96 A ε 0 (a) t I (b) (c) r 0 < r < a 6 [16 ] 1831 (a) Motional Induction Transformer Induction (b) Motional Induction Transformer Induction (c) Φ V V = dφ dt (A.7) E B E = B t (A.8) (d) B = B 0 sin (ωt) (B 0 ω ) l R t = 0 x a (d-1) t (d-2) v x t (d-3) (d-2)