2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
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1 2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec p. 1/30
2 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i i i+1 n 3 2 r 2 F i n i 2 i n 1 r 1 1 ( ) A ) dr = n(r)dr n(r) r Γ Γ i B I(2011), Sec p. 2/30
3 (4) w W q = B A F B q dr = A E dr. ( ) w Γ w Γ 1 ( ) q q B (5) E(r) = q 4πε 0 ˆr r 2. (i) A B ( A A B) q O r A A r A r B A I(2011), Sec p. 3/30
4 w = B A E(r) dr = A A E(r) dr B A E(r) dr 1 A A (ˆr) (dr) E dr = 0. A B dr = ˆrdr (6) w = rb r A q 4πε 0 dr r 2 = q 4πε 0 ( 1 1 ). r A r B (ii) A A A B B q O A A A I(2011), Sec p. 4/30
5 (7) w = ra r A = q 4πε 0 q dr 4πε 0 r 2 rb r A ra r A dr r 2 = q 4πε 0 q dr 4πε 0 r 2 ( 1 1 ) r A r B. rb q dr 4πε 0 r 2 = ( ) w (iii) q O B b x a y r θ c E A I(2011), Sec p. 5/30
6 abc E (E bc abc ) (8) E dr = E y = E rcosθ. (9) a c a b c E dr = E rcosθ. ( E cosθ E ac ) E(r) w = B A E(r) dr I(2011), Sec p. 6/30
7 2.4.2 P 0 φ( ) P 0 A ( A r A ) (10) φ(r A ) w(p 0 A) = A P 0 E(r) dr, B ( : ) A (11) φ(r B ) = B P 0 E(r) dr. A B w(a B) = w(a P 0 )+w(p 0 B) (12) = w(p 0 A)+w(P 0 B) = φ(r B ) φ(r A ). P 0 I(2011), Sec p. 7/30
8 (13) (14) w(a B) = B A B A E(r) dr. E(r) dr = φ(r B ) φ(r A ). φ: ( ) (15) r φ(r) = E(r ) dr. P 0 P 0 q (6) r A, r B = r (16) φ(r) = q 4πε 0 1 r. I(2011), Sec p. 8/30
9 ( ( )) (17) E(r) = i E i (r), (E i i ) (18) r r φ(r) = E(r ) dr = E i (r ) dr P 0 P 0 i = r E i (r ) dr = φ i (r) i P 0 i (φ i i ) = 1 q i 4πε i 0 r r i. I(2011), Sec p. 9/30
10 ( ρ(r)) (19) φ(r) = 1 4πε 0 ρ(r )dv r r. cf. ( ) (15) (N/C)m = J/C 1J/C 1V ( Volt) E V/m (x,y,z) (x+dx,y,z) (x,y,z) (x+dx,y,z) x x + dx (20) w = E dr = E x dx. I(2011), Sec p. 10/30
11 (21) w = φ(x+dx,y,z) φ(x,y,z) = φ x dx. E x = φ x. y z (22) E = (E x,e y,e z ) = ( φ x, φ y, φ z ). = ( x, y, z ) ( ) I(2011), Sec p. 11/30
12 (23) E = φ(r) (= gradφ). (gradient: ) (15) ) f f f f gradient f rotation( ) A(r) ( Az A(r) = y A y z, A x z A z x, A y x A ) x (24), y A rotation( ) ( A = rota = curla ) (23) I(2011), Sec p. 12/30
13 (25) ( E) z = E y x E x y = ( ) φ + ( ) φ x y y x = 2 φ x y + 2 φ y x = 0. x,y (26) E(r) = 0. ) φ A(r) = f(r) (f(r) ) A(r) = 0 (26) (27) E(r) = q 4πε 0 r r 0 r r 0 3 I(2011), Sec p. 13/30
14 (28) = = ( E) z = E y x E x y ( q y y 0 4πε 0 x r r 0 3 y [ q (y y 0 ) x 4πε 0 ) x x 0 r r r r 0 3 (x x 0) y ] 1 r r 0 3 = 0. ) (29) E = 0. ( ) ( ) I(2011), Sec p. 14/30
15 2.4.5 (Stokes) ( )Γ A(r) Γ Γ 1 dr ( ) A t (30) A t (r)dr = A(r) dr. Γ ( 1 A t A ) (circulation) (cf. (1)) Γ Γ A P 1 Γ 1 Γ 2 P 2 I(2011), Sec p. 15/30
16 P 1, P 2 Γ 1 P1 P 2 A(r) dr = P2 P 1 A(r) dr, Γ 2 P2 A(r) dr, P 1 (31) A(r) dr = A(r) dr + A(r) dr. Γ Γ 1 Γ 2 Γ I(2011), Sec p. 16/30
17 (32) =. Γ y 0 + y y ( ) xy y 0 z x 0 x 0 + x x (33) A(r) dr = x0 + x + x 0 x0 x 0 + x A x (x,y 0,z)dx+ y0 + y y 0 A x (x,y 0 + y,z)dx+ A y (x 0 + x,y,z)dy y0 y 0 + y A y (x 0,y,z)dy I(2011), Sec p. 17/30
18 = x0 + x + x 0 y0 + y y 0 [A x (x,y 0,z) A x (x,y 0 + y,z)]dx [A y (x 0 + x,y,z) A y (x 0,y,z)]dy A x (x,y 0 + y,z) = A x (x,y 0,z)+ A x(x,y 0,z) y A x (x,y,z) y ( : 2 y +O(( y) 2 ) y=y0 ) x, y 2 x0 + x x 0 A x(x 0,y 0,z) y [ A ] x(x,y 0,z) y y y0 + y dx+ y 0 y x+ A y(x 0,y 0,z) x [ Ay (x 0,y,z) x y x ] x dy I(2011), Sec p. 18/30
19 = ( Ay x A ) x x y = ( A) z x y = ( A) z ds y ( x y = = ds ) n( ) z ( ) = ( A) nds = ( A) ds. (ds nds ) ( ) (32) S Γ (34) = = ΓA(r) dr A(r) dr ( A) ds = ( A) ds. ( ) S I(2011), Sec p. 19/30
20 (35) A(r) dr = Γ S ( A(r)) ds. : ω r = (x,y,0) v(r) = ( yω,xω,0) v(r) = rω (36) v(r) dr = 2πaaω = 2πa 2 ω. r=a y v r x (37) v(r) = (0,0,2ω) I(2011), Sec p. 20/30
21 (38) r a v(r) ds = 2ωπa 2. ) A = 0 A 0 2 P 1, P 2 Γ 1 P 2 Γ (39) A(r) dr = A(r) dr. P 2 1 Γ 1 Γ 2 Γ = Γ 1 Γ 2 (40) A(r) dr = A(r) dr A(r) dr = 0. Γ Γ 1 Γ 2 I(2011), Sec p. 21/30
22 ( A(r) dr = 0 Γ 2 ) (35) (41) ( A) ds = 0. S S (42) A(r) = 0. ( rotation ) ( A = 0 A dr = 0) A(r) = 0 f(r) A(r) = f(r) 2 A(r) P 0 I(2011), Sec p. 22/30
23 (43) P2 P 1 A(r) dr P0 = A(r) dr + P 1 = f(r 2 ) f(r 1 ). P2 P 0 A(r) dr P 1 r f(r) A(r ) dr P (44). 0 P 0 (45) f(r +dr) f(r) = A(r) dr. (dr = (dx,dy,dz)) r 1 P 2 r 2 P 0 r (46) f(r +dr) f(r) = f x dx+ f y dy + f z dz = f(r) dr. I(2011), Sec p. 23/30
24 (47) A(r) = f(r). (f ) A = f A = 0 ( (25) (26) ) ( ) (5) (9) w = B A E(r) dr E(r) = g(r)ˆr E(r) r (ii) (48) = w = ra r A B A E(r) dr g(r)dr ra r A g(r)dr = rb r A g(r)dr. I(2011), Sec p. 24/30
25 w φ E = φ E = 0 ( ) r (1/r 2 ) E = 0 ( ) (1/r 2 ) ( ) (= 2. 5) gradient (49) f = ( f x, f y, f z ( f) x = f/ x f x f (cf. (46)) f f ). I(2011), Sec p. 25/30
26 ( ) (50) φ(r) = const. ( ) (51) E = φ E φ E E φ = const.? E I(2011), Sec p. 26/30
27 : 1 q(> 0) +q 1: (cf ) z (18) d +q [ 2 φ(r) = q 1 (52) O 4πε 0 x2 +y 2 +(z d/2) 2 d q ] 2 1. x2 +y 2 +(z +d/2) 2 I(2011), Sec p. 27/30
28 (r = x 2 +y 2 +z 2 d) (53) = = = [ ( φ(r) = q ) 1/2 ) ] 1/2 r 2 zd+ d2 (r 2 +zd+ d2 4πε [ ( q 1 1 zd ) 1/2 ( 4πε 0 r r + d2 1+ zd ) ] 1/2 2 4r 2 r + d2 2 4r 2 [ q 1 1+ zd ( 4πε 0 r 2r 1 zd ) ] +O((d/r) 2 ) 2 2r 2 q zd 4πε 0 r, 3 p = (0,0,qd) 1 p r. 4πε 0 r 3 I(2011), Sec p. 28/30
29 E(r) = φ(r) ( r n / x = nx/r n+2 ) E x 1 [ ] px (54) 4πε 0 r p r3x, 3 r 5 E y 1 [ ] py 4πε 0 r p r3y, 3 r 5 E z 1 [ ] pz 4πε 0 r p r3z. 3 r 5 p ] (55) E(r) = 1 4πε 0 [ 3(p r) r 5 r p r 3 p = (0,0,qd) (p r = qdz) = 1 4πε 0 3(p r)r r 2 p r 5. I(2011), Sec p. 29/30
30 (56) E x 1 3qdzx 4πε 0 r 5 E z qd 3z 2 r 2 4πε 0 r 5 = qd 3xz 4πε 0 r, E 5 y qd 3yz 4πε 0 r, 5 = qd 4πε 0 ( 3z 2 r 5 1 r 3 ). z (r = (0,0,z)) (57) ( ) xy (z = 0) (58) (xy ) I(2011), Sec p. 30/30
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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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)
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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
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
2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30
2.5 (Gauss) 2.5.1 (flux) v(r)( ) n v n v n (1) v n = v n = v, n. n n v v I(2012), ec. 2. 5 p. 1/30 i (2) lim v(r i ) i = v(r) d. i 0 i (flux) I(2012), ec. 2. 5 p. 2/30 2.5.2 ( ) ( ) q 1 r 2 E 2 q r 1 E
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9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
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II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
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1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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液晶の物理１：連続体理論（弾性，粘性）
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