213 12 1 21 5 524 3-5465-74 nkiyono@mail.ecc.u-tokyo.ac.jp http://lecture.ecc.u-tokyo.ac.jp/~nkiyono/index.html 3 2 1 3.1 ρp, t EP, t BP, t JP, t 35 P t xyz xyz t 4 ε µ D D S S 35 D H D = ε E B = µ H E B
12 2 E ds = 1 ρdv ε 1 µ D D S S D B d S = 36 E d B l = S d S B d l = S ε E + J d S 4 4 div E = 1 ε ρ div B = rot E = B 1 rot µ E B = ε + J 37 3.2 3.2.1 D E d S = 1 ε D ρdv 72 E E D 36 1.8.4 37
12 3 div EdV = E ds 73 D E D 72 73 D D div EdV = 1 ε E D D ρdv 74 74 div EP, t 1 ε ρp, t P, t div EP, t > 1 ε ρp, t E ρ 38 P D P div EP, t > 1 ε ρp, t t D div EdV > 1 ε D ρdv 74 div E = 1 ε ρ D 73 72 D B d S = div B = 38 ρ ρ
12 4 3.2.2 S E d B l = S d S 75 E S rot E ds = E d l 76 S E S 75 76 S S rot E ds B = S d S 77 E S 77 rot EP, t B P, t P, t t B = F rot EP, t F P, t xyz rot E F R 1 x, y, z, t F 1 x, y, z, t R 2 x, y, z, t F 2 x, y, z, t R 3 x, y, z, t F 3 x, y, z, t
12 5 R 3 x, y, z, t > F 3 x, y, z, t x, y, z P rot E F 39 P D P R 3 x, y, z, t > F 3 x, y, z, t x, y, z P P xy S D S xy S z t rot E ds > S S F d S 77 rot E = F = B S 76 75 1 B d l = ε E + J ds µ S S 1 rot µ E B = ε + J 3.3 39 38
12 6 ρ J div E = 78 div B = 79 rot E = B rot B = ε µ E 81 t rot B = ε 2 E µ 2 4 8 8 81 rotrot E = ε µ 2 E 2 82 xyz E E = E 1 E 2 rotrot E E E 3 E = = y z E 3 y E 2 z E 1 z E 3 = E 2 E1 y E1 + E 2 y + E 3 z E 1 + E2 y + E3 z E1 = E E + E 2 y + E 3 z 2 E 2 y 2 E 1 y 2 E 1 2 z + 2 E 3 2 z 2 E 3 z y 2 E 2 z 2 E 2 2 + 2 E 1 2 y 2 E 1 z 2 E 3 2 E 3 2 y + 2 E 2 2 y z 2 2 2 2 2 2 + 2 y + 2 2 z E 2 1 + 2 y + 2 2 z E 2 2 + 2 y + 2 2 z E 2 3 83 = 2 2 + 2 y 2 + 2 z 2 41 78 E = 1 E = E 82 E = ε µ 2 E 2 84 4 B E B E 4 t rot rot 41 83 grad div rot rot
12 7 8 t 81 79 B B 2 2 B = ε µ 2 B 2 85 E B 2 f fx, y, z, t = ε µ x, y, z, t 86 2 fx, y, z, t 1 ε µ 1 ε µ = c 3. c 42 44. 2 f 2 x, t = 1 2 f v 2 x, t 87 2 fx, t 1 φy ψy fx, t = φx vt + ψx + vt 88 1 X Y X = x vt, Y = x + vt fx, t X, Y gx, Y gx, Y gx vt, x + vt = fx, t 87 X, Y 2 g X, Y = 89 X Y 2 X Y Y X 89 gx, Y X 1 φx Y 1 ψy gx, Y = φx + ψy X = x vt Y = x + vt x t 88 42
12 8. 86 3 87 1 1 x 2 1 88 x xy v > y = φx vt t y = φx v t x y φx vt x = x vt y x, t = x, t y x vt, y vt y t x y = ψx + vt v v y = ψx t v 87 88 1 E B x 3 F x, t 1 f a k v F x, t = f k x vt a k F k v E B E x, t = f k x vt e B x, t = g l x ut b 9 k, l, e, b, v, u f g 84 85 f k x vt e = ε µ v 2 f k x vt e g l x ut b = ε µ u 2 g l x ut b k l. f k x vt e ε f k x vt k x k = f k x vt = fk 1 x 1 + k 2 x 2 + k 3 x 3 vt x 1 f k 1x 1 + k 2x 2 + k 3x 3 vtk 1 k 1 k 2 k 3, x = x 1 x 2 x 3
12 9 x 1 f k 1x 1 + k 2x 2 + k 3x 3 vtk 2 1 x 2 x 3 2 f k x vt = f k x vt k 2 1 + k 2 2 + k 2 3 = f k x vt k E x, t v B x, t u c v = u = 1 ε µ = c 9 u = v = c 78, 79, 8, 81 f k x ct k e = 91 g l x ct l b = 92 f k x ct k e = cg l x ct b 93 cg l x ct l b = f k x ct e 94 45. 78, 79, 8, 81 9 u = v = c 91, 92, 93, 94 f g f x, t g x, t 91 92 k e = l b = k e l b E B 93 b k e 94 e l b e b k, l e, b
12 1 e b k l l = k l = k 93 l f k x ct k e l = cg l x ct b l 94 1 f k x ct k e l = f k x ct e f x, t k e l = e 95 a a = k e k, e, a 95 a, l, e l, e, a l k l = k E B E B 3.4
12 11 3.4.1 F grad φ = F φ rot G = F G F 43 2.7.4 F rot F = 3 2 divrot G = F = div F = F xyz F F x, y, z = F 1 x, y, z F 2 x, y, z F 3 x, y, z G 3 y G 2 z = F 1 G 1 z G 3 = F 2 G 2 G 1 y = F 3 G 1 x, y, z, G 2 x, y, z, G 3 x, y, z G 1 x, y, z = z F 2 x, y, tdt G 1 x, y 1 2 x, y, x, y, z z 1 G 1 G 2 x, y, z = G 1 z x, y, z = F 2x, y, z x F 3 t, y, z + G 1 t, y, z dt y x 1 G 2 x, y, z G 1 y x, y, z = F 3x, y, z G 3 x, y, z = y F 1 x, t, z + G 2 x, t, z dt z 43
12 12 y 1 div F = G 3 = = = = = y y y y y F1 G 3 y G 2 z x, y, z = F 1x, y, z F 1 + F 2 y + F 3 z = F 1 x, t, z + G 2 x, t, z z x, t, z + 2 G 2 x, t, z z F 2 dt dt y x, t, z F 3 z x, t, z + 2 G 2 z 2 G 1 y z x, t, z 2 G 2 z x, t, z + 2 G 1 dt = x, t, z dt y z x, t, z + 2 G 2 z x, t, z dt G 1 z G 3 = G 1 z = F 2 F div F = 3.4.2 79 B A rot A = B A rot A 8 B rot E = rot A rot E + A = E + A grad φ = E + A
12 13 φ φ, A E, B 79 8 E = grad φ A B = rot A φ div A = 1 ε ρ 96 divgrad φ = φ 8 32 µ µ ε = 1 c 2 1 rotrot A µ = ε grad φ ε 2 A 2 + J rotrot A = graddiv A A 83 grad div A + 1 φ c 2 1c 2 A 2 2 = µ J 97 44 96 97 46. 45 46 div A + 1 c 2 φ = 98 ρ = J = φ A c φ = 1 c 2 2 φ 2 A = 1 c 2 2 A 2 84 85 44 1 c 2 2 2 A A 1 c 2 2 A 2 ac + bc = a + bc 45 46
12 14 44 1 ξx, t = x vt, ηx, t = x + vt fx, t = g ξx, t, ηx, t x f g x, t = X = g X ξ ξx, t, ηx, t + g η ξx, t, ηx, t Y g ξx, t, ηx, t + ξx, t, ηx, t Y x 2 f x, t = g ξx, ξ 2 t, ηx, t X X + g ξx, η t, ηx, t Y X + g ξx, ξ t, ηx, t X Y + g ξx, η t, ηx, t Y Y = 2 g 2 g 2 g ξx, t, ηx, t + 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 t f g ξ x, t = ξx, t, ηx, t X = v g X ξx, t, ηx, t + v g Y + g η ξx, t, ηx, t Y ξx, t, ηx, t t 2 f x, t = v g ξx, ξ 2 t, ηx, t X X + v g ξx, η t, ηx, t Y X + v g ξx, ξ t, ηx, t X Y + v g ξx, η t, ηx, t Y Y = v 2 2 g 2 g 2 g ξx, t, ηx, t 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 87 2 g 2 g 2 g ξx, t, ηx, t + 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 = 2 g 2 g 2 g ξx, t, ηx, t 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 2 g ξx, t, ηx, t = X Y ξx, t ηx, t X Y 89 3 89 X Y 1 g X X, Y = g X, Y dy = χx Y X
12 15 X X Y X 1 g gx, Y = X, Y dx = χxdx + ψy X ψy Y Y χx X X 1 φx gx, Y = φx + ψy 45 e, b, l x k E Ex 1, x 2, x 3, t = 78 E 1 x 1, x 2, x 3, t E 2 x 1, x 2, x 3, t E 3 x 1, x 2, x 3, t E 1 1 x 1, x 2, x 3, t + E 2 2 x 1, x 2, x 3, t + E 3 3 x 1, x 2, x 3, t = Ex 1, x 2, x 3, t = fk 1 x 1 + k 2 x 2 + k 3 x 3 ct e 1 e 2 e 3 i = 1, 2, 3 E i i x 1, x 2, x 3, t = k i f k 1 x 1 + k 2 x 2 + k 3 x 3 cte i 78 E = f k s ct e 91 f k 1 x 1 + k 2 x 2 + k 3 x 3 ctk 1 e 1 + k 2 e 2 + k 3 e 3 = f k x ct k e = 79 78 E B f g k l e b 92 g l x ct l b =
12 16 B Bx 1, x 2, x 3, t = 8 B 1 x 1, x 2, x 3, t B 2 x 1, x 2, x 3, t B 3 x 1, x 2, x 3, t E 3 x 1, x 2, x 3, t E 2 x 1, x 2, x 3, t = B 1 2 3 x 1, x 2, x 3, t E 1 x 1, x 2, x 3, t E 3 x 1, x 2, x 3, t = B 2 3 1 x 1, x 2, x 3, t E 2 x 1, x 2, x 3, t E 1 x 1, x 2, x 3, t = B 3 1 2 x 1, x 2, x 3, t Ex 1, x 2, x 3, t = fk 1 x 1 + k 2 x 2 + k 3 x 3 ct e 1 e 2 Bx 1, x 2, x 3, t = gl 1 x 1 + l 2 x 2 + l 3 x 3 ct b 1 b 2 e 3 b 3 k 2 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 3 k 3 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 2 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ctb 1 k 3 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 1 k 1 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 3 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ctb 2 k 1 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 2 k 2 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 1 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ctb 3 f k 1 x 1 + k 2 x 2 + k 3 x 3 ct k 2 e 3 k 3 e 2 k 3 e 1 k 1 e 3 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ct b 1 b 2 k 1 e 2 k 2 e 1 b 3 f k x ct k e = cg l x ct b 93 81 8 E B ε µ = 1/c 2 f g e b k l c 2 c c cg l x ct l b = f k x ct e 94
12 17 46 98 t div A = 1 2 φ c 2 2 96 ρ = φ = 1 c 2 2 φ 2 99 98 97 J = A = 1 c 2 2 A 2 1 1 rot rot A = 1 c 2 2 rot A = B 85 99 grad 1 t B = 1 c 2 2 B 2 2 rot A grad φ = 1 c 2 grad 2 φ 2 A = 1 3 A c 2 3 grad φ + A = 1 2 c 2 2 grad φ + A E = grad φ A 1 E = 1 c 2 2 E 2 84