( ) LAN LAN tex ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1

Transcription

1 ( ) LAN LAN tex ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1

2 Coulomb Lienard-Wiechert potential Laplace

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4 1 2 O B θ A OA = a, AOB θ O r 2 AO BO l θ = l/r θ (rad) 0 θ 2π 2 π OB = b cos θ = a b a b (1.1) 3 ( ) O r S S O Ω Ω = S/r 2 (1.2) 0 S 4πr 2 0 Ω 4π Ω (sr) 1 rad + 2 sr r ds O R ds : ( ) ds ds 4

5 R ds Ω = (1.3) R 3 x, y 2 π ( ) 2 π ( ) 2 GM α R ds θ, θ + dθ 2π R sin θ R dθ [0, α] θ 2α d θ 2π α 0 R 2 sin θdθ = 2πR 2 (1 cos α) (1.4) Ω = 2π(1 cos α) α GM d α = tan 1 d/r α 1 cos α 1 α 2 /2 + α 4 /24 Ω 2πα 2 πd 2 /R 2 GM πd 2 R 5

6 γ GM γ γ GM γ ( ) ( ) x (a, 0, 0) yz (a, 0, 0) (0, y, z) z c dy dz dω x a dz dy b y dω = dy dz a 2 + y 2 + z a (1.5) 2 a2 + y 2 + z 2 0 y b 0 z c Ω = b 0 dy c 0 a dz (1.6) (a 2 + y 2 + z 2 ) 3/2 Ω = T an 1 ( ) bc a a 2 + b 2 + c 2 (1.7) 1 dt (t 2 + a 2 ) = t 3/2 a 2 (t 2 + a 2 ) 1/2 1 ds (s 2 + a 2 ) s 2 + a 2 + b s 2 + a 2 + b 2 = a 2 + b 2 s u s u 2 1 du a 2 + u = 1 2 a T an 1 (u/a) T an 1 p T an 1 q = T an 1 p q 1 + pq (nabla Hamilton grad ) ( ) 6

7 ( ) ( ) ( ) φ φ { φ = ê x x + +ê y y + ê z z } φ (1.8) ê i i (i = 1, 2, 3) φ φ φ ( ) A A(divA) A(rotA curla) A ( ) A A = A x x + A y y + A z z A ( ) ( ) (1.9) ( ) ( ) (r, v, ω) v = r ω r, v ω 7

8 0 i dx, dy, dz x, y, z ρ ρ(x, y, z, t) dx dy dz ρ dx dy dz t x x i x {i x (x + dx, y z t) i x (x, y, z, t)} dy dz = i x dx dy dz (1.10) x y, z ρ t + i = 0 (1.11) X X X X X ( ) X ( ) A A ( ) z ω ω ( ) r = (x, y, z), ω = (0, 0, ω), v = r ω = ω(y, x, 0) (1.12) 2 ω = v y x v x y (1.13) ( 2) v z ( v) z = v y / x v x / y cyclic 2 8

9 ( ) ( ) v z v y x v x y ( v) z ( v) xy 2 (3 (x, y) z 4!) v 2 ( v) z x y A A ( ) ( ) 0 0 A y D A B C x A ABCDA A dr = = B A B A x (x, y)dx + C B A y (x + dx, y)dy + {A x (x, y) A x (x, y + dy)}dx A A D C D A x (x, y + dy)dx + A {A y (x, y) A y (x + dx, y)}dy D A y (x, y)dy = A z (x, y) dx dy (1.14) ( A) z = A y x A x (1.13) y ( 2) 9

10 ( ) rotation rotation curl curl, curl X X X X X X X X X Stokes ( ) F F = 0 F = φ potential φ (x, y, z) (x 1, x 2, x 3 ) = {x i, i = 1, 2, 3} {q i, i = 1, 2, 3} 1 q i {x i ; i = 1, 2, 3} q 1 = const. q 2 = const. q 3 = const. 2 q 1 = const., q 2 = const., q 3 = const. q i (x 1, x 2, x 3 ) q i = q i (x 1, x 2, x 3 ), (i = 1, 2, 3) (1.15) 10

11 dr i ˆx i dr = dx 1ˆx 1 + dx 2ˆx 2 + dx 3ˆx 3 = i dx i ˆx i (1.16) {q i } dr = r q 1 dq 1 + r q 2 dq 2 + r q 3 dq 3 = i r q i dq i = i h iˆq i dq i (1.17) r/ q i = 1 r q i = h iˆq i (1.18) ˆq i = 1 h i r q i (1.19) (dr) 2 ˆq i ˆq j (i j) q i ˆq i ˆq j = δ ij = h 2 i = r q i r q i = { 1 (i = j) 0 (i j) (1.20) ( ) 2 ( ) 2 ( ) 2 x1 x2 x3 + + (1.21) q i q i q i h i {ˆq i } ˆq 1 ˆq 2 = ˆq 3 (1, 2, 3 cyclic ) (1.22) ds 2 {q i } ds 2 = dr dr = ( i h iˆq i dq i ) ( j h j ˆq j dq j ) = i (h i dq i ) 2 (1.23) A {ˆx i } A x i A i A = A 1ˆq 1 + A 2ˆq 2 + A 3ˆq 3 = i A iˆq i (1.24) A i = A ˆq i (i = 1, 2, 3) (1.25) ˆq i A i A i (projection operator) 11

12 ˆq i (ˆq i A) ˆq i A i A ˆq i (ˆq i A) = A(ˆq i ˆq i ) ˆq i (ˆq i A) = ˆq i (A ˆq i ) (1.26) A ˆq i q 1 q 2, q 3 q 1 ˆq 1 (1.18) q 1 = 1/h 1 q 1 = q 1 r = 1 h 1 ˆq 1 (1.27) {q i } ˆq 1 (ˆq 2 ˆq 3 ) 1 1 = ˆq 1 (ˆq 2 ˆq 3 ) = h 1 h 2 h 3 q 1 ( q 2 q 3 ) = h 1 h 2 h 3 q 1 x 1 q 1 x 2 q 1 x 3 q 2 x 1 q 2 x 2 q 2 x 3 q 3 x 1 q 3 x 2 q 3 x 3 (1.28) Jacobian q 1 ( q 2 q 3 ) {q i } h 1 h 2 h 3 (1.23) h i q i q i (x 1, x 2, x 3 ) dq i = j q i x j dxj = j g i j dx j, g i j = (g) ij qi x j (1.29) det(g) 0 dx j g 1 = G dx j = i G j i dqi (1.30) dr = ˆx j dx j = j j ˆx j G j i dqi = i i h i ˆq i dq i (1.31) (1.18) dq i ˆx j G j i = h i ˆq i or ˆq i = 1 ˆx j G j i (1.32) h i j j (1.31) dq i j gi jdx j ˆx j dx j = h i ˆq j gj i dx j ˆx j = h i ˆq i gj i (1.33) j i j i dx j ˆx j (1.30, 1.33)! 12

13 {q i, ; i = 1, 2, 3} ˆq i ˆq j = 1 h i h j dq i = j gi j dx j, dx j = i Gj i dqi h i ˆq i = j Gj i ˆx j, ˆx j = (1.34) i gi j h i ˆq i k ˆx k G k i ˆx l G l j = 1 G k i G k j = δ ij (1.35) h i h j k l k G k i G k j = h 2 i δ ij (1.36) ˆx i ˆx j = k h k ˆq k gi k h l ˆq l gj l = l k h 2 k g k i g k l = δ ij (1.37) Jacobian det(g) = 1/det(G) = 1/h 1 h 2 h 3 φ φ = φ x 1 ˆx 1 + φ x 2 ˆx 2 + φ x 3 ˆx 3 ( ) = ( φ q 1 q 1 x 1 + φ q 2 q 2 x 1 + φ q 3 q 3 x 1 ) ˆx ( φ q 1 q 1 x 3 + φ q 2 q 2 x 3 + φ q 3 q 3 x 3 ) ˆx 3 = φ q 1 q 1 + φ q 2 q 2 + φ q 3 q 3 = φ q 1 1 h 1 ˆq 1 + φ q 1 1 h 2 ˆq 2 + φ q 3 1 h 3 ˆq 3 φ = A = i A iˆq i { i ( (1.27) ) 1 h i } ˆq i φ (1.38) q i A = i (A iˆq i ) ; i 1 (1.39) (A 1ˆq 1 ) = A 1 (ˆq 2 ˆq 3 ) (1.38) = {A 1 h 2 h 3 ( q 2 ) ( q 3 )} = (A 1 h 2 h 3 ) { q 2 q 3 } + A 1 h 2 h 3 {( q 2 ) ( q 3 )} (A 1 h 2 h 3 ) ( q 2 q 3 ) = 1 h 1 (A 1 h 2 h 3 ) q 1 ˆq 1 ( 1 h 2 ˆq 2 1 h 3 ˆq 3 ) = 1 (A 1 h 2 h 3 ) h 1 h 2 h 3 q 1 13

14 ( q 2 q 3 ) ˆq 1 1 ( q 2 q 3 ) = q 3 ( q 2 ) q 2 ( q 3 ) = 0 ( φ = 0 ) (1.40) A = 1 h 1 h 2 h 3 { (A1 h 2 h 3 ) + (h 1A 2 h 3 ) + (h } 1h 2 A 3 ) q 1 q 2 q 3 (1.41) A = φ (1.38) { ( ) 2 1 h2 h 3 φ = ( φ) = + ( ) h1 h 3 + ( )} h1 h 2 φ (1.42) h 1 h 2 h 3 q 1 h 1 q 1 q 2 h 2 q 2 q 3 h 3 q 3 A A = i (A iˆq i ) (1.43) i = 1 (1.27) (A 1ˆq 1 ) = (A 1 h 1 q 1 ) = (A 1 h 1 ) ( q 1 ) + A 1 h 1 q 1 ( = 0 ) q 1 //ˆq 1 (A 1 h 1 ) 1 (1.38) { 1 (A 1 h 1 ) = ˆq } (A 1 h 1 ) ˆq 3 q 1 h 2 q 2 h 3 q 3 = 1 h 1 h 2 (A 1 h 1 ) q 2 ˆq h 1 h 3 (A 1 h 1 ) q 3 ˆq 2 (1.44) A = ˆq { 1 (h3 A 3 ) (h } 2A 2 ) h 2 h 3 q 2 q 3 + ˆq { 3 (h2 A 2 ) (h } 1A 1 ) h 1 h 2 q 1 q 2 h 1ˆq 1 h 2ˆq 2 h 3ˆq 3 1 A = h 1 h 2 h 3 q 1 q 2 q 3 h 1 A 1 h 2 A 2 h 3 A 3 + ˆq { 2 (h1 A 1 ) (h } 3A 3 ) h 3 h 1 q 3 q 1 (1.45) (1.46) (1.23) h 1 = 1, h 2 = r, h 3 = 1 14

15 φ = φ r e r + 1 φ r θ e θ + φ z e z (1.47) 2 φ = 1 r φ (r r r ) φ r 2 θ + 2 φ (1.48) 2 z 2 A = 1 r r (ra r) + 1 A θ r θ + A z (1.49) z ( 1 A z A = r θ A ) ( θ Ar e r + z z A ) z e θ r + 1 ( r r (ra θ) A ) r e z (1.50) θ h 1 = 1, h 2 = r, h 3 = r sin θ φ = φ r e r + 1 φ r θ e θ + 1 φ r sin θ ψ e ψ (1.51) 2 φ = 1 φ r 2 (r2 r r ) + 1 φ (sin θ r 2 sin θ θ θ ) φ r 2 sin 2 (1.52) θ ψ 2 A = 1 r 2 r (r2 A r ) + 1 A = 1 r sin θ + 1 r r sin θ { θ (sin θa ψ) A θ ψ { r (ra θ) A r θ θ (sin θa θ) + 1 } e r + 1 r A ψ ψ r sin θ { 1 A r sin θ ψ r (ra ψ) } e θ (1.53) } e ψ (1.54) Gauss A S V dv A = A ds (1.55) : V S 1 A 1 dx 1 dx 2 dx 3 = dx 2 dx 3 {A 1 (x 1max, x 2, x 3 ) A 1 (x 1min, x 2, x 3 )} (1.56) x 1 V x 2, x 3 x 2, x 3 S x 1max, x 1min x 2, x 3 15

16 ê 1 dx 2 dx 3 = ds ê 1 A 1 (x 1max or 1min, x 2, x 3 ) dx 2 dx 3 = (A ds) 1 (1.57) V dv A ( ) 1 = A ds x 1 s (1.58) 1 ds A1 dx3 x3 dx3 3 P dx2 dx2 2 x2 dl C Q Stokes A C S ( A) ds = A dl (1.59) C S c : 1 S ( A3 A ) 2 dx 2 dx 3 = x 2 x 3 {A 3 (x 1, x 2max, x 3 ) A 3 (x 1, x 2min, x 3 )}dx 3 {A 2 (x 1, x 2, x 3max ) A 2 (x 1, x 2, x 3min )}dx 2 (1.60) 1 x 3 1 C x 3 1 x 2 2 x 2 C ( ) C P(Q) x 2 ( ) Q(P) A 3 A dl 2 Stokes 16

17 2 Coulomb q 1 q 2 F F = q 1 q 2 4 π ε 0 r r 3 (2.1) r/r ( ) ( ) ε 0 µ 0 ( ε 0, µ 0 ) F = q 2 E E q 2 q 2 q 1 H Michel Faraday ds σ q 1 ds σ σ ds q 2 q 1 q 2 = σds (2.2) q 1 σ ds F = q 1 σ ε 0 ds r 4π r 2 r (2.3) q 1 r F r F = q 1σ ε 0 ( ) : 1. dω 4π (2.4) 17

18 ( ) ( ) Coulomb Einstein, Eddington Einstein Eddington Madelung > 2. q 1, q 2 q 1 = α q 1 + (1 α) q 1 q 2 (α q 1, q 2 ), ((1 α) q 1, q 2 ) ( ) 1/4π = ε 0 = (!) Rutherford. 18

19 / / ( or ) ( Fairbank ) Fairbank Q = q/ 4 π ε 0 [ ] 1/2 [ ] ( ) ( ) Lorentz q v B F F = qv B (2.5) v B Lorentz ( ) Lorentz B H H B 19

20 H B H B = µ 0 H + M M B H B H, B Faraday M. Faraday Maxwell H = i H = i + D (2.6) t D t H = 0 i + ( D) = 0 t D = ρ i + ρ t = 0 (2.7) 20

21 Lorentz ( ) MeV ( ) 22 Na Faraday Maxwell Heaviside D = ρ D ρ Maxwell D = ρ, H = i + D t (2.8) µ F µν = µ 0 j ν (2.9) B = 0, E = B (2.10) t λ F µν + µ F νλ + ν F λµ = 0 (2.11) c( ) µ 0 q ( ) v v B qv B 21

22 (!!) Maxwell Coulomb E = ρ e, H = ρ m (2.12) E H ρ e ρ m ρ m Maxwell E, D, H, B 22

23 3 Coulomb Coulomb 2 r 2 + ɛ ɛ F Q Q 1Q 2 r 2 (3.1) a b Coulomb Cavendish Maxwell ɛ SW a, b A, B A B ( ) A ( ) A, B switch ( ) A Q a ( ) Q a = 0 1/r 2 : V (R) = θ a σ R s 2 π a 2 σ a sin θ dθ U(s) r potential U(r) σ a a R potential V (R) s 2 = R 2 + a 2 2aR cos θ 2 s ds = +2aR sin θ dθ = 2 π a 2 σ a 1 a R rmax r min s U(s) ds (3.2) = 2 π a σ a R {f(r max) f(r min )} f(r) = r s U(s) ds potential φ(b) φ(b) = 2πaσ a b {f(b + a) f(b a)} + 2πbσ b {f(2b) f(0)} (3.3) b 23

24 φ(a) φ(a) = 2πaσ a a {f(2a) f(0)} + 2πbσ b {f(b + a) f(b a)} (3.4) a potential f(a) = f(b) σ a = 0 } (3.5) 2 a a{f(0) f(2b)} = b{f(b a) f(b + a)} (3.6) 0 = b{f (b a) f (b + a)} (3.7) a, b (b > a) f (r) r c 0 1 f (r) = c 0 r + c 1 = r U(r) U(r) = c 1 r + c 0 (3.8) r U(r) 0 c 0 = 0 potential 1/r F (r) 1/r 2 B 0 r ( ) 1/r 2 B A C B (B) B ( A )=( C ) 24

25 1/r 2 r 1/r 2 1/r 2 ( 0 ) Coulomb q 1 q 1 r q 2 F F = q 1 q 2 r 4 π ε 0 r 3 = q 2 E (3.9) E q 1 r q 2 q 1 r Coulomb r 0 4 π ε ( 0 r ) E = = {( 1r ) r + 1 } q r 3 3 r r = 3 { 3 r r + 3 } = 0 (3.10) r 5 r 3 E = 0 (3.11) r = 0 E E = ρ ε 0 (3.12) : r q E = q(r r)/{4 π ε 0 R r 3 } R(R r ) E ds = q (R r ) (3.13) ε 0 25

26 c.f. Gauss E dv = q ε 0 = 1 ε 0 ρ dv (3.14) E = 0 (3.15) : (3.9) (3.15) φ φ = 0 (3.15) potential φ E = φ (3.16) E potential φ ( ) (3.9) potential φ = q 1 4 π ε 0 r (3.17) φ 0 1. i = σe (3.18) i σ 2. 1/ε r (ε r > 1) ε r (3.12) (εe) = D = ρ (3.19) 26

27 ε = ε r ε 0 (3.20) Laplace, Poisson D = ρ, E = 0, D = εe (3.21) E ε ( ) E = 0 E = φ 3 E φ (3.17) (3.19) 2 φ = ρ ε (3.22) 2 φ = 0 (3.23) (3.22) Poisson (3.23) Laplace Poisson (3.22) : Laplace Potential Laplace (x, y, z) x, y, z ±h Tayler φ(x, y, z) = 1 3 [ 1 2 {φ(x + h, y, z) + φ(x h, y, z)} + 1 {φ(x, y + h, z) + φ(x, y h, z)} {φ(x, y, z + h) + φ(x, y, z h)} ] + O(h 4 ) (3.24) O(h 4 ) h 4 h 0 ( ) { } h Laplace [ ] 3 x, y, z c.f. ( Laplace ) f(z) = 1 2πi f(ζ) ζ z dζ = 1 2 π 2 π 0 f(z + ɛe iθ )dθ (3.25) 27

28 ζ = z + ɛ e iθ SOR(succesive over-relaxation) Laplace potential ( ) c.f. (x, y, z) φ (x, y, z) φ x = const. φ 2 φ/ x 2 ( 2 / y / z 2 )φ = 2 φ/ x 2 < 0 > 0 φ 2 φ = 0 Dilichlet φ Neuman ( 2 ) Laplace Laplace δ ( φ) 2 dv = 0 (3.26) ( ) Laplace ( ) Laplace Poisson Laplace φ ɛ f ψ = φ + ɛ f ψ 28

29 ( ψ) 2 dv = {( φ) 2 + 2ɛ φ f } dv + O(ɛ 2 ) (3.27) f φ f ( ψ) 2 dv = {( φ) 2 2ɛ f 2 φ} dv (3.28) 2 φ = 0 O(ɛ) ɛ δ ( φ) 2 dv = 0 φ Laplace Laplace ( ) E = 0 ABCD AB BC ( E) ds = E dl = AB E dl E dl = 0 (3.29) DC D A C B 1, 2 AB E DC ( ) E ( ABCD ) E (3.30) D = ρ D dv = D ds + D ds 1 2 = Q = (D 2 ) (D 1 ) = (3.31) ( ) D σ 29

30 E n ɛ ε E n = σ σ ε E n = σ/ε (3.32) a potential 2 φ = 0 φ 2 φ = 1 ( r 2 φ ) + 1 ( sin θ φ ) r 2 r r r 2 sin θ θ θ = 0 (3.33) ε ε z φ z z = 0 φ = 0 φ z = 0 φ cos nθ, sin nθ sin nθ θ = 0 2π 2 n = 1 φ(r, θ) = cos θ f(r) (3.33) f = r m r 2 d2 f df + 2r 2f = 0 (3.34) dr2 dr r m (m 2 + m 2) = r m (m + 2)(m 1) = 0 (3.35) m = 1 m = 2 A, B φ = (A r 2 + B r) cos θ. (3.36) A, B : r φ = Ez = Er cos θ B = E A : r 0 φ = 0 A = 0 B r = a φ = φ φ (r = a) = (A 1 a 2 Ea) cos θ = B a cos θ = φ (r = a) (3.37) D φ ε r (r = a) = ε φ (r = a) (3.38) r 30

31 ε ( 2A a 3 E) = ε B (3.39) (3.37) (3.39) B, A B = 3ε E, ε + 2ε A = ε ε ε + 2ε Ea 3 (3.40) φ = 3 ε E r cos θ (r a) ( ε + 2ε ) ε ε a 3 ε + 2ε r r E cos θ (r > a) 2 (3.41) : E r = φ r = E = (E 2 r + E 2 θ) 1/2 = 3ε ε + 2ε E cos θ, E θ = 1 r φ θ = 3ε E sin θ (3.42) ε + 2ε 3 ε ε + 2ε E = (3.43) : E r = (1 + 2 ) ( a3 ε ε E cos θ, E r 3 θ = 1 + ε ) ε a 3 E sin θ (3.44) ε + 2ε ε + 2ε r 3 E r θ ε > ε ( ) 3ε /(ε + 2ε ) < 1 ( ) E E = (ε ε )/(ε + 2ε ) E ( ) D = ε 0 E + P = ε E 3ε P = (ε ε 0 ) E (3.45) ε + 2ε ε /ε = 4 z 31

32 pot. dist θ=45 deg r θ = 45 potential ( E = 2 ) r = a r ( ε D ) 32

33 Laplace Laplace Laplace a 3 /r 2 potential 0 : = i q i = Q ( i ) 1 : p = i r iq i (r i i ) i q i = Q = 0 -q d r_ r +q r+ ±q d potential p = q (r + r ) = q d potential φ p φ = 1 { q q } 4 π ε 0 r + r : (3.46) (3.47) d 0 ( p = p ) r = (r + + r )/2 φ = 1 4 π ε 0 { } q r + d/2 q r d/2 = { } q 1 4 π ε 0 r2 + r d 1 + O(d 2 ) (3.48) r2 r d 1/ r 2 + r d 1/r(1 r d/2 r 2 ) φ = 1 4 π ε 0 r p r 3 (3.49) r p potential p r potential φ p (r) φ p (r) = + 1 r p = p ( ) 1 4 π ε 0 r 3 4 π ε 0 r (3.50) Coulomb pot. (3.48) Tayler 33

34 (3.50) r ( 1 r 3 ) = 3 r 5 r (r p) = (p )r + (r )p + r( p) + p ( r) = p (3.51) E p = φ p (r) = 1 { 3r (r p) p } 4 π ε 0 r 5 r 3 (3.52) p using 1:2:(abs( $3)) potential x y 34

35 10 5 dipole filed y x y y x 1.3 x x dx/e x = dy/e y E x E y dx/ds = E x / E, dy/ds = E y / E x(s), y(s) s E p --q d θ +q E p U p E d ±q +q q E E d θ = 0 U p = 0 θ +q θ 0 q E cos θ d 2 (dθ) = q E d 2 2 potential U p = 2 q E d 2 sin θ (3.53) sin θ = p E (3.54) 35

36 E p ( ) N N = p E (3.55) 2 p 1, p 2 U 12 p1 r p2 U 12 = p 1 E p2 = 1 { p1 p 2 3 (r p } 1)(r p 2 ) 4 π ε 0 r 3 r 5 dipole-dipole (3.56) p r 0 = ri q i qi (3.57) r i (3.57) q i = Q ρ(r) p = ρ(r) rdv (3.58) p d --p 2 d ±q ±p 2 d p p d potential φ Q φ p 3.48) q potential p φ p d φ Q = d φ p (3.59) 36

37 Poisson delta f(r) r 0 f(r)δ(r r 0 ) dv = f(r 0 ) (3.60) δ(r r 0 ) Dirac delta δ r f(r) r 0 3 ( ) 1 2 = 4 π δ(r) (3.61) r r 0 2 (1/r) = ( r/r 3 ) = (3/r 4 ) (r/r) r 1/r 3 r = 0 ε 0 ( ) ( ) ( ) f(r) 2 dv = f(r) 2 dv f(r) 2 dv = 0 (3.62) V ε r V r ɛ r V f(r) 2 (1/r) dv r = 0 4 π f(0) ɛ f(r) 2 ( 1 r ) dv = f(0) ( r ) dv = f(0) ɛ r 3 ɛ r ds = 4 π f(0) (3.63) r3 Gauss ɛ r/r 3 ds 4π V ( ) 1 f(r) 2 dv = 4πf(0) O.K. (3.64) r (3.61) (3.12) (3.61) 4 π ( ( ) 4 π ) 4 π : ln r 2 Green 4 π 2 π : delta Green Green 37

38 2 φ(r), ψ(r) (φ 2 ψ ψ 2 φ) dv = (φ ψ ψ φ) ds (3.65) Green 2 G(r, r ) = δ(r r ) (3.66) Poisson Green r (3.61) G(r, r ) r Poisson Poisson Green (3.66) 2 φ(r) = ρ(r) ε 0 (3.67) G(r, r 1 ) = 4 π r r φ potential φ ψ (3.68) G (3.65) φ(r) = 1 ρ(r ) 4 π ε 0 r r dv π r r φ(r ) ds 1 4 π φ(r ) (3.68) ( ) 1 ds (3.69) r r ( ) 1 = r r (3.70) r r r r 3 φ(r) = 1 4 π ε 0 ρ(r ) r r dv + 1 { } 1 4π r r φ(r ) + (r r) r r 3 φ(r ) ds (3.71) potential r r 1 Coulomb { } G(r, r ) ρ(r ) dv (3.72) ε 0 (3.66) r 1 potential Green { ρ(r )/ε 0 } ( ) (G(r, r )) 38

39 2 φ = E (3.31) σ(r ) {σ(r )/ε 0 1/ r r } Coulomb ds σ (r ) r ρ( r ) V S potential r ( ) 3 potential φ(r ) r potential (3.71) ρ(r ) potential φ(r) φ = {}ρ dv + {} φds + {} φ ds r r potential Laplace Poisson p r potential φ p (r) (3.50) φ p (r) = 1 4 π ε 0 r p r 3 (3.73) +σ σ τ ds potential ±σ d r d dφ(r) = 1 { σds σds } (3.74) 4 π ε 0 r r + d r d = d 2 1 r + d = 1 r 1 + 2r d r 2 + d2 r 2 1 r (1 r d r 2 ) (3.75) dφ(r) = σ r d σd r ds ds = = τ dω (3.76) 4 π ε 0 r3 4 π ε 0 r 3 4 π ε 0 39

40 r ds (3.71) 3 φ(r ) = τ/ε 0 potential (3.76) potential +σ σ dω 0 2π potential 2π 1/r τ/ε 0 potential jump z potential : (3.50) (3.76) P r P(r )dv r 0 dφ(r 0 ) (3.50) dφ(r 0 ) = 1 (r r 0 ) P(r ) dv = + 1 ( ) 1 P(r )dv (3.77) 4 π ε 0 r r π ε 0 r r 0 r ( ) ( ) 1 1 r A = A + 1 r r A (3.78) dφ(r 0 ) = 1 4 π ε 0 [ { 1 r r 0 P(r ) } ] 1 r r 0 P(r ) dv (3.79) V S V φ(r 0 ) = π ε 0 r r 0 P(r ) ds π ε 0 r r 0 P(r )dv (3.80) (3.80) S 1 S σ = P n ( n ) r P V 40

41 P > 0 P P ds ε/ε 0 = ε r ε = 2ε 0 3/4 ( ε r ) P ρ true ρ i = P ρ true (3.12) E = ρ true /ε 0 E ρ true ρ i E = 1 ε 0 (ρ true + ρ i ) = 1 ε 0 (ρ true P) = ρ total ε 0 (3.81) D D = ε 0 E + P (3.82) D = ρ true, or E = 1 ε 0 ρ total (3.83) E P ( ) E P P χ ( χ electric susceptibility) (3.82) P = χ ε 0 E (3.84) D = εe, ε = ε 0 (1 + χ) = ε 0 ε r, ε r = 1 + χ (3.85) :(3.84) E (3.81) P E P (3.84) 41

42 P? σ (3.31) D = ρ (3.83) ( ) E = 0 E (3.30) Coulomb E = ρ/ε 0 D = ρ E D D = εe = ε r ε 0 E D = ε 0 E + P P = χ ε 0 E ( ) jump E σ/ε 0 D ( ) E ( ) 42

43 4 ( ) i q n v i = n q v (4.1) q v i dv dq ρ dq = ρ dv dq/dt = (dρ/dt)dv dv ( i)dv i + ρ t = 0 (4.2) dv E i i = σ E (4.3) σ ( ) i I 43

44 E V ( ) σ = 1/R V = IR n ( ) v q n q E (4.3) (4.1) (4.3) (4.3) Newton Stokes (4.3) n m dv dt = n q E n α v (4.4) m 1 α v = v 0 exp ( α ) m t + q α E (4.5) t v = qe/α (4.3) n q v = n α q2 E = σ E, α = nq2 σ (4.6) : (4.3) : 1 ( ) 1 m/α Q C R sw d +Q -Q R C SW ε sw exp( t/rc) V = Q/C E = V/d P = (ε r 1)Eε 0 P P 44

45 P / t!? ρ i = P P / t i = / t( P) P P P E P P P 0 / t( P) 0 P v = σ E = µe (4.7) n e µ (mobility) ( ) Coulomb ( ) ρ t + i = 0 Ohm i = σe i = σ E ( ) Coulomb D = εe E = D/ε D = ρ ρ t + σ ε ρ = 0 (4.8) ρ(r, t) = ρ(r, 0) exp ( σ ) ε t (4.9) ρ(r, 0) t = 0 ( ) ε/σ 45

46 σ = 10 8 /1.7 [Ω 1 m 1 ] ε = [F m 1 ] ε/σ = [ ] t = 0 R { ρ 0 r R ρ(r, 0) = 0 r > R r > R ρ = 0 r R (4.10) i = 1 d ( ) r 2 d i r i r 2 r = dr dr + 2 r i r = σ ε ρ (4.11) d i r dr + 2 r i r = 0 i r = const. r 2 (4.12) di r dr + 2 r i r = σ ε ρ 0 (4.13) i r = σ ρ 0 3 ε r (4.14) r = R const. = σ ρ 0 R 3 i r (r, t) = exp ( σ ) ε t σ ρ 0 3 ε r r R σ R 3 ρ 0 3 ε r 2 r > R 3 ε (4.15) r 2 r R r ( ) 4 π r 3 ρ 3 4 π ε r = ρ r2 3 ε 4 π r 2 ρ dr E (4.16) E = R 0 ρ r 2 3 ε 4 π r2 ρ dr = 3 5 Q 2 4 π ε R (4.17) 46

47 Q = 4 π R 3 ρ/3 d E dt = Q d Q 4 π ε R dt = 2σ ε E (4.18) D = ρ r { 1 ( } r R R) 1 V (r) = Q r R 4 π ε r > R r E r r E r (r) = d V dr = Q r R 4 π ε R 3 1 r 2 r > R (4.19) (4.20) P E r i r P = r R r > R P in = R 0 0 E r (r) i r (r) 4 π r 2 dr (4.21) Q r 4 π ε R 3 σ ρ r 3ε 4 π r2 dr = σ 5 ε P out = σ ε Q 2 4 π ε R Q 2 4 π ε R (4.22) (4.23) P = P in + P out = 6 σ Q 2 5 ε 4 π ε R = 2 σ E (4.24) ε P d E/dt Ohmic loss

48 1 mm 2 1A 1/( ) 8 g/cm g.,, /cm 3 (drift) 83 µ m/s 10 m/83 µ m/s = s 48

49 5 Coulomb q m, q m Coulomb F = 1 q m q m r 4πµ 0 r 2 r (5.1) Coulomb H = 1 q m r 4πµ 0 r 3 m = q m d M : B = µ 0 H + M H = 0 H = ρ m µ 0, B = ρ m (5.2) (1) q m = 0 m = 0 M n, M) (2) Coulomb 49

50 m H = 1 [ m ] 3r (r m) + 4πµ 0 r3 r 5 (5.3) H (5.3) m = q m d d q m (3) Ampare (4) ε r (= ε/ε 0 ) (ε r 1) µ r (= µ/µ 0 ) 1/r 2 Coulomb H = 0, B = 0 H = 0 (5.3) H B : H = 0 B = 0 B = 0 H = 0 potential φ m H = φ m M n, M φ m φ m (r) = 1 4πµ 0 V M(r ) dv + 1 r r 4πµ 0 S M(r ) r r ds (5.4) 50

51 6 Coulomb B/ t Faraday i, r ( ) i j r k, (j,k=1,2, 3) Faraday E = B t B S t ds = B ds = dφ t S dt ( E) ds = E dl = V S Φ(= B ds) B Faraday c (6.1) ( ) = ( ) (6.2) 51

52 (6.2) (6.1) (6.2) (6.1) E (6.1) B M (6.2) E + B t = 0 (6.3) E B B = 0 B A B = A (6.4) B A = (B r)/2 (6.4) (6.3) (E + A t ) = 0 (6.5) E = φ A (6.6) t E = 0, E = φ (6.5) (6.3) E φ φ (6.4) A (6.6) E = ρ/ε 0 2 φ + t A = ρ ε 0 (6.7) A A = 0 2 φ = ρ ε 0 (6.8) Coulomb gauge 52

53 7 Coulomb Lorentz B q v B F = qv B (7.1) B H B B H H B E (7.1) qe E B B E (7.1) B Lorentz dw dt dw dt = F v = qv (v B) = q(v v) B = 0 (7.2) Fermi-Chandrasekar (7.1) v B i idv df dv df = idv B (7.3) (7.1) I ds df df = I ds B (7.4) 53

54 Ids/ idv q m r H(r ) qm r H(r ) = q m 4πµ 0 r r 3 Ids H(r ) F F = I ds ( ) qm r µ 4πµ 0 r 3 0 I ds r(= r ) q m F Ids r dh Ids r dh(r) dh = Ids 4π r r 3 (7.5) ds check Biot Savart I 1 ds 1 I 2 ds 2 I1 ds1 I2 ds2 (7.4), (7.5) check (7.5) (7.5) r H = 0 H ( ) C I H C H C H W 54

55 C I C H C H Ids r r d d s d r d S C I C H I ds C H r C H dr W W = H dr = I dr (ds r ) = I (dr ds) r C H 4π C H C I r 3 4π C H C I r 3 = I ds r 4π C H C I r = I dω (7.6) 3 4 π dr ds = ds r ds C I dr C H ds C I C H 4 π H dr = I C H 4π 4π = I = i da (7.7) C H C H (da C H ) Stokes H H = i (7.8) Biot-Savart Biot-Savart (7.8) (7.8) ( H) = 0 i = 0 Maxwell D = ρ, B = 0, E = B ρ, i+ t t = 0 (7.8) Maxwell H = i + D (7.9) t E H D B (7.9) i m 0 55

56 / t D/ t P/ t / t ( P) ε r = 1 Maxwell Coulomb vt R y B r q x r q q r B(r, t) t=0 v v v v v Lorentz qv B v (R, t) v v E v (R, t) = v B v (r + vt, t) Lorentz B v Tayler B v (r + vt + v t, t + t) = B v (r + vt, t) [ B v (r + vt, t) + t (v )B v (r + vt, t) + B ] v(r + vt, t) = B v (r + vt, t) t t (v R )B v (r + vt, t) + B v(r + vt, t) = 0, t R = r + vt (7.10) v R ( v B v (R, t)) = v( R B v ) + (v R )B v 56

57 R B v = 0 B(R, t) E v (R, t) = t v Faraday (7.11) t v i + ρ t = 0 (7.12) Maxwell E = B t H = i + D t (7.14) H = 0 t D = ( ) D = ( H i) = i = ρ t t t ( D ρ) = 0 D ρ = C 1 ( B) = t ( ) B = ( E) = 0 B = C 2 t Coulomb (7.12) (7.13) (7.14) ( D ρ) = 0, t (7.13) (7.14) ( B) = 0 (7.15) t C 1, C 2 C 1, C 2 (7.15) C 1, C 2 Maxwell rank (7.12) (7.14) (7.13) E B D = ρ, B = 0 t = 0 57

58 8 V ρ(r) W E r ρ(r)dv potential energy dw E 1 dw E (r) = ρ(r)dv 4 πɛ 0 ρ(r ) r r dv (8.1) r potential r V r r, r r W E = 1 2 dw E (r)dv = 1 8πε 0 dv ρ(r) dv ρ(r ) r r (8.2) ρ(r) = D ϕ( D) = (ϕd) ( ϕ) D W E = 1 { } dv { D(r)} dv ρ(r ) 8πε 0 r r = 1 { } dv D(r) dv ρ(r ) 1 { } dv D dv ρ(r ) (8.3) 8πε 0 r r 8πε 0 r r D 1 r 2, 1 r r 1 r r 2 r 0 ρ(r )(r r ) r r 3 ρ(r ) r r 1 dv (r r )ρ(r ) = E(r) (8.4) 4 πɛ 0 r r 3 W E = 1 dv(e D) (8.5) 2 (E D)/2 W H W H = 1 2 dv (H B) (8.6) 58

59 q i φ i (i = 1, 2,, n) D = ρ E q i = j C ij φ j (8.7) C ij C ij ε i (φ j = 0; i j), q i = C ii φ i q i > 0 φ i C ii > 0 i, j ground q i δq i φ i δφ i φ j δφ j j δq j = C ji δφ i + C jj δφ j = 0 C jj > 0, δφ i > 0, δφ j > 0 C ji < 0 (8.5) W W E = 1 2 E Ddv = 1 2 ( φ) Ddv = 1 2 {φ D (φd)} dv φ φ i D = ρ q i ( ) φ 1 R D 1 R 2 W E = 1 φ i q i = 1 φ i C ij φ j (8.8) 2 2 i W i, j C ij = C ji ij W H = 1 dv B(r) H(r) 2 (7.5) Biot-Savart W H = 1 2 dv B(r) 1 4π B (i r ) r = i r r 3 r r r r 3 = r r r dv i(r ) r r r r 3 { r r } r r B 3 { 1 } r r 59

60 { } { } 1 B(r) r B(r) = r r r 1 r r r r r B(r) D/ t = 0 W H = µ 0 8π dv dv i(r) i(r ) r r r B(r) = µ 0 i(r) 1 8π dv dv i(r ) r { } B(r) r r r i(r ) i(r ) r { } { } B(r) 1 = r r r r r B(r) i(r ) Gauss dv ds W H = µ 0 8π dv dv i(r) i(r ) r r (8.9) W H = 1 L ij I i I j (8.10) 2 ij L ij = µ 0 1 dl i dl j (8.11) 4π r i r j (8.8) (8.10) I i φ i C ij L ij L ii C ii L ii C, L (8.10) Poynting Vector S S = E H (8.12) (E H) = H ( E) E ( H) Maxwell B = µh, D = εe (E H) + t ( 1 2 E D + 1 ) 2 B H + E i = 0 (8.13) 60

61 Joule Joule E i = 0 (8.13) (E H) = 1 [W/m 3 ], E i = 1 2 [W/m3 ] (8.13) t ( 1 2 E H + 1 ) 2 B H = 3 2 [W/m3 ] m 3 J J E H E = H = 0 (E H) = 0 r ds df x ds x y, z df x = T xx ds x + T xy ds y + T xz ds z df y = T yx ds x + T yy ds y + T yz ds z df z = T zx ds x + T zy ds y + T zz ds z df = T ds (8.14) T T xx,, T zz T xx O T = T yy T xy + T yx T xz + T zx T xy + T yx 0 T yz + T zy (8.15) 2 O T zz T xz + T zx T yz + T zy T xy T yx T xz T zx T yx T xy 0 T yz T zy (8.16) 2 T zx T xz T zy T yz 0,, 61

62 f(x+dx,y,z) x z f(x,y,z) y df x {df x (x + dx, y, z) df x (x, y, z)} + {df x (x, y + dy, z) df x (x, y, z)} + {df x (x, y, z + dz) df x (x, y, z)} = ds x = dydz ( Txx x + T xy y + T xz z ) dv (8.17) dv x y, z (8.17) T x Maxwell E H dv ρ i E, H D B D = εe, B = µh Lorentz f L dv f L = ρe + i B (8.18) x ( Dx (ρe) x = ( D) E x = E x x + D y y + D ) z z [ { } { }] = ε x (1 2 E2 x) + y (E E x xe y ) E y + y z (E E x xe z ) E z z ( Ey {E ( E)} x = E y x E ) ( x Ex E z y z E ) z x = { } { } 1 x 2 (E2 y + Ez 2 E x ) E y y + E E x z z (8.19) 62

63 (E 2 x E 2 y E 2 z )/2 = E 2 x E 2 /2, E = B t (ρe) x = x (E xd x E D/2) + y (E xd y ) + z (E xd z ) (8.18) ( i B = H D ) B t ( D B t ) x (8.20) {( H) B} x (8.19) [ 1 {( H) B} x = {B ( H)} x = µ 2 x (H2 y + Hz 2 H x ) H y y H z H x H y y + H H x z z = y (H xh y ) + z (H xh z ) H x = ( ) H 2 x x 2 ( Hy y + H ) z z + y (H xh y ) + ( z (H Hx xh z ) H x x + H y y + H z z ( Hx µ x + H y y + H ) z = B = 0 z (i B) x = x (H xb x B H/2) + y (H yb y ) + z (H xb z ) (f L ) x = x (E xd x E D/2) + y (E xd y ) + z (E xd z ) + x (H xb x B H/2) + y (H xb y ) + z (H xb z ) 1 c 2 ( ) D t B x ] H x z ) (8.21) t (E H) x (8.22) εµ = 1/c 2 c (8.22) (E H)/c 2 g 1 c 2 S = 1 c 2 E H (8.23) 63

64 (i) Lorentz f L = ρe + i B g t = 1 S c 2 t = 1 E H c 2 t (8.24) (ii) T T E T M T E = T M = T = T E + T M E x D x E x D y E x D z E y D x E y D y E y D z E z D x E z D y E z D z H x B x H x B y H x B z H y B x H y B y H y B z H z B x H z B y H z B z 1 2 E D 1 2 B H (iii) Maxwell T ( f L + g ) = T ix t x + T iy y + T iz z i x, y, z T ix = (T E ) ix + (T M ) ix i (8.25) (8.26) 1 c 2 x ε 2 E2 0 0 T E = 0 ε 2 E ε 2 E2 (8.27) (T E ) xx > 0 (T E ) yy = (T E ) zz < 0 (8.26) ( ) S.F. 64

65 q Maxwell Maxwell E D/2, 65

66 9 φ A B E = B t potential φ B = 0 B = A vector potential (E + A )=0 scalar t E + A t = φ (9.1) D=ρ H = i + D ε µ t D = εe B = µh ε µ = 1/c 2 Maxwell potential (φ, A) Ψ A + 1 φ c 2 t, (9.2) c 2 t 2 d Alembertian E = ( φ + A ) = φ Ψ t t = ρ ε. φ = ρ ε Ψ (9.3) t A A A x { A} x = { Ay y x A } x { Ax y z z A } z x = 2 A y y x 2 A x y + 2 A z 2 z x 2 A x = { } 2 z 2 x A x y + 2 A 2 z 2 x (9.5) (9.2) = { ( A) 2 A } x (9.4) A = ( A) 2 A (9.5) ( A) = ( A) 2 A = { Ψ 1 } φ 2 A c 2 t 66

67 H = i + D t A = µi + Ψ (9.6) Lorentz (9.3) (9.6) Ψ = A + 1 c 2 φ t = 0 (9.7) 2 φ 1 c 2 2 φ t 2 = ρ ε (9.8) 2 A 1 2 A = µi (9.9) c 2 t2 Lorentz Lorentz gauge potential (φ 1, A 1 ) E B potential φ 2 = φ 1 + χ t A 2 = A 1 χ E B χ = 0 (9.10) A 2 = A 1 ( χ) = B, (9.11) A 2 + φ 2 = A 1 t t (φ 1, A 1 ) Lorentz gauge χ t + φ 1 + χ t = E (9.12) Ψ 2 = A c 2 φ 2 t = Ψ 1 χ = χ (9.10) χ = 0 potential Lorentz gauge A 2 = A 1 2 χ = 0 2 χ = A 1 χ (9.3) 2 φ 2 = ρ ε (9.13) potential φ Coulomb potential ( A = 0) gauge Coulomb gauge 67

68 (9.2) Ψ Ψ = A + 1 ρ ρ φ = µ( i + ) + Ψ, i + c 2 t t t = 0 (9.14) gauge potential Maxwell potential (φ, A) potential B = A, E = φ A t Lorentz gauge Maxwell φ = ρ/ε, A = µi potential vector potential Newton Lagrange potential potential E, B φ, A potential potential vector potential potential H = 0 A Lagrangian Hamiltonian Hamiltonian (13.102) ( ) 68

69 i = ρ = 0 potential φ A ( 2 1 ) 2 {φ(r, t) A(r, t)} = 0 (9.15) c 2 t 2 φ φ x y ( 2 scalar ) φ x = φ = 0 y ( 2 z 1 ) ( 2 φ = 2 c 2 t 2 z 1 ) ( c t z + 1 ) φ(z, t) = 0 (9.16) c t 0 ( z + 1 c t ( z 1 c t ) φ = 0 ) φ = 0 (9.17) 2 1 z t c ( ct ) (9.17) (9.16) φ(z, t) = f(z + ct) + g(z ct) (9.18) f (9.17) g f f p = z + ct p = f(p) = p = t dz dt = v z = c. (9.19) f = (9.17) c z (9.15) c c g c ( c) g z c (9.17) (9.15) X Z k Y (9.16) (9.18) (9.15) z ˆk ˆk ˆk x ˆk y ˆk z ˆk 2 x + ˆk 2 y + ˆk 2 z = 1 (9.20) 69

70 z zê z zê z = Xê X + Y ê Y + Zê Z = (r) (9.21) ê X ê Y ê Z X Y Z ê z ê X = ˆk x etc. z = (r) ê z φ(r, t) = f(r ˆk + ct) + g(r ˆk ct) (9.22) (9.15) f(r ˆk + ct) = ˆkf (r ˆk + ct) f(r ˆk + ct) = ˆk 2 f (r ˆk + ct) = f (r ˆk + ct) 2 t 2 f(r ˆk + ct) = c 2 f (r ˆk + ct) (9.23) f f (r ˆk + ct) (9.22) (9.15) ˆk (9.17) (9.15) (9.15) pick up ˆk (9.17) f (9.15) } φ = φ 0 exp[i(k r ωt)] (9.24) A = A 0 exp[i(k r ωt)] φ A (9.24) ( ) (9.24) (9.16) (9.17) k ω ( ) ( ω ) 2 k 2 = (9.25) c Lorentz φ 0 A 0 φ 0 = c2 ω k A 0 = c ˆk A 0 (9.26) (9.15) ( ) φ A E E = A t φ = iωa 0 exp[i(k r ωt)] ik φ 0 exp[i(k r ωt)] { } = i ωa 0 c k(ˆk A 0 ) exp i(k r ωt) = i E 0 exp i(k r ωt) (9.27) 70

71 A0 K E 0 = ω(a 0 ˆk(ˆk A 0 )) (9.28) A0 K(K,A0) E 0 A 0 ω E 0 k E = 0 E = ik E 0 exp[i(k r ωt)] = 0 E 0 k (9.29) i B B = A = +i (k A 0 ) exp[i(k r ωt)] = i B 0 exp[i(k r ωt)] (9.30) B 0 = (k A 0 ) (9.31) B k (9.24) A 0 k A 0 E (9.31) A 0 B (9.26) A 0 φ 0 φ = 0 gauge E K A 0 E 0 E k B E k B E 0 = ωa 0, B 0 = k A 0 (9.32) B 0 = 1 c E 0 (9.33) E B k Poynting vector S = E H E B (9.27) (9.30) E = B = 0 E = B = 0 Coulomb Coulomb (9.33) qe 0 q v B 0 =1 v/c v/c 1 (v/c (9.33) e = ε 0 E 2 mc = E 2 /(µ 0 c) e = mc 2 Einstein 71

72 advanced potential retarded potential (9.8) ( 2 2 / (ct) 2 )φ = ρ/ε c Poisson Coulomb potential φ C (r, t) φ C (r, t) = 1 4 π ε r r ρ( ) r dv ρ(r, t) r r (9.34) r t ρ(r, t) r c r r c t = r r /c ρ(r, t t) potential φ(r, t) φ t (r, t) φ t (r, t) = 1 4 π ε t = t r r /c dv ρ(r, t ) r r (9.35) (9.35) (9.8) r t r 1 r r = (r r ) r r, 3 t = 1 (r r ) c r r, ρ = ρ t t (9.36) r ρ(r, t ) r r = (r r ) r r 3 ρ(r, t ) (r r ) c r r 2 ρ(r, t ) t (9.37) 2 r ρ(r, t ) r r = 1 2 ρ(r, t ) c 2 r r t 2 4πδ(r r ) ρ(r, t ) (9.38) (9.35) (9.8) (9.8) c c ± (9.35) c c (9.22) (9.8) (9.9) φ ± (r, t) = 1 4 π ε A ± (r, t) = µ 4π dv ρ(r, t ±) r r dv i(r, t ±) r r t ± = t ± r r /c (9.39) 72

73 t retarded potential, t + advanced potential advanced potential 1 (9.39) Lorentz Hertz potential (9.8) (9.9) ρ i φ A Lorentz Hertz (9.8) (9.9), i + ρ t = 0, A + ( ) φ = 0 t c 2 i = p t, ρ = p A = Π (9.40) t, φ c = Π 2 Lorentz p (9.9) (9.40) Π p ( 2 1c 2 2 t 2 ) Π = µp (9.41) Π/c 2 Hertz (9.8) (9.9) (9.39) 1? 73

74 10 potential (9.39) retarded potential E B (r, t ) (r, t) E t t t 1 c r r (10.1) φ = 1 4 π ε dv ρ(r, t ) r r r = 1 4 π ε dv { (r r ) r r 3 ρ + 1 r r } ρ t r t (10.2) ρ/ t r ( ) t r i(r, t ) + ρ(r, t ) t = 0 (10.3) r t r ρ = t r i(r, t ) = r i i t r t = r i + (r r ) c r r i (10.4) t φ (10.2) 2 (10.4) r t { r r r i(r, t) + r r c r r i } dv (10.5) t Gauss 0 (10.5) { (i(r, t ) r ) (r } r ) dv 1 (r r ) c r r 2 c 2 r r (r 3 r ) i t dv (10.6) A t = µ 4π dv 1 i(r, t ) r r t (10.7) t = t 74

75 E = φ A t = 1 4 π ε π εc + µ 4π dv ρ(r, t ) (r r ) r r 2 r r dv (i(r, t ) r ) r r r r 2 [ dv 1 i(r, t ) + r r r r t r r ] r r r r i(r, t ) t (10.8) 1 Coulomb r = r 1/r 2 1/r i r i i r i t (r r ) 1/4 π εc 2 = µ/4π B B = A = µ 4π = µ 4π x B = µ 4π dv [( r 1 r r dv r i(r, t ) r r ) i + 1 r r r i(r, t ) [ r i(r, t )] x = i z y i y z = i z t t y i y t t z = ] ( t i ) t x [ dv (r r ) r r 3 i(r, t ) 1 ] 1 (r r ) r r c r r i(r, t ) t (10.9) (10.10) (10.11) Biot-Savart (10.8) ( i / t ) r r 1/r a r = r a E B leading term 1/r 1/ r r 1/r E 1 = µ 1 dv i 4π r t B 1 = µ ( ) 1 r 4πc r r dv i = 1 ( r ) E t 1 c r (10.12) 75

76 E a r B E 1 B 1 r r B 1 = E 1 c (10.13) ρ(r, t ) (t ) p p(t ) = p ρ (r ) dv r ρ(r, t ). (10.14) p = t ( dv r ρ ) = t dv r [ r i(r, t )] = r i(r, t ) ds + dv i(r, t ) (10.15) 0 2 p t 2 = dv i t (10.16) (10.12) dv ( i / t ) p" θ r da E 1 = µ 4 π r p, B 1 = r c r E 1 r da dw dw =(E 1 1 µ B 1) da, da = (r/r) r 2 dω = µ 16π 2 c p 2 dω dw = µ 16π 2 c p 2 sin 2 θ dω (10.17) z θ = θ = π/2 x y 1 E p r 76

77 ( ) E 2 = µ ( p x + p y ) /4π r E 2 r z ( ) r x y ( ) γ r a E exp[i(ˆk r ωt/c)] i/ t iω/c ω = ck k 1/r r k r a, r λ (= λ/2π) (10.18) (wave zone) 1/r 1/r 2 r (10.8) (10.11) 1/ r r 1/r p (10.8) t r Tayler 1/ r r r Tayler (r /r) 1/r 2 ect. 1/r potential φ A ρ ( ) i/ t i/ t ρ m i m p µ µ ( ) 77

78 11 Lienard-Wiechert potential retarded potential φ(r, t) (9.39) φ(r, t) = 1 4 π ε ρ(r, t ) r r dv, t = t r r /c (11.1) r r v (r, t) Doppler r t r 1/(1 α) α α v (r r ) c r r (11.2) q φ L (r, t) = 1 q 4 π ε r r (1 α) A L (r, t) = µ qv 4π r r (1 α) potential Lienard-Wiechert potential (11.3) (11.4) v = dr dt, t = t 1 c r r (11.5) (11.1) r g t g r (t) r (t g ) t g r r (t g ) r r t r g (t g ) + r v g ( ) (t t g ) 78

79 t g =t r r g (t g ) /c r r rg r" t =t r r (t ) /c r (t ) =r g (t g ) + r + (t t g )v g (t t g ) r (t ) r c(t t g ) = r r g (t g ) r r (t ) = r r g (t g ) r r g (t g ) r v g (t t g ) = r r g(t g ) r r g (t g ) (r + v g (t t g )) = n (r + v g (t t g )) r t t g = r (t ) r ρ dr = q etc. n = r r g(t g ) r r g (t g ) (11.6) n r c n v g = n r D, D = c n v g (11.7) r (t ) = r g (t g ) + r + v g n r ρ dr = ( ) r det = r D (11.8) ( ) r ρ(r ) det dr (11.9) r x x x x y z y y y x y z z z z x y z ( ) r 1 det = r 1 n v g c (11.3, 11.4) = 1 + n v g D (11.10) (11.11) r r v (t ) = dr = v dt g + (1 ) t t g v g ( ) r r r det Jacobian r 79

80 Lorentz gauge potential (9.39) potential n r g (t g ) r(t) Lienard-Wiechert potential (r, t ) q v(t ) (r, t) potential φ L (r, t) = q 4 π ε s, A L(r, t) = µ qv(t ) 4π s s = r r β (r r ), β = v(t )/c, t = t r r (t ) /c (11.12) E(r, t) B(r, t) B H B E L = φ L A L φ L t ( ) 1 = 1 s s s = 1 2 s { r 2 r β (r r )} (11.13) t r t r ( t ) t r = r + ( r t ) t (11.14) r r ˆn ˆn = r r r r (11.15) r t r t = 1 c r r(t) r (t ) = r r c r r + (r r ) v c r r r t (11.16) r t = 1 1 β ˆn ( ˆn ) c (11.17) (11.13) = r 1 1 β ˆn ˆn c t (11.18) r r = r r r r 1 1 β ˆn ˆn = 1 β ˆn ˆn c (r r ) ( v) r r = ˆn + 1 ˆn(ˆn β) 1 β ˆn (11.19) 80

81 (11.13) 2 ( 1 (β r) = r 1 β ˆn ( (β r ) = r 1 1 β ˆn 2 = β ) ˆn 1 (β r) = β c t 1 β ˆn ) ˆn (β r 1 ) = c t 1 β ˆn 1 1 β ˆn ˆn c β (r r ) + ˆn c β r ˆn c ( β r + β v) β 2 ˆn (11.20) 1 β ˆn (11.19,11.20) φ L = q { } 1 ˆn 4 π ε s 2 1 β ˆn β + 1 ˆn 1 β ˆn c β (r r β 2 ) 1 β ˆn ˆn = q [ 1 ˆn ˆn β + {β ˆn β π ε s 2 1 β ˆn β r }] r c Coulomb part A L (r, t) t = µq ( v ) 4π t s (11.21) (11.22) v t t t (11.12) t t t s t t t = 1 1 (r r ) ( v) c r r t t A L = µq ( 1 s v + 1 ) t 4π s 2 t s v t t = 1 1 β ˆn s = t t { r r β (r r )} = (r r ) ( v) r r β (r r ) + β v = ˆn v β (r r ) + β v µ 4π = 1 4 π εc A 2 L A L t = q 4 π ε [ { ˆn β β 2 c + β r } r c 2 v s 2 (1 β ˆn) + β s c ] 1 1 β ˆn (11.23) (11.24) (11.25) E = A L t φ L E = q β 4 π ε sc(1 β ˆn) + q [ 1 ˆn β + ˆn β {β ˆn β π ε s 2 1 β ˆn β r }] r c (11.26) 81

82 r r s E = = ˆn, 1 β ˆn = ˆn (ˆn β) 1 β ˆn q [ 1 β 2 4 π ε s 2 (1 β ˆn) (ˆn β) + 1 { (ˆn β)( s c(1 β ˆn) β ˆn) β(1 β ˆn)} ] (11.27) 2 E = q [ 1 β 2 4 π ε s 2 (1 β ˆn) (ˆn β) + 1 { β} ] sc(1 β ˆn) ˆn (ˆn β) (11.28) 2 1/s β B = A L = µq ( v ) 4π s ( ) 1 = 4 π ε s q φ L = µq { 4π ( ) 1 v + 1s } s v v = r v + t v = t ˆn β v = t 1 β ˆn (11.29) (11.30) B = q 4 π εc s 2 [ {ˆn β + q (1 β 2 )β ˆn = + 4 π ε c s 2 1 β ˆn (β ˆn β 2 + β r )} ] r s(ˆn β) β + c c (1 β ˆn) q { (β ˆn)( 4 π ε s(1 β ˆn) 2 c β ˆn) ˆn β ˆn (ˆn β) 2 ˆn 1 β ˆn B [ q 1 β 2 B = 4 π ε c s s(1 β ˆn) (β ˆn) + 1 { c(1 β ˆn) ˆn ˆn 2 (11.28) B } ( (ˆn β) β)} ] (11.31) B = 1 c ˆn E (11.32) (11.28) 1 β ˆn β B = 1 E ˆn c Coulomb field β = 0 r β x r x y ˆn = (cos θ, sin θ, 0) s = r(1 β cos θ) = r(1 ˆn β) E = q 4 π εr 2 1 β 2 (1 β cos θ) 3 {(cos θ β) ê x + sin θ ê y } (11.33) 82

83 { } t ê x ê y x y Coulomb β 0 β 1 0 β < 1 β β β = 0 ˆn ˆn β ˆn E ˆn β = 0.25 θ 15 E E y x (11.28) (11.31) β s 1 Poynting vector r 1/r 2 E B β E r B r q E r = 4 π ε s c(1 β ˆn) ˆn {(ˆn β) β} 2 (11.34) q B r = 4 π ε c 2 s(1 β ˆn) ˆn [ˆn {(ˆn β) β}] 2 (11.35) B r = 1 c E r (11.36) E r B r ˆn Poynting vector S r S r = ˆn µc E r 2 (11.37) 83

84 r r 2 dω dt dw dw = S r r 2 dωdt (11.38) dt p dt dt = 1 β ˆn (11.39) p = dw dt = q2 16π 2 εc 1 (1 β ˆn) 5 ˆn {(ˆn β) β} 2 dω (11.40) ˆn β 1 (11.40) ˆn β ˆn β ˆn 0 (q v sin θ)2 p (11.41) 16πεc 3 q v = p (10.17) X β β β β = 0 ˆn (ˆn β) = β sin θ p = q2 β 2 sin 2 θ (11.42) 16π 2 εc (1 β cos θ) 5 β dipole β sin 2 θ (1 β cos θ) 5 θ = cos β2 1 3β (11.43) (Bremsstrahlung) (11.42) 2π sin θ θ P π 2π 0 sin 2 θ sin θ (1 β cos θ) 5 dθ = 8π 3 P = 1 (1 β 2 ) 3 (11.44) q 2 β2 6πεc(1 β 2 ) 3 (11.45) ( ) 84

85 z θ β y n x β β β β β z x x y z y r = ˆnr ˆn = ( sin θ cos φ, sin θ sin φ, cos θ ) β = ( 0, 0, β ) β = ( β, 0, 0 ) (11.46) (11.40) ˆn {(ˆn β) β} 2 = β 2 { (1 β cos θ) 2 (1 β 2 ) sin 2 θ cos 2 φ } φ π β 2 { 2 (1 β cos θ) 2 (1 β 2 ) sin 2 θ } sin θ θ P P = q2 β2 6 π ε c 1 (1 β 2 ) 2 (11.47) 1. 1s? 2. 1C ( ) 3. KEK 960m 30GeV?? 85

86 12 r 0 E q r 0 potential φ(r 0 ) = (8.8) 4 π ε 0 r 0 E = q 2 φ(r 0) = q2 8πε 0 r 0 (12.1) Einstein E = mc 2 1/2 e m e r e = r e ( cm) e 2 4 π ε 0 m e c 2 (12.2) B r θ q v E z q v( v/c 1) z r E(r) E(r) = qr 4 π ε 0 r 3 (12.3) (11.28) s r ˆn = r/r β = 0 B(r) (11.31) β ˆn B(r) = q 4 π ε 0 c 2 r 2 v r r = 1 c 2 v E (12.4) E B g(r) (8.23) g(r) = 1 c 2 E B µ 0 = 1 c 4 µ 0 E (v E) = 1 c 4 µ 0 { v E 2 (v E) E } (12.5) r 0 r 2 dr π 0 ( ) 2 q 2π sin θdθ E 2 = 4π 1 (12.6) 4 π ε 0 r 0 86

87 z r 2 dr π r 0 0 2π sin θ 1 ( ) 2 q (v r)r r 6 z = q2 4 π ε 0 16 π 2 ε π 3 v r 0 (12.7) z G z = 2 3 q 2 4 π ε 0 c 2 r 0 v (12.8) G x y v ( m em ) e 2 m em = 2 (12.9) 3 4 π ε 0 c 2 r 0 (12.2) 1/2 2/3! (12.6) r 0 0 m em ( ) speed (11.47) β 2 P = e2 v 2 6 π ε 0 c 3. (12.10) F r v F r P W dw dt = P = F r v = e2 6πε 0 c 3 v2 = e2 6πε 0 c 3 { } d (v v) v v dt (12.11) d (v v) 1 0 dt F r F r = e2 v 6 π ε 0 c 3 (12.12) 87

88 (12.12) (12.12) ( ) (12.12) Newton 88

89 13 φ, A 2 φ 1 2 c 2 t φ = A 1 2 c 2 t A = 0 2 Lorentz (9.7) (13.1) A + 1 c 2 φ t = 0 (13.1) φ A ( ) E H(or B), c Newton Newton ( ) Galilei ( ) S z v S y y x x vt P(x y z ) P(x,y,z) z z t = 0 x = x y = y z = z vt t = t (13.2) S F m d2 r = F(r, t) (13.3) dt2 F (r, t ) = F(r, t), d 2 r d(t ) = d2 r 2 dt 2 m d2 r d(t ) 2 = F (r, t ) (13.4) 89

90 ( ) Galilei (13.2) (13.1) x = x, z = z z z + t t t = z, y = y t = t t t + z z t = t v z (13.5) [ { 2 x y 2 + z + 1 ( c t v )} { z z 1 ( c t v )}] φ(x, y, z ) = 0 (13.6) z x y { ( 1 v ) c z + 1 } { ( 1 + v ) c t c z 1 c ( ) z φ = f 1 v/c c t ( )=const. t t ( ) z + g 1 + v/c + c t } φ = 0 (13.7) z 1 v/c c = 0 or z = v z = ±c(1 v/c) = ±(c v) (13.8) z (c v) ( c + v) (13.1) c Michelson Morley Maxwell Galilei Newton O.K. { Galilei Maxwell No! (13.9) Maxwell (13.8) Maxwell R R c 500 v c = 2πR 1 c = 2π 500 = (13.10) 90

91 v/c (v/c) 2 ( ) S v v S v ( Bradley, 1727) 41 c v 41 2 tan 1 v c = 41 (13.11) Fizeau n v c (1853) c = c 0 (1 n ± v 1n ) 2 (13.12) c 0 Galilei c 0 n ± v (13.11) (13.12) (13.9) Maxwell ( ) Voigt Lorentz Galilei (13.2) Maxwell x = x y = y z = γ (z v t) t = γ (t v ) c z 2 (13.13) β = v c, γ = 1/ 1 β 2 (13.14) Maxwell t r r t ( ) Lorentz check ( z = γ z v ) (, c 2 t t = γ v z + ) t 91

92 2 z 2 1 c 2 2 t 2 = 2 z 2 1 c 2 2 t 2 (13.15) Lorentz (13.13) Lorentz Newton Newton ( Galilei ) Maxwell ( Lorentz ) Maxwell (13.13) 1. ( ) 1 r E B 2. B = 0 (1) (2) ( ) ( ) ( ) Dirac (Schrödinger) 2p 3/2 (2p) 2p 3/2 (2p) 2p 3/2 (2p) 1s 1/2 (1s) ( ) 3. c ( ) ɛ µ (1) (2) (1) (2) 92

93 Einstein Maxwell ( ) Lorentz (1) (2) (3) A B C A B C v A B C A B C A B C A B C A C A C ( ) B A C B ( ) B A C A C B ( ) ( A C) ( ) ( ) ( ( ) ) ( ) Lorentz S Z v S S t = 0 x 2 + y 2 + z 2 c 2 t 2 = 0 S S Lorentz S S v v = v Lorentz Einstein (1) ( ) 93

94 A = B A = B Newton Galilei Maxwell Lorentz v/c 0 v/c 1 Poincare Lorentz Poincare Einstein Poincare (2) ( ) Lorentz Galilei Maxwell c Lorentz ( ) S; (x, y, z, t) S Z v S ; (x, y, z, t ) x = x y = y z = γ(z v t) t = γ (t v ) c z 2 γ = 1/ 1 β 2, β = v/c (13.16) (x 0, x 1, x 2, x 3 ) = (ct, x, y, z) (13.17) Lorentz x µ = L µ νx ν ( ν L µ ν x ν ) (13.18) 94

95 (L µ ν) = γ 0 0 βγ βγ 0 0 γ. (13.19) (13.18) ν 0 3 (Einstein ) (13.13) (13.18) ( ) Lorentz t = 0 v Lorentz (1) S S (z 1 = z 2 ) t = t 2 t 1 = γ(t 2 t 1 ) vγ(z 2 z 1 )/c 2 = γ t S S t =0 t=0 after t t t t = t /γ = t 1 β 2 (13.20) 1 β 2 (2) Lorentz S z (= z 2 z 1) S t ( S, t 2 t 1 = 0 ) z = z 2 z 1 = γ (z 2 z 1 cβ(t 2 t 1 )) = γ z 1 β 2 z = z /γ = z 1 β 2 (13.21) t 2 = t 1 z v/(c 2 ) S z 2 z 1 S z 2 t 2 z v/(c 2 ) z 1 z 2 z 1 (3) S u (= dz /dt ) S dz = γ(dz v dt) dt = γ(dt v c 2 dz) } u = dz dt = 95 dz dt v 1 v c 2 dz dt = u v 1 u v c 2 (13.22)

96 S S z (z ) u = dz/dt u = u + v 1 + u v c 2 (13.23) ( ) (13.20) (13.21) β(= v/c) β S S S ( v) z (S S, v v) Fizeau (13.22) (13.22) u < c, v < c u < c (4) S z θ S S y = m z, m = tan θ ( π 2 θ π 2 ) y θ z S y = m γ(z + β c t ) = m (z + β c t ) m = γ tan θ = tan θ y θ = tan 1 (γ tan θ) (13.24) θ z θ 0, θ π 2 θ > θ Lienard-Wiechert potential (5) u Lorentz O r r ut O S O r S O r ) (13.25) r u ( ) r (r ) r β(= u/c) r = r, r = γ(r βc t) (13.26) 96

97 c t = γ (c t β r ) = γ (c t β r) (13.27) r r r = r + r = r + (γ 1)r γβc t (13.28) r = r r r = ( ) β β β r β, γ 1 = β 2 1 β2 (1 + 1 β 2 ) ( ) r 1 = r + γβ 1 + (r β) c t 1 β2 c t = γ(c t β r) (13.29) Lorentz z u α z u y 1. x α R x ( α) 2. z u (13.18) Lorentz L z (β) 3. x ( x ) +α R x (+α) (13.29) β = (0, β y, β z ) (13.29) = R x (+α)l z (β)r x ( α) γ 0 β y γ β z γ = β y γ γ2 γ γ β2 y 1 + γ β yβ z γ 2 β z γ γ β yβ z 1 + γ2 1 + γ β2 z (13.30) (6) Lorentz S, S, S t = t = t = 0 } S S z v S S y, z u 97

98 β = βy 2 + βz 2 S S Lorentz L(u)L z (v) L(u)L z (v) L z (v) L(u) ( u y 0) Lorentz ( ) c β v u L(u) L z (v) Lorentz β (13.29) β (13.30) L(u) L z (v) = L(cβ)R x (θ), (13.31) = R x (φ)l(c β ) R x L(c β), L(c β ) θ = φ β β Lorentz Lorentz Lorentz Lorentz Lorentz Lorentz ( Lorentz ) z α cos α sin α 0 R z (α) = sin α cos α 0 (13.32) α = iθ c t, z R(iθ) cosh θ 0 0 i sinh θ R(iθ) = i sinh θ 0 0 cosh θ (13.33) cosh θ = γ (13.16) Lorentz (13.16) (ict, z)) ( ) ( ) µ km MeV km 98

99 ( ) x(y) β x (β y ) Lorentz L x (β x )(L y (β y )) β y = a δt/c (δt ) L y (a δt)l x (β x ) = R z (α)l(β) (13.34) α β L(β) β Lorentz a v a/2c 2 v z (v z ) (Thomas precession) z Lorentz ( ) (1.13) x, y Lorentz x (y) 1 β 2 x, ( 1 β 2 y) ( ) ( ) m x m y xz y Lorentz m m Maxwell B D (ê i, i = 1, 2,, n) (ê i, i = 1, 2,, n) ê j = i ê i a i j, ê i = j ê j b j i (13.35) a j i bi j ê j = i ê i a i j = i,k ê k b k i a i j (13.36) 99

100 k δ j kê k b k i a i j = δ j k, i a k i b i j = δ j k (13.37) a b Cronecker δ r i r = i ê i x i = j ê j x j (13.38) x i = j a i j x j, x j = i b j i xi (13.39) (13.35) (13.39) (a i j) (b j i ) (13.38) ê j x j x j ê j r = ê j ê i x i = g j i x i (13.40) i i g j i = g i j = ê i ê j (13.41) Cronecker δ 13.40) x i = g i j x j (13.42) j g g (13.37) g j i g i k = g j i g i k = δ j k (13.43) i i ê j g j i ê i (13.44) i (13.40) (13.43) (13.44) ê j x j = g j i ê i g j k x k = g j i ê i g k j x k = δ i k ê i x k = j i j k i j k i k i ê i x i = r 100

101 r r i ê i x i = j ê j x j (13.45) (13.41) g j i = g i j = ê i ê j (13.46) ê i x i ( ) i g? i i (contraction) ( ) Einstein i xi y i x i y j T i j T i j = x i y j x y ê i, (i = 1, 2, 3) { } ê 1 ê 2 ê 3 ê 1 (ê 2 ê 3 ), 1,2,3 cyclic ê 1 = ê 2 ê 3 ê 1 (ê 2 ê 3 ) ( ) (13.40, 13.42) (13.41, 13.46) 101

102 Lorentz (13.45) r (x µ ) = (x 0, x 1, x 2, x 3 ) = (ct, x, y, z) (13.47) metric tensor g µ ν (g µ ν ) = (13.48) (x µ ) = (x 0, x 1, x 2, x 3 ) = ( ct, x, y, z) = ( ν g µ ν x ν ) (13.49) t = 0 t (x µ ) s 2 = x 2 + y 2 + z 2 c 2 t 2 = µ x µ x µ (13.50) Lorentz ( Lorentz ) Lorentz Einstein x µ x ν : x ν = L ν µx µ x µ x µ = x µ x µ xµ (x µ ) (L µ ν) Lorentz Lorentz : L T = , yz : (13.51) Lorentz

103 Lorentz x ρ, x τ x µ x µ = L µ ρ x ρ g µ ν x ν = L µ ρ x ρ g µ ν L ν τ x τ x ρ x ρ = x ρ g ρ τ x τ ) (13.52) L µ ρ L ν τg µ ν = g ρ τ (13.53) L µ ρ Lorentz (13.17) Lorentz (13.49) Lorentz (13.17) (13.49) metric tensor ds dx, dy, dz ds 2 = i (dx i ) 2 = g ij dx i dx j (13.54) g µ ν 4 (1.23) g 11 = h 1, g 22 = h 2, g 33 = h 3 g µ ν g µ ν g µ ν = (13.55) x = x ν µ x ν x = µ (L 1 ) ν µ (13.56) x ν x µ µ (13.57) ν = g νµ µ = 2 x y z 2 2 (ct) 2 = µ µ (13.58) Maxwell Lorentz Maxwell 103

104 : i + ρ t = 0 4 j µ = (j 0, j 1, j 2, j 3 ) = (cρ, i x, i y, i z ) (13.59) µ j µ = 0 (13.60) Lorentz µ 4 j µ 4 Lorentz : 4 E = φ A t : A µ = A + 1 c 2 φ t = 0 ( ) 1 c φ, A x, A y, A z (13.61) µ A µ = 0 (13.62) E x = 1 (cφ 0 ) 0 (ca x ) = c( 0 A 1 1 A 0 ) E y = 2 (cφ 0 ) 0 (ca y ) = c( 0 A 2 2 A 0 ) (13.63) E z = 3 (cφ 0 ) 0 (ca z ) = c( 0 A 3 3 A 0 ) A µ = g µν A ν = ( 1 c φ, A x, A y, A z ) B = A : B x = 2 A 3 3 A 2 B y = 3 A 1 1 A 3 (13.64) B z = 1 A 2 2 A 1 (13.64) (13.65) 2 0 E x /c E y /c E z /c E x /c 0 B z B y f µ ν = E y /c B z 0 B x = µa ν ν A µ (13.65) E z /c B y B x 0 104

105 B 0 E x /c E y /c E z /c f µ ν = g µ ρ f ρ τ g τ ν E x /c 0 B z B y = (13.66) E y /c B z 0 B x E z /c B y B x 0 Maxwell 1. D = ρ : 0 f 00 = 0 1 f f f 3 0 = 1 j 0 ε 0 c c = µ 0 j 0 (13.67) 2. H D t = i B 1 c 2 E t = µ 0 i : µ f µ0 = µ 0 j 0 (13.68) x 2 f 21 3 f 31 0 f 01 = µ 0 j 1 (13.69) 1 f 11 = 0 µ f µ1 = µ 0 j 1 (13.70) (13.67) (13.70) µ f µ ν = µ 0 j ν (13.71) 3. B = 0 : 4. E + B t = 0 : 1 f f f 12 = 0, 1 f f f 12 = 0 (13.72) x : 2 f 03 3 f f 23 = 0, 2 f f f 23 = 0 y : 3 f 01 1 f f 31 = 0, 3 f f f 31 = 0 z : 1 f 02 2 f f 12 = 0, 1 f f f 12 = 0 λ f µν + µ f νλ + ν f λµ = 0 (13.73) (λ, µ, ν) cyclic 2 D = ρ B = 0 D = ρ H D = i 1 t 105

106 B = 0 E + B = 0 1 t q m Faraday i m (2.9) (2.11) (13.66) f µ ν Lorentz E B Lorentz E = E E = γ{e + (v B) } B = B B = γ{b (v E) /c 2 } (13.74) γ = 1/ 1 β 2, β f µ ν f µ ν f ν µ E B Lorentz Lorentz B Lorentz φ = 0 φ = e i(k r ω t) 0 ( ) φ k r ω t x µ k µ k µ Lorentz Doppler f α β j γ f α β j β E, B, i, ρ f α β j β f µ ν ( ) 106

107 Lorentz 3 E, B (13.22) Lorentz (13.1) (13.2) 2 2 Maxwell β ( ) Lorentz 4 ds 2 = g µ ν dx µ dx ν = dx ν dx ν x ν dx i dx i = dx 1 dx 1 + dx 2 dx 2 + dx 3 dx 3 = 0 ds 2 = dx ν dx ν = dx 0 dx 0 = c 2 dt 2 = c 2 dτ 2 (13.75) dτ ( ) dτ x µ x µ τ x µ τ ( ) dx µ dx 0 dτ = vµ = dτ, dx1 dτ, dx2 dτ, dx3 (13.76) dτ ds 2 = c 2 dt (dx 3 ) 2 = c 2 dt 2 (1 β 2 ) = c 2 dτ 2 (13.77) v µ = 1 (c, c β) (13.78) 1 β 2 v µ v µ v µ v µ v µ = c 2 (13.79) v µ Lorentz scalar 107

108 a Lorentz (13.79) τ a µ v µ = 0 ( a µ dvµ ) (13.80) dτ ( ) a µ a µ = a 2 = (13.81) ( ) z β β Lorentz ( ) a µ = (0, 0, 0, a) T (13.78) β Lorentz (a µ) = 1 1 β 2 d dt c 1 β cβ 1 β 2 = γ 0 0 βγ βγ 0 0 γ a = βγa 0 0 γa (13.82) z γ = 1/ 1 β 2 d β dt = a = (13.83) 1 β 2 c β = a t + const. (13.84) 1 β 2 c t = 0 β = 0 const. = 0 t = 0 z = 0 dz = z z 0 = z = c2 a [ ( a c t ) 2 dz dt = a t (13.85) a t ( ( a ) = c2 a ) c t a c t (13.86) ( a c t ) 2 1 ] c2 a 1 ( a ) 2 2 c t a = 2 t2 (13.87) 108

109 ( ) z z t a t/c 1 z = c2 a a c t = c t c! neck ( ) (13.59) cρ Lorentz Lorentz potential ( ) Lagrangian L = 1 2 m v2 L = mc 2 1 β 2 ( mc 2 + m 2 v2 : ) (13.88) Lagrangian (13.88) L v p = L v = mv 1 β 2 (13.89) (13.78) (13.89) p µ = mc (1, β) (13.90) 1 β 2 W W = p v L (13.91) = mc 2 1 β 2 (13.92) mc 2 + m 2 v2 (: ) (13.93) W 2 = p 2 c 2 + m 2 c 4 (13.94) 109

110 p (13.89) 3 (13.93) Einstein (13.90) 4 f α β j β f αβ j β = 0 E x /c E y /c E z /c E x /c 0 B z B y E y /c B z 0 B x E z /c B y B x 0 cρ i x i y i z = E i/c ρe x + (i B) x ρe y + (i B) y ρe z + (i B) z (13.95) ( )/ c Lorentz Newton d dt ( mc β ) = q(e + c β B) = q(e + v B) 1 β 2 (13.96) d dt ( mc 2 ) = qe β c = qe v 1 β 2 (13.96) m/ 1 β 2 m m 1/ 1 β 2 (13.76) Lorentz (13.79) β p 1/ 1 β 2 potential A µ = (1/c, A x, A y, A z ) (13.61) Lagrangian (13.88) j µ A µ potential j µ A µ L = mc 2 1 β 2 + q (A v φ) (13.97) p L (L r v ) p = L v = p (13.89) q A Hamiltonian H H = p v L = mv 1 β 2 + q A (13.98) m c2 1 β 2 + q φ (13.99) 110

111 Hamiltonian β v p (13.98) (p qa) 2 = m2 c 2 1 β 2 m2 c 2 (13.100) (H qφ) 2 = m 2 c 4 + c 2 (p qa) 2 (13.101) H = m 2 c 4 + c 2 (p qa) 2 + q φ (13.102) (13.98) m v/ 1 β 2 q A Biot-Savart ( ) qa Lagrangian (13.97) Euler d dt ( ) L L v r = 0 (13.103) (13.96) L r = L = q (A v φ) = q (v )A + v A φ (13.104) dp dt = d m v dt + q da 1 β 2 dt = d dt m v 1 β2 + A (13.105) t + q (v )A E = 1 A + φ, B = A etc. (13.106) c t E = (0, 0, E) (13.96) dp dt = qe = p x = p y = p z = m v x 1 β 2 = m v y 1 β 2 = m v z 1 β 2 = p z0 + q E t, p z0 (13.107) c 2 β 2 = v 2 x + v 2 y + v 2 z 1 = 1 { (pz0 + q E t) 2 + m 2 c 2 + p 2 1 β 2 x + p 2 1/2 1 { m c y} = (pz0 + q E t) 2 + c 2} 1/2 (13.108) m c 111

112 x y x v x = dx dt = C (pz0 + q E t) 2 + C 2 (13.109) C m 2 c 2 + p 2 x + p 2 y = dx x2 + A = ln x + x 2 + A, ( ) x sinh 1 A ( ) cosh 1 x A A > 0 A < 0 (13.110) z x x 0 = z z 0 = c q E sinh 1 ( ) p0z + q E t C (13.111) c (pz0 + q E t) q E 2 + C 2 (13.112) t W = hν de Broglie p = h/λ (13.94) (13.113) W = p c (13.113) r ( ) h ν 1 (h/2 π) 112

113 ( ) exp( m r) φ π = E π r E π m 0 E π 0 E π Coulomb ρ = 0 E = 0 ( ) = 0 1/r potential ( ) = 0 Coulomb ( = 0) ( ) Michelson-Morley Einstein Michelson-Morley Einstein yaoki 113

114 14 Laplace Laplace Green Laplace eq. : 2 ϕ = 1 r ϕ θ ϕ = ϕ(r) ( 1 d r dϕ ) = 0, r 0 d r dr dr dr ( r ϕ ) ϕ r r r 2 θ = 0 (14.1) 2 ( r dϕ ) = 0 (14.2) dr r dϕ dϕ = c 1 = const. dϕ dr = c 1 r (14.3) ϕ = c 1 ln r + c 2 (14.4) Laplace : 2 ϕ = 1 ( r 2 ϕ ) + 1 r 2 r r r 2 sin θ ( sin θ ϕ ) + θ θ 1 2 ϕ r 2 sin 2 θ ψ = 0 (14.5) 2 θ ψ ( d r 2 dϕ ) = 0 (14.6) dr dr 114

115 r 2 dϕ dr = c 1 dϕ dr = c 1 r 2 (14.7) ϕ(r) = c 1 r + c 2 (14.8) Coulomb potential potential φ = 0 c 1 D = ρ a O d P R D Q a d σ potential d = 0 (14.4) c 2 r = a ϕ = 0 ϕ = σ 2 πɛ ln a (14.9) r 0 < d < a ϕ = 0 D Σ 1 D Σ ( E = 0 E E ) OP = d, OQ = D, OR = r σ E σ = σ r d 2 πɛ r d 2, Σ E Σ = Σ r D 2 πɛ r D 2. E σ + E Σ r d = d ê, D = D ê (E σ + E Σ ê ) = 0 ( r D r d ) 2 = D Σ σ d = const. (σ Σ! ) (14.10) QR PR = ΣD = const. ( r ) R P Q σd D Σ R OQ 115

116 D a a d = a + D a2 D = (14.11) a + d d D D d = a 2 D ( ) 2 D a = ΣD Σ = σ (14.12) a d σd Σ = σ Σ + σ = 0 σ Σ potential potential potential 0 ( potential ) OQ X(OX = x) potential ϕ(x) ϕ(x) = σ 2 π ɛ ln d x a Σ 2πɛ ln D x a + const. (14.13) x 0 σ = Σ 1 a σ + Σ = 0 ϕ(x) = σ 2 π ɛ ln d x D x + c 2 (14.14) ϕ(x) = 0 P, Q E(r) = r ϕ(x) = σ 2 π ɛ ln d r D r (14.15) σ ( ) r 2 π ɛ d r r 2 D r 2 (14.16) d r θ a dθ D (3.32) D = ρ s = ɛ E r (E r r ) 116

117 2 π S = ( σ a dθ ɛ 0 2 π ɛ π dθ 0 a 2 + d 2 2ad cos θ = a 2 d 2 ( = a ɛ σ ) ( 2 a π 2 π ɛ a 2 d 2 a π ) 2 D 2 a 2 = σ ) ( ) a a 2 + d 2 2ad cos θ a a 2 + D 2 2aD cos θ π Σ z σ d σ b D dz x + F = σ2 4 π ε = σ2 b 4 π ε + = σ2 1 2 π ɛ D d F = σ2 d 2 π ɛ a 2 d 2 σ σ ( σ dz) 1 σ( σ dz) z 4 π ε b 2 + z 2 x 1 dz b 2 + z 2 dz (b 2 + z 2 ) 3/2 b b2 + z { 2 + dz (b 2 + z 2 ) = 2 } 3/2 b 2 θ s(θ) = σ ) (1 d2 1 2 π ɛ a 2 a 2 + d 2 2ad cos θ (a2 + d 2 2ad cos θ + z 2 ) x (a cos θ d)/ a 2 + d 2 2ad cos θ + z 2 adθ, dz φ 0 d a 0 potential 2δ( 0) ϕ(x) = σ 2 π ɛ ln d x D x + c 2 (14.17) 117

118 ϕ = ϕ(a) ϕ(d + δ) ϕ = + σ 2 π ɛ ln a d D a ln δ D d δ (14.18) D a = a2 d a(a d) a =, D d = a2 d d d = a2 d 2 d ϕ = + σ { 2 π ɛ ln d (a d) a(a d) 1 } δ a2 d 2 d = σ ( ) a 2 2 π ɛ ln d 2 a δ c d a a2 d 2 a δ c = σ ϕ = = a ) (1 d2 δ a 2 2 π ɛ ln a2 d 2 a δ (14.19) ln a2 d 2 a δ ln a δ d2 a ( ) 4 d (14.20) a d = 0 ln(a/δ) a/δ a 10 m 2 cm a/δ d 0 c (d/a) 2 δ a δ a δ a d 0 d 0 d d 2 d d > 0 d < 0 h h q -q h q potential 0 V q 118

119 D d = a 2 d D h (...) a r r r = a 2 /r ϕ(r, θ, φ) Laplace ( ) 1 a 2 r ϕ r, θ, φ (14.21) Laplace w = a/z a = 1 ( Lord Kelvin ) x, y z = x + i y z f(z) u v f(z) = u(z) + iv(z) (14.22) i (i 2 = 1) x 0 x 0 ( ) z = x + i y z 0 x y 0 ( ) Cauchy-Rieman u x = v y, u y = v x (14.23) (14.23) x y u(x, y) v(x, y) Laplace 2 u x u y 2 = 2 v x v y 2 = 0 (14.24) Laplace (x, y) u(x, y) potential ( u = const. potential u u = x, u ) (14.23) y u x v x + u v y y = 0 (14.25) 119

120 u = v = u = v = u = v = u v y θ const. x ln(r) const. f(z) = ln z = ln r + iθ(z = x + iy = re iθ ) ln r = θ = ln z -q -a { -q b q q -q -q +q q... ln z + {ln(z + 4a) + ln(z 4a)} +... {ln(z + 2a) + ln(z 2a)} {ln(z + 6a) + ln(z 6a)}... z z ± 2b etc (14.26) sin z, cos z sin z z = cos z = n=1 ( 1 + z ) ( 1 z ) n π n π { 1 n=1 1 (2n 1) 2 ( ) } 2 2z π (14.27) (14.28) potential 120

121 f(z) = z 2 = x 2 y 2 + 2ixy (14.29) u = x 2 y 2, v = 2xy u = ± z v = z xz yz quadrupole lens xz, yz x = 0 or y = 0 z m/e quadrupole mass filter v = ± quadrupole lens 121

122 quadrupole magnetic lens KEK quadrupole lens z 2 z 3 (magnetic spectrograph) quadrupole mass filter Schwartz-Christoffel Laplace eq. z pl w pl w = f(z) z pl w pl y dz2 v θ dz1 w0 dw2 z0 x θ dw1 u z-pl w-pl z-pl z 0 w-pl w 0 = f(z 0 ) dw = df dz (14.30) dz z 0 dz 1, dz 2 w dw 1, dw 2 w 0 arg(dw 2 ) arg(dw 1 ) = { arg ( ) df(z0 ) dz } { + arg(dz 2 ) arg ( ) df(z0 ) dz } + arg(dz 1 ) = arg(dz 2 ) arg(dz 1 ) (14.31) 122

123 θ = θ ( ) z 0 df (14.31) dz f(z) (14.30) df = 0 dz 0 dw = 0 (14.30) (14.31) dz f(z) = z β π z0 z-pl x βπ w0 (1- β)π w-pl arg(dw 2 ) arg(dw 1 ) = β{arg(dz 2 ) arg(dz 1 )} (14.32) β w = w 0 df dz zβ 1 z pl π w pl β π (1 β)π z pl z 1, z 2,..., z n w pl α i π(i = 1,..., n) f(z) n df dz = i=1 A (z z i ) α i (14.33) A w pl. Schwartz-Christoffel z w z w z pl w pl z A A z=-1 z=1 B C D A z-pl A C v B u w-pl D w A ± + i0 + i0 B 1 +ia π C 0 + i0 +π D 1 ia π (14.33) dw dz = 1 Az2 z ( = A z 1 ) z (14.34) 123

124 ( ) z 2 w w 0 = A 2 ln z A w 0 ɛ 0 z 1 ɛ z 1 + ɛ A = 2a π, w 0 = a π ia ( a ) w = π ia 1 2 z2 ln z 1 iπ 2 w ia 1 2 z2 ln z 1 2 w ia 2a π ( ) z 2 2 ln z (14.35) (14.36) z pl potential ln z w pl U z pl potential U = V 0 ln z = φ + iψ (14.37) iπ z = re iθ, φ = V 0 π θ, ψ = V 0 ln r (14.38) π z pl φ = 0 (θ = π) φ = V 0 (14.37) (14.36) U ( ) iπ z = exp U (14.39) V 0 ( a ) w = π i a + 2a { i U V )} 0 (2 V 0 2πi exp π i UV0 (14.40) dw du = 2a ( i 1 exp 2 π i U ) = 2 a i { ( 1 exp 2 π ψ ) ( )} 2 π i exp φ V 0 V 0 V 0 V 0 V 0 (14.41) (14.40) potential φ = w pl potential 15 potential θ = π/16 ψ = 0 (r = 0) B D ψ > 0 (r < 1) ψ < 0 (r > 0) (14.41) du dφ dψ du dφ dw/dφ ψ du = dφ + i dψ du = dφ ( dψ = 0 potential dφ dw dw = dw ) dφ dφ 124

125 E = du dw = V 0 2a { ( 1 + exp 4 π ψ ) ( 2 exp 2 π ψ ) cos V 0 V 0 ( )} 1/2 2 π φ (14.42) 0 θ π/4 3π/4 < θ π {} = min θ π/4 θ 3π/4 π/4 θ = 3π/4 V 0 125

126 Laplace 126

127 2 ϕ = 1 ( r 2 ϕ ) + 1 r 2 r r r 2 sin θ ( sin θ ϕ ) + θ θ 1 2 ϕ r 2 sin 2 θ ψ = 0 (14.43) 2 (r, θ, ψ) ϕ ϕ r R(r) (θ, ψ) Y (θ, ψ) ϕ(r, θ, ψ) = R(r)Y (θ, ψ) (14.44) (14.44) (14.43) ϕ(r, θ, ψ) r 1 d R dr ( r 2 dr ) = 1 dr Y { ( 1 sin θ Y ) + 1 } 2 Y sin θ θ θ sin 2 θ ψ 2 (14.45) r (θ, ψ) (θ, ψ) r A ( 1 sin θ Y ) Y sin θ θ θ sin 2 = AY (14.46) θ ψ2 Y θ Θ ψ Ψ sin θ Θ ( d sin θ dθ ) + A sin 2 θ = 1 dθ dθ Ψ d 2 Ψ = B = (14.47) dψ2 d 2 Ψ = BΨ (14.48) dψ2 Ψ ψ 2π m B = m 2, Ψ = Ce imψ + De imψ (14.49) Ψ (14.47) Θ ( 1 d sin θ dθ ) ) + (A m2 sin θ dθ dθ sin 2 Θ = 0 (14.50) θ x = cos θ 1 d sin θ dθ = d dx { d (1 x 2 ) dθ } ) + (A m2 Θ = 0 (14.51) dx dx 1 x 2 m = 0 (14.51) Θ x x = 0 x 0 x 1 l 0 A = l(l + 1) x l (14.51) Θ x = 1 1 P l (x) Legendre 127

128 (14.51) x y = x parity P l (x) x (14.51) x 1 ( Legendre ) P l (x) P l (1) = 1, P l (x) = ( ) l P l ( x) { d (1 x 2 ) dp } l + l(l + 1)P l (x) = 0 (14.52) dx dx l m 0 Pl m (x) (14.51) Pl m (x) = (1 x 2 ) m/2 dm P l (x) (14.53) dx m Laplace Laplace P l m (x) Legendre cos θ sin θ sin θ = 1 x 2 m 0 x (14.53) l m P l (x) l 0 m l (14.51) m m l P l, Pl m m Ψ m (ψ) Ψ m (ψ) ψ m 1 1 dx P m l Ψ m (ψ) 1 2 π e i m ψ (14.54) (x)p m 2 (l + m)! (x) dx = l 2l + 1 (l m)! δ ll (14.55) 2l + 1 (l m)! 2 (l + m)! P l m (x) (14.56) 1 x 1 x Yl m (θ, ψ) = ( ) (m+ m )/2 (2l + 1) (l m )! 4 π (l + m )! P m l (cos θ)e imψ (14.57) (θ, ψ) m 128

129 l ± Yl m (θ, ψ) = (l ± m)(l ± m + 1)Y m±1 l (θ, ψ) { l ± = e ±iφ i cot θ φ ± } θ (14.58) (14.45) (14.46) ( d r 2 dr ) = AR (14.59) dr dr R = u r u r l 0 d 2 u dr 2 = A r 2 u (14.60) u r l+1 u r l } (14.61) A = l(l + 1) (14.62) l (14.52) l l = 0 R = α + β/r Coulomb pot. ( ) l = 1, m = 0 r < a R r r > a R 1/r 2 Laplace eq. φ = lm (A lm r l + B lm r l 1 )Y lm (θ, ψ) (14.63) f(θ, ψ) Y lm Y lm f(θ, ψ) = lm f lm Y lm (θ, ψ) (14.64) f lm f lm = π 0 sin θ dθ 2 π 0 dψ Y lm(θ, ψ)f(θ, ψ) (14.65) Y lm θ, ψ f(θ, ψ) Y lm A x A x x ê x (θ, ψ) ê x Yl, m (θψ) 129

130 (14.65) f lm f LM {L l, M m} f(θ, ψ) l, m f(θ, ψ) f l m f L M (L l, M m) (14.64) f(θ, ψ) 2 dω = f lm 2 (14.66) lm 0 (14.64) (14.64) sin θ, cos θ l r l x, y, z l (14.61) r l 1 1 (x, y, z l r2l+1 ) r = 1 r l r (l+1) 2 φ = 1 r r ( r φ ) φ r r 2 θ + 2 φ 2 z = 0 (14.67) 2 φ(r, θ, z) = R(r)Θ(θ)Z(z) (14.68) Θ l 2 d 2 Θ dθ = 2 l2 Θ (14.69) sin lθ, cos lθ θ = 0 θ = 2 π l Θ = A sin lθ + B cos lθ, l (14.70) Z k 2 k 2 d 2 Z dz 2 = k2 Z (14.71) e kz e kz k 2 < 0 κ 2 = k 2 e iκz, e iκz Z(z) = Ce kz + De kz Ce iκz + De iκz (14.72) 130

131 R d 2 R dr r ρ ) dr (k dr + 2 l2 R = 0 (14.73) r 2 ρ = kr (14.74) 2 R ρ + 1 ) dr (1 2 ρ dρ + l2 R = 0 ρ 2 (14.75) Bessel Bessel J l (ρ) Bessel N l (ρ) Y l (ρ) Neuman Neuman Bessel J l (ρ) = m=0 m=0 ( ) m ( ρ ) l+2m (14.76) m! Γ(l + m + 1) 2 Bessel ρ < l ρ ( l 2 ρ J l (ρ) π ρ cos ρ 2l + 1 ) π 4 0 Neuman N l (ρ) = 2 π J l(ρ) ln ρ 2 1 m ψ(m + 1) + ψ(l + m + 1) ( ρ l+2m ( ) π m!(l + m)! 2) 1 π l 1 m=0 (l 1 m)! m! ( ρ 2 l = 0 di-gamma ) l+2m (14.77) ψ(m + 1) = ψ(m) + 1 m = γ m γ {( n ) } γ = lim n 1/m ln(n) m=1 (14.78) (14.79) Neuman Frobenius Θ Z Bessel k 131

132 k 2 < 0 Bessel Bessel Bessel 2l ρ J l(ρ) = J l 1 (ρ) + J l+1 (ρ) (14.80) { ( ρ exp t 1 )} = 2 t l= cos(ρ cos θ) = J 0 (ρ) + 2J 2 (ρ) cos 2θ + 2J 4 (ρ) cos 4θ + t ρ J l (ρ) (14.81) sin(ρ sin θ) = 2J 1 (ρ) sin θ + 2J 3 (ρ) sin 3θ + (14.82) θ ( J) l (ρ) explicit R l (ρ) = ρ 1/2 f(ρ) d2 f dρ + 1 l2 1/4 f = 0 2 ρ 2 Bessel parity ρ l ρ Bessel a z θ l = 0 2 ϕ = 1 r 2 ϕ = 0 Z ( r ϕ ) + 2 ϕ r r z = 0 (14.83) 2 ϕ(r, z) = R(r)Z(z) (14.84) ( 1 d r dr ) = 1 d 2 Z r R dr dr Z dz = 2 α2 = r, z (14.85) Z = Z 0 e αz + Z 1 e αz (14.86) z = 0 z 0 z Z 0 Z 0 α 0 α 2 R d 2 R dr r ρ = αr Bessel dr dr + α2 R = 0 (14.87) d 2 R dρ + 1 dr 2 ρ dρ + R = 0 (14.88) 132

133 Bessel J 0 (αr) Neuman N 0 (αr) r = 0 N 0 (αr) α ϕ α (r, z) = J 0 (αr)e αz (14.89) Laplace eq. α (α 0) α z = 0, r a ϕ(r, z) ϕ 0 lim z 0 0 e αz sin aα J 0 (αr) α dα = z = 0, r < a V { π/2, ) r < a, r > a sin 1 ( a r (14.90) ϕ(r, z) = 2 V π 0 sin aα dα ϕ α (r, z) α (14.91) z = 0 E z = σ/ɛ 0 z 0 r < a E z (r, 0) = lim z 0 2 V π 0 dα J 0 (α r) sin aα = 0 dα σ(r) = 2 V ɛ 0 π sin aα α z > 0 q σ q = a 0 1 a2 r 2 (14.92) z ϕ α(r, z) = 2 V π a 2 r 2 (14.93) 1 a2 r 2 (14.94) 2πr dr σ(r) = 4 ɛ 0 a V (14.95) Q C = Q V = 8 ɛ 0 a (14.96) Bessel 133

134 Chladoni e imψ, P lm (cos θ), Y lm (θ, ψ), {r l, r (l+1) }, J l (kr), N l (kr) (l, m) ψ, θ, r (a b) f n f n W (x)dx f n (x) n n m, l f n (x) ( ) W (x) x x : ψ, θ, r W (x) : 1, sin θ, r 2 n n n, n 0 (14.65) (14.66) 1. Laplace 2. Green 3. 1 SOR yaoki Poisson 2 φ = ρ (14.97) 134

135 2 φ 1 = 0 2 φ 2 = ρ φ = φ 1 + φ 2 2 φ 1 = 0 2 φ 2 = ρ φ 1 φ 2 Green Green Poisson Laplace eq. Green θ(x) δ { δ(x) = dθ dx, θ(x) = 0 (x < 0) 1 (x > 1) (14.98) θ(x) T(x) x x dθ dx ɛ f(x)dx = lim ɛ 0 = lim ɛ 0 ɛ dθ dx dθ f(x)dx x 0 0 dx } { [θ(x)f(x)] ɛ ɛ ɛdf(0) dx = f(0) (14.99) x = 0 T (x) dt (x) dx m l (x < 0) = m r (x > 0) (14.100) m l, m r dt m l = lim x 0 dx, θ(x) = m dt r = lim x +0 dx, (14.101) 1 m r m l T (x) m l (14.102) θ(x) δ m l x 0 δ(x x 0 ) x 0 135

136 Laplace d { p(x) d } { dx dx p(x) d } dx 1. ( d p(x) dt ) + q(x)t (x) = ρ(x) (14.103) dx dx ρ(x) = 0 (T (x) = 0) T 1 (x), T 2 (x) 2. T 1 (x), T 2 (x) T 1 (x) Laplace eq. T 1 (x 0 ) dt 2(x 0 ) dx dt 1(x 0 ) dx T 2 (x 0 ) = 1 p(x 0 ) (14.104) 3. G(x, x 0 ) T 1 (x < )T 2 (x > ) ( x < = min(x, x 0 ) x > = max(x 1, x 0 ) ) (14.105) Green x < x 0 T 2 (x > ) = T 2 (x 0 ) = G T 1 (x) x > x 0 T 1 (x < ) = T 1 (x 0 ) = G(x, x 0 ) T 2 (x) x x = x 0 (14.104) T (x) = G(x 1, x 0 ) ρ(x 0 ) dx 0 + C 1 T 1 (x) + C 2 T 2 (x) (14.106) C 1, C 2 Green T 1 (x), T 2 (x) G(x, x 0 ) αt 1 (x) + βt 2 (x) Green d dx {p(x) (T 1(d) T 2(x) T 1(x) T 2 (x))} = 0 (14.104) [ { d f(x) p(x) dg(x, x } ] 0) + q(x)g(x, x 0 ) dx = f(x 0 ) dx dx 136

137 L n (r, r 0 ) 1 2n + 1 r /r0 n+1 (r < r 0 ) r0 /r n+1 (r > r 0 ) (14.107) r Green r ω r0 R 1 R = 1 r 2 0 2rr 0 cos w + r = 2 r l l=0 r0 l+1 l=0 P l (cos w) (r < r 0 ) r l 0 r l+1 P l(cos w) (r > r 0 ) (14.108) r Coulomb potential Green r ω r0 R φ(r, w) = q 4 π ε 0 a r 0 q potential r = a φ 0 { φ(r, w) = q } 1 4 π ε 0 R r0r l l a P l(cos w) 2l+1 l=0 (14.109) { } 1 R a 2l+1 r0 l+1 r P l(cos w) l+1 l=0 (14.110) r = a potential 0 1 potential Laplace eq. R r 0, r (r 0, θ 0, ψ 0 ), (r, θ, ψ) w θ 0, ψ 0, θ, ψ P l (cos w) = l (l m)! ɛ m (l + m)! P l m (cos θ)pl m (cos θ 0 ) cos{m(ψ ψ 0 )} (14.111) m=0 Legendre ɛ m = (2 δ m,0 ) : Neuman (14.112) 137

138 15 z V>0 V=0 a σ ɛ r = 1, µ r = 1 V ( ɛ r µ r ) ρ t + i = 0 ρ t = 0 i = 0 (15.1) i (!) 2 s 1 s 2 l (15.1) Gauss 0 I2 I1 s2 s1 i ds + s 1 i ds = 0 s 2 I 1 + I 2 = 0 (15.2) I i = s i i ds s i (15.1) ( Kirchhoff ) 100(120) Hz i E i = σ E z = z 1 z 2 z = z 1 z 2 V (= E(z 2 z 1 )) i ds dz = I (z 2 z 1 ) = σ E a 2 π (z 2 z 1 ) (15.3) V = IR R = z 2 z 1 σπ a 2 (15.4) 138

139 Ohm s low L H = i + D D t t 0 H z r ( ) 0 H θ (r) r Stokes vskip 5mm E H I r a H ds = H θ (r) 2πr = i ds = r 2 (15.5) a I r < a 2 I r a H θ (r) = 2π r (15.6) I 2πa r r < a 2 H z I H E H E H z E H z E H (15.6) E r l z 2πrl E l I r a (E H) ds = r 2 (15.7) a E l I r < a 2 E l V I r > a ( E I) r 2 139

140 ? H θ z H θ (r) (15.6) cos ωt H θ H θ H θ t r ( ) E z ( ) H θ E (1) z (r, t) I(t) = I 0 cos ωt B = µh E = B t θ r E(1) z (r, t) = µ r 2πa 2 t (I 0 cos ωt) = µ ω r 2πa I 2 0 sin ωt (15.8) r E z (1) (r, t) E z (1) (0, t) = µ ωr2 4 π a I 2 0 sin ωt (15.9) r = 0 H θ = 0 E z (1) (0, t) = 0 E z (r, t) E 0 cos ωt E z (r, t) =E 0 cos ωt µωr2 4 π a I µωσ r2 2 0 sin ωt = E 0 {cos ωt 4 ) =Ē(r) cos(ωt + α), (c.f I 0 π a 2 = σe 0 ( ) } µωσ r 2 2 1/2 Ē(r) = E 0 {1 +, tan α = µ ω σ r2 4 4 } sin ωt (15.10) (15.11) Ē r i z = σ E z cos ωt cos(ωt + α) +α U M U M = µ 2 a 0 H 2 θ 2πr dr = µi2 16π = L i 2 I2 (15.12) L i = µ 8π (15.13) 140

141 ( ) r > a U M r > a (15.6) (15.5) (15.6) Maxwell E = B t, H = i + D t Ohm (15.14) i = σe (15.15) D = εe, B = µ H (15.16) H B D i E ( ( E) + µ σ E ) t + E ε 2 = 0 (15.17) t 2 σ = 0 σ 0 ( E) = ( E) 2 E (15.18) Coulomb E = ρ/ɛ (4.9) ρ = 0 ( 2 + 1c 2 2 t 2 + µσ t ) E = 0 (15.19) c 2 = 1/µɛ E e iωt E(r, t) = E ω (r)e iωt (15.20) Fourier z ( E ω (r) ) { 1 ( d r d ) } ω2 r dr dr c iωµσ E 2 ω (r) = 0 (15.21) ( d 2 dr r ) d dr + κ2 E ω (r) = 0 κ 2 = ω2 c 2 + iωµσ (15.22) 141

142 = +iωµh 0 (r)e iωt (15.26) r = 0 0 Bessel J 0 (κr) E ω (r, t) = E ω 0 J 0 (κr)e iωt (15.23) (15.22) κ 2, 1 MHz ω = 2π 10 6 sec 1 c m/sec µ 4π 10 H/m 7 1 ( ω ) ( ) σ Ω m, ωµσ c 100 m ( ω ) 2 c J 0 (z) 2π J 0 (z) = 1 cos(z sin θ)dθ (15.24) 2π 0 ( ) ωµσ z = (1 + i) r 2 cos z = 1 2 (eiz + e iz ) z ωµσ r > 1 ( 2 ) ωµσ E(r) exp r 2 Ohm (15.11) E z (r) δ (skin depth) 2 δ = (15.25) ωµσ TV δ E z ( E) θ ( E) θ = E z r = E ω o e iωt κ dj 0(κr) d(κr) = E ω0 e iωt κj 1 (κr) 142

143 H θ (r) = κ iωµ E ω 0 J 1 (κr) (15.27) J 1 (z) = dj 0(z) 1 Bessel ( ) (15.27) i dz E z H θ κ π/2 I σe z (r, t) I 0 e iωt = a 0 σe ω0 J 0 (κr)e iωt 2π r dr κa = σe ω0 e iωt 2π J 0 (κr)(κr)d(κr) 1 0 κ 2 = 2πaσ ( κ E ω 0 e iωt J 1 (κa), c.f zj 0 (κz)dz = z ) κ J 1(κz) σ = 2π iωµ ae ω 0 e iωt J 1 (κa) (15.28) U E = 1 2 E D U M = 1 2 B H U E U M = ω µσc 2 ( ) 2 J0 (κr) J 1 (κr) 1 (15.29) κ 2 = iωµσ (J 0 /J 1 ) 2 Z E z Z = E I = 1 2πa iωµ σ J 0 (κa) J 1 (κa) (15.30) J 0 1 J 1 κ 2 a Z 1 πa 2 σ 1/σ (15.31) ( (15.17) ω 2 /c 2 ) Biot-Savart 143

144 H θ x z s (0,Y,Z) R r y I ds l a(= l/2π) z = 0 (7.5) yz H(R) = I 2π ds r (15.32) 4π 0 r 3 r = ( a cos θ, Y a sin θ, Z), ds = a dθ( sin θ, cos θ, 0), H(R) = I 4π 2π 0 a dθ (b 2 c 2 sin θ) 3/2 {Z cos θ ê x + Z sin θ ê y + (a Y sin θ) ê z } (15.33) b 2 = a 2 + Y 2 + Z 2 c 2 = 2aY ê x ê y ê z (15.33) sin θ cos θ H x = 0 (15.33) θ = 2ϕ π dθ = 2dϕ (15.34) 2 sin θ = 2 sin 2 ϕ 1 (15.35) π 2 θ 3π 2 3π 2 θ 5π 2 [0, π/2] ϕ H(R) = ai π π/2 0 dϕ {(b 2 + c 2 ) 2c 2 sin 2 ϕ} 3/2 [Z(2 sin2 ϕ 1) ê y +(a+y 2Y sin 2 ϕ) ê z ] (15.36) k 2 = 2c2 b 2 + c 2 (15.37) H(R) = ai π/2 dϕ π(b 2 + c 2 ) 3/2 0 (1 k 2 sin 2 ϕ) [Z(2 3/2 sin2 ϕ 1)ê y +(a+y 2Y sin 2 ϕ)ê z ] (15.38) 144

145 K(k) π/2 dϕ 0 1 k2 sin 2 ϕ E(k) π/2 0 1 k2 sin 2 ϕdϕ (15.39) k k (k) π/2 dϕ 0 (1 k 2 sin 2 ϕ) = 1 3/2 1 k E(k) 2 π/2 sin 2 ϕ 0 (1 k 2 sin 2 ϕ) dϕ = 1 { } (15.40) 1 E(k) K(k) 3/2 k 2 1 k2 1 E(k) t = tan ϕ u = u = tan θ (15.40) 1 k2 t (15.40) H y (R) = H y (R) = sin 2 ϕ = 1 k 2 (1 k2 sin 2 ϕ) + 1 k 2 aiz πc 2 b 2 + c 2 I 2πY (a + Y ) 2 + Z 2 { } b 2 E(k) K(k) b 2 c2 { } a 2 + Y 2 + Z 2 E(k) K(k) (a Y ) 2 + Z2 (15.41) (15.42) Y = 0 k = 0 0/0 { } O(Y 2 ) 0 H y (Y = 0) = 0 z H z (R) = I 2π (a Y ) 2 + Z 2 Y = 0 E(0) = K(0) = π/2 H z ( ) = { } a 2 Y 2 Z 2 E(k) + K(k) (a Y ) 2 + Z2 (15.43) a 2 I 2(a 2 + Z 2 ) 3/2 (15.44) Z = 0 Y = a 0 (15.6) 145

146 1 2 k 2 1 k 2 sin 2 ϕ Tayler K(k) = π 2 { E(k) = π 2 { ( ) 2 1 k ( ) 2 1 k ( ) } k ( ) } k (15.45) (15.46) Gauss Gauss k 2 1 step 0 a 0 = 1, b 0 = k = 1 k 2 step i a i+1 = (a i + b i )/2, b i+1 = a i b i, c i+1 = (a i b i )/2 c i+1 0 ( ) i i + 1 step i K(k) = π 2a N, E(k) = K(k) N step { 1 k } N 2 r Cr 2 r=1 (15.47) Landen k 2 Landen k k (= 1 k 2 ) k 1 k 1 = 1 k 1 + k (15.48) 146

147 K(k) = 1 + k K(k 1 ) (15.49) 2 k 1 k Landen Legendre k 1 k (= 1 k 2 ) ( k 1) k E(k ) K(k ) K(k) Landen E(k)K(k ) + E(k )K(k) K(k)K(k ) = π/2 (15.50) E(k) π Helmholtz coil 2 (z = l 2 ) l Z a z = 0 H z (z = 0) z H z (z) = H z (0)+H z (0)z 2 /2+ H z (4) (0)z 4 /24 + H z(0) = 0 H z (0) H z (0) = 0 l/a? (15.42) (15.43) a 2 I m/πµ 0 a 0 R 2 = Y 2 + Z 2 a 2 + Y 2 + Z 2 (a Y ) 2 + Z E(k) K(k) π 2 2 3a2 Y 2 R 4 a 2 + Y 2 + Z 2 (a Y ) 2 + Z E(k) + K(k) π 2 2 ( Y 2 + 2Z 2 ) a 2 R 4 H y (R) 3mY Z 4πR 5 µ 0, H z (R) m( Y 2 + 2Z 2 ) 4πR 5 µ 0 (15.51) (5.3) r R m m I S m = I S µ 0 (15.52) S = πa 2 147

148 A E B G F D C ABCD Biot-Savart (15.52) BEFGB ABCDA AEFGCDA (15.52) Borda Helmholtz coil 0.5 ( Tesla)? Helmholtz coil yoke g 1.5 Tesla H = i H ds = H dl Hdl + Hdl (15.53) 148

149 B = 0 ( B ) gap B = B H = 1 H H H g µ i ds = NI N I H = NI g + L µ (15.54) L Ohm H (g + L/µ ) µ NI (AT ) ( ) µ gap 10 cm 1 m 2 1 Tesla???? µ ( , 000) ( ) / H?? Lorentz? 149

150 Maxwell E = B t Faraday B S B E ds = ds = dφ (15.55) t dt Φ B ds (15.56) = E dl (15.57) dφ dt < 0 ( ) Kerst Server 2 (1) (2) (1) ( ) R R dφ dt = E dl = E θ 2πR (15.58) E θ W dw = q E θ v = qv dt 2 πr Lorentz dφ dt (15.59) mv 2 = qvb(r) p = qrb(r) (15.60) R W p W = m 2 c 4 + p 2 c 2 t dw dt = pc2 W dp dt 150 (15.61)

151 dφ dt db dt p = mv 1 β 2 W = dφ dt = pc2 W v d dt {2πR2 B(R)} = d dt {2πR2 B(R)} (15.62) mc2 1 β 2 p = qrb (15.62) B(R) Φ = πr 2 B(R) 2 ( dr = 0 ) dt 50 Hz 1/4 B 0 1T R = 0.5 m E θ E = 10 8 V/m B t E θ (15.62) (15.60) V ( ) E H V variety 151

152 16 H M ( ) M = κ m H (16.1) µ = µ 0 (1 + κ m ) = µ 0 µ r κ m (magnetic susceptibility) µ r µ r 1 ( ) κ m ( ) κ m H = 0 M = 0 M H (!) 600 C 1000 C 0.1 ev 600 C 1000 C ( ) B = 0 2 (1) ( ) ( ) (2) ( ) µ d H W = µ H B W = µ B µ H µ B µ W = µ B 152

153 (15.52) r m,e e m ( r v e ) I = v ( ) 2πr S = πr 2 µ µ = ev 2 π r π r2 = e r v 2 L = m r v ( e ) µ = L (16.2) 2 m µ L v ρ e ρ m ( e/m ) ( e/m ) L µ (16.2 ) Lande h 1/2π ( = ) L e m (Bohr magneton) µ B = e (16.3) 2 m m e µ N (nuclear magneton) µ B µ N 2000 ( ) ( ) ( ) J (16.2) µ = g µ B J (16.4) g Lande g µ B µ N 1.0 µ B 2.8 µ N 1.9 µ N ( ) 1/2 ( ) g 2, 5.6, 3.8 g 1 2 Thomas precession Leorentz Lorentz ( ) 0.1 p89 ( QED ) 153

154 ! µ N e ( ) ( ) (16.3) ( ) (16.2) L v v L v Lorentz L θ ( conjugate canonical) Maxwell v p Bohr magneton 1 µ B 1A? Curie 1000 K Larmor precession J (16.4) µ µ H 0 dj dt = = µ H 0 (16.5) dj dt = γ J H 0 (16.6) γ = g µ B ( gyromagnetic ratio ) J J 2 J dj dt = 1 d ( ) J 2 = γj (J H 0 ) = γ H 0 (J J) = 0 (16.7) 2 dt J (16.6) J 2 J (16.6) 154

155 ω daω A(t+dt) ωdt A(t) t t + δt A(t) A(t + δt) (δa) ω δa (δa) ω = δa ωδt A ( ) da = da dt dt ω A (16.8) ω A J (16.6) ( ) dj = dj dt ω dt ω J = J (γ H 0 + ω) (16.9) ω L = γ H 0 (16.10) ( ) dj = 0 ω L dt ω L H 0 (16.10) Larmor frequency ω L H 0 J H 0 H 0 J Stern Gerlach B µ z y x W W = µ B B W F F = W = (µ B) (16.11) µ B = 0 ( ) F = (µ ) B (16.12) (A B) = (B )A + (A )B + A ( B) + B ( A) 155

156 r qm -qm d H(r+d/2) H(r-d/2) 4 µ ±q m d F = q m B(r + d/2) q m B(r d/2) { ( Bx d x = q m B(r) + ê x x 2 + B x d y y 2 + B ) x d ( ) ( ) } z + ê y + ê z + O(d 2 ) z 2 q m { } ( ) = q m d x x + d y y + d z (B x e x + B y e y + B z e z ) z = (µ ) B (16.13) B z y z (µ > 0 ) z Stern Gerlach z z < 0 z > 0 z < 0 ( B z ) µ 2 z ( 1 ) J B z z µ (14.29) magnetic quadrupole field B z 6 ( z φ m r 3 cos 3φ) B z z µ z µ µ Stern-Gerlach (16.11) 156

157 Coulomb E = Z ze2 r v 4 π ε 0 r3 E Lorentz B (13.74) ( B = γ B v ) c E 2 (16.14) B = 0, E E B = + γ v c Ze2 r 2 4 π ε 0 r = γ 3 mc Ze π ε 0 r L 2 r s µ = g s µ B s (16.15) E µ = µ B = γ m Ze 2 g s µ B 4π r 3 L s (16.16) s L E µ 2 Na D 5893 ± 3A 6A ( dipole-dipole (3.56) ) 21cm 21 cm Larmor Hamiltonian (13.102) H = m 2 c 4 + c 2 (p qa) 2 + eφ mc m (p qa)2 + eφ (16.17) mc 2 (6.4) A = 1 2 B r (16.18) H = p2 2m + q 2m B l + q2 8m B2 (r ) 2 + eφ (16.19) 1 4 B = 0 2 (16.2) Bohr 3 l l v 2 B l 3 157

158 ( Hamiltonian B Taylor 1 q ) 2m B l l Taylor ( ) q 2 2 8m B2 2 r 1 B 3 4 (16.19) 3 Taylor 2 B r 2 r 2 B z r 2 x + r 2 y r r2 r 2 M µ B Boltzman B µ W µ B B z µ z ( ) µ z π sin θ dθ ( 2π dφ µ cos θ exp + µb ) 0 0 kt cos θ µ z = π sin θ dθ ( 2π dφ exp + µb ) 0 0 kt cos θ ( ) { = µ coth ( µb kt ) = L kt µb } ( ) µb : Langevin kt L(x) = cosh x sinh x 1 x + ) ) (1 + x + x2 2 + x3 (1 x x + x2 6 2 x3 3 6 (16.20) (16.21) 158

159 µ z µ2 B (16.22) 3kT µ N magnetic susceptibility T? µ = µ B, T = 300K B = 1 Tesla κ m = µ2 3kT N (16.23) ( ) µb kt (16.20) J ( ) J z J z = J, J + 1, J + 2,, J 1, J (2J + 1) cos θ (J z /J), µ = gµ B J J z ( ) J J gµ gµb J z B d J z= J BJ z exp gµ B exp (αj z ) kt dα J µ z q = ( ) = z= J J J exp gµb J z B J J exp (αj z= J z) z= J kt α gµ BB kt {( = gµ B J + 1 ) { α } coth 2 2 (2J + 1) 1 ( α ) } 2 coth 2 x Jα = gmu BJB kt µ z q = gµ B J B J (x) (16.24) B J (x) = 2J + 1 2J ( ) 2J + 1 coth 2J x 1 2J coth x 2J (16.25) (16.25) Brillouin (16.24) (α B/kT ) ( ) 159

160 (!) (1) (2) (magnetic domain) 1 ( ) M c a b H ( a) ( ) ( b) ( c) 160

161 (pinning) H = 0 M 0 ( ) M = 0 H H (hysteresis curve) H = 0 M Mdxdy B M B z y dy z dz x dx -Mdxdy dv = dx dy dz M dx dy, M dx dy (M dxdy) dz = M dv M δq m = δm dxdy δw δw = (δmdxdy) B dz = dv δm B (16.26) H 0 H 1 M M 0 M 1 W W = M1 M 0 B dm (16.27) M 0 M 1 M B ( ) 161

162 17 E, B m, q m d2 r = q (v B + E) (17.1) dt2 2 qe 2 (17.1) (17.1) m d2 r dr + qb dt2 dt = 0 mdv dt v m d dt + qb v = 0 (17.2) ( v 2 ) = 0 (17.3) ( (17.1) E = φ potential φ ) v 2 = v 2 (17.3) (17.2) B B (B v) = 0 d (B v) = 0 (17.4) dt v B ( v ) (17.3) (17.4) v B v v 2 = v 2 v 2 = (17.5) v v = Larmor ω = q m B (17.6) (17.2) ( ) dv = 0 (17.7) dt ω 162

163 B B (17.7) ω (17.6) ω cyclotron frequency ( ) energy Lawrence ( effective mass ) B 2 ê 1 ê 2 ( ê 1 ê 2 = 0) (17.2) r(t) = v t + v (ê 1 cos ωt + ê 2 sin ωt) /ω (17.8) +q B y x t = 0 r (t = 0) = v ê 1 ω q > 0 ê 1 ê 2 x, y B z (17.1) 3 1 v v v v B = 0 (17.1) 1 m d2 r dt 2 = qe (17.9) r (t) = r (0) + v (0) t + qe 2m t2 (17.10) r (0) v (0) v dv dt = 0 v B + E = 0 (17.11) B ˆB v = ˆB ( v ˆB ) v = ˆB E B (17.1) B E = (17.12) ( ˆB ) E E B (17.12) Wien velocity filter (17.1) (17.8) (17.10), (17.12) r(t) = v t + v ω (ê 1 cos ωt + ê 2 sin ω t) + qe 2 m t2 1 B ˆB E t (17.13) 163

164 t E B r vd t r v d (= ) v = v + v d m dv dt = q {(v + v d ) B + E} (17.14) v d B = E (17.15) v d v B v d (17.13) E = 0 B = 0 (17.13) magnetic spectrograph (17.13) v = 0, E = 0 R (17.13) (17.6) R = v = mv ω qb = p qb (17.16) p B acf ±δ 180 f f = 2R (1 cos δ) = Rδ 2 δ f b c d R ds dφ 2δ R dφ = ds R (17.17) a O f R ±δ δ ( ) x x = R(2 cos δ 1), y = 0 164

165 Rutherford B0 B1 B B B 0 B 1 qv B 0 B 1 q v B 1 z B(z) = B(0) + B z z + (17.18) B = 0 B 0 r = B 1 z (17.19) B 1 = B 0 r z + O(z2 ) (17.20) z B 1 z B z = 0 B 1 O(z 2 ) 0 ( ) Newton ( ) m d2 z dt = qv Bz z (17.21) 2 r z AVF yaoki/ z=0 165

166 18 E B r z θ E potential φ(r, z) E = φ(r, z) (18.1) B vector potential A A θ (r, z) A = B B r = A θ z, B z = 1 r r (r A θ) (18.2) vector potential z r Φ(r, z) z B Φ(r, z) = B r r r 0 2πρB z (ρ, z) dρ (18.3) B z (r, z) = 1 Φ(r, z) 2 π r r B = 0 B z B r (18.4) B = 1 r 1 r r (r B r ) r ( r B r + z (18.4) + B z z = 0 (18.5) ) Φ = 0 (18.6) 2 π r r 0 Φ 2π r 2 B z B r 0 B r = 1 Φ 2 π r z (18.7) B Φ (18.2) B A θ B A θ (r, z) = 1 Φ(r, z) (18.8) 2 π r 166

167 (Φ/2 π) (13.102) Hamiltonian q A (18.8) r q A θ z vector potential E ê z r L = r mv r = rê r + zê z m dv dt = q v B (18.9) = (ê z r) m dv dt = mê z (r dv dt ) = mê z d dt (r v) = dl z dt (18.10) = (ê z r) q (v B) = q r ê θ {B z r ωê r + (B r v z v r B z )ê θ r ω B z ê z } (18.11) (18.2) ( ) dz = q r B r dt B dr z dt { (r Aθ ) = q z dz dt + (r A θ) r ê r, ê θ, ê z r, θ, z z } dr = q d dt dt (r A θ) (18.12) ω = dê r dt (18.13) d dt (L z + q r A θ ) = 0 (18.14) L z + q r A θ = L 0 z = (18.15) E r z (ê z r)(= r ê θ ) E ê θ (18.1) E θ = 0 (18.9) ê z z q (v B) ê z = q v (B ê z ) = q B r v (ê r ê z ) (18.16) B = B r ê r + B z ê z ê z ê r = r/r = q B r r v (r ê z) = q B r r B r (18.2) = q L z m r A θ z = ( ê z) (r v) = q B r m r L z (18.17) L z m r 2 z (q r A θ) = L z L z m r 2 z = ( ) L 2 z z 2mr 2 (18.18) 167

168 m r ω 2 m r ω 2 = m d2 z dt = ( ) L 2 z 2 z 2 m r 2 L2 z m r 3 = r ( L 2 z ) 2 m r 2 potential (18.19) (18.20) r potential d 2 r dt = d2 2 dt (rê 2 r + zê z ) = d2 r dt 2 êr + 2 dr dt ê r dê r dt + r d2 ê r dt 2 + d2 z dt 2 êz (18.21) m d2 r dt 2 ê r = m d2 r dt m dr dt êr dê r dt + m rê r d2 ê r dt 2 (18.22) ê 2 r = 1 t 2ê r dê r dt = 0 ( ) 2 dêr + ê r d2 ê r = 0 dt dt 2 (18.9) m d2 r dt 2 ê r = m d2 r dt 2 m r ( ) 2 der = m d2 r dt dt m r 2 ω2 = m d2 r dt 2 L2 z m r 3 (18.23) q (v B) ê r = q r (r v) B = q mr L B = q mr L zb z L r B θ (18.2) B z (18.23) (18.24) = q m r L z 1 r r (r A θ) = L ( z L ) z = ( ) L 2 z L2 z (18.24) m r 2 r r 2 m r 2 m r 3 m d2 r dt = ( ) L 2 z 2 r 2mr 2 (18.1) (18.19) (18.25) q V m d2 r V = q dt2 r ( ) L 2 z = r 2 m r 2 r m d2 z V = q dt2 z ( ) L 2 z = z 2mr 2 z ( q V + ( q V + ) L2 z 2 m r 2 L2 z 2 m r 2 ) (18.25) (18.26) (18.27) 168

169 r z potential Lagrangian magnetic monopole q m B q m qm z R B = q m R (18.28) 4 π R 3 (r, θ, z) R m, q e L z L dl dt = d (R mv) = R mdv dt = q e q m 4 π { v R R ( R dr R 3 dt dt = R q e )} ( v q m 4 π = q e q m 4 π d dt ( R R ) R R 3 ) (18.29) L0 L q e q m 4 π R R = L(0) = (18.30) R L R = 0 φ q e q m 4 π = L0 cos φ (18.31) φ R L 0 (18.31) R L 0 L 0 φ (18.30) L 2 ( L 2 qe q ) 2 ( m + = (L 0 ) 2, L 2 = (L 0 ) 2 qe q ) 2 m = (18.32) 4 π 4 π R L 0 v v m d2 R dt 2 = q e v B (18.33) 169

170 v dv dt = 1 d 2 dt (v2 ) = 0, v 2 = (18.34) R (18.33) R R d2 R dt 2 = d dt ( R dr ) dt ( ) 2 dr = d ( R dr ) v 2 = 0 (18.35) dt dt dt R R dr dt = 1 d 2 dt R2 = v 2 t + A (18.36) R 0 t R 2 = v 2 t 2 + A t + R 2 0 (18.37) A R0 2 A = 0 (R, t) R 0 B (18.30) L 0 L 0 z r L L (0) = L (0) L z = L (0) m r 2 dθ dt (18.38) φ r R0 m r 2 dθ dt = L(0) + q e q m cos φ = = C (18.39) 4 π r 2 = R 2 sin 2 φ = (R v 2 t 2 ) sin 2 φ (18.40) (18.25) (18.26) (18.27) θ = dθ dt = C m sin 2 φ 1 R v 2 t 2 (18.41) C m sin 2 φ 1 ( ) vt R 0 v tan 1 + θ 0 (18.42) R 0 t = 0 θ 0 0 C = m R 0 sinφ v 170

171 θ = 1 sin φ tan 1 ( ) v t (18.37) (18.43) t R θ R 0 (18.43) R θ sin φ R0 R 0 = R cos(θ sin φ) (18.44) ( θ Θ = θ sin φ R R ) (18.45) r (r, θ, z) z (r = 0) V (r, θ, z) r θ potential V (r, z) V (r, z) = V (0, z) + 1 V (0, z) r V (0, z) r 2 + (18.46) 1! r 2! r 2 V (0, z) V r = 0 r V (0, z) = E r (r = 0, z) 0 (18.47) r z 2 V = 0 2 V = 1 r ( r V ) V r r r 2 θ 2 V θ (18.46) + 2 V z 2 = 0 (18.48) } {2 2 r + 2 V (r = 0, z) = 0 (18.49) 2 z 2 r z 171

172 V (r, z) = V (0, z) 1 2 V (0, z) r 2 + O(r 4 ) + (18.50) 4 z 2 z (r = 0, z) potential V (0, z) ( ) r potential Laplace (18.50) r leading term E r (r, z) = V (0, z) r + O(r 3 ) (18.51) 4 z 2 r q 2 V < 0 r z 2 r Laplace potential (18.50) potential (18.3) (18.8) vector potential (18.8) Φ = r 0 2 π r B z (r, z) dr = π r 2 B z (0, z) + O(r 4 ) (18.52) A θ (r, z) = r 2 B z(0, z) + O(r 3 ) (18.53) z z r ( ) ( ) (18.15) L 0 z (18.53) q r A θ = q 2 r2 B z (0, z) + O(r 4 ), L z = m r 2 dθ dt L 0 z O(r2 ) z (18.26, 18.27) q V + L 2 z 2 m r 2 O(r0 ) + (r 2 ) O(r 2 ) dz dt m d2 z dv (0, z) = q dt2 dz t m 2 (18.54) ( ) 2 dz + q V (0, z) = q U 0 = (18.55) dt U 0 dz = 0 potential ( dt dz dt 0 ) (18.55) t z 172

173 d 2 q dt = m (U 0 V (0, z)) d (18.56) dz r 0 z ( ) (18.26, 18.27) m d2 r dt = q d 2 V (0, z) r q2 B z (0, z) 2 r + (L0 z) 2 (18.57) 2 2 dz 2 4 m m r 3 V (r, z) (18.50) L z (18.15) (18.53) L 0 z = 0 z r = 0 L 0 z = 0 L 0 z = 0 (18.57) r r z... B z (0, z) = 0 q(u0 V ) d ( q(u0 V ) dr ) = + q d 2 V (0, z) r (18.58) dz dz 4 dz 2 z q d2 V (0, z) < 0 dz 2 r(> 0) z gap gap gap r gap lens 0V +V 0V z 0 V einzel lens (Unipotential lens) gap Dee 173

174 ( magnetic lens ) (18.57) r z B r q v z B r = F θ θ v θ 0 qv θ B z = F r r v θ B z Bz 2 B B z ( ) z cyclotron freq. r Lens type β-ray spectrometer z dispersion β-ray spectrometer L 0 z 0 (18.15) (18.54) (18.57) A θ (18.53) (18.15) L z = L 0 z q 2 r2 B z (18.59) L z =m r 2 dθ dt dθ dt = L0 z m r + ω 2 L ω L = q B z, Larmor freq. (18.60) 2 m Larmor freq. ( ) (18.60) z (18.54) L 0 z L0 z r (18.57) L 0 z L0 z L0 z = 0 174

175 L 0 z (18.60) ϕ ϕ z θ = ϕ + χ L 0 z = m r 2 dϕ dt x = r cos ϕ, ȳ = r sin ϕ (18.61) x ȳ ( ) d 2 x dt = d 2 r 2 dt L02 z cos ϕ, 2 m 2 r 3 d 2 ȳ dt = r 2 (d2 dt L02 z ) sin ϕ (18.62) 2 m 2 r3 (18.57) m d2 x { q dt = 2 2 m d2 ȳ dt 2 = } d 2 V (0, z) ω dz 2 L (z) 2 x (18.63) } { q d 2 V (0, z) ω 2 dz 2 L (z) 2 { } < 0 ȳ Quadrupole mass filter x y z z x 2 y 2 = ±a 2 z = z 0 z = z 1 z 0 z 1 z potential z (x, y) z potential z φ(x, y, t) = x2 y 2 V a 2 0 (1 + α cos ωt) (18.64) 175

176 z m d2 x dt 2 = 2x a 2 V 0 q(1 + α cos ωt) (18.65) m d2 y dt = +2y 2 a V 2 0 q(1 + α cos ωt) (18.66) m d2 z dt = 0 2 (18.67) z t z = vt (v ) d 2 x + λ(1 + α cos 2s)x = 0 (18.68) ds2 λ q m z V 0 ω 2 a, s ω 2 2v z (18.69) y λ (18.68) (18.68) Mathieu (λ, αλ) b1 a0 λ used αλ λ 0 a 0 = b 1 (λ, αλ) (λ ) (18.69) λ v (q/m) (q/m) (qv 0 m) (q/m) Quadrupole mass filter Mathieu λ = 1 (18.68) λα 0 Mathieu KEK Mathieu AVF radio frequency quadrupole linear accelerator Hill d 2 x dt 2 + (λ 2 N q N cos N t)x = 0 (18.70) 176

177 N > 2 potential Schrödinger Mathieu λ, q d 2 y + (λ 2 q cos 2 x)y = 0 (18.71) dx2 x y x = 0 c(x), s(x) c(0) = 1, c (0) = 0 s(0) = 0, s (0) = 1 (18.72) c(x) s(x) cosine, sine Wronskian x c(x) s(x) W (x) = c (x) s (18.73) (x) (18.71) dw dx = 0 x = 0 W (x) = 1 λ 2 q cos 2 x x x + π π y(x) π u(x) y(x) = e iµx u(x) (18.74) µ cos µ π = c(π) (18.75) (18.74) Bloch ( Floquet ) (18.71) y(x) (18.72) y(x) = a c(x) + b s(x) (18.76) c(x + π) s(x + π) (18.71) (18.76) c(x + π) = (c c) c(x) + (c s) s(x) s(x + π) = (s c) c(x) + (s s) s(x) (c c) (s s) ( ) y(x + π) y(x + π) = a c(x + π) + b s(x + π) = {a(c c) + b(s c)} c(x) + {a(c s) + b(s s)} s(x) (18.77) = e iµπ {a c(x) + b s(x)} = e iµπ y(x) (18.78) 177

178 e iµπ (18.78) (18.74) y(x + π) = e iµ(x+π) u(x + π) = e iµπ e iµx u(x) u(x + π) = u(x) (18.79) u(x) π e iµπ (18.78) a, b ( ) ( ) (c c) e iµπ (s c) a = 0 (18.80) (c s) (s s) e iµπ b (a, b) T (c c) e iµπ (s c) (c s) (s s) e iµπ = 0 (18.81) e iµπ ( (c c) (c s) ) (s c) (s s) (18.82) (18.77) (18.77) x = 0 (18.77) x x = 0 Wronskian x = π c(π) = (c c), s(π) = (s c) (18.83) c (π) = (c s), s (π) = (s s) (18.84) (c c)(s s) (s c)(c s) = 1 (18.85) (18.77) x = π c(x) s(x) (18.83, 18.84) c( π) = c(π), s( π) = s(π), etc c ( π) = c (π), s ( π) = +s (π), etc s(0) = 0 = (s c)c(π) (s s)s(π) = s(π)c(π) s (π)s(π) (18.86) c(π) = s (π) (c c) = (s s) = c(π) (18.87) s(π) 0 (18.82) e iµπ (e iµπ ) 2 {(c c) + (s s)}e iµπ + 1 = 0 (18.88) 178

179 (18.85) (18.87) (e iµπ ) 2 2c(π)e iµπ + 1 = 0 (18.89) e iµπ {2 cos µπ 2c(π)} = 0 (18.90) cos µπ = c(π) (18.91) (18.75) µ (µ : cosmplex) cos µπ 1 = 2c (π/2)s(π/2) µ Bloch y(x + π) = e iµπ y(x) (18.92) x π e iµπ e iµπ < 1 x y e iµπ > 1 x y e iµπ 1 e iµπ = ±1 µ y π (µ ) 2π (µ ) µ (λ, q) λ µ q µ 0 1 µ >

180 a µ (q), b µ (q) (b µ (q) < a µ (q)) Mathieu a 0, b 1, a 1, i a i 1 b i quadrupole mass filter λ q ( ) q = , λ = a 0 b 1 a 0 : λ(q) = q 2 /2 + b 1 : λ(q) = 1 q + q 2 /8 + Abramowitz Stegan Handbook of Mathematical functions with formulas, graphs and mathematical tables ρ Poisson d2 φ dx 2 = ρ ε = (18.93) potential φ φ = 1 ρ 2 ε x2 + A x + B (18.94) ρ x 2 /2ε (18.93) A x + B Laplace ρ carrier 180

181 3/2 Poisson (18.93) ρ x v(x) x 1 2 m v(x)2 + qφ(x) = 0 (18.95) 0 0 i x i = q ρ(x) v(x) = (18.96) (18.93) (18.95) (18.96), ρ(x), v(x) i φ d 2 φ dx = i m 2 q ε 2qφ(x) dφ dx x 1 2 ( dφ dx (18.97) ) 2 = 2i m qε 2q { φ(x) + A} (18.98) x = 0 φ(x) = 0 dφ dx = 0 A = φ3/4 (x) = 4i m x (18.99) qε 2q (18.99) i i = 4 9 qε 2q m 1 x 2 φ(x)3/2 (18.100) e d x I (i ) V (φ(x) x = d ) 3/2 Langmuir- Child dφ(x = 0)/dx 0 181

182 y ra x z 4 B z z 4 + DC 4 DC step 1: ( ) 2 r φ(r) = U a, r 2 = x 2 + y 2 (18.101) r a r a U a ρ = 0 potential ( etc) B = (0, 0, B) z q < 0 m d2 x dt 2 m d2 y dt 2 = q ( = q φ x + B y ) t ( φ y B x ) t (18.102) d 2 x dt + 2ω y 2 B t ω2 c x = 0 d 2 y dt 2ω x 2 B t ω2 c y = 0 (18.103) ω B = qb 2m, ω 2 c = 2 q U a m r 2 a Larmor freq. cyclotron freq. (18.103) ξ(t) = x(t) + iy(t) 2 2 d 2 ξ dt 2 i ω dξ 2 B dt ω2 c ξ = 0 (18.104) ξ = ξ 0 e iωt (18.105) (18.103) Ω Ω 2 2ω B Ω + ωc 2 = 0, Ω = ω B ± ωb 2 ω2 c (18.106) 182

183 2 ω B > ω c 2 ( ) ( ) (18.106) 2 Ω 1, Ω 2 (18.103) 2 x = R 1 cos(ω 1 t + α 1 ) + R 2 cos(ω 2 t + α 2 ) y = R 1 sin(ω 1 t + α 1 ) + R 2 sin(ω 2 t + α 2 ) R 1, R 2 Ω 1, Ω 2 (18.107) E = m 2 (ẋ2 + ẏ 2 ) + qūa (x 2 + y 2 ) (18.108) ra 2 (18.107) ξ = R 1 e iϕ 1 + R 2 e iϕ 2 ; ϕ j = Ω j t + α j (j = 1, 2) r 2 = x 2 + y 2 = ξ ξ = R R R 1 R 2 cos{(ω 1 Ω 2 )t + (α 1 α 2 )} ẋ 2 + ẏ 2 = ξ ξ = Ω 2 1R Ω 2 2R Ω 1 Ω 2 R 1 R 2 cos{(ω 1 Ω 2 )t + (α 1 α 2 )} Ω 2 j = 2x B Ω j ω 2 c, Ω 1 Ω 2 = ω 2 c ( ) (18.108) cos{ } E = m R 2 1 (ω B Ω 1 ω 2 c ) + m R 2 2 (ω B Ω 2 ω 2 c ) (18.109) Ω 1 = ω B + ωb 2 ω2 c, Ω 2 = ω B ωb 2 ω2 c E = m ωb 2 ω2 c {R1 2 Ω 1 R2 2 Ω 2 } (18.110) E j = m ω 2 B ω2 c R 2 j Ω j (> 0) E = E 1 E 2 j = 1 j = 2 E = E 2 (< 0) β (LEAR) 183

184 step 2 ( ) 4 U a ω 4 potential φ(r, θ, t) = 2Ũa π tan 1 ( 2 r 2 a r 2 sin 2θ r 4 a r 4 ) cos ωt (18.111) ~ Ua ~ Ua y ~ Ua ra ~ Ua x potential r = r a φ(r a, θ, t) = Ũa cos ωt, 0 < θ < π/2, π < θ < 3π/2 = Ũa cos ωt, π/2 < θ < π, 3π/2 < θ < 2π r < r a 2 φ(r, θ, t) = 0 (18.111) φ(r) (18.111) Ω 1, Ω 2 (18.103) q φ Ω ω (18.111) (18.104) φ φ + i x y = φ ξ = 2Ũa ξ π tan 1 = 2Ũ0 π cos ωt ξ tan 1 ( 2rar 2 2 ( ) ξ 1 i e 2iθ e 2iθ 2i r 4 a (ξξ ) 2 ( ) 2 ( ξ ξ r a 1 ( ξ r a ξ r a ) ) 2 r a ) 2 cos ωt = 2Ũ0 π = 2Ũa π 2i cos ωt r a 2i cos ωt r a r a ( ξ 1 r a ) 4 ( ) { ( ) ξ ξ 4 ( ) } ξ r a r a r a (18.112) 184

185 ξ (18.107) ξ = R 1 e iω 1t iα 1 + R 2 e iω 2t iα 2 (18.113) cos ωt = (e iωt + e iωt )/2 (18.112) ±ω Ω 1, ±ω Ω 2, ±ω 5Ω 1, ±ω 4Ω 1 Ω 2, ±ω 3Ω 1 2Ω 2, ±ω 2Ω 1 3Ω 2, ±ω Ω 1 4Ω 2, ±ω 5Ω 2 Ω 1 Ω 2 Ω 1, Ω 2 > 0 ω R 1, R 2 R 1, R 2 (18.104) 2 ξ t 2iω dξ 2 B dt ω2 c ξ = 2q φ (18.114) m ξ ξ = R 1 e iω 1t + R 2 e iω 2t, R j = R j e iα j (18.115) (18.115) (18.114)) R j, α j { R 1 + 2i(Ω 1 ω B )Ṙ1}e iω1t + { R 2 + 2i(Ω 2 ω B ) R 2 }e iω2t = 2q φ (18.116) m ξ (18.104) R 1, R 2 (18.115) 1 ξ 2 R 1, R 2 Ṙ 1 e iω 1t + Ṙ2 e iω 2t = 0 (18.117) (18.117) (18.116) 2 (18.117) (18.118) Ṙ1, i(ω 1 2ω B )Ṙ1 e iω1t + i(ω 2 2ω B )Ṙ2 e iω2t = 2q φ (18.118) m ξ Ṙ 2 Ṙ 1 = 2q im Ṙ 2 = 2q im e iω 1t φ Ω 1 Ω 2 ξ e iω 2t φ (18.119) Ω 1 Ω 2 ξ R 2 ω = 2Ω 2 (18.112) { } 1 φ ξ 2Ũa π e iωt + e iωt 2 2i {R 1e iω1t + R 2e iω2t } (18.120) r 2 a 185

186 (18.119) Ṙ1 0 Ṙ2 Ṙ 1 = 0 Ṙ 2 = δ R 2, δ 2qŨa πm ω 2 B ω2 c R 1, α 1 R 2 dr 2 dt = δr 2 cos 2α 2, 1, (Ω ra 2 1 Ω 2 = 2 ) (18.121) dα 2 dt α sin 2α = 1 ln(tan α) 2 α 2 = δ sin 2α 2 (18.122) tan α 2 = tan α 20 e 2δ(t t 0) (18.123) α 20 t = t 0 δ > 0 tan α 20 tan α 2 ± α 2 ± π 2 R 2 α 2 δ cos 2α2 sin 2α 2 dα 2 = 1 2 ln(1 2 sin 2α 2) dr 2 dα 2 = R 2 cos 2α 2 sin 2α 2 (18.124) ( R20 R 2 ) 2 = sin 2α 2 sin 2α 20 (18.125) R 20 t = t 0 R 2 (18.125) (18.123) R 2 = R 20 { sin 2 α 20 e 2δ(t t 0) + cos 2 α 20 e 2δ(t t 0) } 1/2 1 sin 2α 2 = 1 2 ( tan α ) tan α 2 (18.126) R 2 α 20 = nπ ξ = R 1 e iω 1t+iα 1 + R 2 e iω 2t+iα 2 = (x + iy) ω = 2Ω 2 α 2 ±π/2 R 2 186

187 4 4 (18.107) R 1, R 2 187

188 Ω 2 = 1 60 Ω 1 max(r 1, R 2 ) r = 0 R 1 + R 2 > r a r = r a 4 potential φ(r a, θ) = { Ũa, 0 < θ < π/2, π < θ < 3π/2 Ũa, π/2 < θ < π, 3π/2θ < 2π potential 1 r a φ(r a, θ) Fourier φ(r a, θ) π φ(r a, θ) = n=1 φ n sin(2nθ) = odd n ( 4Ũa πn ) sin(2nθ) (18.127) 188

189 sin 2lθ 0 θ 2π 2π π/2 4Ũa, l = φ(r a, θ) sin 2lθdθ = 4Ũa sin 2lθdθ = 0 0 l 0, l 2π = φ n sin 2nθ sin 2lθdθ = φ l π (18.128) n=1 2 φ(r, θ) Fourier 0 φ(r, θ) = v n (r) sin 2nθ (18.129) n=1 Laplace { ( 1 d r dv ) } n 4n2 r dr dr r v 2 n sin 2nθ = 0 (18.130) n=1 { } = 0 v n r r v n = r 2n φ(r, θ) φ(r, θ) = a n r 2n sin 2nθ = 4Ũa ( ) 2n 1 r sin 2nθ (18.131) π n r n a odd n r = r a (1) φ(r, θ) = 4Ũa 2iπ odd n a n r 2n a 1 n = 4 Ũa π n { (re ) iθ 2n ( ) re iθ 2n } ln(1 + x) = x x2 2 + x3 3 = ( x) n ( x < 1) n n=1 n x x 1 2 ln1 + x 1 x = x + x3 3 + x5 5 = x n n odd n φ(r, θ) = Ũa iπ {ln1 + z + z ln1 1 z 1 z } = 2Ũa iπ i arg r a r a ( ) 1 + z 1 z (18.132) (18.133) (18.134) (18.135) ( ) re iθ 2 ( ) re z =, z iθ 2 = (18.136) r a 1 + z 1 z = 1 + z 1 + z r a 189

190 { } φ(r, θ) = 2Ũa π tan 1 ( ) 2 r 2 a r 2 sin 2θ r 4 a r 4 (18.137) 190

191 19 E B vacuum conductor E = B t E = B E t E = 0 B = 0 B = 0 B B skin depth ( 1/ σ) H = i + D E t H H (H ) i s H = i s (19.1) E H z is y x x y x B y E z ( Poynting vector ) k in, k out 191

192 k in ( ) E B out Bout y y kout z Bin θ out θ in O kin O E E in x + E out x = 0, E in y + E out y = 0, E in y = 0 (19.2) E out E in B in = B out E out, B out, k out c B out = E out k out B out θ in, θ out O B O B B in = B out (B in ) sin θ in (B out ) sin θ out = 0 (19.3) (B in ) cos θ in (B out ) cos θ out = 0 (19.4) θ in = θ out (19.5) k out z k in k in, k out, B in, B out E = 0 X (skin depth ) ( etc.) (E = E in + E out = 0) H B B in + B out cos θ in = 2 B in cos θ in B in = B 0 e i(k in r ω t) z = 0 i s = 2 B in µ 0 cos θ in (19.6) i s = 2 B in µ 0 cos θ in cos(k in y sin θ in ω t) (19.7) 192

193 z y i s k in k in sin θ in λ 2 Λ = λ sin θ in (19.8) Λ 2 (19.7) ẏ = ω k in sin θ in = c sin θ in (19.9) ẏ sin θ in 1 ±x i s i ω y c sin θ in ( ) ( ) = c 2 (19.10) B B out, z = 0 B B in y = B out y B in = B out (19.11) E in = E out E E in sin θ in = E out sin θ out E E in cos θ in E out cos θ out = 0 (19.11) (19.11) k ω ( ) ω E, B E, B y Λ = λ/ sin θ in ω λ E, B E TV θ i b 193

194 z λ b θ i Λ y Λ ( ) λ, θ i, b y = y Λ = λ/ cos θ i 2 b Λ Λ = λ = 2 b, cos θ i n n (19.12) λ λ = 2 b cos θ i n λ c = 2 b (19.13) λ c ( ) E x x x = 0, a E 0 < x < a, 0 < y < b ρ = i = 0 x = 0, a y = 0, b Maxwell x a z b y e i ω t z e i kz z z k z = ω/c ρ = i = 0 Maxwell E = B t, H = D t, D = B = 0 (19.14) B = µ 0 H, D = ε 0 E (19.15) x = 0 a H x = E y = 0 y = 0 b E x = H y = 0 } (19.16) 194

195 H z = (A sin k x x + B cos k x x)(c sin k y y + D cos k y y)e i(kz z ω t) (19.21) z e i kz z / z i k z E, H x y E z y z iω µ i k z 0 iω µ 0 i k z 0 0 i k z iω ε 0 0 i k z 0 0 i ω ε 0 H x H y E x E y E y x E x y = i ω µ 0 H z = E z x H z y H z x (19.17) H y x H x y = i ω ε 0 E z (19.18) ( Ez (19.17) (H x, E y ) y, H ) ( z Ez (H y, E x ) x x, H ) z y (E x,, H y ) (E z, H z ) E z x E x ik z 0 0 iω µ 0 E z E y H x = 1 0 ik z iωµ 0 0 y 0 iωε 0 ik z 0 H (19.19) z H y iω 0 0 ik z x H z y ( ω ) 2 = k 2 kz 2 = k 2 z c (19.19) (19.18) E z, H z ( ) ( ) ( ) 2 x + 2 E z = (k 2 k 2 E z 2 y z) 2 H z H z (19.20) i(kz z ω t) E z = (A sin k x x + B cos k x x)(c sin k y y + D cos k y y)e k 2 x + k 2 y + k 2 z = k 2 x + k 2 y + k 2 z = k 2 (19.22) 195

196 A D k x k y (19.16) ( ) TM ( H z = 0), ( ) TE ( E z = 0) E x,, H y k x k x ( k y k y) TM transverse magnetic wave; transverse z H z = 0 H transverse H z = 0 A = B = C = D = 0 (19.21) (19.19) E x = i k x k z (A cos k kx 2 + kz 2 x x B sin k x x)(c sin k y y + D cos k y y)e i(kz z ω t) (19.23) y = 0, b E x = 0 x D = 0, k y b = n π, n H x = i ω ε k y (A sin k kx 2 + kz 2 x x + B cos k x x)(c cos k y y D sin k y y) e i(kz z ω t) (19.24) x = 0, a H x y B = 0, k x a = m π, m AC = E 0 z E x = ik x k z k 2 k 2 z cot k x x E z H x = ωε 0 k z E y E y = ik y k z k 2 kz 2 cot k y y E z H y = + ωε 0 k z E x (19.25) E z = E 0 z sin k x x sin k y y e i(kz z ω t) H z = 0 k x = m π/a, k y = n π/b m > 0, n > 0 m, n E z = 0 TM m, n TE transverse electric field E z = 0 A = B = C = D = 0 A = C = 0, B D = Hz 0 E x E y = iωµ 0k y k 2 kz 2 = + iωµ 0k x k 2 kz 2 tan k y y H z H x = k z ωµ 0 E y tan k x x H z H y = + k z E x ωµ 0 E z = 0 H z = Hz 0 cos k x x cos k y y e i(kz z ω t) (19.26) k x = m π/a, k y = n π/b. m = n = 0 k 2 k 2 z = 0 196

197 k x,, k y ( eigen-value, characteristic value) Laplace Bessel l, m x, y k x = n π/a z e i(kz z ω t) k z < k z k z z e iω t z e ±i kz z e i k z ± e i k z 2 cos k z 2 i sin k z ( ) Bessel J n (r), N n (r) r J n (r) ± i N n (r) r Legendre P l m (x), Q l m (x) P l m (x) ± 2 i/π Q l m (x) x, (y) ( ) ( k ) k 2 = kx 2 + ky 2 + kz 2 (k x, k y, k z ) k x, k y kz 2 > 0 (k x, k y ) n, m TM 01 k, a, b a x x b y b y a a, b a, b TE 0 1 m = 0, n = 1 (19.26) k x = 0, k y = π/b 197

198 E y = E z = H x = 0 E x = iωµ 0 π k 2 kz 2 b sin y π z ω t) b ei(kz Hz 0 H y = i k z π k 2 kz 2 b sin y π z ω t) b ei(kz Hz 0 H z = Hz 0 cos y π z ω t) b ei(kz (19.27) ( ) E x, H y k 2 k 2 z = (π/b) 2 k z = k2 (π/b) 2 a x b y x y = 0, b y E x ( ) sin(y π/b) y y = b/2 H y, H z k z /k y m = n = 1 π/b < k π/a TE 01 H( ) i s ( ) z y y b k y = 2k z H D t H y E x H y E E x E E E x E (y=b/2 ) H y = 0 (H ) 198

199 ρ = i = 0 t iω z i k z Maxwell z 2. r Bessel 3. θ sin, cos 4. x, y (19.14) r, θ (19.17) H z, E z r, θ E r, E θ, H r, H θ E r E θ H r H θ = 1 ik z 0 0 iωµ 0 i k z iωµ 0 0 iωε ik z 0 iω ε 0 0 ik z E z r 1 E z r θ H z r 1 H z r θ (19.28) 199

200 = k 2 kz 2 = (ω/c) 2 kz 2 z { 1 r r (r E θ) E } r = iωµ H z θ { 1 r r (r H θ) H } r = iωεe z (19.29) θ (E z 0, H z = 0), (E z = 0, H z 0) TE TM TE E z = 0 (19.28) (19.29) ( 1 r H ) z H z = (k 2 k r r r r 2 θ z) 2 H 2 z (19.30) H z r θ θ sin nθ, cos nθ θ 2 π ( / ) n θ sin nθ, cos nθ r R(r) Bessel ( r d r dr ) + (k 2 k 2 R dr dr z) r 2 = n 2 (19.31) r = 0 R(r) Neuman H z (r, θ, z, t) = H 0 J n (k r, r) cos nθ e i(kz z ω t) (19.32) H 0 (19.32) (19.28) (E r,, H θ ) k 2 r = k 2 k 2 z E r = + iωµ n k 2 kz 2 r H 0 sin nθ J n (k r, r)e = + iωµ n k 2 kz 2 r H 0 tan nθ H z E θ = iωµ J n(k r, r) k 2 kz 2 J n (k r, r) H z H r = k z ωµ E θ i(kz z ω t) H θ = k z ωµ E r (19.33) r = a E θ = H r = 0 d dr J n(k r, r) = 0 (19.34) r x J n (x) (19.34) Bessel Bessel 200

201 d J n (x) dx = n x J n(x) J n+1 (x) = {J n 1 (x) J n+1 (x)} /2 = J n 1 (x) n x J n(x) (19.35) n = 0 d J 0 (x) = J 1 (x) (19.36) dx (19.33) E r n n = 0 r, θ, z, t E r H θ r = a E θ = H r = 0 H z θ z z Q ( n = 0 ) TM H z = 0 (19.28) (19.29) (19.30) H z E z (19.32) H z E z k 2 r = k 2 k 2 z E r = ik z d J n (k r r) E k 2 kz 2 z dr E θ = ik z n k 2 kz 2 r tan nθ E z i(kz z ω t) E z = E 0 cos nθ J n (k r, r) e H r = ω ε E θ k z H θ = ωε E r (19.37) k z r = a E θ = H r = 0 J n (k r a) = 0 (19.38) n = 0 E θ = H r = 0 (r, θ, z, t) z DTL k z 0 J 0 (0) = 1, J (0) = 0 E z E r 0 z z z TE 01 x z 201

202 E x = 0 (19.27) E x z cos(k z z ω t) t z t z t E x = 0 k z Z = l π, l (19.39) z = 0 ( cavity, resonator) TE 0 1 k 2 y + k 2 z = k 2 f ( f = ω 2 π = ck 2 π = c ) 2 ( ) 2 ( ) 2 m n l + + (19.40) 2 L x L x, L y, L z x, y, z TE 0 1 m = 0, n = 1 L y L z y z l = 1 ( ) ESR ( ) W P Q = 2 π W P (19.41) quality factor (Q ) dw/dt = P/T, (T ) W (t) = W (t = 0) exp ( ω t/q) (19.42) Q/ω ω T/Q ω/(q δ f) δ f = Q/ω (19.41) Q (19.42) ( ) 202

203 203

204 20 etc X γ etc recoil nm 1 204

205 Lamb shift D Na Hg Xe τ cτ wave train c T ct Einstein Max-Planck 205

206 1, (2) E 1 (E 2 ) (E 2 E 1 ) ν hν = E 2 E 1 (20.1) h ( J sec) 1 2 ( ) ν u(ν) 1 N 1 B 1 2 dn(1 2) dt = B 1 2 N 1 u(ν) (20.2) 2 1 (20.2) 2 1 A 2 1 dn(2 1) dt = A 2 1 N 2 + B 2 1 N 2 u(ν) (20.3) A 2 1, B 1 2, B 2 1 Einstein A, B 2 B ( ) 1 (20.2), (20.3) N 1 N 2 Boltzmann B 1 2 N 1 u(ν) = A 2 1 N 2 + B 2 1 N 2 u(ν) ( N 2 = N 1 exp hν ) kt u(ν) = A ( ) B 1 2 hν exp 1 kt (20.4) (20.5) k Boltzmann ( J K 1 ) T (hν) (20.5) Boltzmann Boltzmann Bose-Einstein Max-Planck ρ(ν) L e i(k r ωt) r t k j L = 2n j π, (j = x, y, or z n j ) (20.6) T 1 T T Hermit 2 T 1 = 1 T 2 B 1 2 = B 2 1 detailed balance 206

207 k x, k y, k z (20.6) (n x, n y, n z ) L n j n 1 n n x2 + n y2 + n z 2 n n N(n) = 4π 3 n3 = 4π ( ) 3 L 3 2π k (20.7) ( ω k 2 = c ) ( ) 2 2 2πν = (20.8) c (20.7) ν 2 2 L 3 ρ(ν) = 1 L 3 2 dn(ν) dν = 8πν2 c 3 (20.9) ν 1 hν(= ω) Ē(ν) Boltzmann n=0 ( nhν exp nhν ) kt hν Ē(ν) = n=0 ( exp hν ) = ( ) (20.10) hν exp 1 kt kt (20.9) u(ν) u(ν) = 8πh ν3 c 3 1 ( ) hν exp 1 kt (20.11) (20.5) A ( 2 1 ν ) 3 = 8πh (20.12) B 1 2 c A B (20.11) (20.11) ν 3 f(ν/kt ) Wien Wien ν (0, ) (20.11) 0 u(ν) dν = 8 π h c 3 0 ν 3 e C ν dν 1 e C ν C = h/k T e Cν < 1 = 48 π h 1/n 4 c 3 C 4 n 207

208 Riemann ζ(n) n ζ(4) = π 4 /90 ζ Fourier S(T ) = σ T 4, σ 8 π5 k 4 15 c 3 h 3. (20.13) T Stefan-Boltzmann ( ) ( ) 1 1/ m 2 1 cd ( ) ( 100 W 1/100) ( ) (cosmic back ground) (20.11) T 2.7 K ( 4 ) : λ λ λ λ!? ( 1 ) N 2 (N 1 ) N 2 > N 1 E 2 (E 1 ) hν 12 = E 2 E 1 ν 12 du(ν) Einstein A, B dt du dt = {A + B u(ν)}n 2 BN 1 u(ν) = AN 2 + B u(ν)(n 2 N 1 ) (20.14) B, u N 2 N 1 > 0 N 2 N 1 N 2 > N 1 ν 12 (Light Amplification by Stimulated Emission of Radiation ) 208

209 N 2 > N 1 ( N 2 = exp hν ) 12 (20.15) N 1 kt hν 12 > 0, k > 0 T < Γ γ 13 γ γ Γ 1 γ ij ( j i ) dn 1 dt = (Γ + γ 21 + γ 31 )N 1 +γ 12 N 2 +γ 13 N 3 dn 2 dt = γ 21N 1 (γ 12 + γ 32 )N 2 +γ 23 N 3 dn 3 dt = (Γ + γ 31)N 1 +γ 32 N 2 (γ 13 + γ 23 )N 3 (20.16) N 1 + N 2 + N 3 = N (20.16) dn i = 0 dt γ ij (i > j) γ 12 (γ 13 + γ 23 ) N 1 = γ 12 (γ 13 + γ 23 ) + (γ 12 + γ 23 )Γ N (20.17) γ 23 Γ N 2 = γ 12 (γ 13 + γ 23 ) + (γ 12 + γ 23 )Γ N N 2 > N 1 ( Γ > γ γ ) 13 γ 23 (20.18) Γ (20.18) 209

210 N 2 > N 1 N 2 N Γ ( Γ > exp hν ) ( γ γ ) 13 + γ 03 kt γ 23 (20.19) γ 2 γ 02 + γ 12 Boltzmann factor (4 ) ( γ 10 = exp hν ) γ 01 (20.20) kt α = e 2 4 π ε 0 c 1 ( ) γ ( Γ ) 137 He-Ne 1 m 0.7 L m (1.007) L m 1 ( ) ( ) cm 2 30cm M1 l1 M0 l2 M2 M 0 2 l 1, l 2 M 1, M 2 2(l 1 l 2 ) (l 1 l 2 ) µ m 10 3 km (2π ) ( ) 210

211 X 2 Z = 34 Se 24 34Se (1s) 2 (2s) 2 (2p) 6 Ne 3p 3s 2p 2s 1s (1s) 2 (2s) 2 (2p) 5 (3p) ( ) (3s) (2p) (3p) (2p) (3p) (2s) 73 Ta Å TV i t ( ) X ( ) X X (radiative electron capture) 1 2 ( X ) 2 ( ) β ( ) 211

212 r vt d v v (Coulomb ) v Lorentz (13.74) E B v E = E = q r 4 π ε 0 r v 3 v, B = B = 0 (20.21) v Lorentz v E, v B E = γ{e + (v B ) }, B = γ{b (v E ) /c 2 } (20.22) B E r r E E = 1 q r sin θ (20.23) 1 (v/c) 2 4 π ε 0 r 3 r = (vt) 2 + d 2, sin θ = d r t t = 0 r = d E Fourier E (ω) = 1 2π E e iωt dt = 1 1 q 2π 1 (v/c) 2 4 π ε 0 d (cos ωt i sin ωt) dt (20.24) {(vt) 2 + d 2 } 3/2 t sin ωt Bessel K ν (z) K ν (z) = Γ(ν + 1/2) (2π) ν cos t dt (20.25) π 0 (t 2 + z 2 ) ν+1/2 E (ω) = 1 1 q 2ω ( ω ) 2π 1 (v/c) 2 4 π ε 0 v K 2 1 v d (20.26) E ω (20.26) ω ω 212

213 d d d ( ) E E (ω) sin ωt E (ω) 2 B, B (undulator) wiggler 213

214 wiggler λ W N W λ W N W v wiggler t vt = N W λ W 1 λ L wiggler N W c ct t wiggler 2 v c 3 λ L = 1 β β λ W (c v)t = N W λ L vt = N W λ W (1 β)(1 + β) 2 λ W = 1 β2 λ W = λ W (20.27) 2 2γ 2 2γ 2 wiggler λ L 4 1 N W 2 1 wiggler wiggler wiggler (bunching) bunching 2 2 Doppler 3 β = p γ 2 1/γ 4 coherent coherent 214

215 wiggler wiggler (20.27) 215

216 21 ẍ x... x ẋ (= ) mẍ + 2γẋ + mω 2 0x 2 = 0 (21.1) γ spontaneous emission Thomson x z E x = E 0 e i(kz ωt), E y = E z = 0 B y = B 0 e i(kz ωt), B x = B z = 0 B 0 = 1 c E 0, k = ω c (21.2) E 0 1/c x v x Lorentz v B (8.23) k ω Compton mẍ + 2γẋ + mω 2 0x 2 = qe 0 e iωt (21.3) xy z = 0 (21.2) (21.1) ( ) (21.3) x = Ae iωt 216

217 (21.3) 1 x = ( ) + qe 0 m 1 (ω 02 ω 2 ) 2iωγ e iωt (21.4) (E x ) E x (11.34) n β n β n 0 (10.17) 2 p x = qx (21.4) p x = q2 E 0 ω 2 m e iωt (ω 02 ω 2 ) 2iωγ (21.5) dσ dω = ρ E(scatt.) ρ E (inc.) (21.6) ρ E (scatt.) 1 dω ρ E (inc.) (8.5), (8.6) (9.33) ρ E = 1 2 E D B H = ε 2 E µ B2 = εe 2 = εe 0 2 e 2i(kz ωt) (21.7) c ρ E c z = 0 (ρ E c) (ρ E c) 1/2 = ε E 2 0 θ, ψ x α y x α n θ z dσ dω = µ2 16π 2 ( ) q 2 2 ω 4 m (ω 02 ω 2 ) 2 + (2ωγ) 2 sin2 α (21.8) α sin 2 ω ω 0 ω 4 (Rayleigh ) π sin 2 αdω = 2π sin 2 α sin αdα = 8π 3 1 qe 0 e iωt 2 dw = µ dt 16π 2 c p 2 sin 2 θdω 0 (21.9) 217

218 σ = dσ dω dω = 1 6π ( ) µq 2 2 ω 4 m (ω 02 ω 2 ) 2 + (2ωγ) 2 (21.10) ω 0 = 0 (12.2) Thomson σ T r e = 1 e 2 4 π ε 0 mc = 1 µ 0 e 2 2 4π m σ T = 8π 3 r e 2 (21.11) γ = 0 Bohr Compton x (21.8) x, y ( ) x, y x β E Ê ˆn Ê = (cos β, sin β, 0) ˆn = (sin θ cos ψ, sin θ sin ψ, cos θ) cos α Ê ˆn cos α = Ê ˆn = sin θ(cos β cos ψ + sin β sin ψ) = sin θ cos(β ψ) (21.12) β 0 π π sin 2 α (β ψ) β β 0 π ψ ψ = 0 sin 2 α = 1 π π 0 sin 2 αdβ = 1 π π 0 cos 2 βdβ = π 0 (1 sin 2 θ cos 2 β)dβ = 1 + cos2 θ 2 [ β ] π sin 2β = π (21.13) (21.8) (21.13) ω 0 = 0 dσ dω = 1 + cos2 θ 2 r e 2 Klein-Nishina (21.14) xy (21.4) ẍ = qe 0 m f(ω)e iωt, f(ω) = ω 2 ω 02 ω 2 2iωγ (21.15) 218

219 x (11.34) (ẍ ) β = c, 0, 0 y ρ θ r z n = 1 ( x, y, z) r n β= ẍ (0, z, y) rc n (n β)= ẍ r 2 c ( y2 z 2, xy, xz) (21.16) dx dy = ρ dρ dθ z dζ N de sc = Ndζq 4πε 0 1 rc n (n β)ρ dρ dθ (21.17) θ x = ρ cos θ, y = ρ sin θ y, z 2π y 2 dθ = 2π ρ 2 sin 2 dθ = ρ 2 π 0 0 (21.18) 2π z 2 dθ = 2πz 2 0 (de sc ) x = + Ndζq 1 qe 0 4 π ε 0 r 3 c m f(ω)e iωt π(2z 2 + ρ 2 )ρdρ (21.19) t = t r/c ρ 0 r 2 = z 2 + ρ 2, 2rdr = 2ρdρ (21.20) (de sc ) x = πr e Ndζ E 0 f(ω)e iωt = πr e Ndζ E 0 f(ω)e iωt 2r 2 ρ 2 z z z rdr r 3 eikr (2z 2 + ρ 2 ) rdr r 3 eikr (2r 2 ρ 2 ) e ikr dr = 1 ik lim [ ] e ikr R eikz = R z ik + lim e ikr R ik (21.21) ρ = 0 kr ρ ρ ρ 2 ( ) (de sc ) x = i 2π k r e Ndζ E 0 f(ω)e i(kz ωt) = i A E x dζ (21.22) 2π k = λ A A λr e Nf(ω) (21.23) 219

220 ρ = 0 (λr e ζ) z (de sc ) x = iae x dζ ζ dζ D D z = 0 E x (z) = E x (D = 0) exp i(kz ω t){1 + i A D} E 0 exp i(kz ωt + AD) (21.24) D k(z D) + (k + A)D ωt = k(z D) + k D ωt ( (z D)) k D k = k + A = n k ( 1/n) n 1 + A k = 1 + 2π r e Nf(ω) = 1 + q2 k 2 2ε 0 m 1 ω 02 ω 2 2iωγ (21.25) n ( (21.25) 90 ( ) γ = 0 γ 0 ( ) (21.22) i γ = 0 90 O(dζ) γ > 0 Etot Ex Etot Ex dex dex (γ=0) (γ>0) f(ω) = ω2 {(ω 2 0 ω 2 ) + 2iωγ} (ω 2 0 ω 2 ) 2 + (2ωγ) 2 (21.26) de x i E x γ < 0 f(ω) ω < ω 0 (21.25) q 2 /2m ω 0 > ω n > 1 ( ω 0 ) 220

221 ω ( ) 2. ( ) 3. ( ) 3 polarizability dipole ion freq. electron n ( tan δ = Im ε ) Re ε No air glass air ( ) 219 (21.22) L A B C E L E A, B, C E! 221

222 Snell (21.19) (21.22) L P A B C dx C A E x e ikr 1 1 P dx exp (ik l(x)) ( x ) l(x) (LP (x) + P E(x)) 2 (21.27) x x x x l(x) dl(x) dx = 0 (21.28) x A C Snell L A B C E x l(x) E N 2 N Snell LBE LAE, LCE LAE LCE X X Bragg Laue (21.27) (Fermat) (21.27) k l(x) x 222

223 x δ l(x)dx = 0 k (21.23) (21.25) n(x) ( k) δ n(s)ds = 0 (21.29) Fermat 1 (21.27) kr ωt φ = (kr ωt)ds (21.30) 3 kr ωt = 1 (p r E t) = 1 ( dt p dr ) dt E = 1 Ldt (21.31) Fermat Hamilton (21.31) S Ldt (is/ ) 1 (21.27) p r, E t (21.27) ( ) ( ) Doppler (p106 ) kr ωt Lorentz ( ) S Lorentz Lagrangian 4 3 p v = 2T, E = T + V (T V ) Lagrangian 223

224 1 (21.8) Rayleigh Einstein ( ) C D E CD + DE n + EF = AB n + BF (21.32) A B F ( ) F P n n f = P n 2 /λ X n 1 Fresnel lens (21.4) (21.5) qẍ ṗ (21.1) (21.3) 224