Maxwell ( H ds = C S rot H = j + D j + D ) ds (13.5) (13.6) Maxwell Ampère-Maxwell (3) Gauss S B 0 B ds = 0 (13.7) S div B = 0 (13.8) (4) Farad

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1 13 Maxwell Maxwell Ampère Maxwell 13.1 Maxwell Maxwell E D H B ε 0 µ 0 (1) Gauss D = ε 0 E (13.1) B = µ 0 H. (13.2) S D = εe S S D ds = ρ(r)dr (13.3) S V div D = ρ (13.4) ρ S V Coulomb (2) Ampère C H = B/µ C S 167

2 Maxwell ( H ds = C S rot H = j + D j + D ) ds (13.5) (13.6) Maxwell Ampère-Maxwell (3) Gauss S B 0 B ds = 0 (13.7) S div B = 0 (13.8) (4) Faraday B B E ds = ds (13.9) L S rot E = B (13.10) (1) (2) ρ j (3) (4) D = ε E (13.11) B = µ H (13.12) j = σ E. (13.13) ε µ σ

3 Ampère Maxwell Maxwell Maxwell Hertz 1888 Maxwell ρ =0 j =0 E B Maxwell div E = 0 (13.14) div B = 0 (13.15) rot E = B (13.16) rot B = ε 0 µ 0 B. (13.17) x y z E(x, t) B(x, t) (13.14) (13.15) E x =0, B x = 0 (13.18) (13.16) (13.17) 0= B x, E z = B y, E y = B z (13.19) 0= E x, B z = ε 0µ 0 E y, B y = ε 0µ 0 E z. (13.20) x (13.18)(13.19) (13.20) E x = E x =0, B x = B x =0

14) (13.15) E x =0, B x = 0 (13.18) (13.16) (13.17) 0= B x, E z = B y, E y = B z (13.

4 Maxwell E x B x x t E x = B x = 0 (13.21) y z (13.19) (13.20) B y E z 2 E z 2 = 1 2 E z ε 0 µ B y 2 = 1 ε 0 µ 0 2 B y 2 (13.22) (13.19) (13.20) B z E y 2 E y 2 = 1 2 E y 2 B ε 0 µ 0 2 z 2 = 1 2 B z ε 0 µ 0 2 (13.23) x E z E z (x, t) =f(x vt)+g(x + vt) (13.24) f g f(x vt) x v g(x + vt) x v (13.24) (13.19) s = x vt s = x + vt E z x E z = f + g = f s s + g s s = f s + g s B y t B y = f s + g s B y f(x vt) g(x + vt) a b B y = af(x vt)+bg(x + vt). B y = a f + b g = a f s s + b g s s = av f g + bv s s a = 1 v, b = 1 v

20) B z E y 2 E y 2 = 1 2 E y 2 B ε 0 µ 0 2 z 2 = 1 2 B z ε 0 µ 0 2 (13.23) x E z E z (x, t) =f(x vt)+g(x + vt) (13.

5 B y B y = 1 [ f(x vt) g(x + vt)] v E z B y E z = f(x vt)+g(x + vt) B y = 1 (13.25) v [ f(x vt) g(x + vt)] f g v x electromagnetic wave E z B y x E x B x z y x z E z > 0 y B y < x y E y = E 0 sin(kx ωt) (1) k ω (2) E =(0,E 0 sin(kx ωt), 0). (1) 2 E y 2 = 1 2 E y ε 0 µ 0 2 = c 2 2 E y 2 = ( ω) 2 E 0 sin(kx ωt), = c 2 k 2 E 0 sin(kx ωt) c ω 2 = c 2 k 2 c = ω k

y < 0 13.1 x y E y = E 0 sin(kx ωt) (1) k ω (2) E =(0,E 0 sin(kx ωt), 0).

6 Maxwell (2) x B x =0 z E y = B z E y B z = E y = ke 0 cos(kx ωt) B z = k ω E 0 sin(kx ωt) = 1 c E 0 sin(kx ωt) 0 z H =(0, 0, H z ) H z = H 0 sin(kx ωt), H 0 = 1 µ 0 c E 0 = ε0 µ 0 E x y z 13.1: x 13.2 E Faraday rot ( rot E )= ( ) rot B

7 div E = ρ/ε 0 rot ( rot E ) = grad ( div E ) 2 E = 1 ε 0 grad ρ 2 E, D rot B = µ 0 j + µ 0, D = ε 0 E ( ) j rot B = µ 0 ε 0 µ 2 E 0 2 ρ =0 j =0 ε 0 µ 0 =1/c 2 ( ) 1 2 E c 2 2 = 2 2 E = y z 2 E (13.24) x t 2 ( E z 2 = v 2 2 ) f g 1 2 E 2, z ε 0 µ 0 2 = 1 ( 2 ) f ε 0 µ g 2, ε 0 µ 0 v = 1 ε0 µ 0 ε 0 = 107 4πc 2 [C N 1 m 2 ] µ 0 = 4π 10 7 [N A 2 ] v = 1 ε0 µ 0 = c Maxwell

8 Maxwell Hertz Newton Michelson- Morley 1887 Maxwell Einstein 1905 Maxwell Newton Einstein m Hz Gamma rays X-rays Ultraviolet Visible light Infrared Millimeter waves, Telemetry Radar Microwaves FM radio Television Short waves radio AM radio 13.2:

15 10 14 10 13 10 12 10 11 10 10 10 9 10 8 10 7 10 6 10 5 Gamma rays X-rays Ultraviolet Visible light

9 S poynting vector S = E H (13.26) E H 13.1 x y E =(0,E y, 0), E y = E 0 sin(kx ωt), (13.27) z H =(0, 0, H z ), H z = H 0 sin(kx ωt). (13.28) H 0 = 1 µ 0 c E 0 = ε0 µ 0 E 0 (13.29) S = E H =(S x, 0, 0) x x S x = E 0 H 0 sin 2 (kx ωt). 2π t =0 x =0 x =2π/k 2π λ λ = 2π k. x t =0 x 0 λ λ S x S x = 1 λ λ 0 S x (x, t =0)dx = 1 2 E 0H 0 (13.30) x

10 Maxwell (13.28) (13.27) (13.30) 1 2 ε 0E 2 y = 1 4 ε 0E 2 0, 1 2 µ 0Hz 2 = 1 4 µ 0H0 2 (13.29) (13.30) 1 2 E 0H 0 = 1 2 ε 0E0 2 1 = c 1 ε0 µ 0 2 ε 0E0 2 S x = c ( 1 2 ε 0 E y ) 2 µ 0 H z 2 x x E x = B x =0 E 0 B 0 x E 0 E y (x, t) = E 0y sin(kx ωt + θ y ) θ y = θ z θ y = θ E z (x, t) = E 0z sin(kx ωt + θ z ) E(x, t) =E 0 sin(kx ωt + θ), E 0 =(0,E 0y,E 0z ) x t E 0 E 0 B 0 E 0y = E 0z = E 0 θ z = θ y + π/2 θ y = θ + π/2 E y (x, t) = E 0 cos(kx ωt + θ) = E 0 cos(ωt kx θ) E z (x, t) = E 0 sin(kx ωt + θ) = E 0 sin(ωt kx θ) x = kx + θ =0 x 0 = θ/k E y (x 0,t)=E 0 cos ωt, E z (x 0,t)=E 0 sin ωt

11 ω θ z = θ y π/2 θ y = θ + π/2 E y (x, t) = E 0 cos(kx ωt + θ) = E 0 cos(ωt kx θ) E z (x, t) = E 0 sin(kx ωt + θ) = E 0 sin(ωt kx θ) x = kx + θ =0 x 0 = θ/k E y (x 0,t)=E 0 cos ωt, E z (x 0,t)= E 0 sin ωt E y0 E z0 ( E y,e z )

12 Maxwell 13.3 Maxwell z (0, 0, L/2) q (0, 0, L/2) q I I>0 dq dt = I ω I(t) =I 0 cos ωt. z k j(r,t)=i(t) δ(x )δ(y ) k A(r,t)= µ 0 j(r,t r r /c) dr 4π r r = k µ L/2 0 I(t r z k /c) 4π L/2 r z dz k r = z k x y r r = r L r z k = r 2 2rz cos θ + z 2 r z cos θ θ r z I t r z k c t r z cos θ c I T =2π/ω z cos θ /c z cos θ < L c c T I t r/c λ L λ (13.31)

13 A(r,t) k µ 0 4π φ L/2 L/2 I(t r/c) r dz = k µ 0 L I(t r/c) (13.32) 4πr div A + 1 φ c 2 =0 Lorentz (13.32) φ c 2 µ 0 L 4π φ(r,t) c 2 µ 0 L 4π [ z z I(t r/c)+ r3 r 2 c I(t r/c) [ ] z z q(t r/c)+ r3 r 2 c I(t r/c) qω I λ/r φ(r,t) µ 0 I 0 czl 4π r 2 cos ω(t r/c) (13.33) (13.33) (13.32) ] LC +Q 0 Q 0 t =0 L =20µH C = µf L C 13.3: LC I(t) Kirchhoff L di dt V C =0

14 Maxwell ±Q(t) I(t) = dq(t), V dt C (t) = Q(t) C Q d 2 Q(t) dt 2 = Q(t) LC. a b t t Q(t) =a cos + b sin LC LC Q(t =0) = Q 0, a = Q 0 b =0 Q(t) =Q 0 cos I(t =0) = dq(t) dt t LC =0 t=0 Q(t) V C (t) E(t) D(t) V C (t) = Q(t) C, E(t) =V C (t), D(t) =ε d 0 E(t) d rot H = D ω = 1 LC c λ c λ = ω/(2π) LC = ( ) ( )= [s] λ =( ) 2π ( )= [m]

A. Fresnel) 19 1900 (M. Planck) 1905 (A. Einstein) X (A. Ampère) (M. Faraday) 1864 (C. Maxwell) 1871 (H. R. Hertz) 1888 2.2 1 7 (G. Galilei) 1638 2

1 2012.8 e-mail: tatekawa (at) akane.waseda.jp 1 2005-2006 2 2009 1-2 3 x t x t 2 2.1 17 (I. Newton) C. Huygens) 19 (T. Young) 1 A. Fresnel) 19 1900 (M. Planck) 1905 (A. Einstein) X (A. Ampère) (M. Faraday)