128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds

127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ

128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds S 1 +S 2 = { B( r) n( r)}ds { B( r) n( r)}ds S 1 S 2 = Φ 1 Φ 2 Φ 1 = Φ 2 (3.5) S 3.1.2

3.1 129 3.1.3 (0, 0, B) AB l v x y B A B E y - Q Q C f = -e v B B B v l D x A 3.1 E ( e) = V A V B l ( e) = vb ( e) AB E

130 3 II AB = vbl vbl vl = ds/dt S 3.1 3.1.4 ϕ em = E( r, t) d s (3.6) C C 3.1 E( r, t) d s = B( r, t) ds t (3.7) C E( r, t) ds = B( r, t t) ds (3.8) S S S E( r, t) = B( r, t t) (3.9) S

3.1 131 3.1.5 2 1,2 1 I 1 2 Φ 2 Φ 2 I 1 M Φ 2 = MI 1 I 1 1 2 M 3.2 I 1 Φ 2 2 V 2 V 2 = dφ 2 dt = M di 1 dt M

132 3 II V I V = L di dt L H(= V s A ) L I 3.3 n l S I B = µ 0 ni

3.1 133 BS nl Φ = nlbs = µ 0 n 2 lsi L = µ 0 n 2 ls (3.10) L di R dt V V L di(t) dt = RI(t) I(t) = V R (1 e t/τ ) τ = L/R 3.1.1 10 cm 2000 5 10 5 T 15 ===== ===== Φ = 2000 π 0.05 2 sin( 2π t) 5 10 5 T

134 3 II L R I V/R V S 0 t 3.4 dφ dt [V] = 2000 π 0.052 2π T = 7.4 10 2 cos(30 π t) 2π cos( t) 5 10 5 1/15 3.1.2 I z z-x 1 S) v x-y ===== ===== z-x

3.1 135 Φ(t) = µ 0I 2 π r(t) S ϕ em ϕ em = dφ dt = dφ dr dr dt = µ 0IS dr 2 π r 2 dt = µ 0IS 2 π r 2 v x-y 3.1.3 Fig. 3.1 R 1. 2. AB ===== ===== 1. ϕ em = dφ dt = Blv I = ϕ em R = Blv R

136 3 II I 2 R = ( ) 2 Blv R = B2 l 2 v 2 R R 2. f = IBl = Blv R Bl = B2 l 2 v R fv = B2 l 2 v 2 R 3.1.4 5 cm 20 cm 2000 ===== ===== [H] L = µ 0 n 2 ls = µ 0 2000 2 l 2 l π r 2 7 20002 = 4π 10 0.2π 0.0252 0.22 = 4.9 10 2 3.1.5 a 1, a 2 (a 2 > a 1 ) 2 R I

3.1 137 1. 2. l l (a 2 a 1 ) ===== ===== 1. r a 1 < r < a 2 2 π r B(r) = µ 0 I B(r) = µ 0I 2 π r r > a 2 B(r) = 0 2. a2 Φ = l a 1 = µ 0Il 2 π B(r)dr a2 a 1 1 r dr = µ 0Il 2 π log(a 2/a 1 ) L = µ 0l 2 π log(a 2/a 1 )

138 3 II 3.1.6 2 n 1, n 2 l 1, l 2 S 1, S 2 1 2 S 1 > S 2 1 2 ===== ===== 1 B = µ 0 n 1 I 1 2 Φ 2 = n 2 l 2 BS 2 = µ 0 n 1 n 2 l 2 S 2 I 1 M = µ 0 n 1 n 2 l 2 S 2 3.1.7 2 1 L 1 2 M 1 ϕ 1 (t) 2 ϕ 2 (t) ===== ===== 1 I 1 ϕ 1 (t) L di 1 dt = 0 ϕ 2 (t) = M di 1 dt ϕ 1 (t) ϕ 1 (t)/ϕ 2 (t) = L/M

3.1 139 3.1.8 a 1, a 2 a 1 a 2 1. a I B( r) µ0 I t( r ) ( r r ) = 4π r r 3 ds r = (a cos θ, a sin θ, 0) t( r ) = ( sin θ, cos θ, 0) r = (0, 0, 0) ds = adθ θ 0 2π 2. 1 I 1 2 3. a I xy x 1 r = (x, 0, 0) x a (1 + x) n 1 1 + nx, 1 + x 1 x x 1 4. 2 I 2 1 ===== ===== 1. = B( 0) µ0 I t( r ) ( r r ) 4π r r 3 ds

140 3 II = µ0 ( sin θ, cos θ, 0) ( a cos θ, a sin θ, 0) ( 4π ( a ) 3 Iadθ cos θ)2 + (a sin θ) 2 z z B z ( r) 0 2π µ0 a B z ( 0) = 4π a 3 Iadθ = µ 0 I dθ 4π a = µ 0 I 4π a 2π = µ 0 2a I 2. 1 B = µ 0 2a 1 I 1 2 ϕ 2 = BS = µ 0 2a 1 I 1 πa 2 2 M = πµ 0a 2 2 2a 1 3. (x, 0, 0) x a B( r)

3.1 141 = = µ0 I t( r ) ( r r ) 4π r r 3 ds µ0 ( sin θ, cos θ, 0) (x a cos θ, a sin θ, 0) ( 4π (x ) 3 Iadθ a cos θ)2 + (a sin θ) 2 z z B z ( r) 0 2π = B z ( r) µ0 a x cos θ ( 4π (x ) 3 Iadθ a cos θ)2 + (a sin θ) 2 1 since 1 + nx when x 1 (1 x) n ( ) µ0 (a x cos θ) 1 + 3 2ax cos θ 2 x = 2 +a 2 4π ( x2 + a 2) 3 Iadθ = µ0 4π since a x cos θ + 3 a2 x cos θ x 2 +a 3 ax2 cos 2 θ 2 x 2 +a ( 2 x2 + a 2) 3 Iadθ cos 2n+1 θdθ = 0 4. = µ0 4π a 3 ax2 cos 2 θ x 2 +a 2 ( x2 + a 2) 3 Iadθ since x a µ0 Ia a 3 2 = a 4π x 3 dθ = µ 0 πa 2 I 4π x 3 = µ 0 a 2 I 4 x 3 Φ 1 = a1 0 B z ( r) 2πrdr

142 3 II = a 1 B z ( r) 2πrdr B( r) d s semi sphere r B( r) B(r 0 ) r3 0 r 3 ds = 4πr2 semi sphere 2 Φ 1 = = a 1 a 1 = πµ 0a 2 2I 2 2 = πµ 0a 2 2 2a 1 I 2 B z ( r) 2πrdr ( µ 0 4 a 1 a 2 ) 2I r 3 2πrdr dr r 2 M = πµ 0a 2 2 2a 1 3.1.6 1 2 2 1 1 I 1 B( r) = A( r) A( r) = µ 0I 1 d s 1 ( r 1 ) 4π C 1 r r 1 2 Φ 2 = B( r) ds S 2

3.1 143 = { A( r)} ds S 2 = A( r2 ) d s 2 C 2 A( r) µ 0 I 1 d s 1 d s 2 Φ 2 = C 2 4π C 1 r 2 r 1 (3.11) 1 L 21 = µ 0 d s 1 d s 2 (3.12) 4π C 1 r 2 r 1 C 2 2 L 12 C 1, C 2 d s 1, d s 2 r 1, r 2 3.1.7 u e = 1 2 ε 0E 2 (3.13) u m = 1 2µ 0 B 2 (3.14) = µ 0 2 H2

144 3 II 3.14 I(t) t = 0 I(0) = 0 t = t 1 I(t 1 ) = I ϕ(t) ϕ(t) = L di(t) dt I(t) t W = ϕ(t)i(t) t t = 0 t 1 W = = L t1 0 t1 0 = 1 2 LI2 ϕ(t)i(t) dt di(t) I(t) dt dt 3.10 L W = 1 2 µ 0n 2 lsi 2 = 1 2 µ 0n 2 V I 2 = 1 2µ 0 B 2 V V = ls B = µ 0 ni 3.14

3.2 145 3.2 3.2.1 V (t) = V 0 cos ωt ω f = ω/2π R L C I(t) V (t) = V 0 cos ωt = RI(t) I(t) = V 0 cos ωt R V (t) L di dt = V 0 cos ωt L di dt = 0 I(t) = V 0 cos ωtdt = V 0 cos(ωt π/2) L ωl V (t) Q/C = 0 I(t) = dq dt

146 3 II = C dv dt = C dv 0 cos ωt dt = V 0 ωc cos(ωt + π/2) I R II L III q - q C 3.5

3.2 147 R V (t) = V 0 cos ωt V 2 (t)/r < V 2 (t) R > = 1 T = V 0 2 R = 1 2 T 0 1 T V 2 0 R V 2 0 R cos2 ωtdt T 0 1 + cos 2ωt dt 2 T V e = V 0 2 V 2 e R I e = I 0 2 π/2 π/2 ϕ cos ϕ I e V e cos ϕ 3.2.2 C Q 0 S I(t) I(t) = dq(t) dt Q(t)

148 3 II S Q L - Q C R 3.6 R = 0 Q(t) 0 = L di(t) dt = L d2 Q(t) dt 2 + RI(t) + Q(t) C + R dq(t) dt d 2 Q(t) dt 2 = 1 LC Q(t) + Q(t) C Q(t) = Q 0 cos(ω 0 t + δ) 1 ω 0 = LC Q(t) = Ae αt cos(ω t + δ) α = R 2L 1 ω = LC R2 4L 2

3.2 149 3.2.1 1 H 1 A 0.1 µf? ===== ===== 1 2 LI2 = 1 2 1 1 1 2 CV 2 = 1 2 0.1 10 6 V 2 V = 10 7 = 3.16 10 3 [V] 3.2.2 L I R t = 0 I 0 R ===== ===== L di dt + RI = 0 di dt = R L I I(t) = I 0 e t/t

150 3 II T = L/R Q = = 0 0 RI 2 dt RI 2 0 e 2t/T dt = RI0 2 e 2t/T dt 0 [ = RI0 2 T ] 2 e 2t/T 0 = 1 2 RT I2 0 = 1 2 LI2 0 3.2.3 3.1.5 L 1 2 LI2 ===== ===== U = l 2µ 0 a2 a 1 = 1 µ 2 0I 2 l 2µ 0 2π B 2 (r)2πrdr a2 a 1 1 r dr = 1 µ 0 I 2 l 2 2π log(a 2/a 1 ) L = µ 0l 2π log(a 2/a 1 ) U = 1 2 LI2

3.2 151 3.2.4 L 1, L 2 M 2 1. 1 I 1 U 1? 2. 1 2 2 I 1,2? 3. ϕ 2 2 0 I 2 U 2 4. 2 0 I 2 ϕ 1 1 I 1 U 3 5. U 1, U 2, U 3 ===== ===== 1. 2. 3. U = 1 2 L 1I 2 1 Φ 1 = L 1 I 1 + MI Φ 2 = L 2 I + MI 1 ϕ 2 = L 2 di dt T U 2 = ϕ 2 Idt 0

152 3 II T = = 0 I2 0 = 1 2 L 2I 2 2 L 2 di dt Idt L 2 IdI 4. T U 3 = = 0 T 0 ϕ 1 I 1 dt I2 = MI 1 di = MI 1 I 2 M di dt I 1dt 0 5. U 1 + U 2 + U 3 = 1 2 L 1I1 2 + 1 2 L 2I2 2 + MI 1 I 2 = 1 ( ) ( ) 2 (I L1 M I1 1, I 2 ) M L 2 I 2 3.2.5 1.? 2.? ===== ===== 1. Q(t) = Q 0 cos(ω 0 t + δ)

3.2 153 S Q L - Q C R 3.7 I(t) = dq dt = ω 0 Q 0 sin(ω 0 t + δ) = I 0 sin(ω 0 t + δ) ω 0 t + δ = nπ 1 Q 2 0 2 C 2. ω 0 t + δ = π/2 + nπ 1 2 LI2 0 = 1 2 L(ω 0Q 0 ) 2 = 1 2 Lω2 0Q 2 0 = 1 2 L 1 = 1 Q 2 0 2 C LC Q2 0

154 3 II 3.2.6 C R S Q 0 1. 2. R C ===== ===== 1. Q C + RI = 0 Q C + R dq dt = 0 Q(t) = Q 0 e t/t 2. T = CR Q/C Q 0 T e t/t 0 RI 2 dt = R = R ( Q0 T ( Q0 T ) 2 e 2t/T dt 0 ) 2 [ T 2 e 2t/T ] 0 = 1 2 R Q2 0 T = 1 Q 2 0 2 C

3.2 155 3.2.3 3.2.1 ϕ(t) = ϕ 0 cos(ωt + α) (3.15) { I(t) = I0 cos(ωt + β) (3.16a) Q(t) = Q 0 cos(ωt + γ) (3.16b) ϕ(t) = ϕ 0 e i(ωt+α) = ϕ 0 e iωt ϕ 0 = ϕ 0 e iα Ĩ(t) = I 0 e (ωt+β) iωt = Ĩ0e (3.17c) Ĩ 0 = I 0 e iβ Q(t) = Q 0 e (ωt+γ) = Q 0 e iωt Q 0 = Q 0 e iγ (3.17a) (3.17b) (3.17d) (3.17e) (3.17f) L dĩ(t) dt + RĨ(t) + Q C = ϕ(t)

156 3 II {L di(t) dt + RI(t) + Q C } + i{ldi (t) dt + RI (t) + Q C } = ϕ(t) + iϕ (t) L, R, C { } L di(t) dt + RI(t) + Q C = ϕ(t) (3.18) 3.6 d Q(t) dt = iω Q 0 e iωt = Ĩ0e iωt iω Q = Ĩ dĩ(t) = iωĩ0e iωt dt iωlĩ + RĨ + Ĩ iωc = ϕ e iωt Ĩ = ϕ Z Z = R + i ( ωl 1 ) ωc (3.19) (3.20)

3.3 157 R ω ωl 1 ωc = 0 Z 3.20 ω 0 ω 0 = 1 LC (3.21) Z 3.3 3.3.1 D D t S d Q I = dq dt x D = Q D I = S S t I/S = D t

158 3 II I x q -q 3.8 B( r) d s = µ 0 i( r) ds (3.22) C B( r, t) d s = µ 0 C S S E( r, t) { i( r, t) + ε 0 } ds t (3.23) B( r, E( r, t) t) = µ 0 { i( r, t) + ε 0 } (3.24) t

3.3 159 3.1.5 E( r, t) = E 0 sin ωt σ i( r, t) = σ E 0 sin ωt i d ( r, t) = ε 0 E t = ε 0ω E 0 cos ωt ω σ ε 0 10 18 s 1 3.3.1 E 0 f = ω/2π 1. E 0 = 1 µv m 1 f = 10 6 s 1 2. E 0 = 2 10 6 V m 1 f = 10 15 s 1 ===== =====

160 3 II i d = ε 0 t E( r, t) = ε 0 t E 0 cos ωt = ε 0 ω E 0 sin ωt 1. (8.85 10 12 )(2π 10 6 )(1 10 6 ) = 5.6 10 11 [A] 2. (8.85 10 12 )(2π 10 15 )(2.0 10 6 ) = 1.1 10 11 [A] 3.3.2 (σ) 5.8 10 7 Ω 1 m 1 10 15 Ω 1 m 1 ε 0 ωe 0 σe 0 = ε 0ω σ 1. 50 s 1 2. 10 10 s 1 ===== =====

3.3 161 1. 2. 8.85 10 12 2π50 5.8 10 7 = 4.8 10 17 8.85 10 12 2π50 10 15 = 2.8 10 6 8.85 10 12 2π10 10 5.8 10 7 = 9.6 10 9 8.85 10 12 2π10 10 10 15 = 5.6 10 14 3.3.3 v?? ===== ===== t = t /2 t = t + /2

162 3 II D(t - D/2) DD(t) D(t + D/2) t - D/2 t v t + D/2 D(t - D/2) DD(t) D(t + D/2) D(t + D/2) DD(t) D(t - D/2) 3.9 3.3.2 D ds = (S ) S E d r = d B ds C dt S B ds = 0 S H d r = I j + d C dt j S D d S 4 D = ε 0E B = µ 0H i = σe (3.25a) (3.25b) (3.25c)

3.3 163 D = ρ B = 0 E = B t H = i + D t 3.26d { H} }{{ = } i + D t =0 3.26a (3.26a) (3.26b) (3.26c) (3.26d) i + ρ t = 0 (3.27) x, y, z 6 8

164 3 II 3.26d { H} }{{ = } i + D t =0 3.27 i D t / t t { D ρ( r, t)} = 0 F ( r) D ρ( r, t) = F ( r) 3.26a 3.26a 3.26a 3.26b 6 6 3.3.3 3.26d E 3.26c H 2 H { E + B t } = 0 E { H D t } = E i H B t + E D t + H { E} E { H} = E i

3.3 165 H B t + E D t = t {1 2 ( E D + H B) } (3.28) }{{} u( r,t) H { E} E { H} = { E H }{{ } (3.29) } S( r,t) u( r, t) t + S( r, t) = E( r, t) i( r, t) (3.30) 1 2 E D 1 2 H B 3.14 u E i S 3.30 3.27 3.30 u, S 3.27 ρ, i S (Poynting vector) *1 *1 S/c 2 [2]

166 3 II 3.3.4 R Q 0 R 0 t I(t) t I(t) Q(t) C = R 0I(t) I(t) = dq(t) dt I(t) = I 0 e t/τ τ = RC I 0 t = 0 1. 2. 3. 4. D 5. R 0 l l

3.3 167 6. 7. ===== ===== 1. I(t)/πR 2 2. 3. πr 2 µ 0 I(t) = 2πrB(r, t) πr2 B(r, t) = µ 0r 2πR 2 I(t) Q(t) ε 0 πr 2 µ 0 r 2πR 2 I(t) r Q(t) I(t) 2πR2 ε 0 πr 2 = 1 r 2ε 0 (πr 2 ) 2 Q(t)dQ(t) dt = 1 r dq 2 (t) 4ε 0 (πr 2 ) 2 dt Q(t)

168 3 II 4. 2π D 1 R dq 2 (t) 0 4ε 0 (πr 2 ) 2 Rdθ dt = D 1 R dq 2 (t) 4ε 0 (πr 2 ) 2 R2π dt = D 1 1 dq 2 (t) 2ε 0 πr 2 dt D t = 0 t = D 1 1 dq 2 (t) 0 2ε 0 πr 2 dt dt = 1 D dq 2 (t) 2 ε 0 πr 2 dt 0 dt = 1 1 dq 2 (t) 2 C 0 = 1 Q 2 (0) 2 C C = ε 0 πr 2 /D 5. B(r, t) = µ 0 2πr I(t) E(r, t) = R 0I(t) l

3.3 169 S S = 1 µ 0 E(r, t)b(r, t) = 1 µ 0 R 0 I(t) l µ 0 2πr I(t) Sdlrdθ = 1 µ 0 E(r, t)b(r, t)dlrdθ = 1 µ 0 R 0 I(t) l = R 0 I 2 (t) µ 0 2πr I(t)lr2π 6. 7. 3.3.4 3 3.26c 4 3.26d... ρ = 0 i = 0 D = ε 0E B = µ0h

170 3 II x z y E H E = 0 E = µ 0 H t H = 0 H = ε 0 E t z z t E = 0 = z E z = 0 E = µ 0 t H = H = 0 = z H z = 0 z E y = µ 0 t H x z E x = µ 0 t H y 0 = µ 0 t H z H = ε 0 t E = z H y = ε 0 t E x z H x = ε 0 t E y 0 = ε 0 t E z A B = B A ze z = 0 t E z = 0 E z z t E z = 0 H z = 0

3.3 171 E x E y = 0 E y = 0 t H x = 0, z H x = 0 H x = 0 H y E z = 0 H z = 0 E y = 0 H x = 0 z E x = µ 0 t H y z H y = ε 0 t E x 2 3,4 2 E x = 1 t t ε 0 t H y z t z = 1 H y ε 0 z t H y t 1 E x µ 0 z = 1 2 E x ε 0 µ 0 z 2 2 H y t 2 = 1 2 H y ε 0 µ 0 z 2 c 0 = 1 ε0 µ 0

172 3 II E x = E 0 cos ω(t z ) c 0 E y = 0 E z = 0 H x = 0 H y = H 0 cos ω(t z ) c 0 H z = 0 H 0 = E 0 x ε0 µ 0 E z y H SI 2.99792458 10 8 [m/s] µ 0 = 4π 10 7 [H/m] ε 0 [F/m] 3.3.5

3.3 173 E = 0 B = 0 E + B t = 0 B E ε 0 µ 0 t = 0 (3.31a) (3.31b) (3.31c) (3.31d) 3.31c { E( r, t)} + { B( r, t) } = 0 (3.32) t 1 E = 0 2 E 3.31d 2 ε 0 µ 0 t 2 E 2 E( r, t) ε0 µ 0 2 t E( r, t) = 0 (3.33) 3.3.6 (0, 0, z 0 + cos ωt) (0, 0, z 0 cos ωt) z z z z z z x z y x x

174 3 II 3.10 z 3.3.5 0.1 Vm 1??

3.3 175 ===== ===== H 0 ε0 = E 0 µ 0 ε0 H 0 = E 0 µ 0 8.85 10 12 = 4π 10 7 0.1 = 2.7 10 4 [A/m] B 0 = µ 0 H 0 = 3.3 10 10 [T] [W/m 2 ] 1 µ 0 E 0 B 0 = 0.1 2.7 10 4 = 2.7 10 5 3.3.6 z 2 x E 1 (z, t) = E 0 cos(kz ωt) E 2 (z, t) = E 0 cos( kz ωt)?

176 3 II ===== ===== E(z, t) = E 1 (z, t) + E 2 (z, t) = E 0 (cos(kz ωt) + cos( kz ωt)) = 2E 0 cos kz cos ωt B(z, t) = B 0 cos(kz ωt) B 0 cos( kz ωt) = 2B 0 sin kz sin ωt y z 3.3.7 A( r, t) E = ta B = A Maxwell ===== ===== A = 0 Maxwell 2 A = ε0 µ 0 2 t A D = 0 B = 0 E = B t H = D t

3.4 177 A 1 A ( ε 0 t A) = ε0 t ( A) 1 A = 0 2 A ( A) A 2 3 A ( ta) = t ( A) A 3 4 A ( 1 A) = t ( ε 0 t A) µ 0 1 ( ) 2 A + ( A) = t ( ε 0 t A) µ 0 A = 0 2 A = ε0 µ 0 2 t A 3.4

178 3 II 3.4.1 2 1. 2.

3.4 179 3.11 (a) (b) (c) 3.4.2 l S σ

180 3 II 3.12 (a) (b) s -s C s 0 C 0 -s' s' -s 0 3.13 u ρ σ = ρ u σsl P = (ρ usl)/(sl) = ρ u (3.34) P r S u S ρ u d S P P ds S S

3.4 181 - - - - - -s - 3.14 P l E 0 + + + ++ +s + Q P Q P = P ds (3.35) 3.35 ρ P ( r) P ρ P ( r) = P (3.36) E( r) = 1 {ρ( r) + ρ P ( r)} ε 0 S

182 3 II P {ε 0E( r) + P ( r)} = ρ( r) D( r) = ε 0E( r) + P ( r) D( r) = ρ( r) (3.37) P = χ ee χ e χ e D = (ε 0 + χ e ) E ε = ε 0 + χ e (3.38) D = εe (3.39) ε 0 ε D( r) = ρ( r) E( r) = 0 D( r) = ε 0E( r) + P ( r) (c) (3.40a) (3.40b) (3.40c)

3.4 183 3.4.3 3.15 r 1 3.15 abcd 3.40b E d s = 0 C E d s C = E d s + E d s + E d s + E d s ab cd bc da bc da ab cd E E d s C

184 3 II = E 1 t s E 2 t s = { E 1 t E 2 t} s = 0 t E 1 t = E 2 t (3.41) 3.16 r 1 3.16 3.40a D S ds = ρdv V 1 S 1 2 S 2 D ds = { D1 n} S S 1 S 2 D d S = { D2 n} S

3.4 185 { D 1 n D 2 n} S = 0 D 1 n = D 2 n (3.42) 3.4.1 1 2 A,B 2 0.1 nm 5 10 30 Cm A,B? ===== ===== q l p = ql 5 10 30 = q 0.1 10 9 q 5 10 20 [C] 1.60 10 19 30 % 3.4.2 2 = q a q 1. u? 2. E

186 3 II 3. ===== ===== 1. 4πu 2 E 1 4 = ε 0 3 πu3 ρ ρ = q 4 3 πa3 E = q 4πε 0 a 3 u 2. q E = q2 4πε 0 a 3 u q E ext q E(u) = 0 E ext x qe ext = q2 4πε 0 a 3 u qu = 4πε 0a 3 qe ext q 3. 1 p = qu = αe α = 4πε 0 a 3

3.4 187 3.4.3 a P u a 1. a ρ r E( r) 2. u/2 a ρ u/2 a ρ E( r) 3. E( r) P ===== ===== 1. 2. 4πr 2 E( r) = 1 ε 0 4π 3 r3 ρ r r E( r) = E + ( r) = ρ 3ε 0 r ρ 3ε 0 ( r 1 2 u) E ( r) = ρ 3ε 0 ( r + 1 2 u) E = E + ( r) + E ( r) = ρ 3ε 0 ( r 1 2 u r 1 2 u) = ρ 3ε 0 u

188 3 II 3. E = P 3ε 0 3.4.4 p E 2pE T p p 2pE k B T = 2p2 k B T E k B N 1. χ e 2Np2 k B T 2. 300 K 5 10 30 Cm ε/ε 0 1 0.5 nm k B = 1.38 10 23 JK 1 ===== ===== 1. P = χe E P = N p = N 2p 2 2. ε ε 0 1 = χ e ε 0 k B T E χ e = N 2p2 k B T

3.4 189 = 2Np2 / ε0 k B T = 2(0.5 10 9 ) 3 (5 10 30 ) 2 1.38 10 23 300 11 / ε0 3.4.5 0.1 mm 2.3 0.1 µf? ===== ===== C = ε ε 0 S d = 2.3 8.85 10 12 S 0.1 10 3 = 0.1 10 6 S = 0.49 [m 2 ]

190 3 II 3.4.6 ε θ ===== ===== E n D n εe n = ε 0 E cos θ σ P = P n = εe n ε 0 E n = (ε ε 0 ) ε 0 ε E cos θ D = ε E = ε 0 E + P 3.4.7 ε E E 1. 2. 3. ===== ===== 1.

3.4 191 2. D n E = ε ε 0 E 3. P /3ε 0 E E E = E P 3ε 0 ε E = ε 0 E + P E = 3ε 0 2ε 0 + ε E ε ε 0 E = 3ε 2ε + ε 0 E 3.4.8 ε Q 0 1. 2.? 1 L 1 L d

192 3 II ===== ===== 1. ε/ε 0 2. x C = L d ((L x)ε + xε 0) U(x) = 1 dq 2 2 (εl (ε ε 0 )x) L F = x U(x) = 1 dq 2 2 L (εl (ε ε 0)x) 2 (ε ε 0 )( 1) = 1 dq 2 2 L (εl (ε ε 0 )x) 2 (ε ε 0) ε ε 0 = εl (ε ε 0 )x U(x) 3.4.9 ε a E 0 ===== ===== P /3ε 0 E = E 0 P /3ε 0 ε E = ε 0 E+ P P = 3ε 0(ε ε 0 ) 2ε 0 + ε E 0

3.4 193 4π 3 a3 P 4π 3ε 0(ε ε 0 ) = 3 a3 E 0 2ε 0 + ε = 4πε 0(ε ε 0 ) a 3 E0 2ε 0 + ε 3.4.4 2 1. 2. 1 2 2

194 3 II 3 3.4.5 II B M H( r) H( r) = 1 B( r) M( r) (3.43) µ 0 H( r) = i( r) i( r)

3.4 195 * 2 M( r) = χ mh( r) (3.44) χ m B( r) = µ 0 {1 + χ m } H( r) = µ H( r) (3.45) µ = µ 0 (1 + χ m ) µ µ 0 ε { H( r) = i( r) (3.46a) B( r) = 0 (3.46b) *2 EB EH H( r) = 1 µ 0 { B( r) M( r)} M M(EH) = µ 0M(EB) SI EB M EH EB [3] EB

196 3 II 3.4.6 H 1 ( r) t = H 2 ( r) t (3.47) B 1 ( r) n = B 2 ( r) n (3.48) 3.4.10 4 300 K µ µ 0 8.0 10 24 A m 2 0.5 nm ===== ===== B( T) µ Am 2 =JT 1 µb 2(0.5 10 9 ) 3 (8 10 24 ) 2 1.38 10 23 300 µ 0 = 3.1 10 4 3.4.11 M

3.4 197 ===== ===== B = µ 0 ( H + M) B M B in = µ 0 (H in + M) B out = µ 0 H out B in = B out B H out = 1 µ 0 B H in = 1 µ 0 B M B = 0 H out = 0 H in = M 3.4.12 M 2 N S B = µ 0 M 1. S ===== ===== 1. U(x) x U(x) = 1 2µ 0 B 2 Sx x F F = x U(x) = 1 2µ 0 B 2 S

198 3 II 3.4.13 9.5 10 24 JT 1 3 10 28 m 3? ===== ===== µ 0 9.5 10 24 3 10 28 = 3.6 10 1 [T](Wb/m 2 ) 3.4.14 ===== =====

3.5 199 3.5 3.5.1 D( r, t) = ε 0E( r, t) + P ( r, t) (3.49a) H( r, t) = 1 µ 0 B( r, t) M( r, t) (3.49b) D( r, t) = ρ ( r, t) (3.50a) B( r, t) = 0 (3.50b) E( r, t) + tb( r, t) = 0 (3.50c) H( r, t) td( r, t) = i ( r, t) (3.50d) P ( r, t) = χ e E( r, t) M( r, t) = χ m H( r, t) 3.5.2

200 3 II k u m d2 u dt 2 + k u = 0 (3.51) ω 0 = k/m E = E 0 cos ωt m d2 u dt 2 + k u(t) = e E 0 cos ωt (3.52) u = u 0 cos ωt ( mω 2 + k) u(t) = e E 0 cos ωt u = e m(ω 2 0 ω2 ) E (3.53) 1 z 1 p p = ze u (3.54) = α(ω) E (3.55) α(ω) = ze 2 m(ω 2 0 ω2 ) (3.56) E ω ω 0 N 3.55 P = N p = χ ee (3.57)

3.5 201 χ e (ω) = Nze 2 m(ω 2 0 ω2 ) (3.58) χ e ε(ω) = ε 0 + Nze 2 m(ω 2 0 ω2 ) (3.59) ω = ω 0 3.17

202 3 II m d2 u dt 2 + m τ d u dt + k u = e E (3.60) τ u = e m(ω 2 0 ω2 + iω/τ) E (3.61) ε(ω) = ε 0 + Nze 2 m(ω 2 0 ω2 + iω/τ) (3.62) ε(ω) = ε 0 + Nze2 m ε (ω) = Nze2 m ε(ω) = ε(ω) + iε (ω) (3.63) ω 2 0 ω 2 (ω 2 0 ω2 ) 2 + (ω/τ) 2 (3.64) ω/τ (ω 2 0 ω2 ) 2 + (ω/τ) 2 (3.65) 3.5.3 1 2 D E = 1 2 ε E 2 = 1 2 (ε(ω) + iε (ω)) E 2

3.5 203 3.18 P P P = R{ χ e (ω) E} = R{[χ e (ω) + iχ e(ω)][ E(t) + ie (t)]} = χ e (ω) E(t) χ e(ω) E (t) = χ e (ω) E 0 cos ωt χ e(ω) E 0 sin ωt i P (t) = ωχ e (ω) E 0 sin ωt ωχ e(ω) E 0 cos ωt

204 3 II i P (t) E(t) = ωχ e (ω)e0 2 sin ωt cos ωt ωχ e(ω)e0 2 cos ωt cos ωt 1 2 < i P (t) E(t) > = 1 2 ωχ e(ω)e0 2 (3.66) χ e < 0 3.5.1 ===== ===== m d2 u dt 2 + m τ d/dt iω d u dt + k u = e E mω 2 u + iω m τ u + k u = e E u u = = e E m( ω 2 + iω/τ + k/m) e E m(ω 2 0 ω2 + iω/τ) ω 2 0 = k/m

3.5 205 3.5.2 3.4.2 Z m Z = 6 a = 0.07 nm ===== ===== q E = k q2 4πε 0 a 3 u (Ze) 2 4πε 0 a 3 ω 0 = k/zm Ze = 2 4πε 0 a 3 m 6(1.60 10 = 19 ) 2 4π8.85 10 12 (0.07 10 9 ) 3 9.1 10 31 = 6.67 10 16 [Hz]( [s 1 ]) 3.5.3 4.5 1.04 10 16 s 1 2.5 10 15 s 1 4.5 10 15 s 1 ===== =====

206 3 II Nze 2 ε(ω) = ε 0 + m(ω0 2 ω2 ) α = ε 0 + ω0 2 ω2 ω = 0 ε/ε 0 = 4.5 ε(ω) 3.5 = 1 + ε 0 1 (ω/ω 0 ) 2 ω/ω 0 = 0.240 ω/ω 0 = 0.433 4.71 5.31 3.5.4 2 E( r, t) 1 v 2 2 t E( r, t) = 0 (3.67) v = 1 εµ (3.68) n = c v (3.69) 1,2 n 1, n 2 n 2 /n 1 2 1

3.5 207 3.19 N (( f(t, x) = e ( i N )2 sin k 0 i ) (x ω ) 0 t) N k 0 f d (t, x) = i= N N i= N (( e ( i N )2 sin k 0 i ) (x ω ( 0 1 i ) ) t) N k 0 3N N = 100, x 0 = 2π, ω 0 = 2π 3.5.5 ρ = 0 D = ε E B = µ H E H E = 0 E = µ t H H = 0 H = i + ε t E

208 3 II 4-5 25-4 4-5 25-4 4-5 25-4 3.19 1 t = 0 2 t = 6 (f(t, x)) 3 t = 6 f d (t, x) 2 1 3 z z t E = 0 = z E z = 0 E = µ t H = z E y = µ t H x z E x = µ t H y 0 = µ t H z

3.5 209 H = 0 = z H z = 0 H = i + ε t E = z H y = i x + ε t E x z H x = i y + ε t E y 0 = i z + ε t E z A B = B z A x y z E z = 0 z E z H z = 0 f( k r t) E z (z ct) E x E y = 0 i = σ E σ x i y = i z = 0 E y = 0 t H x = 0, z H x = 0 H x = 0 H y E z = 0 H z = 0 E y = 0 H x = 0 z E x = µ t H y z H y = i x + ε t E x

210 3 II 2 3,4 2 t ( t E x ) = 1 ε t z H y 1 ε ti x 1 t z = 1 ε z th y 1 ε ti x t H y 1 µ ze x = 1 εµ 2 ze x 1 ε ti x i x σe x = 1 εµ 2 ze x σ ε te x 2 ze x εµ 2 t E x σµ t E x = 0 (3.70), σ = 0 Ẽ x (z, t) = Ẽ(z)eiωt (3.71) zẽ(z) 2 + (εµω2 iσµω)ẽ(z) = 0 (3.72) Ẽ(z) = E 0 e ikz (3.73) k = ±ω/v v = 1/ εµ E x (z, t) = E 0 cos k(z vt) (3.74)

3.5 211 σ 0 k Ẽ(z) = E 0e i kz [ k 2 + (εµω 2 iσµω)]e 0 = 0 (3.75) E 0 0 k 2 = εµω 2 iσµω (3.76) 1 2 σ 10 7 [Ω 1 m 1 ] ε 10 11 [F m 1 ] σ ε 1018 [s 1 ] (3.77) ω 10 14 1 2 k 2 iσµω (3.78) k = ± 1 i 2 σµω (3.79) E x (z, t) = E 0 cos( z l ωt)e z/l (3.80) l = 2 µσω (3.81) ω 10 10 2 l 10 6 10 7 10 10 6 [m] 10 cm

212 3 II 3.5.4 3.5.3 1. 7.8 10 6 [m/s] 2. 1 ns 1 ns 2 m 3. 1 ns 1 ns 2 m ===== ===== 1. n = ε/ε 0 2.17 2.30 1.381 10 8 [m/s] 1.303 10 8 [m/s] 7.8 10 6 [m/s] 2. 2/1.382 10 8 = 14.48 10 9 [s] 2/1.303 10 8 = 15.35 10 9 [s] t = 0 1+14.48 1+14.48+1[ns] 15.35 15.35+1[ns] 15.35 16.48[ns] 1.1[ns]

3.5 213 3. t = 0 14.48 14.48+1[ns] 1+15.35 1+15.35+1[ns] 14.48 17.35[ns] 2.9[ns] 3.5.5 ω σ/ε ===== ===== k 2 = εµω 2 iσµω = µεω 2 (1 i σ εω ) k µε(1 i σ 2εω )ω k 3.5.6 m t v + m τ v = e E 0 cos(ωt) i(t) = en v(t) ===== =====

214 3 II d/dt iω v = i i = ne v ne 2 = E m(1/τ + iω) = ne2 (1/τ iω) E m(1/τ 2 + ω 2 ) e E m(1/τ + iω) = ne2 (1/τ iω) m(1/τ 2 + ω 2 ) E 0 (cos ωt + i sin ωt) ( ne 2 ) (1/τ iω) R( i) = R m(1/τ 2 + ω 2 E ) 0 (cos ωt + i sin ωt) = ne2 τe 0 m(1 + ω 2 τ 2 R((1 iωτ)(cos ωt + i sin ωt)) ) = ne 2 τ m(1 + ω 2 τ 2 ) (cos ωt + ωτ sin ωt) E 0 = ne2 τ m ( 1 1 + ω 2 τ 2 cos ωt + ωτ 1 + ω 2 τ 2 sin ωt) E 0

215 [1] I,II. [2] http://homepage2.nifty.com/eman/electromag/contents.html [3] EH EB EB EH B EH EB http://www.fdenshi.com/000tokiwajpn/32denjk/010elc.html monograph http://ir.lib.hiroshimau.ac.jp/metadb/up/81936204/refunit43w.pdf [4] G. Arfken, Mathematical Method for Physicist, ACADEMIC PRESS. New York.