elemag.dvi

Transcription

1 II

2 i (Gauss) (Green) (Stokes) (Earnshaw)

3 ii ( I ) ( I )

4 iii : : : : : : : : A 88 B 91

5 1 I m No m Newton (Coulomb) Ampere (Gauss) Cavendish (Faraday) Maxwell divd = ρ divb =0 rote = B t roth = J + D t F (r,t)=q[e(r,t)+v B(r,t)]

6 2 (Lorentz) I,II V.D., M.G. ( )

7 n a b = a i b i = a b cos θ (1.1) i=1 θ a,b 2,3 (scalar) (vector) a = (a 1,a 2,a 3 ) b = (b 1,b 2,b 3 ) a b = (a 2 b 3 a 3 b 2,a 3 b 1 a 1 b 3,a 1 b 2 a 2 b 1 ) (1.2) a b a b sin θ θ a, b a, b (a b, a) =(a b, b) =0 a b sin θ a b 2 + (a, b) 2 = a 2 b 2 a = a(sin θ 1 cos φ 1, sin θ 1 sin φ 1, cos θ 1 ), b = b(sin θ 2 cos φ 2, sin θ 2 sin φ 2, cos θ 2 )

8 1 4 z θ a x φ y a x b x + a y b y + a z b z = a b (cos θ 1 cos θ 2 +sinθ 1 sin θ 2 cos(φ 1 φ 2 )) cos θ 1 cos θ 2 +sinθ 1 sin θ 2 cos(φ 1 φ 2 ) cos θ 1. a b = b a (1.3) 2. a (b + c) = a b + a c (a + b) c = a c + b c (1.4) (ka) b = k(a b) =a (kb) (1.5) a a = 0 (1.6) Problem 1.1 a (b c) =b (c a) =c (a b) (1.7)

9 1 5 a (b c) =b(a c) c(a b) (1.8) [(a b) c] d = (a b) (c d) (1.9) = (a c)(b d) (a d)(b c) (1.10) 1.2 h(x, y) (v x (x, y),v y (x, y)) f x def f(x +Δx, y) f(x, y) = lim Δx 0 Δx (1.11) Theorem f x y = 2 f y x (1.12) 2 f x y = x lim f(x, y +Δy) f(x, y) Δy 0 Δy [ ] = lim lim 1 f(x +Δx, y +Δy) f(x +Δx, y) f(x, y +Δy) f(x, y) Δx 0 Δy 0 Δx Δy Δy [ ] = lim lim 1 f(x +Δx, y +Δy) f(x, y +Δy) f(x +Δx, y) f(x, y) Δy 0 Δx 0 Δy Δx Δx = 2 f y x

10 x, y f(x, y) Δf = f(x +Δx, y +Δy) f(x, y) = f(x +Δx, y +Δy) f(x +Δx, y)+f(x +Δx, y) f(x, y) = Δy y) f(x +Δx, y)+δx f(x, y x f(x, y) f(x, y) Δy +Δx y x df = f f dx + dy (1.13) x y df = gdx + hdy (1.14) (1.12) g y = h x (1.15) df x, y f(x, y) 1.4 / x, / y, / z f(x, y, z) (gradient) def =( x, y, z ) (1.16) f =( x f, y f, f) (1.17) z f =gradf (1.18)

11 1 7 v =(v x,v y,v z ) a b = a x b x + a y b y + a z b z v def =divv = v x x + v y y + v z z (1.19) div (divergence) a b =(a y b z a z b y,a z b x a x b z,a x b y a y b x ) (1.20) v def =rotv =( v z y v y z, v x z v z x, v y x v x y ) (1.21) rot (rotation) grad div rot rot(gradf) = ( f) 0 (1.22) div(rotv) = ( v) 0 (1.23) Problem 1.2 (1.22),(1.23) Problem 1.3 ( v) = ( v) 2 v (1.24) (v u) =( v) u v ( u) (1.25) 1.5 (Gauss) v v x [v x (x +Δx, y, z) v x (x, y, z)]δyδz v x x ΔxΔyΔz

12 1 8 y, z v y y ΔxΔyΔz, v z z ΔxΔyΔz divvδxδyδz Theorem 1.2 ( ) n(x, y, z) (v n)ds = dv divv (1.26) c v c v ([c v] n)ds = dv div(c v) (1.27) div(c v) = c rotv ([c v] n)ds = dv c (rotv) (1.28) (c v) n = (n v) c (1.29) ds(n v) = dv rotv (1.30) v = cφ (cφ) nds = dv div(cφ) (1.31) div(cφ) = c gradφ (cφ) n = c Φn gradφdv = ΦndS (1.32)

13 (Green) (1.26) v = φ grad ψ div(φgradψ) =φ 2 ψ + ψ φ (1.33) φ ψ n ds = ψ n def = ψ n (1.34) dv (φ 2 ψ + ψ φ) (1.35) φ, ψ Theorem 1.3 [ φ ψ n ψ φ ] ds = n dv (φ 2 ψ ψ 2 φ) (1.36) 1.6 (Stokes) v dr = v x (x +Δx/2,y,z)Δx + v y (x +Δx, y +Δy/2,z)Δy + v x (x +Δx/2,y+Δy, z)( Δx)+v y (x, y +Δy/2,z)( Δy) [ ( Δx y v x x + Δx ) ] [ ( 2,y,z Δy +Δy x v y x, y + Δy ) ] 2,z Δx [ vy ΔxΔy x v ] x =( v) z ΔxΔy y = dsn ( v) x y x y

14 1 10 Theorem 1.4 () n v dr = rotv nds (1.37) Problem 1.4 Problem 1.5 (0,0),(2,0),(2,1),(0,1) v = ( y, x, 0)

15 11 2 I F (r,t)=q[e(r,t)+v B(r,t)] (r,t) E B 2.1 Coulomb q,q 1 F q = 1 qq 1 R 4πɛ 0 R 3 1 = 1 qq 1 1 4πɛ 0 R 2 1 ˆR 1 F q1 = 1 qq 1 R 4πɛ 0 R 3 1 = 1 qq 1 1 4πɛ 0 R 2 1 ˆR 1 (2.1) F q q F q1 q 1 R 1 = r r 1, ˆR1 = R 1 R 1 (2.2) MKSA ɛ 0 = C 2 /N m 2 (2.3) Problem 2.1 1/4πɛ 0 = N m 2 /C 2 k

16 2 12 q F q r R 1 O r 1 q 1 F q 1 q 1 E = 1 q 1 ˆR 4πɛ 0 R 2 1 (2.4) 1 q qe E = 1 q 1 ˆR 4πɛ 0 R q 2 ˆR 1 4πɛ 0 R = 2 i 1 4πɛ 0 q i R 2 i ˆR i (2.5) 2.2 (2.5)

17 2 13 Φ(r) = i 1 4πɛ 0 q i R i (2.6) E = grad Φ(r) (2.7) Problem 2.2 (2.7) (2.6) C C (2.5) m ( m ) r i ΔV i Δq i ρ i =Δq i /ΔV i Φ(r) = 1 4πɛ 0 i Δq i r r i = 1 4πɛ 0 i ρ i ΔV i r r i (2.8) Φ(r) = 1 4πɛ 0 dq(r ) r r = 1 4πɛ 0 ρ(r )dv r r (2.9) dq(r )=σ(r )ds (2.10) dq(r )=λ(r )dr (2.11)

18 2 14 Φ(r) = 1 4πɛ 0 dq(r ) r r = 1 4πɛ 0 σ(r )ds r r (2.12) Φ(r) = 1 dq(r ) 4πɛ 0 r r = 1 4πɛ 0 λ(r )dr r r (2.13) ( (2.7) ) r r = R 1 R = ˆR R 2 (2.14) (2.7) E(r) = 1 4πɛ 0 ˆR R 2 ρ(r )dv (2.15) E(r) = 1 4πɛ 0 E(r) = 1 4πɛ 0 ˆR R 2 σ(r )ds, (2.16) ˆR R 2 λ(r )dr, (2.17) Problem 2.3 ( ) 1 = n ˆR (2.18) R n R n+1 2.4

19 x = r cos φ, y = r sin φ, z = z (2.19) (x, y, z) (r, φ, z) ẑ ˆr ˆφ dφ = dr Φ (2.20) dr = ˆrdr + ˆφrdφ + ẑdz (2.21) dφ = (ˆr Φ)dr +(ˆφ Φ)rdφ +(ẑ Φ)dz (2.22) dφ (r, φ, z) dφ = Φ r dr + Φ φ Φ dφ + dz (2.23) z ˆr = r, ˆφ = 1 r φ (2.24) = ˆr r + ˆφ 1 r φ + ẑ z (2.25) x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ (2.26) (x, y, z) (r, θ, φ) ˆr θ ˆθ ˆr, ˆθ ˆφ ˆφ = ˆr ˆθ (2.27)

20 2 16 dφ = dr Φ (2.28) dr = ˆrdr + ˆθrdθ + ˆφr sin θdφ (2.29) dφ = (ˆr Φ)dr +(ˆθ Φ)rdθ +(ˆφ Φ)r sin θdφ (2.30) dφ (r, θ, φ) dφ = Φ r dr + Φ θ Φ dθ + dφ (2.31) φ ˆr = r, ˆθ = 1 r θ, ˆφ = 1 r sin θ φ (2.32) = ˆr r + ˆθ 1 r θ + ˆφ 1 r sin θ φ (2.33) σ z (2.12) Φ(r) = σ 4πɛ 0 ds r r = σ r dr dφ (2.34) 4πɛ 0 r 2 + z 2 φ 2π r 0 Λ Λ 0 r dr r 2 + z 2 = r 2 + z 2 Λ 0 = Λ 2 + z 2 z (2.35)

21 2 17 Φ r, φ (2.25) z E z Λ σ z>0 2ɛ 0 E z = σ ( Λ2 + z 2ɛ 0 z 2 z ) = z σ 2ɛ 0 z<0 (2.36) Λ 2 + z 2 / z = z/ Λ 2 + z 2 Λ λ z (2.13) Φ(r) = λ 4πɛ 0 dz r r = λ 2πɛ 0 0 dz r2 + z 2 (2.37) z Λ z = r sinh θ, sinh Θ = Λ/r (2.38) Θ Φ(r) = λ 2πɛ 0 Θ (2.39) Θ=ln(Λ+ Λ 2 + r 2 ) ln r (2.40) Φ z,φ (2.25) ˆr E r Λ E r = Φ r = λ 2πɛ 0 r (2.41) Problem Problem 2.5

22 Problem 2.6 M p = kg, m e = kg, G = N m 2 /kg 2

23 19 3 q (2.5) r q/4πɛ 0 r 2 q/ɛ dω = cos θds r 2 (3.1) ds θ cos θ r 2 4π 3.2 E n q cos θ E n ds = E nds = ds (3.2) 4πɛ 0 r 2 cos θ/r 2 ds dω E n ds = q 4πɛ 0 dω (3.3)

24 3 20 dω =4π E n ds = q ɛ 0 (3.4) E n ds = Q, Q = q i (3.5) ɛ 0 i ρ E n ds = ρ ɛ 0 dv (3.6) E n ds = divedv (3.7) dive = ρ ɛ 0 (3.8) λ r h 2πrhE = hλ ɛ 0 (3.9) E = λ 2πɛ 0 r (3.10)

25 ρ a r r >a, r<a 1. r>a 2. r<a 4πr 2 E = Q ɛ 0 = 4πa3 ρ 3ɛ 0 (3.11) E = Q 4πɛ 0 r 2 (3.12) 4πɛ 0 r 2 E = 4πr3 ρ 3 E = (3.13) Q 4πɛ 0 a 3 r (3.14) 3.3 (Earnshaw) Theorem 3.1 ( ) (test charge) ( ) (a, 0, 0), ( a, 0, 0), (0,a,0), (0, a, 0) x, y z

26 dive =0=ρ/ɛ 0 E dl U Φ i i q i U = 1 2 N q i Φ i (3.15) i (2.6) U = 1 2 N N j=1 i=1,i j q i q j 4πɛ 0 R ij (3.16) Problem 3.1 N +e N e a N e2 U = 4πɛ 0 a N 2ln2 (3.15) U = 1 ρ(r)φ(r)dv (3.17) 2 (2.9)

27 3 23 ρφ =ɛ 0 Φdiv E ρφ =ɛ 0 [ E Φ+div(ΦE)] U = 1 2 ɛ 0 dv (E 2 +div (ΦE)) (3.18) 2 0 U = 1 2 ɛ 0 dve 2 (3.19) Problem Q πɛ 0 a Q r r<a E = 4πɛ 0 a 3 Q 1 r>a 4πɛ 0 r 2 (3.14) ( ) Q r 2 + C r < a V = 4πɛ 0 a 3 2 Q r>a 4πɛ 0 r r = a ( ) ( ) Q r 2 V = 4πɛ 0 a a2 r<a 2 Q r>a 4πɛ 0 r (3.20) (3.21) (3.22)

28 3 24 U dv ɛ 0E 2 2 U = 1 2 = 1 2 = = U = πɛ 0 = 1 1 ρ 2 2 4πɛ 0 = 1 2 a 0 a 0 dr4πr 2 ρv (r) (3.23) ( ) dr4πr 2 Q r 2 ρ 4πɛ 0 a a2 2 a ) dr ( r a2 r 2 2 Q 2 3 4πɛ 0 a 3 2a 3 Q 2 3 4πɛ 0 a 5 = ɛ ( ) 2 [ 0 Q a 2 4πɛ 0 0 Q 2 3 = 4πɛ 0 a 5 dvρ a 0 1 8π 2 ρ 2 4πɛ 0 1 8π 2 ρ 2 4πɛ 0 = 1 2 = 1 1 8π 2 ρ 2 2 4πɛ 0 Q 2 3 = 4πɛ 0 a dv 1 ρ r r a π dr4πr 2 dr 2πr 2 a 0 a 0 a 0 0 a 1 r 2 dr r 2 dr 0 1 [ r r 2 dr dr 2r r ) dr (a 2 r 2 r4 3 dr r2 a 6 4πr2 + dr 1 ] a r 4 4πr2 0 sin θdθ r2 + r 2 2rr cos θ dt r2 + r 2 2rr t a ] dr 2r { dt 2 r2 + r 2 2rr t = r <r r 2 r >r r r (3.24) (3.25) (3.26) Problem 3.3 (0, 0, 0) (0, 0,a) q 1,q 2 E 1 E 2 1. E 1

29 E 2 3. ɛ 0 (E 1 + E 2 ) 2 2 = ɛ 0E ɛ 0E ɛ 0 E 1 E 2 2 dv ɛ 0 E 1 E 2 r cos θ = t t r A dr r2 + a 2 2rA 3 = 1 r2 + a 2 2rA + C

30 NO dive 1 = ρ/ɛ 0 1 dive 2 = ρ/ɛ 0 E Φ 1, Φ 2 E 1 = Φ 1, E 2 = Φ 2 E = E 1 E 2, Φ= Φ 1 Φ 2 div(φe) =ΦdivE +( Φ) E =0 E E = E 2 (4.1) div(φe)dv = (ΦE) ˆndS (4.2) 0= div(φe)dv = V V V 1 E 2 dv (4.3)

31 4 27 E =0 1 2 Φ=0, E ˆn =0 3 (ΦE) ˆndS = V i (E 1 E 2 ) ds = 1 V i (Q i 1 Q i i S i ɛ 2) (4.4) 0 i V 0 V = V 0 divgradv = (mirror charge) q 0 d 0 d q

32 4 28 r + = ((z d) 2 + r 2 ),r = ((z + d) 2 + r 2 ), Φ(r) = q 4πɛ 0 ( 1 1 ) r + r (4.5) E =(0, 0, qd 2πɛ 0 R ),R= r 2 + d 2 (4.6) 3 σ = ɛ 0 E z = qd 2πR 3 (4.7) Problem 4.1 σ q a q r q 0 (z ), (0, 0,r I ) q I 2 0 0=4πɛ 0 Φ(r) = q a2 + rq 2 2ar q cos θ + q I a2 + ri 2 2ar I cos θ (4.8) ( a q I = q I r I = q a, r q a = a r I (4.9) r q ) q (4.10) r I = a2 r q (4.11) Φ 0 2

33 4 29 Problem 4.2 Q q Φ= Q q I 4πɛ 0 a 4.3 N i Q i 0 Φ(r) Φ(r) =F (r; Q 1,Q 2,,Q N ) (4.12) F {Q i} Φ (r) =F (r; Q 1,Q 2,,Q N) (4.13) {Q i } = {Q i + Q i} Φ (r) =F (r; Q 1,Q 2,,Q N) (4.14) f(r) =Φ(r)+Φ (r) divgradφ = divgradφ =0 divgradf =0 f i Φ, Φ ɛ 0 dv i dive = ɛ 0 ds i gradf n i = ɛ 0 ds i (gradφ + gradφ V ) n i = Q i + Q i i S i S i (4.15) f Q i + Q i Φ (r) = F (r; Q 1 + Q 1,Q 2 + Q 2,,Q N + Q N ) (4.16) = F (r; Q 1,Q 2,,Q N )+F(r; Q 1,Q 2,,Q N )

34 4 30 j Qj Q1 Q2 Qi n Φ i = p ij Q j (4.17) j=1 Q i =0 0 (x =2x ) 2 (4.17) n Q i = C ij Φ j (4.18) j=1 C ii C ij C ij 1V C ij V j =1 σ j (r) C ij = Q i = dsσ j (r) (4.19) S i V i =1 Φ i (r) =1 i Φ i (r) =0 C ij = Q i = ds k Φ i (r)σ j (r) (4.20) k S k (0 1 ) Φ i (r) = 1 4πɛ 0 n k ds k σ i (r 0 ) r r (4.21)

35 4 31 Q1 V1 Q2 V2 R d C ij = 1 4πɛ 0 n k,k ds k ds k σ j (r)σ i (r 0 ) r r (4.22) i, j C ij C P P ij C ij i Q i C ii i j j j Q j C ij 0, (i j) i Q i 0 Q = CV (4.23) R V = Q 4πɛ 0 R C =4πɛ 0 R (4.24) 2 R d ( Q 1 Q 2 ) ( = C 11 C 12 C 21 C 22 )( V 1 V 2 ) ( = C 11 V 1 + C 12 V 2 C 21 V 1 + C 22 V 2 ) (4.25)

36 Q 1,Q 2 d C 11,C 21 C 12,C 22 Q 1,Q 2 C 11 C 12, C 22 C 21 C 12 = C 21 C 11 C 22 C 12 = C 21 (4.26) Q 1 = Q 2 = C(V 1 V 2 ), C = Q 1 V 1 V 2 (4.27) Problem 4.3 C = ɛ 0A d A = d2 d =0.1μm C e2 2C 4.4 dive = ρ(r) ɛ 0 ΔΦ(r) = ρ(r) ɛ 0 (4.28) divgrad = 2 x y z 2 (4.29) ΔΦ(r) = 0 (4.30) (Poisson) (Laplace) r Φ(r) 0 Φ(r) = 1 ρ(r )dv 4πɛ 0 r r (4.31)

37 4 33 (δ) Δ 1 r =0(r 0) (4.32) Δ 1 r dv = ( r ) div dv =4π (4.33) r 3 0 = δ(x) { 0 x 0 η δ(x) =, δ(x)dx =1, for all η > 0 (4.34) x =0 η Δ 1 r =4πδ(x)δ(y)δ(z) def =4πδ(r) (4.35) δ 0 f(x)δ(x a) =f(a) (4.36) a δ (4.31) f(r) =f(x, y, z) f(r) Problem 4.4 f(r, z) f(x, y, z) = f(r, z) Δf(r, z) = 1 ( r f ) + 2 f (4.37) r r r z 2 Problem 4.5 f(r) f(x, y, z) =f(r) Δf(r) = 1 ( r 2 f ) (4.38) r 2 r r

38 d 2 Φ(z) dz 2 = 0 (4.39) Φ(z) =a + bz (4.40) ( r f ) = 0 (4.41) r r Φ(r) =a + b log r (4.42) ( a, b, b > a) ( r 2 f ) = 0 (4.43) r r r 2 dφ dr = C 1, Φ(r) = C 1 r + C 2 (4.44) V b V a V a = C 1 a + C 2, V b = C 1 b + C 2 (4.45) C 1 =(V b V a ) ab b a, C 2 = V bb V a a b a (4.46)

39 4 35 Problem 4.6 q a = 4πɛ 0 ( Vb V a b a ) ab (4.47) Φ=0 r = b Φ=V b Φ =bv b /r E = bv b /r 2 ( ) q a q b = q a + q b ɛ 0 =4πbV b (4.48) ( ) bvb av a q b =4πɛ 0 b (4.49) b a 4πɛ 0 ab b a 4πɛ 0ab b a 4πɛ 0ab ( b a 4πɛ 0 b 2 b a ( ) (4.47) V a V b ) (4.50) ab C ab = C ba = 4πɛ 0 b a (4.51) 4.5 ( I ) x, y ( 2 x y 2 ) Φ(x, y) = 0 (4.52)

40 4 36 ( ) Φ(x, y) =X(x) Y (y) (4.53) Y (y) d2 X(x) dx 2 X(x)Y (y) + X(x) d2 Y (y) dy 2 = 0 (4.54) 1 d 2 X(x) + 1 d 2 Y (y) = 0 (4.55) X(x) dx 2 Y (y) dy 2 1 x 2 y x, y 0 1 d 2 X(x) = 1 d 2 Y (y) = C (4.56) X(x) dx 2 Y (y) dy 2 C ( X, Y ) 3 C = k 2 d 2 X(x) dx 2 = k 2 X, d 2 Y (y) dy 2 = k 2 Y (4.57) X(x) = A k cos kx + B k sin kx (4.58) Y (y) = C k exp( ky)+d k exp(ky) (4.59) (4.60) k Φ(x, y) Φ(x, y) = k (A k cos kx + B k sin kx)(c k exp( ky)+d k exp(ky)) (4.61) a x a f(x) f(x) = ( A n cos nπx a n=0 3 0 X = a + bx, Y = c + dy + B n sin nπx ) a (4.62)

41 4 37 a a a a dx sin mπx a dx cos mπx a a a A 0 = 1 2a A n = 1 a B n = 1 2a nπx sin a = aδ mn (mn 0) (4.63a) nπx cos a = aδ mn (mn 0) (4.63b) dx sin mπx a a a a a a nπx cos a = 0 (4.63c) dxf(x) (4.64) dxf(x)cos nπx a dxf(x)sin nπx a a (n 0) (4.65) y V =0x V 0 x =0,a Φ=0 exp x Φ(x, y) = k (A k cos kx + B k sin kx)(c k exp( ky)+d k exp(ky)) (4.66) y D n =0 x =0,a V 0 A k =0 k = nπ a Φ(x, y) = n=0 C n sin nπ a exp( nπy/a) (4.67)

42 4 38 Φ=0 Φ=0 0 a Φ=V0 y =0 Φ=V 0 V 0 = n C n sin nπ a (4.68) a dxc n n=1 0 sin mπ a sin nπ a = V 0 a C n (4.66) 4V 0 n =1, 2, 3, 4,... nπ C n = 0 n =2, 4, 6, 8,... 0 dx sin mπ a (4.69) (4.70) Φ(x, y) = 4V 0 π l=1 1 2l 1 (2l 1)πx sin e (2l 1)πy/a (4.71) a

43 =ΔΦ= 1 ( r 2 Φ ) + r 2 r r 1 r 2 sin θ ( sin θ Φ ) + θ θ 1 2 Φ r 2 sin 2 (4.72) θ φ 2 Φ(r, θ, φ) =Φ(r, θ) Φ/ φ =0 0=ΔΦ= 1 ( r 2 Φ ) ( 1 + sin θ Φ ) (4.73) r 2 r r r 2 sin θ θ θ Φ(r, θ) =R(r)Θ(θ) (4.74) ( 1 d r 2 dr(r) ) = 1 ( d sin θ dθ ) (4.75) R(r) dr dr Θ dθ dθ r, θ ( d r 2 dr(r) ) = CR(r) (4.76) dr dr ( d sin θ dθ ) = CΘ(θ) (4.77) dθ dθ R 1 r 2 1 R(r) =A n r n + B n r n 1, C = n(n + 1) (4.78) θ μ =cosθ (4.79) d dθ = sin θ d dμ = 1 μ d 2 dμ d dμ [ 1 μ 2 dθ dμ ] + n(n +1)θ =0 (4.80) (4.81)

44 4 40 Φ(r, θ) = n R n (r)p n (μ) = n (A n r n + B n r n 1 )P n (cos θ) (4.82) P 0 =1, P 1 = μ, P 2 = 1 2 (3μ2 1), P 3 = 1 2 (5μ3 3μ) (4.83) 1 1+r2 2rμ = r n P n (μ) (4.84) (0, 0, 1) q = 4πɛ 0 Φ= 1/ 1+r 2 2rμ n=0 rn P n (μ) n=0 Problem r2 2rμ r =0 P 0,P 1,P dμp n (μ)p m (μ) = r2 2rμ 1+s2 2sμ = [-1,1] μ 2 2n +1 δ mn (4.85) n,m=0 r n s m P n (μ)p m (μ) (4.86) n,m=0 1 1 dμr n s m P n (μ)p m (μ) = = = dμ (1 + r2 2rμ)(1 + s 2 2sμ) 1 ln 1+ rs rs 1 rs n=0 (rs) n 2n +1 (4.87)

45 4 41 a ( σ) r>a π 0 2πa 2 1 σ sin θdθ a2 + r 2 2ra cos θ (4.88) 1 1 2πa 2 1 σdμ r 1+(a/r) 2 2(a/r)μ = 2πa2 σ r 1 dμ 1 n=0 ( a ) n Pn (μ) P 0 (4.89) r n = πa 2 1 σdμ r 1+(a/r) 2 2(a/r)μ = 4πa2 σ r (4.90) a E 0 (4.82) ( Φ(r, θ) = A 0 + B ) ( 0 P 0 (μ)+ A 1 r + B ) ( 1 P r r 2 1 (μ)+ A 2 r 2 + B ) 0 P r 3 2 (μ)+ (4.91) r = a Φ=0 A n = B n a 2n+1 (4.92) r E = E 0 ẑ Φ= E 0 r cos θ A 1 = E 0,A n =0(n 1) (4.93) ( ) Φ(r, θ) =E 0 r + a3 cos θ (4.94) r 2 Q ( Φ(r, θ) = A 0 + B ) ) 0 + E 0 ( r + a3 cos θ (4.95) r r 2

46 4 42 A 0 + B 0 /a = V V Φ σ = ɛ 0 E r = ɛ 0 r (4.96) a σ = ɛ 0B 0 +3ɛ a 2 0 E 0 cos θ (4.97) Q = B 0 Φ(r, θ) =V 1 1 σ2πa 2 dμ =4πɛ 0 B 0 (4.98) Q 4πɛ 0 a + Q 4πɛ 0 r E 0r cos θ + E 0a 3 cos θ (4.99) r 2 3 4

47 ρ(r) (2.9) Φ(r) = 1 4πɛ 0 ρ(r )dv r r (5.1) r r 1 = 1 ) 1/2 2r r (1 + r 2 (5.2) r r r r 2 r [ 2 = ) 2r r ( + r ) 2 2r r ( + r 2 + ] r 2 r 2 r 2 8 r 2 r 2 1/r 1 r r = 1 r + r r r [ ] 3(r r ) r 2 r 2 + (5.3) r 5 Φ(r) = { 1 1 ρ(r )dv + 1 r r ρ(r )dv (5.4) 4πɛ 0 r r [ ] } 3(r r ) 2 r 2 r 2 ρ(r )dv r 5 2 ( 1/r ) 1. : q = ρ(r )dv (5.5)

48 5 44 q/r 2. : p = r ρ(r )dv (5.6) p r (5.7) r 3 ( ) 3. 4 : Q ij = (3x ix j δ ij r 2 )ρ(r )dv (5.8) Φ(r) = q 4πɛ 0 r + p r 4πɛ 0 r + 3 i,j Q ij x i x j + (5.9) 8πɛ 0 r5 q, q (0, 0,d/2), (0, 0, d/2) 0 p = r ρ(r )dv = qdẑ Φ(r) = qdz 4πɛ 0 r 3 = p r 4πɛ 0 r 3 (5.10) Problem 5.1 (5.10) r ± = x 2 + y 2 +(z d/2) 2 Φ(r) = q ( 1 1 ) 4πɛ 0 r + r (5.11)

49 5 45 z r θ θ p 5.2 E(r) = Φ(r) = 1 [ ] 3(r p)r r 2 p 4πɛ 0 r 5 (5.12) p = pẑ p z E r = Φ(r) = p cos θ 4πɛ 0 r 2 (5.13) p 4πɛ 0 2cosθ r 3, E θ = p 4πɛ 0 sin θ r 3, E φ = 0 (5.14) ẑ = ˆr cos θ ˆθ sin θ (5.15) E 2 U = qde cos θ = p E (5.16)

50 κ κ Q 0 κq 0 Q 0 (κ 1)Q 0 κq 0 κ 1 χ χ =(κ 1)ɛ 0 (5.17) ɛ ɛ = ɛ 0 + χ (5.18) 1 r dv dp(r ) P R = r r Φ(r) = 1 dv P R = 1 ( ) 1 dv P (5.19) 4πɛ 0 R 3 4πɛ 0 R A B =div(ab) BdivA (5.20) Φ(r) = 1 4πɛ 0 ( ) P dv div + 1 R 4πɛ 0 1 Φ(r) = 1 ( ) P n ds + 1 4πɛ 0 R 4πɛ 0 dv divp R dv divp R (5.21) (5.22) 1

51 5 47 σ P σ P = P n (5.23) 2 ρ P ρ P = divp (5.24) 5.4 dive = ρ ɛ 0 = ρ f + ρ P ɛ 0 = ρ f divp ɛ 0 (5.25) D ( ) ɛ0 E + P div = ρ f (5.26) ɛ 0 ɛ 0 D = ɛ 0 E + P = ɛe (5.27) divd = ρ f (5.28) D ds = Q f (5.29) Q f S (D 1 D 2 ) n = σ f (5.30) D E dr 0 (E 2 E 1 ) t = 0 (5.31) t

52 5 48 Problem 5.2 d d 0 S C = (5.32) (d d 0 )/ɛ 0 + d 0 /ɛ ( ) Problem 5.3 q P = D ɛ 0 E D = E = Q = q + q P q 4πr 3 r (5.33) q 4πɛr 3 r (5.34) P = (ɛ ɛ 0)q 4πɛr 3 r (5.35) Q = q κ (5.36) O 2 H H-O-H Cm ( I ) z E 0 (4.82) ( Φ(r, θ) = A 0 + B ) ( 0 P 0 (μ)+ A 1 r + B ) ( 1 P r r 2 1 (μ)+ A 2 r 2 + B ) 0 P r 3 2 (μ)+ (5.37) E 0 r cos θ ( Φ(r, θ) = A 0 + B ) 0 + r Φ out (r, θ) = B 0 r + ( A 1 r + B 1 r 2 ) cos θ (5.38) ( E 0 r + B ) 1 cos θ (5.39) r 2

53 5 49 (5.38) Φ in (r, θ) =A 0 + A 1 r cos θ (5.40) (5.30), (5.31) Φ out ɛ 0 r = ɛ Φ in r=a r, r=a Φ out θ = Φ in r=a θ (5.41) r=a ( B 0 =0,ɛA 1 = E 0 + 2B ) 1 ɛ a 3 0,A 1 = E 0 + B 1 (5.42) a 3 Φ out = ( E 0 r + (ɛ ɛ ) 0)E 0 a 3 cos θ (5.43) (ɛ +2ɛ 0 )r 2 Φ in = 3ɛ 0 (ɛ +2ɛ 0 ) E 0r cos θ (5.44) E in = Φ in == 3ɛ 0 (ɛ +2ɛ 0 ) (E 0 r) = 3ɛ 0 (ɛ +2ɛ 0 ) E 0 (5.45) a E E = E in E 0 = ɛ ɛ 0 (ɛ +2ɛ 0 ) E χ 0 = (ɛ +2ɛ 0 ) E 0 (5.46) P P = χe in E = P 3ɛ 0 (5.47) E E χe/ɛ 0 P = χe 3 E = P ɛ 0 (5.48)

54 50 6 I C/s j Problem 6.1 n v j = nev (6.1) 6.1 S j(x) n(x)ds n(x) S ( ) 0 j(x) n(x)ds =0 (6.2) S divj(x)dv =0 (6.3) divj(x) = 0 (6.4) j(x) n(x)ds S j(x) n(x)ds = Q t = ρ(x)dv (6.5) t S

55 6 51 divj + ρ =0 (6.6) t 6.2 m dv dt = ee mv τ (6.7) τ v m v = ee (6.8) τ j = n( e)v j = σe, σ = ne2 τ m σ R ρ l S R = ρ l S (6.9) (6.10) (Kirchhoff) Problem 6.2 (Wheatstone)

56 q F = q(e + v B) (6.11) v I ds db μ 0 db = μ 0 Ids r (6.12) 4π r 3 μ 0 =4π 10 7 [N A 2 ] (6.13) B = μ 0I 4π dr ˆR R 2 (6.14) R 2 = r 2 + z 2, dr = ẑdz, dr ˆR = ˆφ r R dz (6.15) B(r) = μ 0I 4π ˆφ rdz (r 2 + z 2 ) = μ 0I 3/2 4π ˆφ z r(r 2 + z 2 ) 1/2 z = z = (6.16)

57 6 53 B(r) = μ 0I 2πr ˆφ (6.17) I qvb = I B/(nS) ns qvb(ns)=i B = μ 0II 2πr (6.18) 6.6 C r C B = μ 0 4π Idr (r r 0 ) C r r 0 (6.19) 3 C B dr = μ 0 I[dr 0 (r r )] dr 4π r r 0 (6.20) 3 C C [dr (r r )] dr =(dr dr ) (r r ) r r = r C r C B dr = μ 0I 4π C C C I(dr dr ) r r 3 (6.21) dr dr r r 1/r 2 cos θ r 3 B dr = μ 0I cos θds (6.22) 4π r 2 4π C B dr = μ 0 I (6.23)

58 6 54 (6.23) μ 0 j nds S S rotb nds rotb = μ 0 j (6.24) Problem 6.3 a I B z = μ 0 Ia 2 2(a 2 + z 2 ) 3/2 (6.25) Problem 6.4 (6.25) B zdz z a z B z = μ 0 Ia 2 2[a 2 +(z z) 2 ] 3/2 (6.26) N L dz Ndz /L db z = L/2 L/2 B z = μ 0NIa 2 2L = μ 0NI 2L μ 0 NIa 2 dz 2L[a 2 +(z z) 2 ] 3/2 (6.27) L/2 dz (6.28) L/2 [a 2 +(z z) 2 ] 3/2 z L/2 z [a 2 +(z z) 2 ] 1/2 L/2

59 : B z /μ 0 ni z/a L/a =8 B z = μ { } 0NI L/2 z 2L [a 2 +(L/2 z) 2 ] + L/2+z 1/2 [a 2 +(L/2+z) 2 ] 1/2 L a, z (6.29) B z = μ 0NI L = μ 0nI (6.30) n = N/L Problem 6.5 Problem 6.6 a I r>ar <a Problem 6.7 a I r Problem 6.8 rotb = μ 0 j 6.6 Problem 6.9 a b 0 (r<a) μ B = 0 I 2πr ˆφ (a <r<b) 0 (r>b) (6.31) ˆφ

60 B = μ 0 4π C Idr (r r ) r r 3 (6.32) r/r 3 (1/r) A(r) = μ 0I dr 4π r r rota x [rota] x = μ ( 0I dz 4π C y r r ) dy z r r = μ [ ] 0I dz (y y )+dy (z z ) 4π r r 3 C C (6.33) (6.34) x B =rota (6.35) E = grad Φ (6.33) A(r) = μ 0 dr 3 j(r ) 4π C r r (6.36) div rot = 0 divb = 0 (6.37) dive = ρ/ɛ x y a r =(acos θ, a sin θ, 0) A(r) = μ 2π 0Ia (d cos θ, dsinθ, 0) (6.38) 4π r r 0

61 6 57 r r r r = 1 r + r r r 3 + (6.39) A x = μ 0Ia 4π 2π 0 (ax cos θ + ay sin θ)sinθdθ r 3 = μ 0Iπa 2 y 4πr 3 (6.40) 1 n z S = πa 2 rota A y = μ 0Iπa 2 x 4πr 3 (6.41) A = μ 0IS(n r) 4πr 3 (6.42) B x = 3μ 0ISxz 4πr 5 (6.43) B y = 3μ 0ISyz 4πr 5 B z = μ 0IS( x 2 y 2 +2z 2 ) 4πr 5 B = μ 0 4π [ ] 3(r m)r r 2 m (5.12) r 5 (6.44) m = ISn (6.45) 1 2 a+t a dx sin 2 kx = a+t a dx cos 2 kx = T 2 T T = π n, (n : )

62 j m M rotm = μ 0 j m (6.46) rotb = μ 0 (j + j m ) (6.47) ( ) M rot(b M) =μ 0 j (6.48) H def = 1 μ 0 (B M) (6.49) roth = j (6.50) M = χh (6.51) χ H B =(μ 0 + χ)h = μh, μ = μ 0 + χ (6.52) (6.50) H dr = j dr (6.53) C C

63 6 59 H 2, H 1, = j (6.54) j =0 H 2, = H 1, (6.55) divb =0 B 2, = B 1, (6.56) B1 B2 H1 H2 6.2: D, H E B q,ρdv I,jdV F = qq F = μ 0II 4πɛ 0 r 2 2πr, E = grad Φ, B =rota Φ= 1 ρ 4πɛ 0 R dv A = μ 0 j 4π R dv dive = ρ/ɛ 0 divb =0 E 1, = E 2,,D 1, = D 2, H 1, = H 2,,B 1, = B 2,

64 60 7 (5.28),(6.37), (6.50) divd = ρ f, divb =0, roth = j (rote ) 7.1 ( ) E ind Φ E ind = dφ (7.1) dt 1 1 E E ind Φ Φ

65 7 61 (7.1) E ind = C E dx = S rote nds (7.2), n C S Φ = B nds S rote = B t (7.3) 7.2 divd = ρ f, rote = B t divb =0, roth = j (Maxwell) roth = j div div roth =divj (7.4) div rot = 0 0 (6.6) ρ/ t j j + D t div D t = divd t = ρ t (7.5) ρ/ t 0 roth = j + D t (7.6) D/ t -

66 7 62 ( - ) ( ) E = E sin ωt j = σe j D j D = D t = ɛωe 0 cos ωt (7.7) j D /j ωɛ/σ 10 8 Ω 1 m 1 ɛ A 2 s/n m 2 ω <10 18 s V = L di dt (7.8) L A,B A V B = M di A (7.9) dt (6.30) Φ=S μ 0 ni (7.10) a n Φ=πa 2 μ 0 n 2 I = LI, L = πa 2 μ 0 n 2 (7.11) Problem 7.1 L n =10 4 m 1 a =1cm

67 7 63 I C Q -Q L 7.1: LC N B I A n A a Φ=N B πa 2 μ 0 n A I A (7.12) M = μ 0 n A N B πa 2 (7.13) 7.4 (3.19) ɛ 0 E 2 /2 ɛe 2 /2 D = ɛe U = E D (7.14) 2 C L dq dt = I, LdI dt + Q C = 0 (7.15)

68 7 64 L d2 Q dt 2 = Q C (7.16) ω 0 =1/ LC t =0 Q 0 Q(t) =Q 0 cos ωt, I = Q(t) = Q 0 ω sin ωt (7.17) Q 2 2C + LI 2 2 = Q2 0 (7.18) 2C 2 B = μ 0 ni, L =(πa 2 l)μ 0 n 2 (7.11) LI 2 2 = Vμ 0n 2 I 2 2 = VB2 2μ 0 (7.19) V = πa 2 l U = B2 2μ 0 (7.20) U = H B 2 (7.21) Problem 7.2 V 0 CR Q(t) t =0 Q = CV 0 (1 exp( t/rc)) Problem 7.3 V 0 LR t =0 I = V 0 R (1 exp( Rt/L)) Problem 7.4 LC LCR V L d2 Q dt + Q 2 C + RdQ dt = V (t) =V 0 cos ωt (7.22)

69 7 65 Q /CV ω ωτ = ω 0 /ω 7.2: LCR ω L, C ω 0 ω 0 ω ( I 5.2 ) Q =ReQ 0 e iωt ( ) 1 Re C Lω2 +iωr Q 0 e iωt =ReV 0 e iωt Q 0 = CV 0 1 ω 2 /ω 2 0 +iωτ ω 0 =1/ LC, τ = RC Q 0 = Q 0 e iφ Q = Q 0 cos(ωt + φ) I = Q 0 ω sin(ωt + φ) I CV 0 ω 7.5

70 7 66 i Φ i Φ i = n L ij I j (7.23) j=1 L ij i E i ind = n j=1 L ij di j dt (7.24) E i = n j=1 L ij di j dt + R ii i (7.25) E i i n n di j n E i I i = I i L ij dt + R i Ii 2 (7.26) i=1 i,j=1 L ij j i Φ i j = A j (r i ) dr i (7.27) C i j i A j (r i ) (6.33) A j (r i )= μ 0I j dr j (7.28) 4π r j r i C j i=1 Φ i j = L ij I j (7.29) L ij = μ 0 4π C i C j dr j dr i r j r i i, j (7.30) L ij = L ji (7.31)

71 7 67 (7.26) n E i I i = i=1 U = n i,j=1 d dt n i,j=1 ( ) Lij I i I j + 2 d dt ( ) Lij I i I j 2 n R i Ii 2 (7.32) i=1 (7.33)

72 68 8 divd = ρ (8.1a) divb =0 rote = B t roth = j + D t (8.1b) (8.1c) (8.1d) D = ɛe, B = μh E, B ( - ) div div rot = 0 ρ t (6.6) =divj (8.2) (divd ρ) = 0 (8.3) t div divb =0 (8.4) t 0 6 6

73 8 69 (6.35) rota = B ( rot E + A ) = 0 (8.5) t rot grad = 0 E + A t = grad φ (8.6) E = A t A,φ grad φ (8.7) Δφ + diva t = ρ ɛ (8.8) ) ( (Δ ɛμ 2 A grad diva + ɛμ φ ) t 2 t = μj (8.9) φ, A 4 4 (8.8) (8.9) 8.1 φ, A φ = φ χ t, A = A + χ (8.10) φ, A Problem 8.1 φ, A

74 8 70 χ (8.8) (Δ ɛμ 2 diva + ɛμ φ t = 0 (8.11) t 2 ) φ = ρ ɛ (8.12) (8.9) ) (Δ ɛμ 2 A = μj (8.13) t 2 φ, A 1 φ, A ) (Δ ɛμ 2 χ = 0 (8.14) t 2 χ Problem 8.2 A =(0,Bx,0) A =( By,0, 0) B =(0, 0,B) 8.2 dive = 0 divb = 0 rote = B t E rotb = ɛ 0 μ 0 t 3 rot (8.15a) (8.15b) (8.15c) (8.15d) rot rotv =graddivv Δv (8.16) grad dive ΔE = rotb (8.17) t 1 diva =0

75 (Δ ɛ 0 μ 0 2 t 2 ) E = 0 (8.18) c = 1 ɛ0 μ 0 (8.19) x dive =0, divb =0 E x x = B x x = 0 (8.20) E x = B x t t = 0 (8.21) E z x = B y E y, t x = B z t (8.22) B z x = ɛ E y B y 0μ 0, t x = ɛ E z 0μ 0 t (8.23) E x,b x x, t 0 B z ( 2 x 2 ɛ 0μ 0 2 t 2 ) E y = 0 (8.24) E y = f + (x ct)+f (x + ct) (8.25) B z = 1 c [f +(x ct) f (x + ct)] (8.26) c E z,b y c c = m/s (8.27)

76 c E B E, B E(r,t)=E 0 e ik r ωt, B(r,t)=B 0 e ik r ωt, (8.28) E 0, B 0 k E 0 =0 k B 0 =0 k E 0 = ωb 0 (8.29) k B 0 = ω c 2 E k (k E 0 )= ω c 2 E 0 (8.30) A (B C) =B(A C) C(A B) (1.8) ω = ck (8.31) k E, k B 1 2 E B B(r,t)= 1 k E(r,t) (8.32) ω

77 8 73 f1 f1 θ1 θ1 θ2 f : E 1 = E 2, D 1 = D 2 (8.33) H 1 = H 2, B 1 = B 2 (8.34) E,D,B,H f = A cos(ωt k x) f 1 f 2 f 1 f 1 = A cos(ωt k x) (8.35) f 1 = B cos(ω t k x) f 2 = C cos(ω t k x) (8.36) z =0 f 1 (z =0,x,y,t)+f 1 (z =0,x,y,t)=f 2 (z =0,x,y,t) (8.37) A cos(ωt k x x k y y)+bcos(ωt k x x k yy)=ccos(ωt k x x k yy) (8.38)

78 8 74 t, x, y A + B = C, ω = ω = ω (8.39) k x = k x = k x, k y = k y = k y (8.40) k, k, k k = ω/v k x = k x = k x ω sin θ 1 = ω sin θ 1 = ω sin θ 2 (8.41) v 1 v 1 v 2 θ 1 = θ 1 (8.42) sin θ 1 sin θ 2 = v 1 v 2 (8.43) v =1/ ɛμ () μ μ 0 v = c/n n = ɛμ ɛμ 0 (8.44) sin θ 1 sin θ 2 = n 2 n 1 (8.45) Problem 8.3 ( ) (3.19) (7.21) U = 1 dv (E D + H B) (8.46) 2

79 H, B ( ) ( H rote + B ) = 0 (8.47) t ( E roth D ) = E j t E D H B + E (roth) H (rote) =E j (8.48) t div(a B) = (rota) B A (rotb) (1.25) ( 1 2 D E + 1 ) 2 B H = E j +div(e H) (8.49) div(e H) dv div(e H) = (E H) nds (8.50) E H S = E H (8.51) U t = E j +divs (8.52)

80 : 1. a b = n a i b i = a b cos θ (1.1) ( (1.2)) i=1 a = (a 1,a 2,a 3 ) b = (b 1,b 2,b 3 ) a b = (a 2 b 3 a 3 b 2,a 3 b 1 a 1 b 3,a 1 b 2 a 2 b 1 ) a b a b sin θ θ a, b a b = b a (9.1) a (b + c) = a b + a c (a + b) c = a c + b c (9.2) (ka) b = k(a b) =a (kb) (9.3) a a = 0 (1.6) a (b c) =b (c a) =c (a b) (1.7) a (b c) =b(a c) c(a b) (1.8)

81 f x def f(x +Δx, y) f(x, y) = lim Δx 0 Δx (1.12) 3. 2 f x y = 2 f y x def =( x, y, z ) (1.16) U(r) U(r) = gradu(r) (1.19) v def =divv = v x x + v y y + v z z (1.21) v def =rotv =( v z y v y z, v x z v z x, v y x v x y ) rot(gradf) = ( f) 0 (1.22) div(rotv) = ( v) 0 (1.23) ( v) = ( v) 2 v (1.24) (v u) =( v) u v ( u) (1.25)

82 (v n)ds = dv divv (1.26) divv grad ΦdV = ΦndS (1.32) [ φ ψ n ψ φ ] ds = n dv (φ 2 ψ ψ 2 φ) (1.36) v dr = rotv nds (1.37) rotv : 1. ɛ 0 = C 2 /N m 2 (2.3) E = 1 q 1 ˆR 4πɛ 0 R q 2 ˆR 1 4πɛ 0 R = 2 i 1 4πɛ 0 q i R 2 i ˆR i (2.5) C 2. Φ Φ(r) = 1 q i (2.6) 4πɛ 0 R i i E = gradφ(r) (2.7)

83 ρ(r) Φ(r) = 1 4πɛ 0 dq(r ) r r = 1 4πɛ 0 ρ(r )dv r r (2.9) 4. = ˆr r + ˆφ 1 r φ + ẑ z (2.25) = ˆr r + ˆθ 1 r θ + ˆφ 1 r sin θ φ (2.33) : 1. E n ds = Q ɛ 0, Q = i q i (3.5) 2. E n ds = ρdv (3.6) 3. dive = ρ ɛ 0 (3.8) 4.

84 U = 1 2 N N j=1 i=1,i j q i q j 4πɛ 0 R ij (3.16) U = 1 2 ɛ 0 dve 2 (3.19) : 1. 1) 2) 3) 2. () Q i Φ i n Φ i = p ij Q j (4.17) Q i = j=1 n C ij Φ j (4.18) j=1 C ij 5. C ij = C ji,p= C 1 6. R C =4πɛ 0 R (4.24)

85 ΔΦ(r) = ρ(r) ɛ ΔΦ(r) =0 (4.30) 8. δ (4.34) { 0 x 0 η δ(x) =, δ(x)dx =1, for all η>0 x =0 9. η Δ 1 r =4πδ(x)δ(y)δ(z) def =4πδ(r) (4.35) 10. div grad = 2 x y z 2 (4.29) 11. z Δf(r, z) = 1 ( r f ) + 2 f (4.37) r r r z Δf(r) = 1 r 2 r ( r 2 f r ) (4.38) 13. F (x, y) f(x)f(y)

86 a x a f(x) f(x) = n= =ΔΦ= 1 ( r 2 Φ ) + r 2 r r ( A n cos nπx a 1 r 2 sin θ + B n sin nπx ) a ( sin θ Φ ) + θ θ (4.62) 1 2 Φ r 2 sin 2 (4.72) θ φ 2 0=ΔΦ= 1 r 2 r ( r 2 Φ ) + 1 r r 2 sin θ ( sin θ Φ ) θ θ (4.73) : 1. p = r ρ(r )dv (5.6) d ±q p = qd q q 2. Φ(r) = qdz 4πɛ 0 r 3 = p r 4πɛ 0 r 3 (5.10) 3. E(r) = Φ(r) = 1 4πɛ 0 [ ] 3(r p)r r 2 p r 5 (5.12) 4. U = qde cos θ = p E (5.16)

87 ρ f ρ P σ P 6. ρ P = divp (5.24) σ P = P n (5.23) divd = ρ f (5.28) D D = ɛ 0 E + P = ɛe (5.27) 7. (D 1 D 2 ) n = ρ f (5.30) (E 2 E 1 ) t =0 (5.31) n, t : 1. ( ) divj + ρ t 2. j E =0 (6.6) j = σe, σ = ne2 τ m (6.9)

88 (a) 0 (b) 4. F = q(e + v B) (6.11) 5. : I ds db μ 0 db = μ 0 Ids r (6.12) 4π r 3 μ 0 =4π 10 7 [N A 2 ] 6. qvb(ns)=i B = μ 0II 2πr (6.18) 7. B dr = μ 0 I (6.23) C rotb = μ 0 j (6.24) 8. B z = μ 0NI L = μ 0nI (6.30) 9. B =rota (6.35) A(r) = μ 0 dr 3 j(r ) 4π r r (6.36) C

89 B = μ [ ] 0 3(r m)r r 2 m 4π r 5 (6.44) 11. m = ISn (6.45) 12. E D j m M rotm = μ 0 j m (6.46) H def = 1 μ 0 (B M) (6.49) roth = j (6.50) 13. M = χh (6.51) χ B =(μ 0 + χ)h = μh, μ = μ 0 + χ (6.52) 14. H 2, H 1, = j (6.54) B 2, = B 1, (6.56) 15.

90 9 86 q,ρdv I,jdV F = qq F = μ 0II 4πɛ 0 r 2 2πr, E = grad Φ, B =rota Φ= 1 ρ 4πɛ 0 R dv A = μ 0 j 4π R dv dive = ρ/ɛ 0 divb =0 E 1, = E 2,,D 1, = D 2, H 1, = H 2,,B 1, = B 2, : 1. E ind = dφ dt rote = B t (7.1) (7.3) 2. roth = j roth = j + D t (7.6) 3. U = H B 2 (7.21) 4. E i ind = n j=1 L ij di j dt (7.24) L ij = L ji L ii n =2L 12 = M

91 : 1. 4 divd = ρ (8.1a) divb =0 (8.1b) rote = B t (8.1c) roth = j + D t (8.1d) 2. φ = φ χ t, A = A + χ (8.10) χ 3. ) (Δ ɛμ 2 φ = ρ t 2 ɛ (8.12) ( ) Δ ɛμ 2 A = μj (8.13) t 2 4. c =1/ ɛ 0 μ 0 c (a) (b) (c) S = E H (8.51)

92 88 A SI (international system of units, ) 1[kg], 1[m], 1[s] divj + ρ t F 1 = k 1 qq =0 (6.6) r 2 (A.1) 2 k 1 q 1[C] k 1 F = q E E = k 1 q r 2 (A.2) F 2 L =2k II 2 d (A.3) k 1 k 2 k 1 k 2 = c 2 (A.4) c

93 A 89 d B =2k 2 α I d (A.5) k 2 α k 2 ( ) B rot E = k 2 (A.6) t E + k 3 v B (A.7) div E =4πk 1 ρ (A.8a) div B =0 rot B =4πk 2 αj + k 2α E k 1 t B rot E + k 2 t = 0 2 B = k 3 k 2 α k 1 k1 c = k 3 k 2 α (A.8b) (A.8c) (A.8d) 2 B t 2 = 0 (A.9) (A.10) (A.4) k 3 = 1 α (A.11) (A.4),(A.11) k 1,α

94 A 90 k 1 k 2 k 3 α SI 1 4πɛ 0 =10 7 c 2 μ 0 4π = E + v B cgs esu 1 c E + v B cgs emu c E + v B cgs 1 c 2 c c 1 E + 1 c v B cgs Heaviside 1 4π 1 4πc 2 c c 1 E + 1 c v B A.1: k 1,k 2,k 3,α

95 91 B II rot(gradf) 0 2. div(rotf) 0 y 3. v =( x2 + y, x 2 x2 + y, 0) v dr 2 (a) a (0,a,0) (0, a, 0) ( ) (b) 4. V (r) = 1 r gradv (r = x 2 + y 2 + z 2 ) n 5. E = gradv (r) =(xy 2,x 2 y y, z) V (r) V = a Q r 2. a Q r 0 3 (0, 0, 0) (0, 0,a) q 1,q 2 E 1 E 2

96 B E 1 (r) 2. E 2 (r) 3. ɛ 0 (E 1 + E 2 ) 2 2 = ɛ 0E ɛ 0E ɛ 0 E 1 E 2 2 dv ɛ 0 E 1 E 2 r cos θ = t t r A dr r 2 + a 2 2rA 3 = 1 r 2 + a 2 2rA + C

97 B 93 II (a) x = a sin θ, y = a cos θ v =( cos θ, sin θ, 0), dr =(acos θ, a sin θ, 0)dθ π v dr = C 1 0 adθ = πa (b) v dr + v dr = rotv nds C 1 C 2 C 2 v dr C 2 v dr 0 v dr = rotv nds = (0, 0, 1 1 C 1 r ) (0, 0 1)dS = rdrdθ = πa r V (r) = E dr = (x,0,0) E dr (x,y,0) E dr (x,y,z) (0,0,0) (x,0,0) (x,y,0) E dr V (r) = x2 y y2 2 z E r = E r = 0 (r a) Q (r>a) 4πɛ 0 r 2 Qr 4πɛ 0 a 3 (r a) Q 4πɛ 0 r 2 (r>a) V (r) = V (r) = Q 4πɛ 0 a Q 4πɛ 0 r Q 4πɛ 0 a Q 4πɛ 0 r (r a) (r >a) ( ) 3 2 r2 2a 2 (r a) (r >a)

98 B E 1 = q 1 4πɛ 0 r r 3 2. a =(0, 0,a) E 2 = q 2 r a 4πɛ 0 r a 3 3. ɛ 0 dv E 1 E 2 = ɛ 0q 1 q 2 (4πɛ 0 ) 2 = q 1q 2 4πɛ 0 1 a = q 1 q 2 (4π) 2 ɛ 0 2π 0 1 π2π sin θdθ drr 2 r 2 ra cos θ 0 r 2 r 2 + a 2 2ra cos θ 3 r at dt dr 0 r2 + a 2 2rat 3 1

99 B 95 II v =( y, x, 0) O (a, b, 0)(a, b > 0) P 1. rotv 2. 0 P O v dr (B.1) (a) (0, 0, 0) (a, 0, 0) (a, b, 0) (b) (0, 0, 0) (a, b, 0) ( x = at, y = bt, 0 <t<1 ) 3. 2 b a(a <b) ( ) (a <r<b ) () q V (r) =C + C r (B.2) 1. r >bv (b) =V ( ) =0 C = C = a<r<b 0 r 0 r<a

100 B r<a a <r<br >b ( ) 3 h Q(> 0) h (h <h) Q Q = 50C,h =6km,h =3km 4 δ 1. a dxδ(x), a>0 a 2. a dxδ(x), a>0 a 3. a dxδ(x b), a>b> a dxδ(x b), 0 <a<b 0 5 p E U = p E U Φ U = dv ρ(r )Φ(r ) r Φ(r ) Φ(r)+(r r) gradφ(r) U dv [Φ(r)+(r r) gradφ(r)]ρ(r )

101 B 97 U Φ(r) dv ρ(r ) + gradφ(r) dv r ρ(r ) gradφ(r) r dv ρ(r ) 1,3 gradφ(r) = dv r ρ(r )= U = p E

102 B rotv =(0, 0, 2) II (a) (a,0,0) v (0,0,0) xdx + (a,b,0) v (a,0,0) ydy = a v 0 xdx + b ady = ab 0 (b) v =( bt, at, 0), dx =(adt, bdt, 0) v dx = (a) -( (b) )= v dr (a) (a, b, 0) (b) C = (a) -( (b) )= C v dr rotr nds = 2dS = ab , 2 0, 3 q, 4 q r 4πɛ 0 r 3, 5 q r 4πɛ 0 r 3, 6 q 4πɛ 0 r 1. Φ(r) = 1 4πɛ 0 ( Q x2 + y 2 +(z h) 2 Q x2 + y 2 +(z h ) + 2 Q x2 + y 2 +(z + h) 2 ) Q x2 + y 2 +(z + h ) 2 1 Q 2 3 Q 4 2. grad ( E(r) = Q (x, y, z h) 4πɛ 0 x2 + y 2 +(z h) (x, y, z + h) 23 x2 + y 2 +(z + h) 23 ) (x, y, z h ) x2 + y 2 +(z h ) + (x, y, z + h ) 23 x2 + y 2 +(z + h ) 23

103 B z =0 E(x, y, 0) = ( ) Q (0, 0, 2h) 4πɛ + (0, 0, 2h ) 0 x2 + y 2 + h 23 x2 + y 2 + h 23 S σ σs/ɛ 0 = ES ( ) σ = Q h 2π + h x2 + y 2 + h 23 x2 + y 2 + h (x, y, z) =(0, 0, 0) E z (0, 0, 0) = Q ( 2 4πɛ 0 h + 2 ) 2 h V/m 50C km 4 1. a dxδ(x) =1,a>0 a 2. a dxδ(x) = 1, a>0 a 3. x b = y a b b 1 dyδ(y), a>b>0 4. x b = y a b dyδ(y), 0 <a<b b δ(y) , 2 E, 3 p, 4 p E

104 B Maxwell II divd = ρ divb =0 rote = B t roth = j + D t 1. E, D, B, H (B.3a) (B.3b) (B.3c) (B.3d) Maxwell dive = (B.4a) divb =0 (B.4b) rote = B (B.4c) t rotb = E (B.4d) t 5. Maxwell (B.4c) rot rot rotv =graddivv Δv grad dive Δ = rotb t (B.5) Δ 2 / x / y / z 2 Maxwell (B.4a) (B.5) (B.4d) (B.5) (Δ ɛ 0 μ 0 2 t 2 ) E = 0

105 B I a Biot-Savart B(r) = μ 0 dr 4π I (r r ) r r 3 z r =(0, 0,z) r r =(a cos θ, a sin θ, 0) dr = dθ dr (r r )= dθ r r = (0,0,z) B B =(, ) 3 4 LC L C 1. dq dt L di dt + Q C =0 = I Q t =0 Q 0 Q(t),I(t) 4. I C Q -Q L

106 B II V/m, C/m 2 T( ) A/m ɛ 0 μ 0 5. E 0 ɛ 0 μ 0 2 E t 2 1 ɛ0 μ m/s 2 ( a sin θ, a cos θ, 0) (az cos θ, az sin θ, a 2 ) a 2 + z μ 0Ia 2 2 a 2 + z 23 3 B dr = μ 0 I r 2πrB B = μ 0I 2πr 4 1. L d2 Q dt + Q 2 C =0 2. T =2π LC 3. Q(t) =Q 0 cos ωt, I(t)= Q(t) = Q 0 ω sin ωt 4. Q 2 /2C LI 2 /2 Q 2 2C + LI 2 2 = Q2 0 cos 2 ωt 2C + LQ2 0ω 2 sin 2 ωt 2 = Q2 0 2C (cos2 ωt +sin 2 ωt) = Q2 0 2C

107 B 103 II , rot(gradf) 0 2. div(rotf) 0 3. rot(rotf) grad(divf) Δf df dr 4. gradf(r) = r r x 5. E =( x2 + y 2 + z, 23 y x2 + y 2 + z 23, z x2 + y 2 + z 23 ) E dr (a) (x 1,y 1,z 1 ) x (x 2,y 1,z 1 ) (b) (x 2,y 1,z 1 ) y (x 2,y 2,z 1 ) (c) (x 1,y 1,z 1 ) (x 2,y 2,z 2 ) 2 a Q (1. ) 1. r V (r) = (α >0) q exp( αr) 4πɛ 0 r (B.6) r

108 B r 4. r 5. 4 ****************************************************************************** Δf = ( 2 f x, 2 f y 2 f z ) Δf = x 2 y 2 z 2 (( )f x 2 y 2 z 2 x, ( )f x 2 y 2 z 2 y, ( )f x 2 y 2 z 2 z ) (x 1 x ) (a) (b) 1 x 2 2 +y2 1 +z2 1 1 x 2 2 +y2 2 +z2 1 1 (c) x 2 2 +y2 2+z2 2 1 x 2 1 +y2 1 +z2 1 1 x 2 2 +y2 1 +z2 1 1 x 2 1 +y1 2+z

109 B E = gradv (r) E = q 1+αr 4πɛ 0 r exp( αr) r 3 α (αr ) 2. Q ɛ 0 =4πr 2 E r, E r Q = q(1 + αr)exp( αr) 3. r +0 q, r /4πr 2 dq/dr = ρ ρ(r) = q α 2 4π r exp( αr) dive = ρ/ɛ q q 4 4 0

110 B 106 II a ρ r 1. r>a 2. r<a 2 a x q,+q x R, R R, q, q/r 2 =2πɛ 0 E E V (> 0) 1,2 Q, q 2 1 V (> 0) 1,2 Q,q Q, Q,q,q 3. Q, Q,q,q 4 δ f(x) 1. a dxδ(x), a>0 a 2. a dxδ(x b), 0 <a<b 0 3. a dxδ(x b), 0 <b<a 0 4. dxf(x)δ(x b).

111 B p 2. p E d q, q (a) (b) 6 (HCl) H Cl 1.27 H +0.17e Cl -0.17e HCl e ****************************************************************************** 1 r h E r 2πrh = Q ɛ 0 = πa2 ρh ɛ 0 E r 2πrh = Q ɛ 0 = πr2 ρh ɛ 0 E r = a2 ρ 2ɛ 0 r E r = rρ 2ɛ q = ±aq/r a 2 /R 2. q/4πɛ 0 R 2 2=E x

112 B a 2 R aq R =4πɛ 0Ea 3 E = 3(p r)r r2 p 4πɛ 0 r 5 (5.12) r = a E = 3Ex a 2 r, r r 3 1. ( ( Q /V Q/V q /V q/v Q 1 Q 2 ) ) ( = Q /V q /V Q/V q/v )( V V ) 2. Q,q >0 q,q<0 3. Q = q f(b) qd E 3. p E = [C m]

113 B 109 II 1 1. E B Maxwell 2. e ɛ 0 μ a I r r r >a,r<a 3 N, L I 4 1. A 1 = B 0 (0,x,0), A 2 = B 0 ( y, 0, 0), A 3 = B 0 ( y/2,x/2, 0) B 1, B 2, B 3 2. A f A A = A +gradf A, A 3. A 1 A 3 f 5 L R V I L di dt + RI = V t<0 t =0

114 B I I,L,R,V

115 B Maxwell e = [C],ɛ 0 = [C 2 /N m 2 ],μ 0 =4π 10 7 [N A 2 ] 3. c =1/ ɛ 0 μ 0 4. c = [m/s] 2 1. j = I/πa 2 2. r { { μ 0 I r > a μ 0 I 1 r>a 2πrB = B = 2πr μ 0 I r2 r a μ a 2 0 I r r a 2πa B 0 h Bh = μ 0 (N/L)hI B = μ 0 (N/L)I 4 B =rota 1. B 1 = B 2 = B 3 =(0, 0,B 0 ) 5 2. rotgradf = 0 3. gradf =( y/2, x/2, 0) A 3 = A 1 +gradf f = B 0 xy 2 1. LI 2 /2 2. di/dt =0 I = V/R 3. I = V (1 exp( Rt/L)) R

116 B I; L; R; V ; II rot(gradf) 0 2. div(rotf) 0 3. E =(y 2, 2xy, 0) E dr (a) (0, 0, 0) x (x, 0, 0) (b) (x, 0, 0) y (x, y, 0) (c) (x, y, 0) z (x, y, z) (d) E =gradf(x, y, z) f(x, y, z) 2 a Q 1. r σ 2. λ 4 q d 1. 2.

117 B ****************************************************************************** (a) 0, (b) xy 2,(c)0,(d)f(x, y, z) =xy Q 2 r 8πɛ 0 a 3 4 q 1. 2d q2 16πɛ 0 d 2 2. R E z = qd 2πɛ 0 R 3 σ = ɛ 0 E z = qd 2πR 3 3. q

118 B 114 II a x q x R 1. q I r I 2. q I 3. 2 δ f(x) 1. a a dxδ(x), a>0 2. a dxδ(2x), a>0 a 3. a dxδ( 3x), a>0 a 4. dxf(x)δ(x b). 3 z ΔΨ(r) = 1 ( r Ψ ) =0 r r r 1. C 1 + C 2 log r 2. a b V a 0 3. C 1,C 2

119 B a<r<b r Φ U = dv ρ(r )Φ(r ) r Φ(r ) Φ(r)+(r r) gradφ(r) U dv [Φ(r)+(r r) gradφ(r)]ρ(r ) U Φ(r) dv ρ(r ) + gradφ(r) dv r ρ(r ) gradφ(r) r dv ρ(r ) 1,3 gradφ(r) = dv r ρ(r )= U = p E 2. p y xy

120 B 116 ****************************************************************************** 1 1. q I = qa/r,r I = a 2 /R 2. 0 q I = qa/r 3. R q r I q I 0 q I Φ=q I /4πɛa = q/4πɛr /2 3. 1/3 4. f(b) r f r = = C 2 r 2. E r = dψ/dr = C 2 /r 3. C 1 + C 2 ln a = V a,,c 1 + C 2 ln b = V b 1 ln(b/a) r 4. E = Va C 1 = V a ln b ln(b/a),c 2 = ln(b/a), Φ= V a ln(r/b) ln(b/a) V a 5. 2πrE = λ /ɛ 0 V a λ =2πɛ 0 ln(b/a) , 2 E, 3 p, 4 p E 2.

121 B 117 II Maxwell dive = 0 (B.7a) divb =0 (B.7b) rote = B (B.7c) t E rotb = ɛ 0 μ 0 (B.7d) t Maxwell 3 rot rot rotv =graddivv Δv grad dive Δ 1 = rotb t (B.8) Δ 2 / x / y / z 2 Maxwell 1 (B.8) 2 4 (B.8) 3 (Δ ɛ 0 μ 0 2 t 2 ) E = 0 2. E =(A sin(kz ωt), 0, 0) k ω 3. v = λf = ω/k 4. ɛ 0 = C 2 /N m 2, μ 0 =4π 10 7 [N A 2 ] 5. 3GHz 2 a I r

122 B r>a B(r) 2. r<a B(r) 3 n, l I C Q Q 2 /2C ɛ 0 E 2 /2 U V L = Vμ 0 n 2 U = LI 2 /2 U = B 2 /2μ 0 V 4 LCR R L C RI + L di dt + Q C =0 1. dq dt = I Q 2. t =0 Q 0 Q(t)

123 B 119 ****************************************************************************** 1 Maxwell E, 2 0, 3 ɛ 0 μ 0 2 E t 2 2. k = ɛ 0 μ 0 ω 3. v =1/ ɛ 0 μ 0 4. v =1/ ɛ 0 μ [m/s] μm f = /s,v= m/s 0.1m = 10cm 2 r { { μ 0 I r > a μ 0 I 1 r>a 2πrB = B = 2πr μ 0 I r2 r a μ a 2 0 I r r a 2πa B = μ 0 ni 2. LI 2 /2=Vμ 0 n 2 I 2 /2VB 2 /2μ 0 = Vμ 0 n 2 I 2 /2 1. d 2 Q dt 2 + R L dq dt + Q CL =0

124 B d 2 x dx +2γ dt2 dt + ω2 0x =0 γ<ω 0 γ >ω 0 t =0 0 ( ) R C/L =2