(C) Kazutaka Takahashi 2018

Transcription

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2 (C) Kazutaka Takahashi 2018

3 i

4 I 11 2 Coulomb Coulomb [ ] Gauss Gauss Gauss Gauss Gauss Gauss [ ] ii

5 Stokes * Poisson * [ ] Poisson ** [ ] ** Cavendish ** II Ohm Ohm Ohm Ampère Ampère iii

6 Biot Savart Lorentz Lorentz Gauss Ampère * * ** Coulomb III Maxwell Maxwell iv

7 12.4 ** ** A (1) 155 B (2) 156 C (3) 157 D (4) 158 v

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9 Einstein Newton Newton

10 GPS Galilei Newton 6 R. P. Feynman Feynman Global Positioning System Feynman 1965 Nobel J. S. Schwinger 8 III

11 J. Kepler G. Galilei R. Descartes I. Newton Kepler T. Brahe Galilei Descartes Newton Aristotélēs M. Faraday J. C. Maxwell N. L. S. Carnot W. Thomson R. J. E. Clausius 19 Newton Newton

12 Newton 1687 Kepler A. Einstein Hilbert Kepler 16 Kepler Newton 2 4

13 Newton Newton ma = F (1.1) m a F F 2 Kepler Newton (1.1) Kepler Newton (1.1) 17 Newton (1.1) Newton Coulomb 2 electric charge Coulomb 2 18 Newton

14 19 Maxwell 4 E(r, t) = 1 ϵ 0 ρ(r, t) (1.2) B(r, t) = 0 (1.3) E(r, t) + B(r, t) = 0 t (1.4) B(r, t) ϵ 0 µ 0 t E(r, t) = µ 0j(r, t) (1.5) Maxwell Maxwell

15 5 6 9 Maxwell [1] 2000 [2] I 5 7

16 [3] [ ] [2] [4] I II [5] pdf Feynman Jackson 1992 ( )( ) : [1] * 24 **

17 1 I 2 Coulomb Coulomb 3 Gauss II 7 Ohm 8 Ampère Biot Savart Lorentz 9 10 III 11 Faraday 12 Maxwell [ ] 9

18

19 I

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21 2 Coulomb Coulomb Coulomb Coulomb 2.1 Coulomb Newton Newton universal gravitation 2 r 1 r 2 2 F 12 = G m 1m 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) 2 mass (2.1) m 1 m G gravitational constant G ( ) ( ) 2 ( ) 2 ma = F ( ) ( ) ( ) 2 G ( ) 3 ( ) 1 ( ) 2 gravitational mass ma = F inertial mass 1 1kg 1m/s 2 1N G N m 2 /kg 2 Newton 1 Newton 13

22 base unit ( ) ( ) 1 dimension 3 (x, y, z) l l = 1.0 cm = 0.01 m l cm ( ) = ( ) ( ) ( ) = ( ) 2 2 r 1 r 2 2 r 1 r Coulomb 0 (2.1) 1 2 r 1 r 2 r 1 r F 21 = G m 1m 2 r 2 r 1 2 r 2 r 1 r 2 r 1 (2.2)

23 2.1: m 1 F 12 m 2 F 21 = F : m F m 1 F 1 m 2 F 2 action-reaction law F 12 = F 21 (2.3) point mass 4 (2.1) superposition principle 2.2 F (r) = i r r i Gmm i r r i 3 (2.4) m r m i r i

24 2 5 vector r = (x, y, z) (2.5) E E = (E x, E y, E z ) (2.6) E x E x E x E x 1 3 ar 1 + br 2 = a(x 1, y 1, z 1 ) + b(x 2, y 2, z 2 ) = (ax 1 + bx 2, ay 1 + by 2, az 1 + bz 2 ) (2.7) a b 2.2 r 1 r 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 (2.8) 1 scalar r = r 2 = x 2 + y 2 + z 2 (2.9) r 2 = r r 1/r r 3 r 2 r equation of motion m d2 r(t) dt 2 = F (r(t)) (2.10)

25 r 2 r(t) t 0 r(t 0 ) t=t0 dr(t) dt Newton Coulomb W. Gilbert Gilbert electricity 17 Newton Coulomb Newton Coulomb Coulomb force q 1 q 2 r 1 r 2 F 12 = k r 1 r 2 2 r 1 r 2 (2.11) Coulomb Coulomb s law 2 2 (2.1) Gm 1 m 2 G q Coulomb (2.11)

26 7 k 8 k ( ) ( ) 2 ( ) 2 = ( ) ( ) 3 ( ) 2 ( ) 2 ma = F m m m k C.-A. de Coulomb [C] 8 k k N m 2 C 2 (2.12) 1C 1kg Coulomb 1C 1C 1kg C kg Coulomb 12 r i i = 1, 2,... q i r q F (r) = kq i q i r r i r r i 3 (2.13) 7 8 k = 1 ϵ 4πϵ (2.1) (2.11)

27 q q 0 ( r r1 F (r) = kqq r r 1 3 r r ) ( 2 r r r 2 3 kqq r 3 r ) r 3 = 0 (2.14) 13 q r 1 q r 2 r Q r r 1 r r q q 1 x 1km y x y x y (2.11) point charge 1 1 Coulomb 1 r 3 V (r) q(r) q(r) = ρ(r) V (r) P.-S. Laplace I Coulomb 19

28 r charge density ρ(r) = lim V (r) 0 q(r) V (r) (2.15) Dirac 17 r q ρ(r) F (r) = kq lim { V (r i)} 0 = kq i ρ(r i ) V (r i ) r r i r r i 3 (2.16a) d 3 r ρ(r ) r r r r 3 (2.16b) r i ρ(r i ) V (r i ) 0 2 volume integral 18 (2.16) 2 r r r q F (r) r q r r F (r) = kq dv (r ) ρ(r ) r r r r 3 = kq dv ρ(r ) r r r r 3 (2.17) d 3 r dv (r ) dv dv r r r d 3 r f(r) = dv f(r) = dx dy dz f(r) (2.18) (2.13) (2.16)

29 Coulomb 2 2 q Coulomb F = qe (2.19) E electric field 19 r E(r) = k d 3 r ρ(r ) r r r r 3 (2.20) field E(r) r E r E : r + = (a, 0, 0) r = ( a, 0, 0) q r = (x, y, z) E(r) = ( r r+ kq r r r r ) r r 3 = x a kq [(x a) 2 + y 2 + z 2 ] 3/2 y z + kq [(x + a) 2 + y 2 + z 2 ] 3/2 x + a y z (2.21a) (2.21b)

30 2.3: z = 0 x y xy z = x = 0 z = 0 y = 0 z = 0 E(x, 0, 0) = E(0, y, 0) = kq x a 3 kq (a 2 + y 2 ) 3/2 x a y 0 kq x + a 3 (2.22) x + a 0 0 (2.23) r + = (a, 0, 0) q r = ( a, 0, 0) q r = (x, y, z) r E(r) = ( r r+ kq r r + 3 r r ) r r 3 = x a kq [(x a) 2 + y 2 + z 2 ] 3/2 y z kq [(x + a) 2 + y 2 + z 2 ] 3/2 x + a y z (2.24a) (2.24b) x = 0 z = 0 kq E(0, y, 0) = (a 2 + y 2 ) 3/2 2a 0 0 (2.25) 21 Gauss 22

31 2.4: x y = 0 z = 0 y x = 0 z = 0 y = 0 z = 0 E(x, 0, 0) = kq x a 3 x a 0 0 kq x + a 3 x + a 0 0 (2.26) xy 2.3 q q x > a x y : 2.5 x x = na n q x y r r = (0, r, 0) na kq E(0, r, 0) = (r 2 + n 2 a 2 ) 3/2 r = n= 0 n= kqr (r 2 + n 2 a 2 ) 3/ (2.27) λ = q/a a 0 q 0 λ linear density (2.15) [2 3] 23 23

32 2.5: 2 2 x x lim af(na) = dx f(x) (2.28) a 0 n= f(x) f a (2.16) V (r i ) 24 r = (0, r, 0) [2 3] E(0, r, 0) = kλ dx r (x 2 + r 2 ) 3/ = 2kλ r (2.29) 2 1 Coulomb 2 3: a 2.6 z = 0 xy 2.7 z z z r = (0, 0, z) z Q λ = Q 2πa 2πa xy x θ 25 adθ 2.8 E z (0, 0, z) = k 2π λadθ 0 2π 1 z a 2 + z 2 (2.30a) (a 2 + z 2 ) 1/2 z (a 2 + z 2 ) 3/2 (2.30b) = k Q dθ 2π 0 z = kq (2.30c) (z 2 + a 2 ) 3/2 2π z a a E z (0, 0, z) kq z 2 (2.31)

33 2.6: 2 3 a 2.7: (a) (b) (c) Q 2 4: 2 xy xy z z z r = (x, y, z) E(r) = 0 0 zf( z ) (2.32) 25

34 2.8: a θ θ + dθ adθ dθ 26 f σ xy 2.9 a a+da da 2 3 Q = 2πσada z de z (r) = 2πkσada (2.33) (z 2 + a 2 ) 3/2 a [ za z E z (r) = 2πkσ da = 2πkσ (z 2 + a 2 ) 3/2 (z 2 + a 2 ) 1/2 0 ] 0 = 2πkσ z z (2.34) z z/ z 2 5: a σ = Q Q 4πa 2 (2.35) 26 z = z 2 f z 2 26

35 2.9: : 2 5 r = (0, 0, r) xy 2.10 z θ dq = Q 4πa 2 2πa sin θadθ = Q sin θdθ (2.36) 2 a sin θ adθ r = (0, 0, r) r a cos θ r = (0, 0, r) z E z (0, 0, r) = kq 2 π 0 r a cos θ dθ sin θ [(r a cos θ) 2 + a 2 sin 2 (2.37) θ] 3/2 [2 6] E z (0, 0, r) = { kq r 2 r a 0 r < a (2.38) r = (0, 0, r) z r = (x, y, z) E(r) = { kq r r 2 r r a 0 r < a (2.39) Q r a

36 symmetry 2 4 z z + z 0 z z z 2.4 [ ] Cartesian coordinate r = x y z (2.40) 3 29 x y z 2.11 polar coordinate x = r sin θ cos φ (2.41) y = r sin θ sin φ (2.42) z = r cos θ (2.43) r = x 2 + y 2 + z 2 (2.44) 28 René Descartes Renatus Cartesius Cartesian 29 28

37 2.11: 3 r (x, y, z) (r, θ, φ) x2 + y tan θ = 2 z tan φ = y x (2.45) (2.46) r θ z φ xy x x y z 30 0 r < (2.47) 0 θ π (2.48) 0 φ < 2π (2.49) 1 e x = 0 0 r = xe x + ye y + ze z (2.50), e y = , e z = (2.51) e x e x = e y e y = e z e z = 1 (2.52) e x e y = e y e x = e y e z = e z e y = e z e x = e x e z = 0 (2.53) e µ e ν = δ µν (µ, ν = x, y, z) (2.54) 30 θ 2π π θ φ φ π

38 µ ν x y z Kronecker Kronecker delta 1 0 δ µν = { 1 µ = ν 0 µ ν (2.55) 1 x y z 3 1 e x e y e z e r e θ e φ e r e r (r) = r e r sin θ cos φ sin θ sin φ cos θ (2.56) r = re r (2.57) [2 11] cos θ cos φ sin φ e θ (r) = cos θ sin φ, e φ(r) = cos φ (2.58) sin θ 0 r = (x, y, z) r = (r, θ, φ) (2.50) (2.57) r = re r + θe θ + φe φ 32 r = (r, θ, φ) f(r) = f(x, y, z) π 2π dv f(r) = dx dy dz f(r) = dr dθ dφ J(r, θ, φ)f(r) (2.59) J(r, θ, φ) = (x, y, z) (r, θ, φ) = r2 sin θ (2.60) 33 f(r) r r dω f(θ, φ) = π 0 dθ 2π 0 dφ sin θ f(θ, φ) (2.61) f θ φ dω 2 Ω

39 2.5 Coulomb Coulomb ρ(r)d 3 r r ρ(r) (2.20) 2 (2.20) (2.32) Coulomb Coulomb k k 31

40 E q F = qe E(r) (2.20) Newton Newton 35 electrostatic field Coulomb 35 32

41 Coulomb [2 1] ( a, 0, 0) q 1 (0, 0, 0) q 2 (2a, 0, 0) q 3 a > 0 (a). (b). 0 (c). (b) [2 2] 2.3 xy f(x, y) dy = f(x, y) (2 2.1) dx [2 3] 2 2 (2.29) 2 2 r = (0, r, 0) [2 4]

42 2.12: [2 8] [2 5] xy a Q z [2 6] (2.37) [2 7] a Q [2 8] 2.12 (a) (b) z = 0 [2 9] (2.8) r 1 r 2 = r 1 r 2 cos θ (2 9.1) θ [2 10] (r, θ, φ) e r e θ e φ (2.58) 34

43 [2 11] cylindrical coordinate (x, y, z) z x y 2 x = ρ cos φ (2 11.1) y = ρ sin φ (2 11.2) (ρ, φ, z) 3 35

44 3 Gauss Gauss Gauss Gauss E(r) r line of electric force [2 2] 33 1 q R E = kq/r 2 2 4πR 2 E = 4πkq (3.1) 4πR 2 R Gauss (3.1) (3.1) (3.1) 1 2 q E E 36

45 3.1: 0 3.2: ds = dsn ds n xy z n 2 E ds = E nds E ds 0 7 S Φ = ds E(r) (3.2) S

46 surface integral ds ds(r) 9 Φ = ds E(r) (3.3) S 10 q S R (3.1) Φ flux of electric force ( ) flux Gauss Gauss Gauss Gauss s law ( S ) = 4πk ( S ) (3.4) S= V ds E(r) = 4πk dv ρ(r) (3.5) V Gauss 12 S V S ds S V S = V 13 V Gauss solid angle 10cm 1cm 2 100m 1cm 2 8 ds Φ ϕ φ π 3 r3 r 4πr 2 38

47 3.3: 1 θ θ r rθ 3.4: r 2 Ω r radian 2 1 1m 1cm 3.3 θ θ r θ rθ θ 0 2π 2π 2πr r θ Ω Ω steradian 14 r Ω r 2 Ω Ω 0 4π 4πr Gauss q V Gauss (3.5) V (3.5) Φ Gauss 14 sr 39

48 3.5 Φ ds ds ds ds ds E(r) = kq ds r 2 (3.6) ds E(r ) = kq ds r 2 (3.7) 0 15 ( ) ds dφ = kq r 2 ds r 2 0 dω = ds r 2 (3.8) = ds r 2 (3.9) Gauss ds r 2 Coulomb 1/r 2 2 Coulomb 3.3 Gauss Gauss Coulomb Gauss 3 1: Gauss a Q 2 5 r E(r) r Gauss 15 Gauss 40

49 3.5: ds r ds r r r ds ds ds 3.6: 3 2 Gauss (3.5) S ds E(r) = 4πr 2 E(r) (3.10) 4πr 2 r a Q 0 { 4πkQ r a 4πk dv ρ(r) = V 0 r < a (3.11) E(r) = { kq r 2 r a 0 r < a (3.12) :

50 2 a πa 2 E(z) πa 2 E( z) = 4πk πa 2 σ (3.13) z E(z) E(z) 2 3 E(z) = 2πkσ z z z ± (3.14) πa 2 E(z + ) πa 2 E(z ) = 0 (3.15) E(z + ) = E(z ) Gauss Coulomb Gauss [1] 3 1 E(r) = E( r ) r r 3 2 (2.32) 25 [2] V S = V [3] Gauss (3.5) [4] [5] V [1] [2] Gauss Gauss [1] 42

51 3.7: r = (x, y, z) dxdydz 3.4 Gauss Gauss (3.5) 1 Gauss 3.7 r 1 dx dy dz (x, y, z) (x + dx, y, z)... (x + dx, y + dy, z + dz) a ā 16 dydz ( ds E(r) dydz E x x + dx, y + dy 2, z + dz ) ( dydz E x x, y + dy 2 2, z + dz ) (3.16a) 2 a,ā dxdydz E x(x, y, z) x (3.16b) 2 E (x, y, z) 1 ( Ex (x, y, z) ds E(r) dxdydz + E y(x, y, z) + E ) z(x, y, z) x y z (3.17) 4πk dv ρ(r) 4πkdxdydz ρ(r) (3.18) 16 43

52 E(r) = 4πkρ(r) (3.19) E(r) divergence E(r) = E x x + E y y + E z z nabla ( = x, y, ) z (3.20) (3.21) (3.19) Gauss r = x 2 + y 2 + z 2 a r > a E(r) = kqr/r 3 ( x E(r) = kq x r 3 + y y r 3 + ) z z r 3 (3.22a) [ ( 3 = kq r 3 + x r x + y r y + z r ) ] 1 z r r 3 (3.22b) [ ( ) ] 3 x 2 = kq r 3 + r + y2 r + z2 3 r r 4 (3.22c) = 0 (3.22d) r < a 0 17 [2 7] 34 [3 5]

53 grad div rot = ( x, y, ) z (3.23) = r = r r ϕ(r) gradient ( ϕ(r) grad ϕ(r) = ϕ(r) = x, ϕ(r) y, ϕ(r) ) z (3.24) (3.25) ϕ(r) E(r) div E(r) = E(r) = E x(r) x + E y(r) y + E z(r) z E(r) rotation ( Ez (r) rot E(r) = E(r) = E y(r), E x(r) E z(r) y z z x, E y(r) E ) x(r) x y (3.26) (3.27) ϕ(r) (x, y, z) (r, θ, φ) 2.4 e r e θ e φ ϕ(r) = ϕ r e r + 1 ϕ r θ e θ + 1 E(r) = 1 r 2 r (r2 E r ) + 1 r sin θ r sin θ ( 1 E(r) = e r r sin θ θ (E φ sin θ) 1 +e φ ( 1 r r (re θ) 1 r ϕ φ e φ (3.28) θ (E θ sin θ) + 1 θ E r r sin θ ) φ E θ r sin θ φ E φ (3.29) ) ( 1 + e θ r sin θ φ E r 1 ) r r (re φ) (3.30) E(r) = E r e r + E θ e θ + E φ e φ 45

54 3.5 Gauss Gauss Gauss Gauss s theorem 18 Gauss S= V ds E(r) = V dv E(r) (3.31) E(r) S S V Gauss divergence theorem Gauss Gauss (3.5) dv E(r) = 4πk V V dv ρ(r) (3.32) V Gauss (3.19) Gauss 0 [3 7] 3.6 [ ] 3 A = A 1 A 2 A 3 = A 1e x + A 2 e y + A 3 e z (3.33) A B inner product scalar product A B = A 1 B 1 + A 2 B 2 + A 3 B 3 (3.34) 19 A B = A B cos θ AB (3.35) [2 9] A A A = A A2 2 + A2 3 θ AB A B A B 0 18 J. C. F. Gauss 1813 M. B. Ostrogradsky dot product 46

55 outer product vector product A 2 B 3 A 3 B 2 A B = A 3 B 1 A 1 B 3 (3.36) A 1 B 2 A 2 B 1 20 A B = A B sin θ AB (3.37) A B A x B y A B z 21 0 A B A B = B A (3.38) A = B A A = 0 (3.39) 0 22 (2.51) 29 e x e y = e z (3.40) e y e z = e x (3.41) e z e x = e y (3.42) x y z x A (B C) = B (C A) = C (A B) (3.43) A (B C) = B(A C) C(A B) (3.44) ( A) = ( A) 2 A (3.45) A B A B C A (B C) (A B) C A (B C) (A B) C 20 cross product

56 3.8: Gauss Φ = ds E(r) S= V q = dv ρ(r) 0 V 3.7 Gauss Gauss Gauss 3.8 Gauss III Gauss

57 II Gauss Gauss Coulomb Coulomb Coulomb Coulomb Gauss Gauss Gauss (3.31) f(x + ) f(x ) = x+ x dx df(x) dx (3.46) Gauss (3.31) 24 Gauss Gauss 2526 Gauss 0 Coulomb 2 2 Coulomb 2 Gauss 24 win win Gauss 5 49

58 3.9: [3 1] r dω Gauss 3.8 [3 1] a Q r E(r) = kq 4πa 2 ds(r 1 r r ) S r r 2 r r (3 1.1) ds r 3.9 r A B dω A B [3 2] Gauss Gauss [3 3] a Gauss σ 50

59 3.10: [3 2] q [3 4] 2 z = a σ z = a σ [3 5] [2 7] 34 Gauss [3 6] a Q ρ(r) E(r) [3 7] Gauss Gauss (3.31) 3.7 [3 8] (a). (A B) = B ( A) A ( B) (3 8.1) (b). (a) (A B) (3 8.2) 51

60 4 Gauss (3.5) 38 (3.19) 44 Coulomb 4.1 work F s W = F s ( ) ( ) = ( ) ( ) 2 ( ) 2 energy 1 2 Coulomb E(r) q C W = dr F (r) = q dr E(r) (4.1) C F (r) = qe(r) 3 line integral C N N C dr E(r) = lim N i=1 C N r i E(r i ) (4.2) q q q q ϕ = dr E(r) (4.3) C

61 [4 1] [4 1] Q 4.1 N r i r i θ i r i l i ϕ = lim = lim N N i=1 N N i=1 rb = dr kq r A r 2 = kq r B kq r A l i kq r 2 i r i kq r 2 i cos θ i (4.4a) (4.4b) (4.4c) (4.4d) l i r i r i = l i cos θ i r A r B r ϕ(r) r A r B ϕ = dr E(r) = ϕ(r B ) ϕ(r A ) (4.5) A B ϕ(r) electric potential electrostatic potential 0 C dr E(r) = 0 (4.6) C 0 0 ϕ(r) 53

62 4.1: Q q A B C C dr E(r) (4.7) C r = r(s) s 0 1 r r(0) r(1) s 1 dr E(r) = ds dr(s) E(r(s)) (4.8) ds dr(s) ds C 0 s (4.7) r = r(s, t) S ds E(r) = 1 0 ds 1 0 dt r(s, t) s r(s, t) t E(r(s, t)) (4.9) 4.2 q E(r) E = kq r r 3 = kq r (4.10) 54

63 1 r = 1 r r = r r 1 r r = r r 1 r 2 = r r 3 (4.11) r r = x 2 + y 2 + z 2 4 E(r) = ϕ(r) (4.12) ϕ(r) = kq r + const. (4.13) 2 r ρ(r) ϕ(r) = k d 3 r ρ(r ) r r + const. (4.14) (4.12) C dr E(r) = dr ϕ(r) (4.15) C 0 s 1 s r(s) dr E(r) = C = ds dr(s) ds ds d ds ϕ(r(s)) ϕ(r(s)) (4.16a) (4.16b) = ϕ(r(1)) ϕ(r(0)) (4.16c) (4.5) r(0) = r(1) (4.6) conservation force q

64 4 1: r + r q (4.14) ( ) 1 ϕ(r) = kq r r (4.17) r r 0 0 r + q r q ( ) 1 ϕ(r) = kq r r + 1 r r (4.18) [4 3] 4 2: a Q E(r) = { kq r r r a 3 0 r < a (4.19) r = r r a ϕ(r) = kq r + C 0 (4.20) C 0 0 C 0 = 0 r < a 0 ϕ(r) = C 1 (4.21) C 1 r = a δ r = a δ r = a + δ ϕ(r) = { kq r r a kq a r < a (4.22) 4 3: xy σ z E z (z) = 2πkσ z z (4.23) ϕ(r) = 2πkσ z + const. (4.24) 0 z = 0 z = const. 56

65 r r (4.5) E(r) r = ϕ(r + r) ϕ(r) = 0 (4.25) r [4 2] (4.6)

66 Coulomb Coulomb Laplace Poisson Poisson Laplace Poisson Coulomb [4 1] Q q 4.2 (a). (r, 0, 0) (r, 0, 0) (b). (r, 0, 0) ( r, 0, 0) (c). (r, 0, 0) ( r, 0, 0) [4 2] (4.6) (4.6) 9 III II 58

67 4.2: [4 1] Q xy (r, 0) q (a). x (b). (c). x y (b) [4 3] 4 1 r + = (a, 0, 0) r = ( a, 0, 0) xy ϕ dy = g(x, y) (4 3.1) dx g [2 2] 33 [4 4] a σ z E(r) ϕ(r) [4 5] 2 (a). x = a y = 0 λ x = a y = 0 λ 0 (b). ϕ xy 59

68 dr E(r) = 0 (5.1) C C 0 C dr dr E(r) 1 1 (5.1) Gauss (3.19) Stokes Stokes theorem 1 C= S dr E(r) = S ds E(r) (5.2) Stokes = r E(r) Stokes E(r) Gauss Stokes W. Thomson Kelvin 1850 G. G. Stokes Stokes H. Hankel 60

69 Stokes S S E(r) = 0 (5.3) E(r) = 0 (5.4) 2 E(r) = ϕ(r) (5.5) ϕ(r) (5.4) (5.4) (5.5) ϕ(r) ϕ(r) = 0 (5.6) ϕ 5.2 S= V ds E(r) = 4πk dv ρ(r) E(r) = 4πkρ(r) (5.7) V dr E(r) = 0 E(r) = 0 (5.8) C V C Gauss Coulomb 3 Gauss Gauss 3 1 Gauss (2.20) 21 Gauss

70 (2.20) Gauss Gauss y E(r) E(r) + C x (5.9) 0 C 2 0 Gauss 2 (5.9) 2 xy C > 0 1 y dr x = 2πr2 (5.10) 0 [5 1] r y x 0 = (5.11) C = 0 Gauss 5.3 Stokes * Stokes dr E(r) = ds E(r) (5.12) C= S S S C C S E(r) r 3 S C 1 1 C S Stokes Gauss V S = V

71 5.1: 5.2: Gauss (5.12) dxe x (r) + dye y (r + dxe x ) dxe x (r + dye y ) dye y (r) ( Ey (r) dxdy E ) x(r) (5.13) x y = dxdy ( E(r)) z (5.14) xy Stokes 5.4 Poisson * (5.5) Gauss (5.7) Poisson Poisson equation 2 ϕ(r) = 4πkρ(r) (5.15) 2 = = 2 x y z Laplacian 2 2 (5.5) Gauss Poisson Poisson 0 2 ϕ(r) = 0 (5.16) Laplace Laplace equation S. D. Poisson Poisson 2 Poisson 5.5 Poisson (4.13) 55 Poisson

72 5.5 [ ] Poisson ** ϕ 1 ϕ 2 Poisson 2 ϕ 1 (r) = 4πkρ(r) (5.17) 2 ϕ 2 (r) = 4πkρ(r) (5.18) 2 (ϕ 1 ϕ 2 ) = 0 (5.19) ϕ 1 ϕ 2 Laplace 0 ϕ 1 = ϕ 2 Poisson Gauss (3.31) 46 E = ϕ ϕ 4 ds ϕ(r) ϕ(r) = dv ( ϕ(r)) 2 + dv ϕ(r) 2 ϕ(r) (5.20) S= V V V Green Green s theorem 5 ϕ = ϕ 1 ϕ dv ( ϕ(r)) 2 = ds ϕ(r) ϕ(r) (5.21) V S= V 0 0 ϕ(r) = 0 (5.22) 0 ϕ(r) 0 ϕ 1 (r) = ϕ 2 (r) ϕ 1 (r) = ϕ 2 (r) [ ] ** q r = r 0 ρ(r) = q 4π 2 1 r r 0 (5.23) (4.13) 55 Poisson (5.15) 4 E ϕ Green 5 ϕ ψ E = ϕ ψ ϕ = ψ ψ 64

73 ( r r 0 = x y z 2 ) 1 (x x0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 (5.24) 0 0 r = r 0 regularization 2 1 r r 0 ( ) 2 = lim ϵ 0 x y z 2 (5.25) (x x0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 + ϵ 2 ϵ r = r r r 0 3ϵ 2 = lim ϵ 0 [(x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 + ϵ 2 ] 5/2 (5.26a) = { 0 r r0 (5.26b) 3 lim ϵ 0 ϵ r = r d 3 r 2 1 = lim r r 0 ϵ 0 = lim ϵ 0 0 4π ϵ 0 = lim d 3 3ϵ 2 r [(r r 0 ) 2 + ϵ 2 ] 5/2 (5.27a) dr 4πr 2 3ϵ 2 (r 2 + ϵ 2 ) 5/2 (5.27b) (5.27c) = 4π (5.27d) 6 1 ρ(r) = q 4π 2 1 r r 0 (5.28) r r 0 0 r = r 0 q Dirac δ 3 (r r 0 ) = 1 4π 2 1 r r 0 (5.29) 0 0 r ρ(r) = qδ 3 (r r 0 ) 6 r = ϵ tan θ 7 Kronecker 8 P. A. M. Dirac 65

74 5.7 Gauss Gauss Stokes 9 Poisson Gauss Poisson 10 Coulomb Poisson 2 I Coulomb II 5.8 [5 1] (5.10) 9 Stokes Stokes

75 5.3: [5 4] C x 0 > a [5 2] 0 0 (5.9) [5 3] C dr E(r) = 0 (5 3.1) C E(r) = 0 (5 3.2) Gauss 43 [5 4] q C 5.3 dr E(r) 0 C 67

76 6 6.1 conductor II Coulomb

77 electrostatic induction 6 1: 6.1 E 0 E σ σ 0 0 = E 0 2πkσ 2πkσ (6.1) σ = E 0 4πk (6.2) E 0 + 2πkσ 2πkσ = E 0 (6.3) electric discharge 69

78 6.1: E 0 6.2: 6.3: 6.4: Gauss E (r) = 4πkσ(r) (6.4) E (r) r Coulomb Gauss S 5 Coulomb 70

79 6 2: 0 q S 1 S 2 S 3 Gauss S 1 ds E(r) = 4πkq (6.5) S 2 ds E(r) = 0 (6.6) S 3 ds E(r) = 4πkq (6.7) S 1 S 3 S q 0 q q electric shielding x < 0 0 earth 0 a r = (a, 0, 0) q x = 0 0 q Gauss q r + = (a, 0, 0) r = ( a, 0, 0) r + q r q x = 0 Laplace 71

80 6.5: x 0 ( ) 1 ϕ(r) = kq (x a)2 + y 2 + z 1 2 (x + a)2 + y 2 + z 2 (6.8) x < 0 ϕ(r) = 0 q ( a, 0, 0) 6.5 method of images σ(y, z) = 1 4πk ϕ(r) x = qa x=0 2π 1 (a 2 + y 2 + z 2 ) 3/2 (6.9) σ σ 72

81 6.6: 6.7: z = ±d/2 z ( z d/2 E z (r) = 2πkσ z d/2 z + d/2 ) = z + d/2 0 z > d 2 4πkσ d 2 < z < d 2 0 z < d 2 (6.10) condenser, capacitor 6 z < d 2 ϕ(r) = 4πkσz + const. (6.11) ϕ ± ϕ + ϕ = 4πkσd (6.12) b a a > b Q Q Gauss r E r = kq r 2 (6.13) ϕ = kq r + const. (6.14) 6 capacitor 1745 E. G. von Kleist 1746 P. van Musschenbroek Leiden jar van Musschenbroek 7 73

82 6.8: p 2 ( 1 ϕ b ϕ a = kq b 1 ) a (6.15) ±Q ±Q Q /Q C = Q ϕ + ϕ (6.16) capacitance 2 C = 1 k ab a b (6.17) a b 2 S σ = Q/S C = 1 S 4πk d (6.18) r + = (0, 0, d/2) q r = (0, 0, d/2) q r x kq E(r) = [ x 2 + y 2 + ( ) z d 2 ] 3/2 y 2 z d 2 kq [ x 2 + y 2 + ( z + d 2 ) 2 ] 3/2 x y z + d 2 (6.19) 74

83 0 asymptotic form r = x 2 + y 2 + z 2 d 8 d/r d Taylor d 0 0 d E(r) kq ( d r 3 e z + 3dz ) r 5 r (6.20) r 3 z r + q r q 9 ( E(r) k p r 3 + 3p r ) r 5 r (6.21) p = q(r + r ) (6.22) ±q electric dipole electric dipole moment p r F 0 10 x f(x) = x + e x (6.23) g(x) = x2 + 1 x + 2 h(x) = 1 + ln x + 1 x (6.24) (6.25) f x g h x 1 f(x) e x (6.26) g(x) x (6.27) h(x) ln x (6.28) x

84 6.9: Cavendish a b a > b (i). (ii). (iii). (ii) 6.5 Cavendish ** Coulomb Coulomb 2 J. Priestley H. Cavendish Maxwell Cavendish 2 2 [6 7] u f Coulomb Coulomb 6.9(i) a b a > b Q σ a = Q 4πa (ii) 0 Coulomb σ a σ b [6 7] ( ) ϕ(r) = 2πσ a f(r + a) f( r a ) ar ( ) + 2πσ b f(r + b) f( r b ) br (6.29) 10 Priestley Coulomb 11 Cavendish Coulomb Ohm Cavendish

85 r ϕ(a) = ϕ(b) ϕ 0 ( ) ϕ 0 = 2πσ a a 2 f(2a) f(0) ϕ 0 = 2πσ a ab + 2πσ b ab ) ( f(a + b) f(a b) ( ) f(a + b) f(a b) ( ) + 2πσ b b 2 f(a + b) f(0) (6.30) (6.31) ϕ 0 σ a σ b ( ) ( ) ( ) pa p ab σ a = ϕ 0 1 2π 1 p ab p b σ b (6.32) p a = f(2a) f(0) f(2b) f(0) f(a + b) f(a b) a 2, p b = b 2, p ab = ab (6.33) (6.32) ( σ a σ b ) = ϕ 0 2π = ϕ 0 2π = ϕ 0 2π ( pa p ab p ab 1 p b ) 1 ( 1 1 ) ( pb p ab p a p b p 2 ab p ab p a ( ) 1 pb p ab p a p b p 2 ab p a p ab ) ( 1 1 ) (6.34a) (6.34b) (6.34c) ϕ 0 f 6.9(iii) 0 (6.29) 0 = 2πσ ap a + 2πσ bp ab (6.35) ϕ(b) = 2πσ ap ab + 2πσ bp b (6.36) σ a σ b σ a = p ab p a σ b (6.37) σ a σ b ( ) p a p2 ab σ b ϕ(b) = 2π ( = 1 p ab p a ( = 1 a b p a ) ϕ 0 f(a + b) f(a b) f(2a) f(0) ) ϕ 0 (6.38a) (6.38b) (6.38c) (iii) (ii) ϕ 0 f 77

86 f Coulomb [6 7] Coulomb u(r) = C R (6.39) (4.13) 55 Coulomb (6.38) ϕ(b) = 0 Coulomb u(r) = C R 1+δ (6.40) δ = 0 Coulomb δ (6.38) [ (a ϕ(b) = 1 a ) 1 δ ( ) ] 1 δ + b a b ϕ 0 b 2a 2a (6.41) δ 0 a b Coulomb Coulomb k 13 Cavendish δ < 1 50 Cavendish Maxwell δ < Gauss [6 3] [6 4]

87 +q q II 15 Wikipedia

88 6.7 [6 1] x = 0 (6.9) [6 2] r r = (a, 0, 0) q a > r (a). 0 r = (a, 0, 0) q a q (b). [6 3] a b (a). E(r) ϕ(r) Q (b). C [6 4] a b [6 5] (a). (6.21) (b). (a) p E 0 (c). (b) r [6 6] (a). (6.21) (b). [6 5](b) 80

89 [6 7] a σ a (a). r r = r ϕ(r) = σ a S a dω(r a ) u( r r a ) (6 7.1) r a u(r) (b). f(r) Ru(R) ϕ(r) = 2πσ ( ) a f(r + a) f( r a ) (6 7.2) ar f(r) = dr Ru(R) (6 7.3) 81

90

91 II

92

93 7 Ohm electric current t r ds J(r, t) ds J(r, t) current density S I(t) I(t) = S ds J(r, t) (7.1) ( ) ( ) 1 ( ) ( ) 2 ( ) 1 Gauss J(r, t) ρ(r, t) V S I(t) = S= V ds J(r, t) (7.2) I(t) Q(t) = dv ρ(r, t) (7.3) V 2 I(t) = d Q(t) (7.4) dt 1 I 2 Gauss Gauss 85

94 7.1: J(r, t) S I(t) d dv ρ(r, t) + ds J(r, t) = 0 (7.5) dt V S= V Gauss (3.31) 46 3 V dv ( ) ρ(r, t) + J(r, t) = 0 (7.6) t Gauss ρ(r, t) + J(r, t) = 0 (7.7) t continuity equation (7.5) (7.7) conservation law E r(t) E(r(t)) = E(r(0)) Q(t) Q(t) III 3 3 (7.7) Gauss Stokes 86

95 7.2: S I = ds J(r) S ρ(r, t) J(r, t) II steady state 1 steady current J(r) = 0 (7.8) J = J(r) (7.8) Gauss ds J(r) = 0 (7.9) S S [7 2] I = ds J(r) (7.10) S S J S I = JS (7.11) (7.10) 87

96 0 7.2 Ohm Ohm Ohm Ohm s law 1826 G. S. Ohm G. R. Kirchhoff 4 voltage V I V = RI (7.12) R electrical resistance Ohm Ohm I V 5 R = ρ l S (7.13) l S ρ electrical resistivity 6 7 (7.11) Ohm V = ρlj (7.14) V V = El (7.15) 8 E Ohm 4 Cavendish

97 Ohm E = ρj (7.16) Ohm E(r) = ρj(r) (7.17) J(r) = σe(r) (7.18) σ = 1/ρ electrical conductivity (7.17) (7.18) Ohm Ohm Ohm 10 0 (7.8) 0 = J(r) = σ E(r) (7.19) Gauss 0 J(r) = 0 (7.20) electromotive force J(r) = σ (E(r) + E emf (r)) (7.21) 2 E emf (r) V emf = dr E emf (r) (7.22) C 12 III 9 Ohm Ohm 12 89

98 7.3 Ohm Ohm + I Ohm Coulomb (!?) 90

99 14 Ohm Ohm ρ v J = ρv (7.23) 15 Ohm m dv dt = qe (7.24) 0 16 m dv dt = q ( E nq σ v ) (7.25) nq 2 /σ Ohm n 0 Ohm 0 17 Ohm [7 1]

100 18 Ohm Ohm Ohm Ohm Coulomb Ohm Ohm Ohm 19 Ohm Ohm Ohm 2 Ohm 20 I Ohm 19 Brown Einstein 20 steady state stationary state steady stationary steady state 21 92

101 7.3: [7 3] N 22 II 7.4 [7 1] ρ(r, t) = ρ 0 (r vt) (7 1.1) v ρ 0 (r) (7.23) [7 2] S (7.10) [7 3] Kirchhoff 7.3 N 0 Kirchhoff Kirchhoff s current law N I i = 0 (7 3.1) i=1 (7.10) I i 22 93

102 [7 4] a b b > a σ Ohm J(r) = σe(r) R [7 5] (7.25) v i (t) = dri(t) dt r i (t) t E 94

103 8 Ampère 8.1 Ampère H. C. Ørsted Ørsted F. Arago A.-M. Ampère Ampère 2 I 1 I 2 r f = 2k m I 1 I 2 r (8.1) 2k m Ampère Ampère force Coulomb 1 Cavendish Coulomb 1785 Poisson 1812 Ohm 1826 Ohm 2 2 Coulomb Coulomb 1750 J. Michell 1760 T. Mayer Coulomb 10 3 J.-B. Biot F. Savart 4 2 (8.2) 95

104 8.1: Ampère d 2 F 12 = k m I 1 I 2 dr 1 dr 2 r 1 r 2 r 1 r 2 3 (8.2) 2 dr 2 1 dr 1 r 1 r 2 (8.1) [8 1] dr 1 dr 2 (8.1) [A] I 1C = 1A s Ampère (8.1) 1m 1m N 1A k m = N A 2 = N s 2 C 2 Coulomb k N m 2 C 2 k m m 2 s 2 2 SI m kg s A MKSA C A A C 5 d 2 F

105 A (B C) = (A C)B (A B)C (8.3) (8.2) ( d 2 F 12 = k m I 1 I 2 [dr 1 dr 2 r ) ] 1 r 2 r 1 r 2 r 1 r 2 3 dr 1 r 1 r 2 3 dr 2 2 r 1 0 C dr 1 (8.4) r 1 r 2 r 1 r 2 3 = 1 dr 1 r1 C r 1 r 2 = 0 (8.5) 0 0 Ampère d 2 F 12 = k m I 1 I 2 dr 1 ( dr 2 r ) 1 r 2 r 1 r 2 3 Ampère H. G. Grassmann 1845 Ampère (8.6) magnetic field (8.6) d 2 F 12 = I 1 dr 1 db (8.7) B static magnetic field 7 db = k m I 2 dr 2 r 1 r 2 r 1 r 2 3 (8.8) r 2 r 1 r 1 8 N A 1 m 1 [T] Tesla electrostatic field magnetostatic field Magnetostatics 8 Ampère 9 B B = µ 0 H = 4πk m H H H 10 97

106 8.3 Biot Savart I r B B(r) = k m I dr r r r r 3 (8.9) C Biot Savart Biot Savart law Biot Savart Ørsted Ampère 11 J(r) B(r) = k m d 3 r J(r ) r r r r 3 (8.10) Idr J(r )d 3 r 0 (8.10) (8.10) (2.20) : z I z r x B(r, 0, 0) = k m I dz e z re x ze z re x ze z 3 = k mi dz r (r 2 + z 2 ) e 3/2 y = 2k mi e y (8.11) r Ampère (r, 0, 0) y 8.2 e φ (r) 8 2: a I xy z (x, y, z) = (ρ cos φ, ρ sin φ, z) 11 98

107 8.2: 8 1 I B 8.3: 8 2 I B B(0, 0, z) = k m I 2π 0 adφ e φ ze z ae ρ ze z ae ρ 3 = k mi 2π 0 dφ a (z 2 + a 2 ) 3/2 (ze ρ + ae z ) (8.12) e φ = ( sin φ, cos φ, 0) e ρ = (cos φ, sin φ, 0) z 8.3 a 2 B(0, 0, z) = 2πk m I (z 2 + a 2 ) e 3/2 z (8.13) z [8 4] 8.5 Lorentz Lorentz Ampère (2.19) 21 (8.7) J (7.23) 91 Idr Idr B ρvdv B = ρdv v B 12 ρdv q qv B 12 (7.23) 91 99

108 m d2 r(t) dt 2 = qe(r(t)) + q dr(t) dt Lorentz Lorentz force 13 B(r(t)) (8.14) : m q m d2 r(t) dt 2 = qe (8.15) E r 0 t = 0 v 0 t = 0 r(t) = qe 2m t2 + v 0 t + r 0 (8.16) 8 4: m d2 r(t) dt 2 = q dr(t) dt 15 dr(t) dt B (8.17) = q m (r(t) r 0) B + v 0 (8.18) r 0 v z dx(t) = q dt m (y(t) y 0)B + v 0x (8.19) dy(t) = q dt m (x(t) x 0)B + v 0y (8.20) dz(t) = v 0z dt (8.21) 13 H. Lorentz Lorentz

109 z z(t) = v 0z t + z 0 (8.22) x y y x d 2 x(t) dt 2 = qb [ q ] m m (x(t) x 0)B + v 0y = q2 B 2 ( m 2 x(t) x 0 m ) qb v 0y (8.23) x(t) x 0 m ( ) ( ) qb qb qb v 0y = C 1 sin m t + C 2 cos m t (8.24) C 1 C 2 r(t) = r 0 dr(t) dt = v 0 t = 0 t=0 1 t = 0 m qb v 0y = C 2 (8.25) v 0x = qb m C 1 (8.26) C 1 = v 0x ω C 2 = v 0y ω ω = qb m (8.27) (8.28) (8.29) (8.19) y(t) x(t) = v 0x ω sin ωt + v 0y ω (1 cos ωt) + x 0 (8.30) y(t) = v 0y ω sin ωt v 0x ω (1 cos ωt) + y 0 (8.31) xy ω Larmor Larmor frequency z ( ) 2 dr(t) = v0 2 (8.32) dt 101

110 8.6 Ampère Biot Savart Lorentz 16 Biot Savart Coulomb Ampère 10 Grassmann 1845 (8.6) 1876 Clausius Grassmann Grassmann 17 Grassmann J. A. Fleming Stokes Grassmann Grassmann 102

111 k m [8 1] Ampère (8.2) (8.1) [8 2] Ampère (8.6) (8.2) [8 3]

112 8.4: [8 3] I I 0 I 0 [8 4] 8 2 [8 5] a b c z a < b < c I I B(r) [8 6] [8 7] [8 8] (8.30) (8.31) f(x, y) = 0 z [8 9] m d2 r(t) dt 2 = qe + q dr(t) dt B (8 9.1) 104

113 0 0 (a). B = 0, E = 0 (b). B = B 0 0 B, E = E E 0 0, r(0) =, r(0) = ,, dr(t) dt = t=0 dr(t) dt = t=0 v E B v 0 [8 10] 2 xy m q v F = m τ v τ (a). E (b). (a) B = Be z x x (c). E = 0 B = Be z 105

114 9 Coulomb Gauss Poisson 9.1 Biot Savart B(r) = k m d 3 r J(r ) r r r r 3 (9.1) Gauss Gauss line of magnetic force S magnetic flux Φ = ds B(r) (9.2) S [Wb] Weber 1Wb = 1T m 2 S 0 Gauss S ds B(r) = 0 (9.3) Gauss B(r) = 0 (9.4) 106

115 (9.1) B(r) = k m d 3 r J(r ) r r r r 3 (9.5) (3 8.1) 51 B(r) = k m d 3 r J(r ) ( r r ) r r r 3 (9.6) r r r r r 3 = r r 1 r r = 0 (9.7) 1 Gauss Gauss Ampère 0 0 C dr B(r) (9.8) C a Ampère dr B(r) = 2πa 2k mi = 4πk m I (9.9) a C I = ds J(r) S C= S dr B(r) = 4πk m S ds J(r) (9.10) 2 1 r = r

116 0 C S Ampère Ampère s law Stokes ds B(r) = 4πk m ds J(r) (9.11) S S S B(r) = 4πk m J(r) (9.12) Ampère (9.1) [9 2] C= S ds B(r) = 0 B(r) = 0 (9.13) S dr B(r) = 4πk m ds J(r) B(r) = 4πk m J(r) (9.14) S S S S= V ds E(r) = 4πk dv ρ(r) E(r) = 4πkρ(r) (9.15) V dr E(r) = 0 E(r) = 0 (9.16) C 0 / / Gauss 9.2 * * Gauss Poisson 108

117 0 ϕ(r) ϕ(r) = 0 0 Gauss Biot Savart 3 B(r) = k m d 3 r J(r 1 ) r (k r r = m d 3 r J(r ) ) r r (9.17) r r 0 A(r) = 0 Gauss B(r) = A(r) (9.18) Gauss 4 A(r) vector potential 5 Biot Savart A(r) = k m d 3 r J(r ) r r (9.19) 3 3 B = 0 Poisson (3.45) 47 B(r) = ( A(r)) = ( A(r)) 2 A(r) (9.20) (9.14) ( A(r)) 2 A(r) = 4πk m J(r) (9.21) Poisson 1 (9.19) (9.21) 1 A(r) = k m d 3 r J(r ) r r = k m d 3 r J(r 1 ) r r r = k m d 3 r J(r 1 ) r r r = k m d 3 r ( r J(r 1 )) r r k m d 3 r J(r ) r r r (9.22a) (9.22b) (9.22c) (9.22d) 3 A ϕ(r) = Aϕ(r) A 4 A (9.18) B = 0 Helmholtz 5 ϕ(r) scalar potential 109

118 r r 1 (7.8) Gauss (3.31) 46 A(r) = k m ds J(r ) r r (9.23) S 0 2 A(r) = 4πk m J(r) (9.24) 3 Poisson B(r) = A(r) (9.25) 2 A(r) = 4πk m J(r) (9.26) ** (9.26) (9.19) A(r) = 0 6 A(r) A (r) = A(r) + A 0 (r), A 0 (r) = 0 (9.27) 0 ϕ(r) ϕ (r) = ϕ(r) + ϕ 0 (r), ϕ 0 (r) = 0 (9.28) ϕ 0 (r) 0 A 0 A 0 (r) = χ(r) (9.29) χ(r) A(r) A (r) = A(r) + χ(r) (9.30) 6 110

119 gauge transformation ϕ(r) A(r) ϕ (r) A (r) gauge invariance A (r) = 0 χ(r) 2 χ(r) = A(r) (9.31) χ(r) 7 Poisson 8 A (r) (9.26) A(r) = 0 (9.26) A(r) = 0 (9.26) (9.19) A(r) = 0 (9.19) : Gauss : Ampère (9.18) (9.19) 7 gauge fixing 8 Poisson 111

120 III 9.4 [9 1] (9.13) (9.14) Biot Savart (9.1) [9 2] (9.1) (9.12) (5.27) 65 [9 3]

121 [9 4] (7.19)

122 10 magnet a I B(r) = k m I dr r r r r 3 (10.1) C C r a 0 B(r) A(r) B(r) = A(r) A(r) dr A(r) = k m I C r r (10.2) r = r r = r 1 r r = = 1 (r r ) 2 1 r2 2r r + r 2 (10.3a) (10.3b) = 1 (10.3c) r 1 2 r r r + r 2 2 r 2 1 r ( ) (10.3d) 1 r r r 2 1 (1 + r ) r r r 2 (10.3e) r 2 A(r) k ( mi dr 1 + r ) r r r 2 C (10.4) 1 B(r) = A(r) A(r) (10.2) 114

123 xy cos θ sin θ r = a sin θ, dr = adθ cos θ (10.5) 0 0 sin θ A(r) k mia r = k mia r 2π 0 y x 0 dθ πa r 2 cos θ 0 [ 1 + a ] r 2 (x cos θ + y sin θ ) (10.6a) (10.6b) = k m ISe z r r 3 (10.6c) S = πa 2 m = ISe z = IS (10.7) S 2 (10.6) B(r) = (k m m r ) r 3 = k m m r r 3 k mm r ( r 3 = k m m r 3 + 3m r ) r r 5 (10.8a) (10.8b) (10.8c) (3 8.2) 51 3 (6.21) 75 magnetic dipole moment m magnetic dipole m = I ds (10.9) S S 10.2 Coulomb 2 115

124 10.1: (a) (b) (c) (d) Coulomb p = qd d 0 1 q q 10.1 r 0 r (6.21) 75 dϕ(r) = k p (r r 0) r r 0 3 (10.10) p = qdr 0 r 0 r r 0 ϕ(r) = kq dr 0 r r 0 3 = kq 1 dr 0 r0 r r 0 = kq C C ( ) 1 r r + 1 r r (10.11) 3 Coulomb 0 magnetic charge Coulomb magnetic potential r 0 m = q m dr 0 ( ) r r 0 ϕ m (r) = k m q m dr 0 C r r 0 3 = k 1 mq m r r + 1 r r ( r r+ B(r) = ϕ m (r) = k m q m r r + 3 r r ) r r 3 Coulomb q m 3 C (10.12) (10.13) 116

125 10.2: 10.3: 10.3 N S N S 4 solenoid magnetic moment [8 3]

126 0 magnetic monopole Gauss (9.4) [10 1] (10.9) m = I r dr (10 1.1) 2 C S C [10 2] (10.8) 7 8 spin 118

127 [10 3] n 119

128

129 III

130

131 11 III 11.1 II Lorentz Lorentz B v q B v qv B 1 0 = qe + qv B (11.1) E = v B (11.2) σ Ohm 11.2 J E + v B = 1 σ J (11.3) V emf = dr v B (11.4) C 1 v 2 (11.3) 7 123

132 11.1: 11.2: ab abcd ab 11.3: dr v v dr v dr (11.4) (3.43) 47 V emf = dr v B (11.5) C 11.3 v dr V emf (t) = d ds B (11.6) dt S(t) 11.2 abcd S(t) (9.2) 106 Lorentz Lorentz S(t) 124

133 II Faraday Lorentz Faraday (11.6) 4 V emf (t) = d ds B(r, t) (11.7) dt S S Faraday electromagnetic induction Faraday Faraday s law of induction 5 [11 2] C 3 J. Henry Faraday Henry [H] 4 B(r, t) V emf (t) Faraday Henry 1830 (11.7) F. E. Neumann 1845 (11.7) E. Lenz Lenz 1834 Lenz Neumann Faraday Neumann 125

134 C= S dr E(r, t) = d dt S ds B(r, t) (11.8) S C C Faraday Stokes (5.12) 62 ds ( E(r, t) + t ) B(r, t) = 0 (11.9) S 6 Faraday E(r, t) + B(r, t) = 0 (11.10) t Faraday 7 Faraday 6 7 Faraday Lorentz 126

135 d V emf (t) = k emf ds B(r, t) (11.11) dt S k emf Faraday E(r, t) + k emf B(r, t) = 0 (11.12) t k emf k m k emf Lorentz (8.14) 100 Lorentz (11.6) k emf 1 principle of relativity k emf = 1 Faraday (11.8) Faraday 11.4 [11 1]

136 11.4: [11 2](a) oa oabcde 11.5: [11 2](b) o abcdo [11 2] (a) (b)

137 12 Maxwell Maxwell ds E(r) = 4πk dv ρ(r) E(r) = 4πkρ(r) (12.1) S= V V dr E(r) = 0 E(r) = 0 (12.2) C ds B(r) = 0 B(r) = 0 (12.3) S dr B(r) = 4πk m ds J(r) B(r) = 4πk m J(r) (12.4) C= S Faraday C= S dr E(r, t) = d dt S S ds B(r, t) E(r, t) + B(r, t) = 0 (12.5) t (12.2) 7 ρ(r, t) + J(r, t) = 0 (12.6) t (12.1) (12.4) t E(r, t) = 4πk ρ(r, t) t (12.7) B(r, t) = 4πk m J(r, t) (12.8)

138 0 E(r, t) = 4πkρ(r, t) (12.9) B(r, t) k m k t E(r, t) = 4πk mj(r, t) (12.10) 2 1 ds E(r, t) = 4πk dv ρ(r, t) (12.11) S= V V ( dr B(r, t) = 4πk m ds J(r, t) + 1 ) E(r, t) (12.12) 4πk t C= S S displacement current J d (r, t) = 1 4πk te(r, t) Maxwell (12.3) (12.2) (12.5) (12.5) E(r, t) + B(r, t) = 0 (12.13) t (12.3) ds B(r, t) = 0 B(r, t) = 0 (12.14) S Faraday (12.5) z < 0 z Q(t) = It Gauss (12.9) E(r, t) = kit r r 3 (12.15) 1 130

139 J d (r, t) = 1 4πk I E(r, t) = t 4π (12.12) r r 3 (12.16) B(r, t) = B(ρ, z, t)e φ (12.17) z ρ z e ρ 0 Gauss (12.14) e z 0 2 z ρ k m I S 2πρB(ρ, z, t) = ds 4πk m I + k m I S ds S r r 3 z > 0 r r 3 z < 0 (12.18) r ρ 2π ds r 3 = ρ dρ z dφ (12.19a) (ρ 2 + z 2 ) 3/2 0 ρ 2 0 dρ 2 = πz (12.19b) 0 (ρ 2 + z 2 ) ( 3/2 ) = 2π z z 1 (12.19c) z ρ2 + z 2 B(ρ, z, t) = k mi ρ ( 1 ) z ρ2 + z 2 (12.20) z > 0 z < 0 z = Maxwell ds E(r, t) = 4πk dv ρ(r, t) (12.21) S= V V dr E(r, t) = d ds B(r, t) C= S dt S (12.22) ds B(r, t) = 0 (12.23) S ( dr B(r, t) = 4πk m ds J(r, t) + 1 ) E(r, t) 4πk t (12.24) C= S 2 [12 1] S 131

140 12.1: z < 0 I Q(t) = It E(r, t) = 4πkρ(r, t) (12.25) E(r, t) + B(r, t) = 0 t (12.26) B(r, t) = 0 (12.27) B(r, t) k m k t E(r, t) = 4πk mj(r, t) (12.28) Maxwell Maxwell equation Maxwell (ρ, J, E, B) 2 Maxwell Maxwell Maxwell 4 (12.28) k m k t E(r, t) = 4πk m J(r, t) = 4πk m ρ(r, t) (12.29) t ( ) E(r, t) 4πkρ(r, t) = 0 (12.30) t 132

141 Maxwell 1 (12.25) 0 2 (12.26) B(r, t) = 0 (12.31) t 3 0 Maxwell 1 3 Maxwell Maxwell (3.45) 47 ( E(r, t)) 2 E(r, t) = B(r, t) (12.32) t ( B(r, t)) 2 B(r, t) = k m k Maxwell E(r, t) + k m k 2 B(r, t) + k m k 0 2 t E(r, t) + 4πk m J(r, t) (12.33) t 2 E(r, t) = 4πk ρ(r, t) 4πk m J(r, t) t (12.34) 2 t 2 B(r, t) = 4πk m J(r, t) (12.35) 2 E(r, t) + k m k 2 B(r, t) + k m k 2 E(r, t) = 0 t2 (12.36) 2 B(r, t) = 0 t2 (12.37) wave equation k v = (12.38) k m 3 133

142 k k m 1/ Maxwell v c m/s 1849 H. Fizeau 1856 R. Kohlrausch W. E. Weber k k m Maxwell v c 4 k k m k = 1 4πϵ 0 (12.39) k m = µ 0 4π ϵ 0 permittivity µ 0 permeability 5 (12.40) ϵ 0 µ 0 = 1 c 2 (12.41) Maxwell 2 Maxwell E(r, t) = 1 ϵ 0 ρ(r, t) (12.42) E(r, t) + B(r, t) = 0 t (12.43) B(r, t) = 0 (12.44) B(r, t) 1 c 2 t E(r, t) = µ 0J(r, t) (12.45) 7 Ohm Maxwell Maxwell t t E(r, t) = 1 ϵ 0 ρ(r, t) (12.46) E(r, t) B(r, t) = 0 t (12.47) B(r, t) = 0 (12.48) B(r, t) + 1 c 2 t E(r, t) = µ 0J(r, t) (12.49) ρ(r, t) = ρ(r, t) (12.50) Faraday 5 134

143 Maxwell J(r, t) = J(r, t) (12.51) E(r, t) = E(r, t) (12.52) B(r, t) = B(r, t) (12.53) 12.3 law of conservation of energy v m q 1 2 mv2 Lorentz 6 v(t) = dr(t) dt ( ) d 1 dt 2 mv2 (t) = mv(t) v(t) (12.54a) = qv(t) (E(r(t), t) + v(t) B(r(t), t)) (12.54b) = qv(t) E(r(t), t) (12.54c) d ( ) 1 dt 2 m ivi 2 (t) = q i v i (t) E(r i (t), t) = dv J(r, t) E(r, t) (12.55) i i (7.23) 91 J(r, t) E(r, t) Maxwell 4 (12.45) ( ) 1 J E = B ϵ 0 µ 0 t E E = ϵ 0 E t E + 1 E B (12.56) µ 0 (3 8.1) 51 E B = B E E B (12.57) 6 Lorentz 135

144 Maxwell 2 (12.43) J E = ϵ 0 E t E + 1 µ 0 B E 1 µ 0 E B = ϵ 0 E t E 1 B µ 0 t B 1 E B µ 0 = ( ϵ0 t 2 E2 + 1 ) B 2 1 E B 2µ 0 µ 0 (12.58a) (12.58b) (12.58c) ( ϵ0 t 2 E2 (r, t) + 1 ) ( ) 1 B 2 (r, t) + E(r, t) B(r, t) = J(r, t) E(r, t) (12.59) 2µ 0 µ 0 (12.55) V ( d ϵ0 dv dt 2 E2 + 1 ) B 2 + dv 1 E B = d 2µ 0 µ 0 dt T (12.60) V T 2 Gauss ( d ϵ0 dv dt 2 E2 + 1 ) B 2 + 1µ0 ds E B = d 2µ 0 dt T (12.61) V 0 E E(t) = dv V V S d (T + E) = 0 (12.62) dt ( ϵ0 2 E2 (r, t) + 1 ) B 2 (r, t) 2µ 0 (12.63) (12.59) E(r, t) + S(r, t) = J(r, t) E(r, t) (12.64) t E(r, t) = ϵ 0 2 E2 (r, t) + 1 2µ 0 B 2 (r, t) (12.65) S(r, t) = 1 µ 0 E(r, t) B(r, t) (12.66) E(r, t) S(r, t) Poynting Poynting vector 7 Poynting (12.64) Poynting J. H. Poynting Pointing 136

145 Poynting (12.64) [12 4] (12 4.1) 12.4 ** I II E(r) = ϕ(r) (12.67) B(r) = A(r) (12.68) ϕ A electromagnetic potential Maxwell (12.44) B(r, t) = A(r, t) (12.69) 2 (12.43) E(r, t) = ϕ(r, t) + Ẽ(r, t) (12.70) 2 Ẽ(r, t) + A(r, t) = 0 (12.71) t Ẽ(r, t) = A(r, t) (12.72) t 137

146 Maxwell 2 3 E(r, t) = ϕ(r, t) A(r, t) t (12.73) B(r, t) = A(r, t) (12.74) ϕ(r) t A(r, t) = 1 ϵ 0 ρ(r, t) (12.75) ( A(r, t)) 2 A(r, t) + 1 c 2 t ϕ(r, t) + 1 c 2 2 t 2 A(r, t) = µ 0J(r, t) (12.76) 2 ϕ(r, t) c 2 t 2 ϕ(r, t) ( 1 t c 2 t 2 A(r, t) + 1 c 2 2 A(r, t) + t2 ) ϕ(r, t) + A(r, t) = 1 ρ(r, t) (12.77) ϵ 0 ( 1 c 2 ϕ(r, t) + A(r, t) t ) = µ 0 J(r, t) (12.78) Maxwell 9 A = 0 1 c 2 ϕ(r, t) + A(r, t) = 0 (12.79) t 8 2 ϕ(r, t) c 2 t 2 ϕ(r, t) = 1 ρ(r, t) ϵ 0 (12.80) 2 A(r, t) c 2 t 2 A(r, t) = µ 0J(r, t) (12.81) (12.74) A(r, t) A (r, t) = A(r, t) + χ(r, t) (12.82) 0 (12.73) ϕ(r, t) ϕ (r, t) = ϕ(r, t) χ(r, t) (12.83) t χ(r, t) (12.79) 12.5 Maxwell 8 Lorenz L. Lorenz Lorentz 138

147 12.2: Maxwell Maxwell Maxwell 12.2 Faraday Faraday Faraday 10 Einstein 139

148 I II Gauss Coulomb Biot Savart Maxwell Maxwell 1 Maxwell 1 2 Newton Newton Maxwell r t 11 0 Maxwell Maxwell

149 12.3: [12 2] V (t) I(t) S 1 S [12 1] e ρ 0 e z [12 2] 12.3 (12.12) S 1 S 2 (6.18) 74 [12 3] Maxwell [12 4] (a). ϵ 0 dv E 2 (r) = 1 dv ρ(r)ϕ(r) (12 4.1) 2 2 V V ϕ(r) (b) V 141

150 ρ = 0 J = 0 Maxwell E(r, t) = 0 (13.1) E(r, t) + B(r, t) = 0 t (13.2) B(r, t) = 0 (13.3) B(r, t) 1 c 2 E(r, t) = 0 t (13.4) 2 E(r, t) + 1 c B(r, t) + 1 c 2 2 E(r, t) = 0 (13.5) t2 B(r, t) = 0 (13.6) t2 Maxwell Maxwell electromagnetic wave 2 Maxwell Maxwell z E(z, t) separation of variables E(z, t) = E 0 Z(z)T (t) (13.7) 142

151 E 0 E 0 ( Z (z)t (t) + 1 ) c 2 Z(z) T (t) = 0 (13.8) Z Z z T T t Z (z) Z(z) + 1 c 2 T (t) T (t) = 0 (13.9) 1 z t 2 t z z t 0 Z (z) Z(z) = C 1 (13.10) 1 T (t) c 2 T (t) = C 1 (13.11) 2 C 1 = k 2 Z (z) = k 2 Z(z) (13.12) T (t) = c 2 k 2 T (t) (13.13) k 2 2 k C 1 < 0 k [13 1] Z(z) = Z 0 sin(kz kz 0 ) (13.14) T (t) = T 0 sin(ckt ckt 0 ) (13.15) 0 4 Z 0 T 0 E 0 E = E 0 sin(k(z z 0 )) sin(ck(t t 0 )) (13.16) 1 k ck c E = 0 ke 0z cos(k(z z 0 )) sin(ck(t t 0 )) = 0 (13.17) z t E 0z = 0 k = 0 E = 0 E 0z = 0 z k k z 1 Fourier 143

152 Maxwell E = E 0x E 0y 0 sin(k(z z 0)) sin(ck(t t 0 )) (13.18) B = 1 c E 0y E 0x 0 cos(k(z z 0)) cos(ck(t t 0 )) (13.19) plain wave z t 2 ξ = z ct η = z + ct ξ η E = 0 (13.20) E(z, t) = E + (ξ) + E (η) = E + (z ct) + E (z + ct) (13.21) E ± 1 E + (z ct) c 2 c E(r, t) = E + (k r ckt) + E (k r + ckt) (13.22) k k E(r, t) = 0 k (E +(k r ckt) + E (k r + ckt) ) = 0 (13.23) E ± E ± k 0 k E(r, t) = 0 (13.24) [13 2] Maxwell E 1 E 2 E 1 + E 2 2 z 144

153 k k ** (13.46) (13.47) ϕ(r) = 1 d 3 r 1 4πϵ 0 r r ρ(r ) (13.25) A(r) = µ 0 d 3 r 1 4π r r J(r ) (13.26) I II ϕ A 5 ( r r ρ ϕ(r, t) = 1 d 3 r 1 4πϵ 0 A(r, t) = µ 0 d 3 r 1 4π r r J r, t r ) r c ( r, t r ) r c (13.27) (13.28) (12.73) (12.74) 138 r r r c r r /c t t r r c 3 Fourier 4 Feynman If it gets difficult in places, well, that s life there is no other way. Feynman Lectures on Physics, Volume II 5 Lorenz 145

154 c retarded potential causality z t = 0 I 8 0 z (ρ, φ, z) E(r, t) = E ρ (ρ, t)e ρ + E φ (ρ, t)e φ + E z (ρ, t)e z (13.29) B(r, t) = B ρ (ρ, t)e ρ + B φ (ρ, t)e φ + B z (ρ, t)e z (13.30) x = ρ cos φ (13.31) y = ρ sin φ (13.32) e ρ = 1 x cos φ ρ y = sin φ, 0 0 e φ = 1 ρ y x 0 = sin φ cos φ 0, e z = (13.33) z ρ E(r) = 0 (13.34) B(r) = µ 0I 2πρ e φ (13.35) 0 7 t = z Gauss ρ 0 Maxwell 1 3 E ρ = 0 (13.36) B ρ = 0 (13.37) 6 (13.39) (13.39) 146

155 13.1: z t l 2 < z < l 2 (13.28) ρ 13.1 l A(r, t) = µ 0I 4π l 2 = c 2 t 2 ρ 2 (13.38) z z < l 2 dz 1 r r e z (13.39) A(r, t) = µ 0I dz 1 4π r r e z (13.40a) = µ 0I 4π = µ 0I 4π z z < l 2 z z < l 2 l/2 l/2 dz 1 ρ2 + (z z ) 2 e z dz 1 ρ2 + z 2 e z = µ [ 0I ( ln ρ2 + z 4π 2 z )] l/2 = µ 0I 4π ln ρ 2 + ( ) l l 2 ρ 2 + ( ) e z l 2 2 l 2 ( = µ 0I 4π ln ct + ) c 2 t 2 ρ 2 e z ct c 2 t 2 ρ 2 l/2 e z (13.40b) (13.40c) (13.40d) (13.40e) (13.40f) B(r, t) = A(r, t) (13.41a) A z = y Az x (13.41b) 0 147

156 = A z ρ e φ = µ 0I ct 2πρ c2 t 2 ρ e 2 φ (13.41c) (13.41d) Maxwell 2 7 A(r, t) E(r, t) = = µ 0I c t 2π c2 t 2 ρ e 2 z (13.42) Maxwell 4 8 z ρ S dr B(r, t) 1 d c 2 ds E(r, t) = µ 0 I (13.43) dt C S C S ρ ct dr B(r, t) = 2πρB φ (ρ, t) = µ 0 I (13.44) c2 t 2 ρ 2 C 1 d c 2 ds E(r, t) = 1 d dt S c 2 dt = µ 0I 2c = µ 0I 2c = µ 0 I d dt d ( dt ρ 0 ρ 2 2πρ dρ E z (ρ, t) dρ 2 0 c2 t 2 ρ 2 (2ct 2 ) c 2 t 2 ρ 2 1 ) ct c2 t 2 ρ 2 (13.45a) (13.45b) (13.45c) (13.45d) Maxwell 4 E(r, t) = µ 0I 2π c c2 t 2 ρ 2 e z (13.46) B(r, t) = µ 0I ct 2πρ c2 t 2 ρ e 2 φ (13.47) ρ 13.2 ρ < ct ρ = ct 0 (13.35) Poynting S(r, t) = 1 µ 0 E(r, t) B(r, t) = µ 0I 2 4π 2 c 2 t ρ(c 2 t 2 ρ 2 ) e ρ (13.48) 13.3 Maxwell 7 Maxwell z =

157 13.2: (13.46) (13.47) ( ): t = t 1, t 2 t 1 < t 2 ρ ( ): ρ = ρ 0 t ρ = ct Maxwell Maxwell Maxwell Maxwell Lorentz Newton 13.2 Einstein 149

158 (12.15) 130 Gauss Gauss 13.4 [13 1] Poynting Poynting k 2 < 0 [13 2] (13.22) k [13 3] r = r t re(r, t) [13 4] ρ(r, t) J(r, t) E(r, t) = k B(r, t) = k m Maxwell d 3 r ρ(r, t) r r r r 3 (13 4.1) d 3 r J(r, t) r r r r 3 (13 4.2) 150

159 Poisson Coulomb Fourier Maxwell Fourier

160 Maxwell E. P. Wigner The unreasonable effectiveness of mathematics in the natural sciences Ohm Maxwell Ohm Ohm 152

161 Newton 4 5 Lorentz Lorentz Coulomb Ampère Faraday Coulomb C. Huygens 6 Newton k

162 19 Einstein Einstein Newton 154

163 A (1) 1. x = a y = 0 λ x = a y = 0 λ [1 1] E(r) ϕ(r) [1 2] xy dy = f(x, y) dx dy = g(x, y) dx f g xy 2. 2 z = a/2 z = a/2 σ σ [2 1] E(r) [2 2] [2 3] b 2b < a ϕ(r) [2 4] [2 5] [2 3] 3. z I m q 155

164 B (2) 1. ρ(r) E(r) [1 1] (a). r = x 2 + y 2 + z 2 : ρ = ρ(r) (b). z : ρ = ρ(x, y) (c). r = x 2 + y 2 : ρ = ρ(r) (d). xy r = x 2 + y 2 : σ = σ(r) [1 2] a E(r) Q 2. z < 0 I Q(t) = It 3. [3 1] [3 1] Maxwell [3 2] Gauss Coulomb [3 3] [3 4] [3 5] 156

165 C (3) 1. xy a x 2 + y 2 a 2 z = 0 Q [1 1] E(r) [1 2] z [1 3] [1 2] 2. R 0 Q [2 1] E(r) [2 2] R 1 R 2 R 0 < R 1 < R 2 Q Q < Q [2 3] [2 2] [2 4] [2 2] ϕ(r) 0 3. [3 1] E(r) = k C dr E(r) = 0 C d 3 r ρ(r ) r r r r 3 [3 2] (a). Gauss (b). Gauss (c). (d). 157

166 D (4) 1. I z [1 1] (ρ, φ, z) B(r) = B ρ (r)e ρ + B φ (r)e φ + B z (r)e z e ρ,φ,z Maxwell (a). B ρ B φ B z ρ (b). B ρ = 0 (c). B z = const. [1 2] Ampère B z = 0 [1 3] 2. oa o ω oabcdeo B oa ocde oa a [2 1] Φ(t) = ds B S(t) S(t) [2 2] bcdeo R Ohm oa ab 3. [3 1] (a). A(r) = 0 (b). B(r) = 0 (c). t C(r, t) + D(r, t) = 0 (d). 2 E(r, t) 1 2 v 2 E(r, t) = 0 t2 [3 2] E B (a). B = 0, 0 (b). E = 0, (c). E B, 0 [3 3] (a). Ampère Maxwell 4 M4 (b). Ohm Maxwell (c). 158

167 Ampère, 95 Ampère, 108 Ampère, 95 Arago, 95 Biot, 95 Biot Savart, 98 Cavendish, 76 Coulomb, 18 Coulomb, 17, 70, 76 Coulomb, 17, 40 Dirac, 65 Faraday, 125 Faraday, 125 Feynman, 2 Fizeau, 134 Fleming, 102 Gauss, 46 Gauss, 46, 86 Gauss, 38, 44, 48, 107 Gilbert, 17 Grassmann, 97, 102 Green, 64 Hankel, 60 Henry, 125 Huygens, 153 Kepler, 4 Kirchhoff, 88 Kirchhoff, 93 Kohlrausch, 134 Kronecker, 30 Laplace, 19 Laplace, 63 Larmor, 101 Lenz, 125 Lorentz, 100 Lorentz, 100, 123 Lorenz, 138 Lorenz, 138, 145 Maxwell, 134 Maxwell, 6, 132, 142 Mayer, 95 Michell, 95 MKSA, 96 Neumann, 125 Newton, 3 Newton, 5 Ohm, 88 Ohm, 88, 123 Ørsted, 95 Ostrogradsky, 46 Poisson, 63 Poisson, 63 Poynting, 136 Poynting, 136 Priestley, 76 Savart, 95 Stokes, 60 Stokes, 60, 62 Thomson, 60 van Musschenbroek, 73 von Kleist, 73 Weber, 134 Wigner, 152,

168 , 146, 149, 106, 60, 16, 52, 135, 135, 35, 45, 47, 45, 60, 15, 18, 13, 89, 123, 72, 28, 18, 96, 111, 111, 138, 111, 45, 55, 1, 73, 137, 15, 116, 116, 115, 115 Coulomb, 116, 117, 14, 52, 114, 117, 106, 124, 15, 13, 97, 13, 13, 106, 16, 46, 109, 137, 145, 39, 118, 97, 108, 65, 71, 32, 61, 53, 69, 71, 75, 52, 54, 23, 127, 117, 28, 20, 14, 118, 146, 87, 87, 28, 97, 65, 88, 53, 55, 5, 17, 21, 20, 85, 75, 115, 75, 88, 88, 89, 74, 36, 57, 38, 142, 137, 145,

169 , 19, 64, 21, 85, 85, 134, 68, 45, 46, 44, 21, 32, 140, 44, 45, 46, 133, 142, 13, 134, 102, 79, 144, 16, 28, 46, 47, 109, 114, 137, 145, 130, 142, 69, 86, 55, 53, 102, 37, 54, 30, 54, 134, 73, 39, 63, 30, 38, 38, 85,