sin.eps

Transcription

1 9 ( ) :

2

3 ( ). 3 ( ) (Maxwell).3 (= )

4 4.4 x y(x) dy =3 y(x) =3x dx y(x) =3x + y(x) =3x + y(x) =3x + /m OK.5

5 ( ) ( ) +3 m+3kg m 3kg ( ) I(MKA ) m,kg,s(sec),a 4 4 ( ) m kg s 3 A F( ) m kg s 4 A ( ) F( ) ( ) 4.7 (): 3 ( ) X m( ) Y ( ) kg kg ( )

6 6 3 (): 3 3 OK (4): (3): (x, y, z) (4): ( ) ( ) ( ) ( ) = 3 ( )

7 7... (A, b ), A A = A x ˆx + A y ŷ + A z ẑ (.) ˆx, ŷ, ẑ x, y, z ( i, j, k (x, y, z) ˆ ) A A A x + A y + A z ˆ( ) Â, ˆn A A Â A A A A Â A = A A = A Â (.) A.. 3 ( ˆx, ŷ, ẑ) ˆx ˆx =, ˆx ŷ =, ˆx ẑ = A = A x ˆx + A y ŷ + A z ẑ B = B x ˆx + B y ŷ + B z ẑ A B = (A x ˆx + A y ŷ + A z ẑ) (B x ˆx + B y ŷ + B z ẑ) = A x B x ˆx ˆx + A x B y ˆx ŷ + A x B z ˆx ẑ + A y B x ŷ ˆx + = A x B x + A y B y + A z B z (.3) A x = Ax x A a A a A â.:

8 8 A x ˆx A ˆx = A x A x â A â A â (. )..3 +x +y +z (. ) a b a b a b a b a b (.3 ) ˆx ŷ ˆx ŷ = ẑ ŷ ẑ = ˆx, ẑ ˆx = ŷ ŷ ˆx = ẑ ˆx ˆx A = A x ˆx + A y ŷ + A z ẑ B = B x ˆx + B y ŷ + B z ẑ x z O O:origin.: a b y A B =(A x ˆx + A y ŷ + A z ẑ) (B x ˆx + B y ŷ + B z ẑ) = A x B x ˆx ˆx + A x B y ˆx ŷ + A x B z ˆx ẑ + A y B x ŷ ˆx + A y B y ŷ ŷ + A y B z ŷ ẑ + A z B x ẑ ˆx + A z B y ẑ ŷ + A z B z ẑ ẑ = A x B y ẑ A x B z ŷ A y B x ẑ + A y B z ˆx + A z B x ŷ A z B y ˆx =(A y B z A z B y )ˆx +(A z B x A x B z )ŷ +(A x B y A y B x )ẑ (.4) A (B ) A, B, 3 (.4 ) ( B θ <θ<π B A ϕ <ϕ< π ) () A (B ) =B ( A) = (A B) (.5) a b a b =.3:.4: ( ) A (B ) = B(A ) (A B) (.6) : 3: a, b, c 3 b a, c a (b a) (c a) r = xˆx + yŷ + zẑ r a (r a) {(b a) (c a)} = r (a b + b c + c a) =a (b c) ϕ A θ B. ( ) s a(s) a(s) =a x (s)ˆx + a y (s)ŷ + a z (s)ẑ

9 .. ( ) 9 3 t r(t) 3 x, y, z 3 A(x, y, z) ( ).. r(s) r(s) s s s s r(s ) r(s ) (.5 ) x r( s ) z r s ) ( y.6:.5: -. r(s) =sˆx +sŷ, s -. r(s) =sˆx + s ŷ, s (.6 ) -3. r(s) =3cosψˆx +3sinψŷ, ψ π -4. r(s) =sˆx + sŷ + sẑ, s s x r = xˆx +xŷ, x y s z r (s) s x. a b b a r(s) =a+s(b a), s.

10 .. ( : x = y, y ➀: r(s) = sˆx sŷ, s, ➁: r(s) =sˆx + sŷ, s r(s) =s ˆx + sŷ, s ) 3. ( ).. r(u, v) r(u, v) u, v (u u u,v v v ) ( v u ).7 u, v u u u,v (u) v v (u) r ( u,v ) x z r ( u,v ) r ( u,v ) r ( u,v ).7: -5. r(u, v) =uˆx + vŷ + ẑ, u, v -6. r(u, v) =uˆx + vŷ +( u v)ẑ, u, v u -7. r(u, v) =3cosv sin uˆx +3sinvsin uŷ +3cosuẑ, u π, v π 3 u θ v φ r(θ, φ) =3cosφsin θ ˆx +3sinφsin θŷ + 3cosθẑ, θ π, φ π : (.8 ) r(θ, φ) = a cos φ sin θ ˆx+ a sin φ sin θŷ + a cos θẑ, θ π, φ π a z z z y x y x y x y (a) (b) (c) (a) r(u, v) =r + u cos vẑ + u sin v ˆx, u a, v π (b) r(u, v) =r + a sin u cos v ˆx + a sin u sin vŷ + a cos uẑ u π, v π (c) r(u, v) =r + a sin u cos v ˆx + a sin u sin vŷ + a cos uẑ u π, v π.8: ( ) ( )

11 .. ( )..3 ( ) r(u, v, w) ( u, v, w ) r(u, v, w) ( ) u, v, w -8. r(u, v, w) =uˆx + vŷ + wẑ, u, v, w, ˆx, ŷ, ˆx + ŷ, ẑ, ˆx + ẑ, ŷ + ẑ, ˆx + ŷ + ẑ (u, v, w) (x, y, z) r(x, y, z) =xˆx + yŷ + zẑ, x, y, z -9. r(u, v, w) =u cos w sin v ˆx + u sin w sin vŷ + u cos vẑ u, v π, w π r(u, v, w) (u, v, w) (r, θ, φ) r(r, θ, φ) =r cos φ sin θ ˆx + r sin φ sin θŷ + r cos θẑ r, θ π, φ π z z θ r y x y x φ.9:.:..4 (). P (). Q P (3).3 (4).4 (5).5 (6).6 z 3 P x 4 y Q z 3 P x 4 y z 3 x 4 y.:.:.3:

12 z 3 z 3 z 3 x 4.4: y x y x y.5:.6: (= ) ( ) ( ) z ˆx, ŷ, ẑ (x,y,z ) r r = x ˆx + y ŷ + z ẑ +x +y z P y y +z ( ) ˆx ŷ = ẑ x x.7:.3. ( ) ρ, φ, z z ρ: xy (z = ) z ρ φ: +x ρ φ +x +y P y z: z ( ) z φ ρ ρ :,φ : π, z : x.8:

13 .3. 3 x = ρ cos φ, y = ρ sin φ, z = z (.7) ρ = x + y, φ =tan y x, z = z (.8) ( ˆρ, ˆφ, ẑ).9 P ˆρ φ, z ρ (. ) ˆρ =cosφˆx +sinφŷ, ˆφ = sin φˆx +cosφŷ, ẑ = ẑ (.9) ˆρ ˆφ = ẑ ( ˆφ ẑ = ˆρ, ẑ ˆρ = ˆφ) x z P ẑ ˆρ ˆφ φ z ρ ( ) y z φ x cosφx^ sinφ^x φ ρ^ φ sinφy^ φ ^ cosφy^ y z φ x sin φ^ x^.:.9:.: φ = ˆρ ˆρ = ˆx φ = π ˆρ = ŷ ˆρ, ˆφ φ ˆρ(φ), ˆφ(φ) (. ) φ φ cosφ^ρ ^y sinφ^ρ cosφ φ^ ˆx =cosφ ˆρ sin φ ˆφ, ŷ =sinφ ˆρ +cosφ ˆφ, ẑ = ẑ (.) (ρ,φ,z ) r r r = ρ ˆρ(φ )+z ẑ (r = ρ ˆρ(φ )+φ ˆφ(φ )+z ẑ φ φ ) φ ˆρ, ˆφ ( ) (ρ, φ, z) ˆρ(φ), ˆφ(φ), ẑ ρ, z φ. φ = ˆρ = ˆx, ˆφ = ŷ. φ = π 3 6 ˆρ = ˆx + ŷ, ˆφ = ˆx + 3. φ = π ˆρ = ŷ, ˆφ = ˆx y 3 ŷ

14 4 4. a =3ˆx +4ŷ +5ẑ φ = ˆρ, ˆφ, ẑ a =3ˆρ() + 4 ˆφ() + 5ẑ φ = π 6 a =(3 3 +)ˆρ( π 6 )+( 3 + 3) ˆφ( π 6 )+5ẑ φ =tan 4 3 = α a =5ˆρ(α)+5ẑ.3.3 r, θ, φ r: θ: r z +z z φ: xy x φ( ) r, θ, φ r :,θ : π, φ : π z x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, (.) r = x x + y + z, θ =tan + y, z φ =tan y x, (.) ρ = r sin θ, φ = φ, z = r cos θ, (.3) r = ρ ρ + z, θ =tan + z, z φ = φ (.4) ˆr, ˆθ, ˆφ x z φ θ r P y ˆr = sinθ cos φˆx +sinθsin φŷ +cosθẑ, (.5) ˆθ = cosθ cos φˆx +cosθsin φŷ sin θẑ, (.6) ˆφ = sin φˆx +cosφŷ (.7) r θ ( ) ˆr ˆθ = ˆφ ( ˆθ ˆφ = ˆr, ˆφ ˆr = ˆθ) x φ ˆx = sinθ cos φˆr +cosθ cos φ ˆθ sin φ ˆφ, (.8) ŷ = sinθ sin φˆr +cosθ sin φ ˆθ +cosφ ˆφ, (.9) ẑ = cosθ ˆr sin θ ˆθ (.) ˆr, ˆθ θ, φ ˆφ φ ˆr(θ, φ), ˆθ(θ, φ), ˆφ(φ) θ, φ ˆr ˆθ ˆφ y.3.4 ( ) z z

15 .4. 5 ( ) Q [] r = xˆx + yŷ + zẑ E(r) E(r) = Q xˆx + yŷ + zẑ 4πε (x + y + z ) x + y + z, E(r) = Q ˆr (.) 4πε r ˆr ( ).4 ˆ : A(r) dr, : A(r) d, 3 : φ(r)d, A(r)d, (.).4. OK A(x, y, z) A(r) =A x (r)ˆx + A y (r)ŷ + A z (r)ẑ ˆ A x = A x x ˆx + A y x ŷ + A z x ẑ (.3) (ˆ ) (ˆ ) (ˆ ) Adx = A x dx ˆx + A y dx ŷ + A z dx ẑ (.4).4. dr dr s r(s) s ( s ) s (ds ) r(s) dr ( ) dr (. ) dr dr

16 6 r(s) r(s) = X(s) ˆx + Y (s)ŷ + Z(s)ẑ (.5) X(s), Y (s), Z(s) s s ds X(s), Y (s), Z(s) dx = dx ds ds, dy = dy ds dz ds, dz = ds ds dr(s) = dx dy dz dsˆx + dsŷ + dsẑ (.6).: ds ds ds dr ds = dx dy ˆx + ds ds ŷ + dz dr ẑ dr = ds ds ds dr s (dr(s) ) dr dr dr = dr x z r(s) dr r(s+ds) y ˆ A(r) A dr s ( s ) r(s) A(r(s)) = A(x(s),y(s),z(s)) A(r(s)) dr(s) ( ) ˆ ˆ s A dr = A(r(s)) dr(s) (.7) s s s s s s. ˆ A(r) =xy ˆx + y ŷ + zẑ.3 A dr ( ) s r(s) =ˆx + sŷ + ẑ, s ( s ds r ) dr = dsŷ (r x, z dr y s r ) r(s) x y s z r(s) A(r) A(r(s)) x y s z A(r(s)) = s ˆx + s ŷ + ẑ (.8) s A s A(r(s)) dr = {s ˆx + s ŷ + ẑ} (ds ŷ) =s ds (.9) r(s) (s ds ) dr(s) s ˆ ˆ ˆ A dr = A(r(s)) dr(s) = s ds = (.3) 3

17 .4. 7 ˆ. A(r) =xy ˆx + y ŷ + zẑ.3 A dr ( ) 3 () r(s) =ˆx sŷ + ẑ, s r(s) =ˆx +( s)ŷ + ẑ, s OK 3. ˆ A(r) =y xˆx + axyŷ + zẑ.4 A dr ( ).4 θ θ = π θ =x ξ = π θ r(ξ) =a cos θ ˆx + a sin θŷ = a cos(π ξ)ˆx + a sin(π ξ)ŷ = a cos ξ ˆx + a sin ξŷ (.3) dr =(asin ξ ˆx + a cos ξŷ)dξ A x = a cos ξ, y = a sin ξ, z = A = a sin ξ( a cos ξ)ˆx + a( a cos ξ)a sin ξŷ = a 3 (sin ξ cos ξ ˆx +sinξ cos ξŷ) (.3) ˆ ˆ π A dr = a 3 (sin ξ cos ξ ˆx +sinξcos ξŷ) (a sin ξ ˆx + a cos ξŷ) dξ ˆ π ˆ π = a 4 {sin 3 ξ cos ξ +cos ξ sin x i} dξ = a 4 {( cos )cosξ +cos ξ}( sin x i) dξ ˆ ˆ = a 4 {( t )t + t } dt = a 4 {t 3 t t}dt = 3 a4 (.33) z y P r a x y ξ θ a P a.4: 3 x.3: 4. : z = E(r) = xˆx + yŷ q [] r = sˆx +(+s)ŷ, s 4πε (x + y ) 3

18 8 [ ] ( ) OK [ = ] s r(s) =x(s)ˆx + y(s)ŷ = sˆx +(+s)ŷ E (r(s)) = 4πε x(s)ˆx + y(s)ŷ [{x(s)} + {y(s)} ] 3 = 4πε sˆx +(+s)ŷ {s +(+s) } 3 = 4πε sˆx +(+s)ŷ (5s +4s +) 3 (.34) qe(r(s)) qe(r(s)) { } dr dr = dx(s) dy(s) ˆx + ds ds ŷ ds =(ˆx +ŷ)ds dw dw = qe(r(s)) dr = q sˆx +(+s)ŷ (ˆx +ŷ)ds = q 5s + ds (.35) 4πε (5s +4s +) 3 4πε (5s +4s +) 3 dw ( s ) ˆ W = dw = q 4πε ˆ 5s + ds (.36) (5s +4s +) 3 ( ) t =5s +4s + dt =(5s +)ds s : t : W = q 4πε q 4πε ˆ 5s + ds = q (5s +4s +) 3 4πε ( ) ˆ dt t 3 = q ( ) 4πε (.37).4.3 u, v r(u, v) u, v A(r) A(r) d (.38)

19 .4. 9 d (.38) d d u v u u r = r du u v u v r = r dv d v (.5 ) d = u r v r = r r r du dv = u v u r du dv (.39) v x z d r( u, v) d v r u r y : r(θ, φ) =a cos φ sin θ ˆx + a sin φ sin θŷ + a cos θẑ (θ, φ ) d = θ r φ r = a.5: sin θdθdφ(sin θ cos φˆx +sinθsin φŷ +cosθẑ) = a sin θdθdφˆr(θ, φ) : d = r v r du dv u OK A(r) d r(u, v) d = u r v r ˆ v ˆ u A d = A(r(u, v)) r u r du dv (.4) v v u..6 A(r) =xˆx + yŷ + x yzẑ A d d +z ( ) r(x, y) =xˆx + yŷ + ẑ, x, y x r = dx ˆx, y r = dy ŷ d = x r y r = dx dy ˆx ŷ = dx dy ẑ r(x, y) (x, y, ) A(r(x, y)) = xˆx + yŷ + x yẑ A d = ˆ ˆ (xˆx + yŷ + x yẑ) ẑ dx dy = ˆ ˆ x ydxdy= 6 (.4)..7(a) A(r) =xˆx+yŷ +x yzẑ A d d +z ( ) r(x, y) =xˆx + yŷ + ẑ, x, y x y x x y x (.7(b)) y x ˆ ˆ x ˆ ˆ x A d = (xˆx + yŷ + x yẑ) ẑ dy dx = x ydydx= (.4)

20 3. x y x r(x, y) =xˆx + yŷ + ẑ, y x, y (.7(c)) A d = ˆ ˆ y (xˆx + yŷ + x yẑ) ẑ dx dy = ˆ ˆ y x ydxdy= (.43) z z y y y x y y = x y=x y x.6: x x x (a) (b) y (c) x.7: 4. E(r) = xˆx + yŷ + zẑ 4πε (x + y + z ) 3 E(r) d E(r) a ( ) a r(θ, φ) =a cos φ sin θ ˆx+a sin φ sin θŷ+a cos θẑ, θ π, φ π d d = a sin θdθdφ(sin θ cos φˆx+sinθ sin φŷ+cosθẑ) (θ, φ) r(θ, φ) x(θ, φ) =a cos φ sin θ, y(θ, φ) =a sin φ sin θ, z(θ, φ) =a cos θ E(r) = 4πε a cos φ sin θ ˆx + a sin φ sin θŷ + a cos θẑ ({a cos φ sin θ} + {a sin φ sin θ} + {a cos θ} ) 3 = {cos φ sin θ ˆx +sinφsin θŷ +cosθẑ} 4πε a (.44) E(r) d = 4πε a {cos φ sin θ ˆx +sinφsin θŷ +cosθẑ} a sin θdθdφ(sin θ cos φˆx +sinθsin φŷ +cosθẑ) = sin θ dθ dφ (.45) 4πε E(r) d = ˆ π ˆ π φ= θ= sin θ dθ dφ = ˆ π ˆ π dφ sin θdθ= π = (.46) 4πε 4πε 4πε ε E(r) d Q /ε a [m] ε ( ).4.4 u, v, w r(u, v, w) ϕ(r) ϕ(r)d (.47)

21 .5. ( ) d (.47) d d u, v, w u r, v r, w r : d = u r v r w r = r u r v r du dv dw (.48) w OK : r(r, θ, φ) =r cos φ sin θ ˆx + r sin φ sin θŷ + r cos θẑ r r = (cosφ sin θ ˆx +sinφsin θŷ +cosθẑ)dr (.49) θ r = (cosφ cos θ ˆx +sinφcos θŷ sin θẑ)rdθ (.5) φ r = ( sin φ sin θ ˆx +cosφsin θŷ)rdφ (.5) r r θ r =( sin φˆx +cosφŷ)rdrdθ (.5) r r θ r φ r = r sin θdrdθdφ (.53) (u, v, w) r(u, v, w), u u u,v u v,w u w d = r u r v r wdu dv dw ˆ w ˆ v ˆ u φd = φ(r(u, v, w)) r u r v r du dv dw (.54) w w ˆ w A d = w v u ˆ v ˆ u v u A(r(u, v, w)) r u r v r du dv dw (.55) w.5 ( ).5. sin x cos(sin x ) f(x) x = f(x) =a + a x + a x + a 3 x 3 + (.56) f(x) a,a,a, a,a,a, ( x =) f() f (),f (), f () f(x) x df dx x =

22 x df dx (x) =a +a x +3a 3 x + x = df dx () = a d f dx (x) =a +3 a 3 x +4 3a 4 x + x = d f dx () = a f(x) =f() + df dx ()x + d f dx ()x + 3 d 3 f dx 3 ()x3 + d 4 f 4 3 dx 4 ()x4 + = x = ( ) n= d n f n! dx n ()xn (.57) : (.57) ++3+ n ( ) ++ A. e x x = e x =+x + x + 3! x3 +. sin x x = sin x = x 3! x3 + 5! x5 (.8 ).5 f(x) =sinx x =. rad sin x y = x y = x-x ( /3! -.5 y = x-x 3 /3!+x 5 /5! 9th order x radian degree( ).8: sin x, 3, 5, 9 ) n= sin.. 3!.3 + 5! x5 =.. 6 y (.58)

23 .5. ( ) 3 ( )sin x x =. x =. x = ( sin ) (.57) f(x) =sinx x = N =, 3, 5, x =.,.3,. (.).: sin x x =. x =.3 x = N =..3. N = N = N = sin x x =. N =5 8 x =.3 N =5 7 7 ( ) x = N =5 N = 8 ( ) ( x =) x = ( x = x ) OK f(x) =f(x )+ df dx (x )(x x )+ d f dx (x )(x x ) + d 3 f 3 dx 3 (x )(x x ) 3 + (.59) 3. sin x x = π 4 sin x = + (x π 4 ) (x π 4 ) 3! (x π 4 )3 + : sin x x =x = sin x x = π 3 x =.5.

24 4 f(x, y, z) =f(r) r = x ˆx + y ŷ + z ẑ f(r) = f(r )+ f x (r )(x x )+ f y (r )(y y )+ f z (r )(z z ) + f x (r )(x x ) + f y (r )(y y ) + f z (r )(z z ) + f x y (r )(x x )(y y )+ f y z (r )(y y )(z z )+ f z x (r )(z z )(x x )+ = n! {Dn f}(r ) (.6) n= x, y, z D D =(x x ) x +(y y ) y +(z z ) z (.6) f(x, y, z) x y z.6 f(x) df (x) dx = lim Δx f(x + Δx Δx ) f(x ) f(x +Δx) f(x) f(x) f(x Δx) = lim = lim Δx Δx Δx Δx Δx (.6) f(x, y) f x, f y ( ) â = a x ˆx + a y ŷ r â Δa f(r +Δaâ) f(r) Δa = f(x +Δaa x,y+δaa y ) f(x, y) Δa Δa â f a f a = lim f(r +Δaâ) f(r) f(x +Δaa x,y+δaa y ) f(x, y) = lim Δa Δa Δa Δa f(x +Δaa x,y+δaa y ) f(x, y) lim Δa Δa = lim Δa = lim Δa f(x +Δaa x,y+δaa y ) f(x +Δaa x,y)+f(x +Δaa x,y) f(x, y) Δa f(x +Δaa x,y+δaa y ) f(x, y +Δaa y ) a x + lim Δaa x Δa = f x a x + f y a y f(x, y +Δaa y ) f(x, y) a y Δaa y (.63) (.64) (.65)

25 .7. 5 f a = f x a x + f y a y (.66) f x a x + f ( ) f y a f y = ˆx + x y ŷ â (.67) f = f f ˆx + x y ŷ (.68) f a = f â (.69) f f f f(r) = f f ˆx + x y ŷ + f z ẑ (.7) f f â f â = df f â da df â f f da f : r f(r) = f(r) f r f(r) f.7 ( ).7. Δ f(r) : f(r)d (.7) Δ Δ Δ r f gradf

26 6 Δ f(r)d Δ Δ f(r)d Δ Δ f(r) lim f(r)d = f(r) (.7) Δ Δ Δ r = xˆx + yŷ + zẑ =(x, y, z) Δ =ΔxΔyΔz f(r) f(r)d = f(r)δx Δy Δz = f(r) (.73) Δ Δ f(r) r (f(r ) ) r (.6) f(x,y,z )=f(x, y, z)+ f x (x, y, z)(x x)+ f y (x, y, z)(y y)+ f z (x, y, z)(z z) + f x (x, y, z)(x x) + + f x y (x, y, z)(x x)(y y)+ f + f y (x, y, z)(y y) + f z (x, y, z)(z z) y z (x, y, z)(y y)(z z)+ f z x (x, y, z)(z z)(x x) r =(x, y, z) Δ =Δx Δy Δz ˆ z+ Δz z Δz ˆ y+ Δy y Δy = f(x, y, z) ˆ z+ Δz z Δz ˆ x+ Δx x Δx f(x, y, z)dx dy dz ˆ z+ Δz z Δz ˆ y+ Δy y Δy = f (x, y, z) x ˆ y+ Δy y Δy ˆ x+ Δx x Δx ˆ z+ Δz z Δz ˆ x+ Δx (.74) x Δx dx dy dz = f(x, y, z)δx Δy Δz (.75) f x (x, y, z)(x x)dx dy dz ˆ y+ Δy y Δy ˆ x+ Δx x Δx (x x)dx dy dz = (.76) (x ) f f (x, y, z) (x, y, z) x x f(x) =x x x x df df (x) =x dx dx (x )=x x x x f(x )=x df dx (x )=x ( df dx (x )= df dx (x ))

27 .7. 7 ˆ z+ Δz z Δz ˆ y+ Δy y Δy = f (x, y, z) x ˆ x+ Δx x Δx ˆ z+ Δz z Δz f x (x, y, z)(x x) dx dy dz ˆ y+ Δy y Δy ˆ x+ Δx x Δx (x x) dx dy dz = 4 f x (x, y, z)δx3 Δy Δz (.77) f 4 y (x, y, z)δy3 Δz Δx, f 4 z (x, y, z)δz3 Δx Δy ˆ z+ Δz z Δz + 4 ˆ y+ Δy y Δy ˆ x+ Δx x Δx { f x (x, y, z)δx f f(x,y,z )dx dy dz = f(x, y, z)δx Δy Δz y (x, y, z)δy + f (x, y, z)δx z Δ =Δx Δy Δz Δ lim Δ } Δx Δy Δz + (.78) ˆ f(r )d z+ Δz ˆ y+ Δy ˆ x+ Δx = lim f(x,y,z )dx dy dz Δ Δ Δx Δy Δz Δx Δy Δz z Δz y Δy x [ Δx = lim f(x, y, z)+ { } ] f Δx Δy Δz 4 x (x, y, z)δx + f y (x, y, z)δy + f (x, y, z)δx + z = f(x, y, z) (.79) Δx, Δy, Δz Δ.7. Δ Δ A(r) : A(r) d (.8) Δ 3 Δ 3 (.8) x x ± Δx y,z

28 8 Δ r ( Δ ) lim A(r) d = A x Δ Δ Δ x + A y y + A z (.8) z r Δ =ΔxΔyΔz x + Δx x Δx ˆn ˆx, ˆx A d = A(x + Δx z r (x Δx,y,z) x d Δy Δ x Δ z y (x + Δx,y,z).9: Δx,y,z) ˆxΔy Δz + A(x,y,z) ( ˆx)Δy Δz (.8) 6 A(r) d Δ = A(x + Δx = + A(x, y + Δy + A(x, y, z + Δz Δx,y,z) ˆxΔy Δz + A(x,y,z) ( ˆx)Δy Δz Δy,z) ŷδz Δx + A(x, y,z) ( ŷ)δz Δx Δz ) ẑδx Δy + A(x, y, z ) ( ẑ)δx Δy { Ax (x + Δx,y,z) A x(x Δx,y,z) + A y(x, y + Δy Δx + A z(x, y, z + Δz ) A z(x, y, z Δz Δz ),z) A y(x, y Δy,z) Δy } Δx Δy Δz (.83) Δ =ΔxΔyΔz Δ lim A(r) d = A x Δ Δ x + A y y + A z z Δ (.84) A x (r) x + A y(r) y + A z(r) z A(r) (.85) A A; A ( ) diva (.84)

29 x Δ x A(r) +x : A(r) d (.86) Δ x Δ x z (x, y, z + Δx Δy d Δ z ) r y Δ x r A x (r) x (x, y, z Δx ) lim A(r) d = A x (r) (.87).3: Δ x Δ x Δ x Δ x ˆn = ˆx Δy Δz (.3 ) A d = A ˆxΔy Δz = A x (.88) Δ x Δ x Δ x +y, +z Δ y, Δ z lim A(r) d = A y (r) (.89) Δ y Δ y Δ y lim A(r) d = A z (r) (.9) Δ z Δ z Δ z.7.4 +x Δ x Δ x +x A(r) : A(r) dr (.9) Δ x Δ x Δ x r ( ) lim A(r) dr = A z Δ x Δ x Δ x y A y (.9) z.3 Δy Δz ˆn = ˆx A A(r) dr = { A(x, y + Δy Δy Δ x Δ x ΔyΔz,z) ẑδz + A(x, y,z) ( ẑδz) + A(x, y, z + Δz Δz ) ( ŷ)δy + A(x, y, z ) ŷδy} = A z(x, y + Δy,z) A z(x, y Δy,z) A y(x, y, z + Δz ) A y(x, y, z Δz ) (.93) Δy Δz Δ x lim Δ x Δ x A(r) dr = A z Δ x y A y z (.94)

30 3 +y, +z Δ y, Δ z Δ y, Δ z lim Δ y lim Δ z A(r) dr = A x Δ y Δ y z A z x A(r) dr = A y Δ z Δ z x A x y (.95) (.96) (.94), (.95), (.96) x, y, z A ( Az y A ) ( y Ax ˆx + z z A ) ( z Ay ŷ + x x A ) x ẑ = A (.97) y A A A.8 :.8. ϕ = ϕ ϕ ˆx + x y ŷ + ϕ z ẑ (.98) A = A x x + A y y + A z (.99) z ( Az A = y A ) ( y Ax ˆx + z z A ) ( z Ay ŷ + x x A ) x ẑ (.) y.8. ϕ = ϕ ρ ˆρ + ϕ ρ φ ˆφ + ϕ z ẑ (.) A = (ρa ρ ) + A φ ρ ρ ρ φ + A z (.) z ( A z A = ρ φ A ) ( φ Aρ ˆρ + z z A ) ( z (ρa φ ) ˆφ + ) A ρ ẑ (.3) ρ ρ ρ ρ φ

31 .9. : ϕ = ϕ r ˆr + ϕ r θ ˆθ + ϕ r sin θ A = (r A r ) r + (sin θa θ ) r r sin θ θ ( (sin θaφ ) A = A ) θ ˆr + r sin θ θ φ r + r ( (raθ ) r A r θ φ ˆφ (.4) + A φ (.5) r sin θ φ ( A r sin θ φ (ra ) φ) ˆθ r ) ˆφ (.6).9 : ϕ(r),ψ(r) A(r), B(r) ( A) = (.7) ( ϕ) = (.8) (ϕψ) = ψ ϕ + ϕ ψ (.9) (ϕa) = A ϕ + ϕ A (.) (A B) = ( A) B ( B) A (.) (ϕa) = ϕ A + ϕ A (.). :. ( ) A d = A d (.3) d ( ). A d = A dr (.4) d ( d )

32 3 3. ( ϕ ψ + ϕ ψ)d = ϕ ψ d = ϕ ψ d (.5) n ψ ψ d = ˆnd ϕ ˆn = n n ) (ϕ ψ) = ϕ ψ + ϕ ψ (ϕ ψ)d = ϕ ψ d = ( ϕ ψ + ϕ ψ)d 4. (ϕ ψ ψ ϕ)d = (ϕ ψ ψ ϕ) d = ) ϕ ψ ( ϕ ψ n ψ ϕ n ) d (.6)

33 33 3 ( ),,,,,,,,,,, ( ) ( ) (positive charge) (negative charge).6 9 ( ) [m] 3. : : ( ; =A s) ( ): ( ) r Δ [m 3 ] Δ ΔQ [] ρ(r) [/m 3 ] ρ(r) = ΔQ lim Δ Δ [/m 3 ] (3.) r Δ : : λ [/m] : σ [/m ]

34 34 3 : ( ) : ( ) ( ): ( ) / t /4 r Q [] r q [] q q qq (r r ) F = 4πε r r r r = qq r r (r r ) 4πε r r 3 (3.) Q r k = r I ε 4πε O ε = 4π 7 c [F/m] 3.: c c = [m/s] c =3 8 [m/s] =9 9 4πε 3.. ( ) 3 (q, Q,Q,,Q N [] ) q Q,Q,,Q N ( ) : (+δ) 77 δ. δ 5 5 () 936 Plimpton Lawton δ < 9 97 δ = (.7 ± 3.) 6 ( Jackson, lassical Electrodynamics ) ( Hooke Newton )

35 q r Q,Q,,Q N r, r,, r N F = F + F + = qq (r r ) 4πε r r 3 + qq (r r ) N 4πε r r 3 + = qq n (r r n ) 4πε r r n 3 (3.3) n= ( ; action at distance) (electric field) ( ; action through medium ) (3.) F = q Q (r r ) = qe(r) (3.4) 4πε r r 3 E(r) r Q r q F = qe(r) q Q ( ) r ( ) r Q E(r) E(r) = Q 4πε (r r ) r r 3 (3.5) (3.4) [N/] [/m] 3.3. : r, r,, r N Q,Q, Q N N (3.3) N Q n (r r n ) E(r) = 4πε r r n 3 (3.6) n= ρ(r) : ρ(r) [/m 3 ] r r d dq dq = ρ(r ) d

36 36 3 r de(r) de(r) = dq(r r ) 4πε r r 3 = ρ(r )d (r r ) 4πε r r 3 ( ) ˆ E(r) = ρ(r )(r r ) de(r) = 4πε r r 3 d (3.7) ρ( r ) d d σ(r )[/m ] r O r 3.: r d = d dq = σ(r )d r de(r) de(r) = σ(r )d (r r ) 4πε r r 3 (3.8) σ ( r ) d r r ˆ E(r) = de(r) = σ(r )(r r ) 4πε r r 3 d (3.9) O r 3.3: λ(r ) r dr = dr r de(r) dr de(r) = λ(r )dr (r r ) 4πε r r 3 (3.) O r ˆ E(r) = de(r) = ˆ λ(r )(r r ) 4πε r r 3 dr (3.) 3.4: d r r r r r λ( r ) dr r r r q =μ, Q = 3μ q (x, y, z) =(,, ) m Q (x, y, z) =(,, ) m q ( ) (μ 6 ) a [m] ( Q []) x [m] a [m] σ [/m ] x [m] σ [/m ] x [m]

37 a 3.5: x a z σ [/m ] 3.6: () A: () B: (3) : 3.7: A 3.8: B 3.9: () ( )

38 : A 3.: B 3.: 3 ( ) ( ) = () 4 ( ) 5

39 Q [] Q ε E(r) d = Q ε (3.) Q ρ(r) [/m 3 ] E(r) d = ρ(r)d (3.3) ε ( ) ( ) E(r) ( ) OK (3.3) Δ Δ lim E(r) d = lim ρ(r)d (3.4) Δ Δ ε Δ Δ Δ Δ E(r) = ρ(r) ε (3.5) E(r) ( ) E(r) E r { E}(r) ( ) E(r) E(r) E(r) ρ(r) ε (3.3) = : Q [] a [m] : ( )

40 4 3 a [m] Q ε E(r) E(r) = Q xˆx + yŷ + zẑ (3.6) 4πε (x + y + z ) 3 : θ, φ r(θ, φ) =a(cos φ sin θ ˆx +sinφsin θŷ +cosθẑ) =aˆr θ π, φ<π d = r r dθ θ φ dφ = a (cos φ sin θ ˆx +sinφsin θŷ +cosθẑ)sinθdθdφ= a ˆr sin θdθdφ (3.7) r(θ, φ) E = Q a(cos φ sin θ ˆx +sinφsin θŷ +cosθẑ) 4πε (a cos φ sin θ + a sin φ sin θ + a cos θ) 3 = Q ˆr 4πε a (3.8) E d = Q 4πε ˆ π ˆ π ˆr a a ˆr sin θdθdφ= Q ε (3.9) ε Q [] z ( h [m] a [m]) 3.4. ( ) 6 ( ). 3.3 a [m] Q [] r r 6

41 r>a E(r) d = E r (r)ˆr ˆrd =4πr E r (r) = Q ε E r (r) = Q 4πε r E(r) = Q 4πε r ˆr = Q 4πε r 3 r (3.) r a ρ(r) = Q 4πa 3, (r a) /3 E(r) d =4πr E r (r) = ε E r (r) = Qr 4πε a 3 E(r) = Qr 4πε a 3 ˆr = ρ(r)d = Q 4πr 3 ε 4πa 3 /3 3 Q 4πε a 3 r (3.) Q [] a 3.3: a. σ [/m ] d (, d) E(r) d = E(d) = ρd = σd= σ, E(d) = σ (3.) ε ε ε ε d a λ [/m] ρ [m] a ( a, a ) () Q [] () Q [] r λl [] l ρ a a a 3.4: 3.5: (?) E(r) dr = (3.3)

42 4 3 7 (3.3) (3.3) (3.3) () E z y E y z =, E x z E z x =, E y x E x y = (3.4) : E = (3.5) [ ] P, Q A, B ( : P A Q : Q B P ) ˆ ˆ E(r) dr = E(r) dr + E(r) dr = (3.6) : P B Q ˆ ˆ Q, E(r) dr = E(r) dr (3.7) ˆ A ˆ E(r) dr E(r) dr = B ˆ ˆ P E(r) dr = E(r) dr (3.8) P Q ˆ E(r) dr P Q E(r) r A r B q [] 7

43 r r dr E(r) qe(r) qe(r) dr dw dw = qe(r) dr W [J] ˆ ˆ W = dw = q E(r) dr (3.9) y Q [] E(r) 3.6 q [] ( ŷ, 3ŷ ŷ z = ) (a), (b), W,W W = W x 3.6: 3.5. ( ) r Z r Z r : r Z ϕ(r) = ˆ r r Z E(r) dr (3.3) ϕ(r) r Z r [J/]=[] r P r Q q [] r Z ˆ rq ˆ rz ˆ rq q E(r) dr = q E(r) dr q E(r) dr r P r P r ˆ Z rq ( ˆ rp ) = q E(r) dr q E(r) dr = q{ϕ(r Q ) ϕ(r P )} (3.3) P, Q (= ) () r Z

44 44 3 q [] q ( ) OK Q r = x ˆx + y ŷ + z ẑ ( ) E(r )= Q r 4πε r 3 = Q r 4πε (r ) 3 (3.3) r r s r(s) = s(sin θ cos φˆx +sinθ sin φŷ +cosθẑ) = sˆr, s r (3.33) θ, φ r r = r (sin θ cos φˆx+sinθ sin φŷ +cosθẑ) s dr = ˆr ds r(s) E(r(s)) = Q sˆr 4πε sˆr 3 = Q sˆr 4πε s 3 = Q ˆr 4πε s ( s<) (3.34) s s 3 = s 3 ˆ ϕ(r ) = E dr = Q ˆ r ˆr 4πε s ( ˆr ds) = Q 4πε ˆ r s ds = Q 4πε r = Q 4πε r (3.35) Q [] r ϕ(r) = Q 4πε r (3.36) r Q ϕ(r) = (3.37) 4πε r r

45 λ(r) [/m] σ(r) [/m ] ρ(r) [/m 3 ] ˆ λ(r )dr ϕ(r) = (3.38) 4πε r r σ(r )d ϕ(r) = (3.39) 4πε r r ρ(r )d ϕ(r) = (3.4) 4πε r r l λ [/m] r a [m] ( Q []) x a 3.7: x E(r) ϕ(r) ϕ(r) = ˆ r r Z E(r ) dr (3.4) ϕ(r) E(r) ϕ(r) E(r) r = xˆx + yŷ + zẑ +x Δx (x Δx Δx,y,z) (x +,y,z) r = x ˆx + yŷ + zẑ ( y, z ) dr = dx ˆx ˆ r+ Δx ˆx E(r ) dr = r Δx ˆx ˆ x+ Δx x Δx ˆ x+ Δx E(x,y,z) ˆxdx = E x (x,y,z)dx (3.4) x Δx Δx E x (x,y,z) x E x (x,y,z) E x (x, y, z)+ E x x (x, y, z)(x x) (3.43)

46 ˆ x+ Δx E x (x,y,z)dx x Δx { ˆ x+ Δx = E x (x, y, z) x Δx ˆ x+ Δx x Δx = E x (x, y, z)δx = ϕ(x + Δx E x (r) { E x (x, y, z)+ E } x x (x, y, z)(x x) dx ˆ } x+ Δx (x x)dx dx + E x (x, y, z) x x Δx,y,z) ϕ(x Δx Δx Δx ϕ(x +,y,z) ϕ(x,y,z) Δx Δx E x (r) = lim Δx Δx ˆ r+ Δx ˆx E(r ) dr = ϕ(r) r Δx ˆx x,y,z) (3.44) y, z Δy, Δz Δy,Δz E y (r) = ϕ(r) y, E z(r) = ϕ(r) z (3.45) (3.46) (3.47) ϕ(r) x, y, z x, y, z E(r) { ϕ(r) ϕ(r) E(r) =E x (r)ˆx + E y (r)ŷ + E z (r)ẑ = ˆx + ŷ + ϕ(r) } ẑ (3.48) x y z.6 E(r) = ϕ(r) (3.49) ϕ(r) (3.4) r (3.4) (3.37) ϕ(r) ϕ(r) r Q [] r : E(r) 3 (x, y, z ) ϕ(r) 8 Ex x (x, y, z) Ex x (x, y, z) x x x x 9 Δx E x(x,y,z) E x(x, y, z)

47 ( ) (dipole; ) ( 3.8 ) r d ±q [] + d ẑ, d [m] ẑ r d r ϕ(r) [] ( ) r d d r ζ = d r ζ ζ ζ = +q d + θ q r 3.8: r z θ + d ẑ +q [] r r d { ( ẑ = r cos θ d { = r d r cos θ + 4 ) +(r sin θ) } ( ) } d r = { r rd cos θ + = r { ζ cos θ + 4 } ζ ( )} d (3.5) r d { ζ cos θ+ ẑ 4 ζ } ζ = ) ( ζ cos θ + ζ ζ ζ =+ cos θ (3 cos θ ) + (3.5) ζ + d +q [] r ẑ q ϕ (r) = 4πε r d q (+ dr ) ẑ 4πε cos θ (3.5) d q [] r ẑ q ϕ (r) = 4πε r + d q ( dr ) ẑ 4πε cos θ (3.53) r ϕ(r) ϕ(r) = q (+ dr ) 4πε cos θ q ( dr ) 4πε cos θ = qdcos θ 4πε r (3.54)

48 48 3 d ( d = dẑ) p = qd [ m] p ϕ(r) ϕ(r) = qdcos θ 4πε r = p ˆr 4πε r = p r 4πε r 3 (3.55) r r E(r) (3.55) A, B, 3 3.9: A 3.: B 3.: A, B A B Δr (Q []) A B W = QE Δr A, B W = E Δr = (3.56) (Δr ) (E ) ( ) ( ) 3.6 (3.37), (3.38), (3.39), (3.4)

49 ρ(r) E(r) E(r) ϕ(r) (3.58) (3.57) E(r) = ρ(r) ε (3.57) E(r) = ϕ(r) (3.58) { ϕ(r)} = ρ(r) ε ϕ(r) = ρ(r) ε (3.59) (3.59) ϕ(r) ϕ A(r) ϕ = ϕ ϕ ˆx + x y ŷ + ϕ z ẑ (3.6) A = A x x + A y y + A z z (3.6) ϕ(r) A A x ϕ x A y ϕ y A z ϕ z ϕ = ( ) ϕ + ( ) ϕ + ( ) ϕ = ϕ x x y y z z x + ϕ y + ϕ z (3.6) ϕ x + ϕ y + ϕ z = ρ (3.63) ε (Laplacian) ϕ = ϕ x + ϕ y + ϕ z (3.64) ϕ = ρ ε (3.65) (Poisson) ρ(r) ϕ = (3.66) (Laplace) ( ) Q [] (3.36) ϕ(r)

50 E(r) = E(r) = r r = r r r r 3 (3.67) ϕ(r) = f(r) f(r) = ρ(r )(r r ) 4πε r r 3 d (3.68) ρ(r ) 4πε r r d = ϕ(r) (3.69) ρ(r ) 4πε r r d (3.7) E(r) = { ϕ(r)} = (3.7)

51 ( ) ( ) ( ) ( ) ( ) () ( )

52 E Q Q (a) (b) (c) ±Q 4.: [] ( 4.) +Q [] Q [] ( ) Q [] ±Q [] Q = (4.) [F](Farad; ) =[/] ( [F] ) 4.. +Q [] Q [] Q/ () : ( [m ]) d [m] z ±Q [] ±Q/ E = Q ε ẑ ˆ ˆ d Q Qd = E dr = ẑ ẑdz = ε ε, = Q = ε d (4.) +Q d Q 4.:

53 () : a [m] a [m] a 3 [m] +Q [] Q [] r [m] E(r) = Q 4πε r ˆr = Q 4πε r 3 r ( ) r(s) =(a s)ˆr, s (a a ) ˆ a a = ˆ Q(a s)ˆr a ( ˆrds) = 4πε (a s) 3 a Qrˆr 4πε r 3 ˆrdr = = Q = 4πε a (a s = r ) Q 4πε [ ] a = r a a 3 + a a 4.3: Q 4πε + ( a a a (4.3) (3) : ( ) +Q [] [] Q [] a [m] +Q [] = Q [] 4πε a =4πε a [F] (4.3) a ( ) (4) : a [m] a [m] +λ [/m] λ [/m] ρ [m] E(ρ) = λ πε ρ ˆρ = λ πε ρ ρ ρ =(a s) ˆρ, s (a a ) ) ˆ a a = ˆ λ(a s) ˆρ a ( ˆρds) = πε (a s) a λρ ˆρ πε ρ ˆρdρ = λ πε log a a (4.4) l [m] lλ [] l = lλ = πε l log a [F] (4.5) a a a 4.4: = λ = πε log a a [F/m] (4.6)

54 ( Q [] ) [F] ( [], []) q [] v [] (q = v) dq [] dw [J] dw = vdq = q dq q [] Q [] ˆ W = dw = ˆ Q q dq = Q = [J] (4.7) 4.3. w [J/m 3 ] w = ε E [J/m 3 ] (4.8) [m ] d [m] = ε E = Ed d W W = = ε d E d = ε E (d) (4.9) d ε E 4.4 () 4.4. N N Q [],,, N [] = p Q, = p Q,, N = p N Q (4.)

55 4.4. () 55 Q [],,, N = p Q, = p Q,, N = p NQ (4.) Q [] Q [],,, N = p Q + p Q, = p Q + p Q,, N = p NQ + p N Q (4.) ( ) Q,Q,,Q N,,, N p p p N Q. = p p p N Q (4.3). N p N p N p NN Q N p ij [F ] p ij j ( ) i : () j j p jj p ij (4.4) () j i p ij p ij (4.5) (3) p ij = p ji (4.6) (j i i j ) Green G. Green 4,, ( ) p. 68

56 ( ) Q [] = Q Q = Q, Q =, = Q = 4πε a 3 4πε a 3 ] [ ] [ ] p p = (4.7) p p Q [ Q 4πε a 3 Q 4πε a 3 p = p = 4πε a 3 Q [] Q [] +Q [] = Q Q ( ) 4πε a 3 4πε a a [ ( )] [ ] [ ] Q 4πε a a + a 3 p p Q = (4.8) p p p = ( 4πε [ + a a a 3 ] [ = p p p p ) p = Q 4πε a 3 4πε a 3 ] [ ] [ ( ) ] [ ] Q 4πε = a a + a 3 4πε a 3 Q Q 4πε a 3 4πε a 3 Q (4.9) 5. (, ) Q [], 6, Q,Q ( ) p ij [ ] [ ] [ ] p p Q = (4.) p p Q = p Q + p Q = p Q + p Q (4.) Q = Q + Q Q = Q = p p Q p + p p p (4.) p p Q p + p p p (4.3) 4.4.,,, N [] Q,Q,,Q N [] Q q q q N Q. = q q q N (4.4). Q N q N q N q NN N 5 OW I 6

57 4.4. () 57 q ii q ij (i j) q ij j ( ) j i q ij ( 4.5) : j j q jj > (4.5) () j q ij (4.6) j + i () 4.5: j j N q jj i=(i j) q ij N q ij (4.7) i= q ij = q ji (4.8) ( ) Q 3 Q = 3 (4.9) Q Q [] 3 Q [] 3 7 OW I

58 58 4 Q 3 Q = = (4.3) 3 3 =, 3 = (4.3) 3 33 = 3 (4.3) 4.6: ij = ji Q Q Q 3 = (4.33) Q = Q 3 Q 3. [F] ( ) [ Q Q ] [ = q q q q ] [ ] (4.34) Q = Q, Q = Q, = + = Q = q q q q +q + q (4.35)

59 ( ) () E ( ) () 5.: 5. ( [m ] d [m]) ±Q [] ±σ = ± Q [/m ] E = σ [/m] ε 5.: (

60 6 5 ) ±σ p ±(σ σ p ) E = σ σ p [/m] ε 5. P (r) [/m ] ρ p (r) [/m 3 ] ( ) p [ m] ( ) ( ) ρ [/m 3 ] ( ) N(r) [/m 3 ] P (r) =Np = ρ(r)d(r) [/m ] (5.) r d(r) [m] r d P (r) d ( ) P d ρ P [/m 3 ] ρ P d = P d (5.) d d ρ p (r) = P (5.3) 5. P P = σ p = P ˆn 5.3

61 ρ(r) ρ p (r) E = ρ + ρ p = ρ + ρ p = ρ P (5.4) ε ε ε ε ε (ε E + P )=ρ (5.5) D(r) =ε E(r)+P (r) [/m ] D(r) =ρ(r) (5.6) ( ) D(r) d = D d = ρ(r) d (5.7) ( ρ [/m 3 ]) E [/m] D [/m ] P (r) [/m ] E(r) [/m] P = χe = χ r ε E (5.8) χ [F/m] χ r D = ε E + P = ε E + χ r ε E =(+χ r )ε E = εe = ε r ε E (5.9) ε [F/m] ε r E D D = εe (5.) ε r = [] [] E = [/m] d E = d

62 6 5 ε [F/m]( ε r ) [] Q [] D D = Q E E = Q ε = Q ε r ε = Ed = Qd ε = Qd ε r ε [F] = Q = ε ε r d = ε r (5.) [F] ε r ε r μ F m m mm () 9 3 μ F ( F ) z z E D ( ) ε ε x y 5.3: : D d = ρd (5.) E dr = (5.3) 5.3 (ɛ,μ ) (ɛ,μ ) z z = z x, y z E x (x, y, z +) = E x (x, y, z ), E y (x, y, z +) = E y (x, y, z ) (5.4) D z (x, y, z +) = D z (x, y, z ) (5.5) E x (x, y, z ±) δ > lim E x(x, y, z ± δ) δ

63 (5.5) : D d = ρd (5.6) 5.4 ( ) r = xˆx + yŷ + z ẑ A Δd D d = D(x, y, z + Δd ) ẑa ( ẑa)+ + D(x, y, z Δd ) D d (5.7) x z z A Δd ε 5.4: Δd (5.7) 3 ( ) (5.6) ( ) Δd ρd = D z (x, y, z +) = D z (x, y, z ) (5.8) Δd ρd = σa σ D z (x, y, z +) D z (x, y, z )=σ (5.9) z <z D z (x, y, z +) = σ ( ) (5.4) : 5.5 = hl ( ) : E dr = (5.) ε y E dr = E(x, y, z h ) (ŷl)+e(x, y + l,z ) (ẑh) + E(x, y, z + h ) ( ŷl)+e(x, y l,z ) ( ẑh) (5.) h (5.) 4 (5.) z l z ε x h ε y E y (x, y, z )=E y (x, y, z +) (5.) ( ) 5.5:

64 ε [F/m] ( ε r ) w = ED [J/m3 ] (5.3) E [/m] D [/m ] ( [m ] d [m] ε [F/m]( ε r ) [] E [/m] D [/m ] W [J] E = d, D = εe = ε d, (5.4) W = = ε d = ε d d d = ED = w (5.5) w (5.3). 5.6 ( [m ] d [m]) ε [F/m]( ε r ) [F] σ p [/m ] ( ) ( ) D [/m ] [] Q [] D ( )D = Q E d = D = Q ε r ε ε r ε [/m] ( ) E = D = ε Q [/m] ε d = E d + E d d d = Q d ε r ε + Q d ε = Q d ( ) +, = Q ε ε r = ε ε d ε r + (5.6) = Q d ε r ε + Q ( d ε = Q + ) 5.6: (5.7) d [F] d d [F] ε [F/m] ( = + ) (5.8) d (5.3)

65 , d σ p σ p = P ˆn = P D = ε E d + P σ p = P = D ε E d = Q ε Q ε r ε = Q ) ( εr (5.9). 5.7 ( [m ] d [m]) ε [F/m] ( ε r ) [F] ( ) [] E [/m] E = d D = ε E = ε d D d = εe = ε rε d σ = ε d d [/m ] σ d = ε ε rε [/m ] Q [] d Q = σ + σ d = ε d = Q = ε d + ε rε d + ε rε d = ε +ε r d (5.3) (5.3) 5.7: ( a [m] a [m]) a + a [m] r [m] r < a + a ε =ε [F/m] r > a + a ε =4ε [F/m] [F] ( ) ε,ε D, D, E, E D = D = Q 4πr ˆr, E = D ε = Q 4πε r ˆr, E = D ε = Q 4πε r ˆr, ˆ a Q = a +a 4πε r ( Q = 4πε a + a a ˆ a +a ˆr ( ˆr)dr ) + Q 4πε a ( a Q 4πε r a + a ˆr ( ˆr)dr ) = Q { 4πε (a + a ) + 4a a } (5.3) = Q = 4πε (a +a ) + 4a (5.33) a z = z z>z ε =ε z<z ε =3ε

66 66 5 z z>z lim E(z + δ) =E ( δ >) E =ŷ +3ẑ [/m] δ D = lim D(z + δ) E = lim D(z δ) D = lim D(z δ) δ δ δ ( ) D = ε E D = ε (4ŷ +6ẑ) D ( z ) E ( x, y ) D = D y ŷ+ε 6ẑ E =ŷ+e z ẑ D y = ε E y E z = D z D = ε (6ŷ+6ẑ) E =ŷ +ẑ ε a a 5.8:

67 67 6 ( ) H(r) [A/m] B(r) [T] B [T] (Tesla; ) B(r) 6.. F ( )E [/m] q [] B ( ) F [N] ( I E E = F/q ) B [T] 6.: I [A] Δr [m] F [N] B [T] B B = F (6.) I Δr [T]( ) [T] = [N/A m] = [kg/a s ]

68 68 6 F = IΔrˆl B ( ) ˆl 6. = B(r) [T] ( ) = 6.. = I [A] s r (s) ( s I ) dr = dr ds ds I Idr [A m] r db(r) [T] db(r) = μ Idr (r r ) 4π r r 3 (6.) = (Biot=avart s law) μ I μ =4π 7 H/m [H] (Henry) ( ) ˆ B(r) = ˆ db(r) = μ Idr (r r ) 4π r r 3 (6.3) O I dr r r r r I db 6.: Idr db B(r) = () ( s ) r (s) =x (s)ˆx + y (s)ŷ + z (s)ẑ I (s ± ) ( r r r ) () dr = dr ds ds Idr (3) = B(r) = μ I 4π ˆ dr {r r (s)} r r (s) 3 : :

69 6.. = 69 z +z I [A] y r = yŷ r (z ) = z ẑ Idr = I dr dz dz r r = yŷ z ẑ, r r =(y + z ), Idr (r r )=Idz ẑ (yŷ z ẑ)=idz y(ẑ ŷ) = Iydz ˆx db(r) = μ Idr (r r ) 4π r r 3 = μ 4π = Idz ẑ r db(r) Iydz (y + z ) ˆx (6.4) 3 B(r) ˆ ˆ μ Iydz B(r) = db(r) = 4π (y + z ) ˆx = μ ˆ Iy dz 3 4π (y + z ) = μ I 3 πy ˆx (6.5) z = y tan θ x z r I r r r B( r ) 6.3: z y 6.. = = r ( ) J (r ) [A/m] d [m ] J (r )d [A m] = ˆ B(r) = db(r) = μ J (r )d (r r ) 4π r r 3 (6.6) MKA J [A/m] w [m] ( ) I [A] z B(,,z) r = x ˆx + y ŷ ( x, w y w) z = w y w J (r )= I w ˆx [A/m] J (r )= d = r r x y dx dy = dx dy J (r )d = I w dx dy ˆx ( = l) r = zẑ r r = x ˆx y ŷ +zẑ r r =(x +y +z ) B(,,z) = ˆ w w = μ I w 4π ˆ ˆ w w = μ I πw ( ẑ) μ 4π I w ˆxdx dy ( x ˆx y ŷ + zẑ) (x + y + z ) 3 ˆ dx ( y ẑ zŷ) (x + y + z ) 3 ˆ w w y y + dy z μ I πw zŷ ˆ w w x d w r r z r r J 6.4: dy = μ ˆ w I ( y ẑ zŷ) 4πw w y + dy z y + dy z = μ I w πw tan z ŷ (6.7) y

70 7 6 ( ) r J(r ) [A/m ] d [m 3 ] J(r )d [A m] = ˆ μ J(r )d (r r ) B(r) = db(r) = 4π r r 3 (6.8) 6.3 ( ) ( ) ( ) 6.3. B [T] I [A] μ B(r) dr = μ I = μ J(r) d (6.9) B(r) =μ J(r) (6.) I I I I I I I I I I (a) μ (I + I ) (b) μ I (c) μ (I I ) (d) μ ( I + I ) (e) μ I (f) μ I 6.5: B dr ( ) (a) B(r) dr : B(r) [T] ( )

71 r ( ) dr B(r) dr B(r) dr ( B(r) dr ) B(r) dr ( ) (b) μ J(r) d : J(r) d (d ) r ( )d d d ˆn(r) ( r ) d = ˆn(r)d r J(r) [A/m ] J(r) d d I [A] I = J(r) d μ ) B(r) B dr B(r) z = xy B(x, y, ) = μ I( y ˆx + xŷ) π(x + y ) z = xy a ( +z ) r x φ r φ r(φ) =a cos φˆx + a sin φŷ = x(φ)ˆx + y(φ)ŷ z φ φ<π ( I π φ< 5π ) r(φ) dr = dr a a a y dφ = adφ( sin φˆx +cosφŷ) dφ r φ x a B( r(φ)) dr = μ I( a sin φˆx + a cos φŷ) πa adφ( sin φˆx+cos φŷ) = μ I π dφ 6.6: ˆ π μ I B(r) dr = π dφ = μ ˆ π I dφ = μ I (6.) π I μ :

72 B(r) ( ) : z +z I [A] φ (B(ρ, φ) =B φ (ρ) ˆφ) ρ B dr = B φ πρ = μ I B φ = μ I πρ, B = μ I πρ ˆφ (6.) : a [m] I [A] ρ < a φ (B(ρ, φ) =B φ (ρ) ˆφ) r ρ [m] ρ > a ρ <a I πa [A/m ] 6.7: { { I μ πρ, ρ < a μ ρi, ρ < a B(r) dr = B φ πρ = πa B φ (ρ) = πa μ μ I, ρ a I πρ, ρ a (6.3) N ( I [A]) A D AD B AB D A' B' B AB l B D l = B AB = B D (6.4) D' A D B D = B AB = B =A B D D B A B l B D l = μ lni B D = B A B = μ NI B = μ NI 6.9 ( N ) I ρ ρ I l ' B 6.8: B φ πρ = μ NI B φ = μ NI πρ B φ = (6.5) 6.9:

73 B(r) B(r) d = (6.6) B(r) B(r) = (6.7) ( ) E(r) E(r) dr = E(r) d = ε ( ) B(r) B(r) dr = μ J d = μ I ρ(r)d = Q B(r) d = ε : E(r) = N Q i (r r i ) 4πε r r i= i 3 : E(r) = ˆ λ(r )dr (r r ) 4πε r r 3 : B(r) = μ 4π : : E(r) = 4πε : E(r) = 4πε σ(r )d (r r ) r r 3 B(r) = μ 4π : ρ(r )d (r r ) r r 3 B(r) = μ 4π Idr (r r ) r r 3 J s (r )d (r r ) r r 3 J(r )d (r r ) r r 3 () a [m] I [A] ( ) ) xy ( ) +x +y φ r = zẑ r (φ )=a cos φ ˆx+a sin φ ŷ, φ π r r = a cos φ ˆx a sin φ ŷ + zẑ r r = (a + z ) Idr = Ia( sin φ ˆx +cosφ ŷ)dφ Biot-avart db = μ I 4π = μ I 4π a( sin φ ˆx +cosφ ŷ)dφ ( a cos φ ˆx a sin φ ŷ + zẑ) (a + z ) 3 z(cos φ ˆx +sinφ ŷ)+aẑ adφ (a + z ) 3

74 74 6 B( r(z)) =B(z) [T] ˆ B(z) = db = μ ˆ I a π {z(cos φ 4π (a + z ) 3 ˆx +sinφ ŷ)+aẑ}dφ = μ I a ẑ (a + z ) 3 r = zẑ, r (φ )=aˆρ(φ ), r r (φ )= aˆρ+zẑ, r r = (a + z ),Idr = I dr dφ = Iaˆφ(φ )dφ, dφ B(z) = μ ˆ π Iaφ(φ ˆ )dφ ( a ˆρ(φ )+zẑ) = μ ˆ I a π (z ˆρ + aẑ)dφ 4π (a + z ) 3 4π (a + z ) 3 = μ I a ẑ (a + z ) 3 ˆ π ˆρ(φ )dφ = ( ) () z +z a x a, b y b I [A] r = xˆx B(r) { (Ans.) : B(r) = μ I b log 4πab b 4 +(x + a ) b 4 +(x a ) +4(x + a )tan b x + a 4(x a )tan } b x a ŷ B(r) r Idr df df = Idr B(r ) [N] (6.8). : I [A] +z r = aŷ y = a l z l y = a l z l z r = aŷ + z ẑ ( l z ) l Idr = Idz ẑ r B(r) = μ I( y ˆx + xŷ) π(x + y ) r B(r )= μ I ˆx πa 6.: Idr z I x l/ I z r y a l/ 6.:

75 df df = Idr B(r )=Idz ẑ l z l ( μ ) I πa ˆx = μ I πa dz ŷ (6.9) ˆ F = df = ˆ l l μ I πa dz ŷ = μ I l πa ŷ (6.) y l [m] : A( ) : I A( ) 7 A l =m I =A y =m F = μ π = 7 N A m m 7 N A. : 6. B(r) =B ŷ a, b I x φ : r (z )= b cos φˆx + b sin φŷ + z ẑ( a z a ) Idr = Idz ẑ df = Idz ẑ B ŷ = B Idz ˆx dn ( b dn = r df = cos φˆx + b ) sin φŷ + z ẑ ( B Idz ˆx) = ˆ a F = B I ˆx dz = B Iaˆx (6.) a ( ) b sin φẑ z ŷ B Idz (6.) ˆ ˆ a ( ) b N = dn = B I sin φẑ z ŷ dz = B I ab sin φẑ (6.3) a r (s) = b cos φˆx b sin φŷ sẑ, ( a s a ) Idr = Idsẑ df = Idsẑ B ŷ = B Idsˆx F = B Iaˆx dn = r df = ( b sin φẑ sŷ) B Ids N = dn = a a B I ( b sin φẑ sŷ) ds = B I ab sin φẑ

76 r (s) = ( b s) cos φˆx+ ( b s) a sin φŷ + ẑ, ( s b) Idr = Ids(cos φˆx + sin φŷ) df 3 = Ids(cos φˆx+ sin φŷ) B ŷ = B Idscos φẑ F 3 = B Ibcos φẑ dn 3 = r df 3 = ( b s) cos φb Idsŷ ( b s) sin φ cos φb Idsˆx ˆ ˆ b ( ) b N 3 = dn 3 = B I(cos φŷ sin φ cos φˆx) s ds = x z φ a a 4 B = B ŷ 3 I y ( b cos φ, b sin φ, ) 6.: (6.4) 4 r (s) =s cos φˆx + s sin φŷ a ẑ, ( b s b ) Idr = I(cos φˆx+sinφŷ)ds df 4 = Idr B = I(cos φˆx+sinφŷ)ds B ŷ = B I cos φdsẑ F 4 = B Ibcos φẑ dn 4 = r df 4 =(scos φˆx + s sin φŷ a ẑ) B I cos φdsẑ = B Is( cos φŷ +sinφcos φˆx)ds N 4 = b B b Is( cos φŷ + sin φ cos φˆx)ds = N = 4 i= N i = B Iabsin φẑ, F = 4 i= F i = () φ = 6.4. B [T] q [] v [m/s] F [N] F = qv B(r) (6.5) r q ρ(r) r 6.3: f(r) =ρ(r)v(r) B(r) (6.6) v(r) r ρ(r) F = f(r)d = ρ(r)v(r) B(r)d (6.7) : B(r) =B ẑ [T](B [T] ) t =s q [] ( m [kg]) v = v ˆx [m/s] r = t> r(t) v(t) m dv(t) = qv(t) B ẑ = qb {v x (t)( ŷ)+v y (t)ˆx} (6.8) dt m dv x dt = qb v y (t), m dv y dt = qb v x (t) (6.9)

77 v x () = v,v y () = ( ) ( ) qb v x = v cos m t qb, v y = v sin m t v(t) =v {cos ( ) ( ) } qb m t qb ˆx sin m t ŷ t r(t) r() = ˆ t r(t) =r() + v(t )dt = mv [ ( ) { ( ) } ] qb sin qb m t qb ˆx + cos m t ŷ (6.3) (6.3) (6.3) ( ) ( ) ( ) () v B F L = qv B (6.33) ( q> q<) E h F E = qe h F L qe h + qv B = (6.34) N [m 3 ] J J = qnv (6.34) N qne h + J B = E h = qn J B = RJ B (6.35) 6.4: R = qn [ m 3 ] (6.36) 9 Hall ( )

78 N [m 3 ] q [] v = v ˆl [m/s] Δt [s] Δ [m ] ΔQ [] ΔQ = qnv Δt Δ (6.37) v Δt = Δr I [A] I = ΔQ Δt = qnv Δ (6.38) 3 N[m ] Δ q v Δr = ˆlΔr F A [N] F A = Idr B = Iˆl Δr B = qnvδ Δr B =(NΔ Δr)qv B (6.39) 6.5: (NΔ Δr)

79 79 7 (= ) (a) (b) ( 7.: ) (Fe) (Ni) (o) (Gd) ( ) 7. ( ) B(r) [T] B = μ J, B = (7.) H(r) [A/m] H = J, B = (7.) B H μ [H/m] B = μh μ H (7.)

80 A(r) ( A) = B = B = A A [Wb/m = H A/m] B A A B A(r) = μ ˆ Idr (7.3) 4π r r μ 4π ˆ Idr r r = μ ˆ 4π = μ ˆ 4π Idr r r = μ ˆ 4π r r Idr Idr r r = B(r) (7.4) r r 3 = J(r) [A/m ] A(r) = μ J(r )d 4π r r (7.5) 7.. a [m] r = xˆx + yŷ + zẑ r a r = a cos φ ˆx + a sin φ ŷ, Idr = Ia( sin φ ˆx +cosφ ŷ)dφ (7.6) r r =(x acos φ )ˆx +(y a sin φ )ŷ + zẑ (7.7) x z r r r y r r = {(x a cos φ ) +(y a sin φ ) + z } = {x + y + z a(x cos φ + y sin φ )+a } a r = {r a(x cos φ + y sin φ )+a } I φ { = r a ( x r r cos φ + y } ) r sin φ + a r (7.8) 7.: a/r ( ) { r r r + a ( x r r cos φ + y { )} r sin φ = r + a sin θ } (cos φ cos φ +sinφsin φ ) r { = r + a sin θ } cos(φ φ ) = r r + a sin θ r cos(φ φ ) (7.9)

81 7.. 8 A(r) = μ ˆ π { Ia sin φ ˆx +cosφ ŷ + a sin θ } ( sin φ ˆx +cosφ )cos(φ φ ) dφ (7.) 4πr r, 3 4 ˆ π ˆ π sin φ cos(φ φ )dφ = cos φ cos(φ φ )dφ = ˆ π ˆ π {sin φ +sin(φ φ)}dφ = π sin φ (7.) {cos φ +cos(φ φ)}dφ = π cos φ (7.) A(r) = μ Iπa sin θ 4πr ( sin φˆx +cosφŷ) = μ Iπa sin θ ˆφ (7.3) 4πr Iπa ẑ = m ẑ ˆr =(cosθˆr sin θ ˆθ) ˆr =sinθ ˆφ A(r) = μ m ˆr 4πr = μ m r 4π r 3 (7.4) m = Iẑ [A m ]( ) ( ) r i M(r) = lim m i [A/m] (7.5) Δ Δ r Δ [m 3 ] M [A/m] r M(r) = Δ m m = MΔ IΔl F IΔl F I B 7.3: F m = μ I [Wb m] B I F B 7.4:

82 8 7 M A(r) = μ M(r )d (r r ) 4π r r 3 (7.6) 7..4 ( 7.5) B I m [A] J m [A/m ] = B Im B dr = μ (I + I m ), B = μ (J + J m ) (7.7) 7.5: I [A] B J m M ( ) M(r ) (7.8) r r r (ϕa) =ϕ A+ ϕ A ϕ = r r, A = M ( ) M(r ) = r r r r M(r )+ r r M(r ) = M(r ) r r + (r r ) M(r ) r r 3 (7.9) M(r ) (r r ) r r 3 = ( ) M(r ) M(r ) (7.) r r r r (7.6) M(r ) A(r) = r r M(r ) = r r Ad = d d M(r ) d r r ( ) M(r ) d r r (7.) d A M = ( ) M(r ) A(r) = d (7.) r r (7.5) M(r ) J m J m (r) = M(r) (7.3)

83 7.3. B, H H : B = μ (J + J m ), J m = M (7.4) B = J + J m = J + M B ( ) B M = M = J (7.5) μ μ μ H(r) = B μ M(r) H(r) =J (7.6) H(r) [A/m] J m J (7.6) ( ) H M M H M = χh (7.7) χ B = μ (H + M) =μ ( + χ)h (7.8) μ ( + χ) =μ μ [H/m] μ μ r = μ/μ =+χ μ μ r B = μh = μ μ r H (7.9) μ r = 5 ( ) 6 ( ).4( ).99999( ) 7.3 B, H H(r) =J(r), B(r) = (7.3) H dr = I = J d, B d = (7.3) B = μh ( ) I ( ) 7.6 z = z z l z ε μ x Δt ε μ y 7.6: z = z

84 84 7 : H y (x, y, z + Δt )l + H y(x, y, z Δt )l + H z(x, y + l,z )Δt H z (x, y l,z )Δt = (7.3) Δt y H y (x, y, z ) = H y (x, y, z + ) (7.33) H t (x, y, z ) = H t (x, y, z + ) (7.34) ( ) J x [A/m] H y (x, y, z + Δt )l + H y(x, y, z Δt )l + H z(x, y + l,z )Δt H z (x, y l,z )Δt == J x (x, y, z ) l (7.35) Δt H z H y (x, y, z ) = H y (x, y, z +)+J x (x, y, z ) (7.36) : ( 7.7) z = z Δh Δ B z (x, y, z + Δh )Δ B z(x, y, z Δh )Δ + B d = (7.37) Δh B z (x, y, z +)=B z (x, y, z ) (7.38) z z Δh Δ ε μ ε μ y z B n (x, y, z +)=B n (x, y, z ) (7.39) ( ) x 7.7: z = z H B 7.4 ()

85 () () ( ) (3) ( ) ( ) [] l [m] E E dr = (7.4) (σ: ) E = E l l J [A/m ] J = σe σ [/m] 7.8: (μ: ) I [A] I = J d ( ) J J I = J NI = E dr = E l = J σ l = I ( ) l σ l = I = RI (7.4) σ l R = l [Ω] σ 7.9: N I [A] ( B [T], H [A/m]) H dr = N I (7.4) H NI = H l B B = μh = μ r μ H μ [H/m] μ r μ r = Φ[Wb] Φ= B d ( ) B B Φ =B N I = H dr = H l = B μ l = Φ ( ) l μ l = Φ=R m Φ (7.43) μ R m = l μ [A/Wb = H ] (7.4) (7.43)

86 N I Φ R m : () () μ r (3) (4) ( ) (B H ) B-H 7.4. ( ) (7.43) OK Φ ( B ) H m H a N I = H dr = H m l + H a δ = B μ l + B δ = Φ μ μ l + Φ μ δ ( l = μ + δ ) Φ=(R m + R a )Φ (7.44) μ 7.: R m R a R a δ N I J = σe J = I = J d = E dr R = l σ B = μh B = Φ= B d N I = H dr R m = l μ. 7. N = 4cm cm mm. I =. A T.. T. 7. ( μ r = μ r = 5) ( δ = mm) l =8,l =5cm 4cm NI = 4 A 7.:

87 () Ad = d A (A ) = A (A )d = ( A)d = ( A)d = (A ) d = (d A) = (d A) () r r = r r r r 3 ( A)d = (d A) (7.45) ( r r ) x = x {(x x ) +(y y ) +(z z ) } x x = = x x {(x x ) +(y y ) +(z z ) } r r (7.46) ( r r ) x = r r ( r r ) x = r r x x r r = x x r r 3 (7.47) y,z r r = ( r r ) x ˆx + ( r r ) y ŷ + ( r r ) ẑ z = (x x )ˆx +(y y )ŷ +(z z )ẑ r r 3 = r r r r 3 (7.48)

88

89 ( ) ( B(r) [T] ) Φ[Wb] B(r,t) d [Wb]( ) Φ [] ( ). ( A) ( ) ( B) ( 8.) B W ON B ( B) A A A ( ) Φ A () I W B A 8.: B A B A A (=). (N ) 8. ( N ) 8.: N ( ). Φ(t) [Wb] B(r) [T=Wb/m ] Φ(t) = B(r,t) d (8.) [] (t) = dφ(t) dt (8.)

90 9 8 () = E(r,t) dr (8.3) E(r,t) dr = d dt B(r,t) B(r,t) d = d (8.4) t ( d ) (8.4) E(r,t)= B(r,t) t (8.5) 3. I(t) =I sin ωt B(x, y, t) = μ I(t)( y ˆx + xŷ) π(x + y (8.6) ) Φ(t) = B(r,t) d r = yŷ + zẑ,s y s + w, a z a r = yŷ + zẑ d = dy dz ˆx dφ(t) dφ(t) =B(r,t) d = μ I(t)( y ˆx) πy ˆ Φ(t) = dφ(t) = = μ I(t)a π ˆ a a ˆ s+w s log s + w s dy dz ˆx = μ I(t) dydz (8.7) πy μ ˆ s+w I(t) πy dy dz = μ I(t)a dy π s y (8.8) x z a I z a s dz d dy y y s + w 8.3: = dφ dt = μ a π log s + w di(t) = μ ai ω log s + w cos ωt (8.9) s dt π s 4. B(r) =B ẑ v(t) =v ŷ t = y = b

91 8.. 9 v y (t) = dy dt = v y(t) =v t + y() = = b y(t) =v t + b (t) (t) =a(v t + b) +z Φ(t) =B (t) =B a(v t + b) z B y = dφ dt = B av (8.) ( ) x a 8.4: v(t) =v ŷ I [A] Φ[Wb] [H](Henry) Φ=LI (8.) L [H] I [A] Φ [Wb] Φ = M I (8.) M [H] I [A] Φ [Wb] L, M M = M ( a [m] N ) I [A] B = μ NI [T] l [m] Nl B = B =

92 9 8 Φ=(Nl)(πa )(μ NI)=μ N πa li, L = μ N πa l (8.3). ( ) r = lŷ B = μ I a ŷ (8.4) (a + l ) 3 Φ= B d = μ I a πb = M (a + l ) 3 I, M = μ πa b = M (a + l ) 3 (8.5) z oil I a a I a l x oil y 8.5: b( a) t [s] L [H] I [A] Δt [s] ΔI [A] = L ΔI Δt ΔQ = IΔt [] ΔW = L ΔI IΔt = LIΔI [J] (8.6) Δt (ΔI di) I =A I [A] ˆ ˆ I W = dw = LI di = LI [J] (8.7) L I LI [J] 8.3. ( a [m] N ) l [m] L = μ N πa l [H] (8.8) B = μ NI [T] (8.9) W = LI = μ N πa li = μ (μ NI) πa l (8.)

93 πa l W = B = B d (8.) μ μ B = μ H B(r), H(r) W = BHd (8.) w(r) = B(r) H(r) [J/m3 ] (8.3). a [m] l [m] L i I [A] ρ [m] W [J] W = HB d = ˆ l z= ˆ π ˆ a φ= ρ= πρh = πρ Iρ I, H = πa πa (8.4) ( ) Iρ μ πa ρdρdφdz = μ I l 4πa 4 ˆ a ρ 3 dρ = μ I l 6π = L ii (8.5) L i = μ l 8π (8.6). a, b [m] I [A] a ρ b B = μ I πρ, H = I (8.7) πρ l [m] W = BH d = L [H] ˆ l ˆ π ˆ b a μ I (πρ) ρdρdφdz = μ I l 4π log b a = LI (8.8) L = l μ π log b a (8.9)

94 94 8 : d dφ dw [J] W = B Hd d dw = H Bddr = H (dφdr) = dφ H dr = dφi (8.3) I ( ) 7. () (8.3) ρ [m] d = dρ dz dφ dφ =BdΦ= μ Iρ dρdz (8.3) πa I = ρ a I dw = dφi = ρ a I μ Iρ πa dρdz = μ I ρ 3 dρdz (8.3) 4πa4 l dz dρ ρ W = ˆ l ˆ a dz μ I ρ 3 4πa 4 dρ = μ I l 6π = L ii (8.33) L i = μ l 8π 8.6:

95 95 9 ( ) (E, B, D, H) ( ) ( ; James lerk Maxwell; ) ( ) ( ) ( ) : E(r) dr = E = B(r,t) E(r,t) dr = d, E(r,t)= B(r,t) (9.) t t : D(r,t) d = ρ(r,t)d D(r,t)=ρ(r,t) (9.) : H(r) dr = I = J(r) d H(r) =J(r) (9.3) : ( )B(r) B(r,t) d = B(r,t) = (9.4)

96 96 9 ( ): ( ) D(r,t)=ε(r)E(r,t), B(r,t)=μ(r)H(r,t) (9.5) 9.. Q(t) = ρ(r,t)d ( dq dt ) Δt [s] ΔQ [] ΔQ = J(r,t) ( d)δt ΔQ = J(r,t) d (9.6) Δt d ΔQ d Δt dq dt = J(r,t) d = d ρ(r,t) ρ(r,t)d = d (9.7) dt t ρ(r,t) d = J(r,t) d (9.8) t ρ(r,t) = J(r,t) (9.9) t 9..3 = : H(r) dr = I = J(r) d (9.) 9.:

97 H(r) dr (9.) ( 9.) J(r) d J(r) d (9.) I +Q -Q I +Q -Q (9.8) (a) (b) 9.: (a) (b) ( 9.) H = J J = H = (9.3) J = (9.9) : D(r,t)=ρ(r,t) (9.4) ( D) t = D t = ρ t (9.5) D t = ρ t = J ( ) D t + J = (9.6) J D t H = D t + J (9.7) = D(r,t) [A/m ] t 9..4 ( ) E(r,t)= B(r,t) t (9.8) D(r,t)=ρ(r,t) (9.9) H(r,t)= D(r,t) + J(r,t) t (9.) B(r,t) = (9.)

98 ( ) = E(r,t)= B(r,t) t (9.) D(r,t)=ρ(r,t) (9.3) H(r,t)= D(r,t) + J(r,t) t (9.4) B(r,t) = (9.5) (J,ρ) ( ) ( ) 9.. x, y ( x, y ) x (E x) ( ) E x (z,t) z = B y(z,t), H y(z,t) = D x(z,t) t z t (9.6) B y = μ H y,d x = ε E x E x z = B y t z = μ H y t z = μ D x t = ε μ E x t E x (z,t) t ε μ E x (z,t) z = (9.7) E x (z,t) =f(z c t), c = ε μ (9.8) ( ) f(u) ε F/m, μ =4π 7 H/m c = ε μ m/s 3 8 m/s (9.9) f(u) (9.8) f(u) u = a ( 9.3) u = z c t t ( ) (z) 9.3: f(u) a

99 t = u = z f(z) z = a Δt [s] u = z c Δt f(z c Δt) z c Δt = a z = a + c Δt f(z) f(z c Δt) f(z c t) Δt c Δt z z ( c ) a a+ c Δt c 3. 8 m/s ( 9.4: ( : t =, : t =Δt) ) c 9.. H y (z,t) t ε μ H y (z,t) z = (9.3) H y (z,t) =g(z c t) (9.3) E x H y x, y z ( ) ε μ c c = με (9.3) μ>μ,ε >ε ( ε r (> ) μ r (> ) εr μ r ) c = fλ λ c f λ ( ) ε r μ r (9.3) (9.8) (9.6) f (z c t)=c μ g (z c t) (9.33) ( ) μ E x (z,t) =c μ H y (z,t) = H y (z,t) (9.34) ε

100 9 μ Z = (9.35) ε Z Z π 377 [Ω] μ Z = ε 9..3 (9.8), (9.34) +z ( ˆd ) r ˆd ˆd r E = E f( ˆd r c t), H = H f( ˆd r c t) (9.36) E, H ˆd E, H (ρ =) E =, H = (9.37) (9.36) ˆd E =, ˆd H = (9.38) H E = μ t ˆd E = Z H E E E E ˆd E = ˆdE = Z E H, E H = E Z ˆd (9.39) E H ( ) H ˆd 9..4 ( ) (Poynting s vector) [W/m ] (r,t)=e(r,t) H(r,t) (9.4) Z E(r,t) (r,t)= E (r,t) Z (9.4)

101 9.. ( ) {E H} {E H} = H E E H = H B t E D t E J (9.4) Gauss {E H}d = {E H} d = d D = εe, B = μh { H B t + E D t (H H) t } d + E Jd (9.43) =H H t { H B t + E D } d = d { μ t dt H + ɛ E } d (9.44) w m = μ H [J/m 3 ] w e = ɛ E [J/m 3 ] ( ) Δ Δ E Δ Δl E J E JΔ = EΔl JΔ = I (9.45) [] Δl I [A] Δ Δ H} d + {E d (w e + w m )d = (9.46) dt E H E H E H d = E H [W/m ] Poynting Maxwell E, H