120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2

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1 9 E B 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 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0 ( I 1 + I 2 ) 119

2 120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2 I 1 B ds = µ 0 I 1 I 1

3 (f) I 2 0 I B ds =2µ 0 I. I n nµ 0 I 9.2 I O r B B = µ 0I 2πr µ 0 I B ds = 2πr ds = µ 0I ds = µ 0I 2πr 2πr 2πr = µ 0I r µ 0 B θ I O r ds B O dφ φ ds 9.2: 9.2 I B ds B θ B ds = B ds cos θ O ds O dφ B ds = B ds cos θ = Br dφ (9.3)

4 122 9 r O r B = µ 0I 2πr B ds = B ds cos θ = 2π 0 µ 0 I 2πr r dφ = µ 0I 2π dφ = µ 0 I 2π 0 O ds O dφ (9.3) µ 0 I Stokes Ampère Stokes A(r) S n rot A n ds = A ds. (9.4) S Gauss S i 9.3: Stokes 9.3 S N S i S i n i i S i (rot A) n i S i = A ds ( i =1, 2,,N) i

5 S N S i N N (rot A) n i S i = A ds i=1 i=1 i N Stokes (9.4) Stokes Stokes X, Y, Z x, y, z (Xdx + Y dy + Zdz) = [ l ( Z y Y z ) + m ( X z Z x l, m, n ds ) ( Y + n x X )] ds (9.5) y Ampère Ampère (9.1) Stokes (9.4) Ampère S rot B ds = µ 0 S j ds S rot B = µ 0 j. Ampère (9.2) 9.1 I r B(r) I B r B(r) r Ampère B(r) B(r) 2πr = µ 0 I B(r) = µ 0I 2πr.

6 a (1) I (2) I a ( 1) B I r B ( 2) 0 a r B 0 a r 9.4: r Ampère B(r) (1) r<a r>a I B(r) 2πr =0 B(r) =0 (r<a) B(r) 2πr = µ 0 I B(r) = µ 0I ( r>a) 2πr (2) a I j = I πa 2 r<a I r>a I Ampère B(r) 2πr = µ 0 j πr 2 = µ 0 I πr2 πa 2 B(r) = µ 0Ir 2πa 2 B(r) 2πr = µ 0 I B(r) = µ 0I 2πr ( r<a) ( r>a)

7 (1) (2) (1) a L n I (2) (1) E B out F z θ 2 θ θ 1 D B in z 2 dz P z 1 9.5: (1) 9.5 z z z P z z z +dz P db 8.3 z db = nµ 0Ia 2 dz 2(a 2 + z 2 ) 3/2 dz P z θ z = a tan θ dz = a dθ sin 2 θ, 1 sin θ (a 2 + z 2 = ) 1/2 a P B z B = nµ 0Ia 2 2 = nµ 0I 2 z2 z 1 θ2 θ 1 dz (a 2 + z 2 ) 3/2 sin θ dθ = nµ 0I 2 ( cos θ 2 cos θ 1 )

8 126 9 θ 1 θ 2 P (2) (1) θ 1 π θ 2 0 B = nµ 0I 2 ( cos 0 cos π )=nµ 0 I. Ampère 9.5 DEF D EF D EF EF DEF Ampère F DE EF 0 Ampère B ds = B in = nµ 0 I DEF B in = nµ 0 I EF DEF Ampère B out Ampère B ds = B in B out = nµ 0 I DEF B in = nµ 0 I B out = R a N I B r B(r) r Ampère B(r) 2πr B(r) N I NI Ampère B ds =2πr B(r) =µ 0 NI

9 a r 9.6: r B(r) = µ 0NI 2πr Ampère P A A φ = B ds P A P 9.7 P A I S 0 S 1 S A φ 0 = B ds φ 1 = P( 0 ) A P( 1 ) B ds. µ 0 I P 1 A A 0 P

10 128 9 A 1 S 0 I P 9.7: I Ampère A P B ds + B ds = µ 0 I P( 1 ) A( 0 ) φ 1 0 P A P A B ds = B ds = φ 0. A( 0 ) P( 0 ) φ 1 φ 0 = µ 0 I φ 1 = φ 0 + µ 0 I A B ds = P( 1 ) A P( 0 ) B ds + µ 0 I P A µ 0 I 1 I S S 0 P φ m = A P B ds (9.6) A φ m magnetic potential B = grad φ m (9.7)

11 Gauss Gauss S B 0 B ds = 0 (9.8) S div B = 0 (9.9) Maxwell j B Ampère rot B = µ 0 j rot B Ampère B Ampère Gauss Gauss div E = ρ/ε 0 rot E =0 Gauss 9.8 I I ds db db = µ 0 4π I ds r r 3 Biot-Savart ds db Biot-Savart r ds I ds 9.8 S S S S ds 1 ds 2 db 1 db 2 ds 1 ds 2 dφ 1 =db 1 ds 1, dφ 2 =db 2 ds 2,

12 130 9 db S ds 2 I ds I ds 1 9.8: Gauss dφ 1 dφ 2 dφ 1 +dφ 2 =db 1 ds 1 +db 2 ds 2 =0. S S db i ds i =0 i I ds I ds S 0 db ds =0. I ds, S S I I ds I B I ds db S B ds =0 S j Gauss Gauss div B = 0

13 B Gauss A B = rot A (9.10) vector potential j A(r) = µ 0 4π j(r ) r r dv (9.11) div (rot A) =0 A A z y A y z A x rot A = z A z x A y x A x y rot A div (rot A) = (rot A) x + (rot A) y + (rot A) z x y z = ( Az x y A ) y + ( Ax z y z A z x ( 2 ) ( A z = x y 2 A z 2 ) A x + y x y z 2 A x z y ) + ( Ay z + x A ) x y ) ( 2 A y z x 2 A y x z 0 A div (rot A) =0 div B =0 B B = rot A (9.10) Ampère (9.2) rot (rot A) =µ 0 j (9.12)

14 132 9 x [ rot (rot A)] x = y (rot A) z z (rot A) y = ( Ay y x A ) x ( Ax y z z A ) z x = 2 A x y 2 2 A x z 2 + ( Ay x y + A ) z z (9.12) x 2 A x y 2 2 A x z 2 + ( Ay x y + A ) z z = µ 0 j x 2 A x / x 2 2 A x / x 2 2 A x x 2 2 A x y 2 (9.12) x 2 A x z 2 = 2 A x, ( Ax x x + A y y + A ) z z 2 A x + x (div A) =µ 0 j x y z = (div A) x 2 A k + k (div A) =µ 0j k ( k = x, y, z ) (9.13) B (9.10) A ψ A A = A + grad ψ (9.14) rot (grad ψ) =0 rot A = rot A + rot (grad ψ) = rot A = B ψ A = A + grad ψ A A ψ grad ψ Poisson (9.14) B = rot A A div A =0

15 A (9.13) 0 Poisson 2 A x = µ 0 j x, 2 A y = µ 0 j y, 2 A z = µ 0 j z. (9.15) Poisson 2 φ = ρ ε 0 Poisson φ(r) = 1 ρ(r ) 4πε 0 r r dv ε 0 1/µ 0 ρ j k j A Poisson (9.15) A(r) = µ 0 4π j(r ) r r dv A B = rot A 9.5 I r P z +L z B dz O r P L A 9.9: 9.9 z I r P AB O z z = L A z = L B

16 134 9 AB P (9.11) j(r )dv = I ds = I dz e z z A(r) = µ 0I 4π L L dz e z (x x ) 2 +(y y ) 2 +(z z ) 2 x y 0 z A x = A y =0, A z = µ 0I r 4π log 2 + L 2 + L r 2 + L 2 L. L L r r 2 + L 2 = L ( r 2 ) L 2 + log r 2 + L 2 + L r 2 + L 2 L log (2L)2 + r 2 r 2 A z log (2L)2 r 2 = 2 log 2L 2 log r A z µ 0I 2π log 2L µ 0I 2π log r L A z µ 0 I/(2π) log 2L P A x = A y =0, A z = µ 0I 2π log x 2 + y 2 + z 2 B = rot A 9.6 Q(r ) I ds P(r) Q(r ) I ds P(r) j A (9.11) j(r )dv = I ds da(r) = µ 0I 4π ds r r db da db = rot (da) = µ 0I 4π rot ds r r.

17 rot (da) P(r) r =(x, y, z) r =(x,y,z ) ds =(dx, dy, dz ) ds/ r r z y ( dz ) y r r = ( y ( = dz 1 ) 2 = (y y )dz r r 3 dz ) (x x ) 2 +(y y ) 2 +(z z ) 2 2(y y ) [(x x ) 2 +(y y ) 2 +(z z ) 2 ] 3/2 x [ ] ds rot r r = ( dz ) x y r r ( dy ) z r r = (z z )dy (y y )dz [ ds (r r ] ) r r 3 = r r 3 (db ) x = µ [ 0 I ds (r r ] ) 4π r r 3 x y z db = µ 0 4π I ds (r r ) r r 3 Bio-Savart 9.7 xy a I x, 9.10 z xy Q I ds P da Q a r = a xy θ = π/2 x φ I ds = Iadφ P yz φ = ϕ = π/2 Q P R P da (da) r = µ 0Ia 4πR sin(ϕ φ) sin θ dφ = µ 0Ia cos φ sin θ dφ 4πR (da) θ = µ 0Ia 4πR sin(ϕ φ) cos θ dφ = µ 0Ia cos φ cos θ dφ 4πR (da) ϕ = µ 0Ia 4πR cos(ϕ φ)dφ = µ 0Ia sin φ dφ 4πR

18 136 9 z P e r I θ r R e θ x O φ dφ Q Ids y 9.10: ϕ = π/2 A φ 0 2π R φ P r a 1/R a/r 1 R = 1 r 2 + a 2 2ar cos(ϕ φ) sin θ 1 r 2 + a 2 2ar sin φ sin θ = = 1 [1+ a sin φ sin θ + ]. r r a/r φ 0 2π A r = A θ =0, A ϕ = µ 0Ia 2 4r 2 sin θ z k p m = Iπa 2 k A ϕ = µ 0p m 4πr 2 sin θ, A = µ 0 4π r P p m r r 3

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e 7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z

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