i

Transcription

1

2 i flux div div

3 ii rot rot grad,rot,div I

4 iii B H E D B H

5 iv

6 magnet elektron electrics cgs Gb cgs 3 (classical) (quantum) 4 5

7 one piece 6 6

8 IC [ ] [ ] 8 (interaction) 7 8

9 ( non-local non-local 10 [] 9 10 EPR

10 F F F F r r r Q q F = Qq > 0 Qq 4πε 0 r (1.1) Qq < (1.1) SI q, Q C ε 0 SI F m W (100W 1 100J )1 100J J t ct c F( ) ε 0 = Qq 4πr F ε 0 [C /N m ] F C /N m 5 100W 100J

11 6 1 r 4πr 100J 4πr πr 1 4πε π ε 0 1 SI 1 4πε 0 SI πε 0 = k k = [N m /C ] 6 SI SI (177 ) C 1m N 10 1C kg kg = C = ( ) 1 = [N] (1.) 6 k m/s

12 ( ) 1 = N (1.3) 9 0 e e 0 (1.4) 1.1. (1.1) Q x Q q x q q Q x Q x q 10 Q F q Q = Qq 4πε 0 x Q x q 3 ( x Qq Q x q ) = 4πε 0 x Q x q e q Q (1.5) () 3 x Q x q () x Q x q e q Q = 1 x Q x q ( x Q x q ) (1.6) x Q x q x Q x q 1 e 11 F q Q = Qq 4πε 0 x Q x q 3 ( x Qq Q x q ) = 4πε 0 x Q x q e q Q (1.7) O F q Q q Q Q q F Q q F Q q = Q q qq 4πε 0 x q x Q 3 ( x qq q x Q ) = 4πε 0 x q x Q e Q q (1.8) kg 3.45m/s 10 x q x Q x q x Q 11 e 1

13 8 1 z e r e z e y F q Q = F Q q 1 q x q = 0 Q r e e e x y (r, θ, φ) x Q x q = r e r (1.9) e r (r, θ, φ) r 13 e θ, e φ θ φ x 14 r e r r z z z e r e r e e y y y e e x x x e x Q x q = r F q Q = Qq 4πε 0 r 3 r e r = Qq 4πε 0 r e r (1.10) Qq > 0Qq < 0 q>0 q<0 q >0 qq >0 q >0 qq < x x e x r e r e x 14 e ρ, e φ, e z

14 Q i (i = 1,,, N) q Q i ( )

15 10 1 q x q Q 1, Q,, Q N x Q1, x Q,, x QN q F q = N F Qi q = i=1 = N i=1 N i=1 Q i q 4πε 0 x q x Qi 3 ( x q x Qi ) Q i q 4πε 0 x q x Qi e Q i q (1.11) q F E = F q x q x q x q qq 4πε 0 x x e x x E( x F ) = q q q q = 4πε 0 x x e x x (1.1) 16 electric field field field

16 SI F = q E F [N]q [C][N/C] V E = V [V] [1/m] [V/m] Q r Q 4πε 0 r 1m Q 4πε 0 r r r 4πr Q 4πε 0 r 4πr = Q 19 ε 0 r Q[C] Q ε 0 1 ε C 1130 FAQ π SI Q 4πQ

17 cos θ θ n E E ns ns S E S 0 1 r r Q 4πε 0 r 4πr = Q ε 0 cos θ (1) () (3) (4) (5) (6) Q[C] Q ε 0 Q Q ε 0 0

18 N S N S N N 1

19 d Q (x, y, z) z ρ 3 dz ρdz z = L z = L L z = 0 z r = x + y Step 1. Step. Step 3. Step z Step 1. dz ρdz Step. (x, 0, 0) (0, 0, z) (0, 0, z + dz) ρdz ρdz 4πε 0 (x + z ) (1.13) z z x z 0 x x ρdz 4πε 0 (x + z ) x x + z = ρxdz 4πε 0 (x + z ) 3 (1.14) 3 1m 1m 1m 3 m cm

20 Step 3. z ( L, L) ρxdz L ρxdz z 4πε 0 (x + z ) 3 L 4πε 0 (x + z ) 3 dz dz L Qρx dz z = x tan θ L 4πε 0 (x + z ) 3 θ x + z = x cos θ 4 z = L tan θ = z α x = α ρx ) 3 x cos θ dθ ( α 4πε x 0 cos θ ρ α cos θdθ = 4πε 0 x α ρ 4πε 0 x [sin θ]α α = α = π E = ρ πε 0 x Q ρ L Q 4πε 0 xl sin α = Q 4πε 0 x x sin α (1.16) L ρ πε 0 x sin α L R (1.15) sin α = L R x L sin α = x R 1 Q Q 4πε 0 x z x x dz ρdz E = 4πε 0 (x + z ) (cos θ e x sin θ e z ) (1.17) cos θ e x sin θ e z x θ z = x tan θ, dz = x cos θ dθ x + z x = cos θ cos θ e x sin θ e z θ tan θ = 1 cos θ E = ρdθ 4πε 0 x (cos θ e x sin θ e z ) (1.18)

21 16 1 Z α α (cos θ e x sin θ e z) dθ (1.19) e z e x Z α α cos θ e x dθ = [sin θ] α α e x = sin α e x (1.0) (1.15) σ 6 (r, θ) 7 r r + dr θ θ + dθ r 0 r 0 θ 0 π r dr θ rdθ dθ rdrdθ σrdrdθ z P σrdrdθ 4πε 0 R σrdrdθ 4πε 0 R z R (1.1) r 0 z P d R r rd dr R R = z + r σz 4πε 0 r0 π () 0 0 rdrdθ (z + r ) 3 dθ θ π σz ε 0 r0 0 rdr (z + r ) 3 r = z tan φ dr = σz ε 0 φ0 0 z tan φ ( z + z tan φ ) tan φ = 1 cos φ 1 ( 1 + tan φ ) 3 φ0 z cos φ dφ = σ ε 0 z cos dφ φ φ0 0 tan φ ( 1 + tan φ ) 3 = cos 3 φ (1.) (1.3) 1 cos dφ (1.4) φ σ sin φdφ = σ [ cos φ] φ 0 0 = σ [ cos φ 0 + cos 0] = σ (1 cos φ 0 ) (1.5) ε 0 0 ε 0 ε 0 ε 0 z cos φ 0 = z + (r 0 ) E = σ z + (r 0 ) z Q = ε 0 z + (r 0 ) πε 0 (r 0 ) z + (r 0 ) z (1.6) z + (r 0 ) Q = π(r 0 ) σ z (r, θ, z)

22 r 0 z + (r 0 ) z z z + (r 0 ) r = 0 r 0 r 0 cos φ 0 = 0 φ 0 σ ε 0 r =1 r = r =3 0 r 0 z (r 0) z r 0 z r 0 z 1 r 0 1E z z + (r 0 ) z z z + (r 0 ) z + (r 0 ) z z + (r 0 ) = 1 + x x r = ( r 0 z ) ( r 0 z ) (1.7) z + (r 0 ) z = 1 + ( 1 r0 ) ( z r0 ) ( z + (r 0 ) r0 ) = z + ( z r0 ) (1.8) z + z z + (r 0 ) z z + (r 0 ) 1 ( r0 ) (1.9) z E (1.6) Q πε 0 (r 0 ) z + (r 0 ) z z + (r 0 ) Q πε 0 (r 0 ) 1 ( r0 ) Q = z 4πε 0 z (1.30) z z Q z r 0

23 18 1 z e z r e r σrdrdθ P z e z r e r (1.1) E( x q ) = ` x x x q x 4πε 0 x x 3 σ 4πε 0 Z r0 Z π 0 0 rdrdθ z e z r e r 3 (z e z r e r ) (1.31) e r σ 4πε 0 Z r0 Z π 0 e r 8 e r r Z π 0 0 dθ e r = 0 (1.33) e r (1.3) z e z r e r = p z + r θ e r e r (1.33) 0 e z σ 4πε 0 Z r0 Z π 0 0 rdrdθ 3 z ez (1.34) z e z r e r 0 rdrdθ 3 r er (1.3) z e z r e r π steradian Ω Ω = dθdφ sin θ (1.35) sin θ θ θ + dθ φ φ + dφ dθ sin θdφ 8 e x, e y, e z

24 Q Q ε 0 4π Ω QΩ 4πɛ 0 a : b a b 0

25 (1.35) θ [0, π] φ [0, π] 4π z α, β γ α β α + β 1-3 q 1 cos θ 1 + q cos θ = (1.36) θ 1, θ P A (1) A A A () A 1 A ( ) θ π(1 cos θ) (3) q 1, q, θ 1, θ σ ε 0 (1 cos φ 0 ) σ 4πε 0 ( P ) (1.37)

26 (1) () z z 1-5 r σ z z < r z 0 z R r + z rz cos θ σr sin θdφdφ z π cos θ = t 0 F (cos θ) sin θdθ 1-6 ρ z R r

27

28 3.1 (Gauss).1.1 flux Q E( x) = 1 4πε 0 r e r R 4πR Q ε 0 1 E ds = E nds (.1) n nds d S n x yz ds = dydz n y ds = dzdx z ds = dxdy n = (n x, n y, n z ) d S = n x dydz e x + n x dzdx e y + n x dxdy e z (.) E d S E n θ E cos θ n 1 cos θ 1.3. E d S (flux) ρ v d S (.3) cos θ S Sv cos θ flux E d S 1 S

29 4 r Q 4πε 0 r r sin θdθdφ Q 4πε 0 (r + dr) (r + dr) sin θdθdφ E ds 0 E d S = Q ε 0 A B A B flux E ds = 0 0 Q E ds ε = Q 0 ε 0

30 E ds flux θ cos θ flux cos flux ds E ds...1 z Q Q ε 0 Q ε 0 rdrdφ Q 4πε 0 (r + z ) θ Q 4πε 0 (r + z rdrdφ cos θ (.4) )

31 6 θ r r θ r = 0 r = r θ θ = 0 θ = π r = z tan θ 1 dr = z cos θ dθ Q cos θ tan θ dθ 4πε 0 cos dφ cos θ = Q dφdθ sin θ (.5) θ 4πε 0 π π dθ cos θ = 1 dφ = π E = Q 0 0 ε Q a r e r a e r r d S r sin θdθdφ e r r r sin θdθdφ Q 4πε 0 r e r a 3 (r e r a) e r r sin θdθdφ (.6) e r a = a cos θ a a Q 4πε 0 (r + a ar cos θ) 3 (r a cos θ) r sin θdθdφ (.7) cos θ = t sin θdθ = dt θ (0, π) t ( 1, 1) sin θdθ 3

32 Qr 1 dt 4πε 0 1 π 0 r at dφ (r + a art) 3 φ π t r at (r + a art) 3 = r ( 1 (r + a art) 1 5 t r ( ) Qr 1 1 dt ε 0 r 1 (r + a art) dt 1 (r + a art) 1 1 [ 1 1 ( dt = r + a art ) ] 1 1 = 1 ( (r a) ) (r + a) 1 (r + a art) 1 ar 1 ar ) (.8) (.9) (.10) (.11) r, a (r + a) r + a (r a) r > a r a r < a a r r > a 1 (r a (r + a)) = ar r (.10) r Qr ε 0 r < a 1 ar (a r (r + a)) = a r > a Q ε 0 r < a 0 0 Q ε 0.3 Z π Z 1 4 sin θdθ dt E ds = 1 Q i (.1) ε 0 E ds = 1 ρdv (.13) ε 0 V V

33 8 V V V V d S ρdxdydz.3.1 z = 0 xy σ S σs z (0, 0, z) z xy V x y z S σs σs ε 0 S σ ε r 0 Q πε 0 (r 0 ) z + (r 0 ) z z + (r 0 ) Q = π(r 0 ) σ r σ ( P 4πε 0 ) π E = σ ε 0 ε 0

34 z z z ρ ρ z ρ ε 0 z πr z ρ ε 0 z πr z = ρ πε 0 r ρ R 4π 3 R3 ρ 0 R Q ε 0 4πr Q 4πε 0 r e r 4π 3 r3 ρ E = 4π 3 r3 ρ 1 4πε 0 r e r = 1 ρr e r (.14) 3ε 0 r R r Q Q

35 30 S S Q Q Q ε 0 Q ε 0 S Q ε 0 S +Q -Q.4 (.13) (.13).4.1 flux E E (E x, E y, E z ) x y z z x+ x y+ y dx dy E z (x, y, z + z) (.15) x y x, y 0 y+ y dx dy x y x+ x x y E(x) E z (x, y, z + z) x y (.16) E z 6 E z x y ( E z 6 d S E z > 0 E d S

36 z z (E z (x, y, z + z) E z (x, y, z)) x y (.17) 7 E z (x, y, z + z) E z (x, y, z) lim = E z(x, y, z) z 0 z z (.18) E z x y z (.19) z z x y x, y, z ( Ex x + E y y + E ) z x y z (.0) z x y z E x x + E y y + E z z dive div 8 A div diva = A x x + A y y + A z z (.1) div (divergence) 9 10 div div (.13) E ds x y z V (.13) ρdv ρ x y z V ρ x y z x y z dive div E = ρ ε 0 (.) div E = 0 div E = 0 (.13) divedv = E ds (.3) V 7 Z x+ x Z y+ y dx dy (E z(x, y, z + z) E z(x, y, z)) x y x y (.17) 8 A( x) 9 10 (; ;) V

37 3 ( ) flux divedv V E ds V.4. dive = 0 ρ (0, 0, ) ε 0 dive = 0 Q E = Q 4πε 0 r e r e r x, y, z ( x x + y + z, y x + y + z, z x + y + z ) 11 ( ) Qx Qy Qz E =,, 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 3 (.4) E x x E x x = = = ( ) Qx x 4πε 0 (x + y + z ) 3 Q 3 Qx x 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 5 Q Q 3 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 5 x (.5) x, y, z Q 4πε 0 (x + y + z ) 3 Q 3 4πε 0 (x + y + z ) 5 y Q Q 3 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 5 3 x, y, z dive Q Q ( = 3 3 x + y + z ) (.6) 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 5 0 div z 11 e r (x, y, z) 1 p x + y + z e r

38 div V x x + V y y + V z z div diva = A r r + A θ θ + A φ φ div (r + r) sin θ θ φ r sin θ θ φ r, r θ, r sin θ φ r sin θ r θ φ r sin θ θ φ (r + r) sin θ θ φ flux (r + r) V (r + r, θ, φ) sin θ θ φ flux r V (r, θ, φ) sin θ θ φ V r > 0 (r + r) V r (r + r, θ, φ) sin θ θ φ r V r (r, θ, φ) sin θ θ φ = ( (r + r) V r (r + r, θ, φ) r V r (r, θ, φ) ) sin θ θ φ (.7)

39 34 r sin θ r θ φ div 1 (r + r) V r (r + r, θ, φ) r V r (r, θ, φ) r r r 0 1 ( r r V r (r, θ, φ) ) (.8) r r V r V r r r V r sin θ div 1 r sin θ θ (r sin(θ + θ)v θ(r, θ + θ, φ) r sin θv θ (r, θ, φ)) 1 r sin θ θ (sin θv θ) (.9) 1 r sin θ φ (r sin θv φ(r, θ, φ + φ) r sin θv φ (r, θ, φ)) div 1 r sin θ φ V φ (.30) div A = 1 r ( r ) 1 A r + r r sin θ θ (sin θa θ) + 1 A φ r sin θ φ (.31).4.4 div diva = A A = e x x + e y y + e z z A = A x e x + A y e y + A z e z e i e j = δ ij 1 «A = e x x + e y y + e z (A x e x + A y e y + A z e z ) = A x z x + A y y + A z (.3) z div A e x e x = e y e y = e z e z = 1 e e F ( x) = lim h 0 F ( x + h e) F ( x) h (.33) h F ( x) h h 0 e F (x) d F (x + h) F (x) F (x) = lim dx h 0 h (.34) x h F (x) h x e x F = F x e y F = F y e e F F x F F = e x x + e F y y + e F z z 1 e x, e y, e z 1 F y (.35)

40 F = e x x + ey y + ez (.36) z e r e r F ( x) F θ r e θ F ( x) 1 F r θ θ h θ h r (.37) 1 r φ h φ e θ F F (r, θ + h, φ) F (r, θ, φ) r (r, θ, φ) = lim h 0 h F (r, θ + h r, φ) = F (r, θ, φ) + h r F θ F θ h r sin θ e φ F = 1 r sin θ F φ F F = e r r + e 1 F θ r θ + e 1 F φ r sin θ φ = e r r + e 1 θ r θ + e 1 φ r sin θ φ e r = r, e θ = 1 r θ, e φ = 1 r sin θ φ A = A r e r + A θ e θ + A φ e φ «A = e r r + e 1 θ r θ + e 1 φ (A r e r + A θ e θ + A φ e φ ) r sin θ φ = Ar r + 1 A θ r θ + 1 (.40) A φ r sin θ φ e r, e θ, e φ e r e r = 0, θ e r = e θ, φ er = sin θ e φ, r e θ = 0, θ e θ = e r, φ e θ = cos θ e φ, r e φ = 0, θ e φ = 0, φ e φ = sin θ e r cos θ e θ (.37) (.38) (.39) (.41) «A = e r r + e 1 θ r θ + e 1 φ (A r e r + A θ e θ + A φ e φ ) r sin θ φ = A r r + 1 A θ r θ + 1 A φ r sin θ φ r e θ B e Ar {z} θ = e θ = A r r + 1 r +A θ e θ θ A θ θ + 1 r sin θ {z} = e r C A + 1 r sin θ e φ Ar A φ φ + r A r + cos θ r sin θ A θ e r φ {z} =sin θ e φ +V θ e θ φ {z} =cos θ e φ +V φ e φ φ {z} sin θ e r cos θ e θ 1 C A (.4) (.31) e

41 div div div (.31) A E = Q 4πε 0 r e r 0 V r Q 4πε 0 r div r V r 0 div E = 0 E θ E φ div E = 0 1 ( r ) r E r r ( r ) E r r r E r = 0 = 0 = C (.43) E r = C r C div E = ρ ε 0 ρ 1 r r r ( r E r ) ( r E r ) = ρ ε 0 = ρ r ε 0 r E r = ρ 3ε 0 r 3 + C E r = ρ r + C 3ε 0 r (.44) C 0 r = 0 E r FAQ E r = Q 4πε 0 r C r = R E = C r E = ρ r 3ε 0 C R = ρ R C = ρ R 3 (.45) 3ε 0 3ε 0 4π 3 R3 ρ = Q E = ρr3 4π 3ε 0 r = 3 ρr3 4πε 0 r = Q 4πε 0 r (.46) dive = ρ (.47) ε 0

42 (r, φ, z) div div div A = 1 r r (ra r) + 1 r φ A φ + z A z (.48) - r kr n n R n -3 r 1 r r 1 < r < r ρ z r z -4 (.41) (1) () e x, e y, e z e r = sin θ cos φ e x + sin θ sin φ e y + cos θ e z (.49) e θ = cos θ cos φ e x + cos θ sin φ e y sin θ e z (.50) e φ = sin θ sin φ e x + sin θ cos φ e y (.51) z e r e z e y e r z cos e r r e e e x y z r e e e r sin e cos z sin x -5(.41)

43

44 m x F (x) m d x dt = F (x) x t 1 x 1 t x x 1 x 1 F(x) x d x x m x 1 dt dx = F (x)dx (3.1) x 1 F (x) dx dx dt x t dt dx dt d dt ( (dx ) ) t 1 t dt = d x dx dt dt (3.) ( 1 t (dx ) ) m d dt = F (x)dx (3.3) t 1 dt dt ( ) 1 dx m 1 ( ) dx x dt t=t m = F (x)dx = dt t=t 1 x 1 F (x)dx F (x)dx (3.4) x=x x=x1 x 1 t 1 x t ( ) 1 dx m dt t=t 1 F (x )dx x = U(x) F (x)dx x=x1 = 1 ( ) dx m dt t=t F (x)dx (3.5) x=x ( ) 1 dx m + U(x 1 ) = 1 ( ) dx dt t=t 1 m + U(x 0 ) (3.6) dt t=t 0 t 1 x 1 t 0 x 0 ( ) 1 dx m + U 1 ( ) dx dt m dt 1

45 40 3 U(x) U(x) F (x) U(x) = F (x)dx F (x) = d U(x) (3.7) dx mg g mg x F (x) = mg U = mgx M m GMm r r U(r) = GMm r r = x kx x = 0 x F (x) = kx 1 kx F (x) x 0 mg U(x) F(x)= - mg x F (x) d U(x) dx U(x) = 1 kx r x

46 Q Q 4πε 0 r F (r) = Qq 4πε 0 r U(r) = Qq F (r) = 4πε 0 r d dr U(r) 3 U(r) F (r) q F E = 1 q F 1C q U V = U q 4 V U F F = d dr U E = d dr V [V] 5 1.5V 1.5V q 1.5 q[j] q[c] 1.5q[J] q > 0 1.5V 1.5 q [J] 3 GMm r GMm Qq > 0 r 4 5

47 4 3 6 Q Qq 4πε 0 r Q r V = Q 4πε 0 r z F (x) F (x) (F x, F y, F z ) F x dx + F y dy + F z dz = F d x (3.8) F d x U(x, y, z) F x = U x, F y = U y, F z = U z (3.9) 6 q ( 1C ) 1.5q[J]

48 F = du dx U F x e x + F y e y + F z e z = e x x e U y y e U z z = e x x + e y y + e z z U ( Φ Φ( x) x, Φ y, Φ ) (gradient) z grad grad gradφ = Φ Φ = e x x + e Φ y y + e Φ z z (3.10) (3.11) grad div rot div grad grad e (gradφ) = e Φ = lim h 0 Φ( x + h e) Φ( x) h (3.1) x Φ h x + h e Φ h h Φ x h e h gradφ Φ h grad gradφ e e Φ h 3.. F = gradu E = V (3.13) V E z V

49 44 3 grad V grad V gradv gradv

50 3.3. rot 45 0 F (x) F U F = x e y = (0, x, 0) F y = x U = xy U x = y F x 0 F = U 3..3 F = U B A C D A B C A D C A B C D C D A C () F = U U 3.3 rot A(x, y) D(x + x, y) C(x + x, y + y) A(x, y) B(x, y + y) C(x + x, y + y) x, y

51 46 3 B C AB BC AD A D DC (A D C ) A B C ( ) ( ) F x (x, y) x + F y (x + x, y) y F y (x, y) y F x (x, y + y) x x F y x y y F x x y (3.14) x F y y F x = 0 (3.15) xy yz zx y F z z F y = 0 (3.16) z F x x F z = 0 (3.17) xy z yz x zx y rot ( rotf = y F z z F y, z F x x F z, x F y ) y F x (rotation) curl (3.18) 3.3. rot rot rot rot

52 3.3. rot 47 V x x V x x div y y V x (x, y y, z) x + V x (x, y, z x }{{}}{{} (3.19) V x (x, y + y) = V x (x, y) + V x (x, y) y + y V x x y y ( ) V y x + x Vy x y x ( Vy x V ) x x y (3.0) y V y x V x y rot rot rot rot rot rot x y z rot yz zx xy x yz 7 rot nonzero F = x e y rot nonzero F y x = 1 rot F = x e y rot ( rotf = y F z z F y, z F x x F z, x F y ) y F x ( ) a b = a y b z a z b y, a z b x a x b z, a x b y a y b x x z zx rot yz rot y x y xy rot z ( (3.1) rot = x, y, ) F z = (F x, F y, F z ) rot F = F (3.) 7

53 grad,rot,div div,rot,grad grad rot div grad rot 0 Φ gradφ rot rot(gradφ) = 0grad rot grad rot rot grad ( )-() rot div 0 V div(rotv ) = 0 C B A D div rot rot div rot rot div 6 rot rot rot rot ( rot)

54 rot div (Stokes) rot S (rotv ) ds = S S V d x (3.3) S S S divv dv = V ds V V rot E = 0 0 S ( ) ( ) ( ) Qq F = grad = 4πε 0 r Qq Qq = e r 4πε 0 r r 4πε 0 r U = Qq 4πε 0 r V = Q 4πε 0 r E = gradv V C gradc = 0 E = grad(v + c) (3.4) 3.4. dive = ρ ε 0 rot E = 0 E = gradv div(gradv ) = ρ ε 0 rot E = 0 grad rot 0 div E = ρ ε 0 div(gradv ) = ρ ε 0 (3.5) grad div gradv ( V x, V y, V z ) div (A x, A y, A z ) A x x + A y y + A z z ( div (gradv ) = x + y + z x + y + z ) V = ρ ε 0 (3.6)

55 50 3 V = ρ ε 0 (3.7) f = j j ρ (source) 0 ε 0 f = 0 V = Q 4πε 0 r r = p x + y + z! 1 p x x + y + z = x 1 x (x + y + z ) (x + y + z ) 3! = x x (x + y + z ) 3 1 = + 3 (3.8) x x (x + y + z ) (x + y + z ) 3 (x + y + z ) 5 1 x = + 3 (x + y + z ) 3 (x + y + z ) 5 ««1 1 x y x r y r z «1 x z r ««x + y x = + 3 z r (x + y + z ) 3 (x + y + z ) 5 1 y + 3 (x + y + z ) 3 (x + y + z ) 5 1 z + 3 (x + y + z ) 3 (x + y + z ) 5 3 = + 3 x + y + z (x + y + z ) 3 (x + y + z ) 5 = 0 «1 = 0 r gradv ( V r, 1 r div div A = 1 r div(gradv ) = 1 r V θ x y z «1 r «1 r «1 r (3.9) 1 V ) r θ φ r sin θ φ `r 1 A r + r r sin θ «r V + r r 1 r sin θ θ (sin θa θ) + 1 A φ r sin θ φ sin θ V «+ θ θ (3.30) 1 V r sin (3.31) θ φ V V = 1 V θ, φ 0 r r = 1 «r r r V 1 = 1 0 = 0 r r r 0 r = grad div

56 f = 0 d f = 0 0 dx dy dx = lim y(x + x) y(x) x 0 x (3.3) d y dx = lim (y(x + x) y(x)) (y(x) y(x x)) x 0 ( x) = lim x 0 y(x + x) + y(x x) y(x) ( x) (3.33) y(x + x) + y(x x) y(x) y(x + x) + y(x x) y(x) = x + y x x y y f(x, y) = 0 x y f(x, y) = log(x + y ) x y f = 0 f(x, y, z) = 0 x, y, z ( 8 ) V 9 z y x x y 8 x, y, z f 9

57 5 3 x y, z E = V V Q 1, Q,, Q N N x 1, x,, x N x V ( x) = Q 1 4πε 0 x x Q1 + Q N 4πε 0 x x Q + = Q i 4πε i=1 0 x x Qi (3.34) grad grad V ( x) = N Q i N 4πε 0 x x Qi = Q i 4πε 0 x x Qi e x Qi x (3.35) i=1 ρ( x) V ( x) = 1 4πε 0 i=1 ρ( x ) x x d3 x (3.36) x ρ( x )d 3 x x V ( x) x

58 ρ R ρr 3 E = 3ε 0 r e r r > R ρr e r r R 3ε 0 (3.37) 0 R E r V r E = E r e r E r = dv dr V V = ρr 3 3ε 0 r V 0 ρr 6ε 0 r > R r R (3.38) 0 R E V 0 r = 0 10 V 0 r > R r = R r R r = R V 0 ρr = ρr 6ε 0 3ε 0 (3.39) V 0 = ρr ε 0 ( 1) r = R r = R 10 r > R V + ρr3 0 3ε 0 r

59 54 3 (r ) sin θdr dθdφ ρ(r ) sin θdr dθdφ dv = 4πε 0 r + (r ) rr cos θ (3.40) φ π θ V = = ρ ε 0 ρ ε 0 π 0 R 0 R 0 sin θdθ dr [ 1 r rr + (r ) rr t 1 dt (r ) dr dt r + (r ) rr t dr (r ) dr [ 1 rr r + (r ) rr t ] 1 ] 1 1 (3.41) 1 1 rr ( r r r + r ) r > r 1 rr ( r ) r < r 1 rr ( r) r 0 R R < r r > r V = ρ R dr (r ) dr 1 ε 0 r = ρr3 (3.4) 3ε 0 r r < R V = 0 ( ρ r dr (r ) dr 1 ) R ε 0 r + dr (r ) dr 1 r 0 ( [(r = ρ ) 3 ] r [ (r ) + ε 0 3r 0 = ρ ( ) r ε R r r ] R r ) = ρ ε 0 R ρ 6ε 0 r r R r (3.43) 1 d (r ddr ) r dr V (r) = ρ (3.44) ε 0 r > R 1 d (r ddr ) r dr V (r) = 0 (3.45) d dr (r ddr ) V (r) = 0 r d dr V (r) = C 1 d dr V (r) = C 1 r V (r) = C 1 r + C r (3.46)

60 r = V = 0 C = 0 d (r ddr ) dr V (r) = ρ r ε 0 r d dr V (r) = d dr V (r) = V (r) = ρ r 3 + C 3 3ε 0 ρ r + C 3 3ε 0 r ρ r C 3 6ε 0 r + C 4 V C 3 = 0 C 1, C 4 V dv r = R dr V (r) ρ R + C 4 = C 1 6ε 0 R dv ρ (r) R = C 1 dr 3ε 0 R (3.49) C 1 = ρ R 3 3ε 0 C 4 = ρ R ε 0 (3.47) (3.48) ρ R + C 4 = ρ R (3.50) 6ε 0 3ε 0 r > R Q Q 4πε 0 r

61 56 3? 0 V = Q 4πε 0 r Q = 4π 3 R3 ρ R 0 ρ 3ρ R 0 4πR3 «j Q r = 0 = (3.51) 4πε 0 r 0 r 0 r = 0 ε 0 Q Q ε Dirac δ δ( x) Q ε 0 «Q = Q δ( x) (3.5) 4πε 0r ε 0 11 δ 1 «1 11 = δ( x) 4πr 1 δ

62 z = d z = d d ρ 0 x = x = y x, y x y d dz φ(z) = 1 ε 0 ρ(z) (3.53) { ρ0 d < z < d ρ(z) = 0 (3.54) d < z < d z= - d z=0 z=d dφ(z) dz = ρ 0 ε 0 z + C 1 φ(z) = ρ 0 ε 0 z + C 1 z + C (3.55) (C 1, C ) z = 0 dφ(0) = 0 C 1 0 dz z = 0 z < 0 C = 0 0 ( 1) 0 z > 0

63 V (x, y, z) = Na + Cl 0 13 (0, 0, d) +q (0, 0, d) q q 4πε 0 x + y + (z d) q (3.56) 4πε 0 x + y + (z + d) d 0 qd p 14 d 0 p p ( ) V (x, y, z) = p 1 1 4πε 0 d x + y + (z d) 1 x + y + (z + d) (3.57) d 0 ( ) 1 1 lim d 0 d x + y + (z d) 1 = d x + y + (z + d) dz ( 1 x + y + z ) = z (x + y + z ) 3 (3.58) z d z + d x + x x x d V (x, y, z) = p 4πε 0 z (x + y + z ) 3 = p cos θ 4πε 0 r (3.59) x + y + z = r, z = r cos θ ( ) E = p z 4πε 0 (x + y + z ) 3 ( p xz yz z = 3 e x + 3 e 4πε 0 (x + y + z ) 5 y + 3 e (x + y + z ) 5 z (x + y + z ) 5 ( ) p z 1 = 3(x e x + y e y + z e z ) e 4πε 0 (x + y + z ) 5 z (x + y + z ) 3 ) 1 e z (x + y + z ) 3 (3.60) C m( )

64 = e r r + e 1 θ r θ + e 1 φ r sin θ φ ( ) p cos θ sin θ E = e r 4πε 0 r 3 + e θ r 3 (3.61) ( ) (3.60) (3.61) r e r = x e x + y e y + z e z e z = cos θ e r sin θ e θ z +z z p p x V = 4πε 0 x 3, E p x = 3 4πε 0 x 4 x 1 p (3.6) 4πε 0 x 3 ( ) ( p x) = p, 1 x 3 = 3 x x U = Qq Q Qq 4πε 0 r 4πε 0 r Q q q 4πε 0 r Qq q Q 4πε 0 r 15 Q 1 4πε 0 r Qq 4πε 0 r q 1 Qq 4πε 0 r Qq 4πε 0 r x k 1 kx k x 1 kx 16 1 ε 0 E 15 Q q U = Qq 4πε 0 r (double-counting) 16

65 ( q, Q) r Qq 4πε 0 r q Q r 1 q r Q q q 1 q + q q (3.63) 4πε 0 r 1 4πε 0 r q 1q 4πε 0 r q 1 q + q q + q 1q 4πε 0 r 1 4πε 0 r 4πε 0 r 1 3 i=1 3 j=1 j i q i q j 4πε 0 r ij = 1 (3.64) q 1, q,, q N N q i q j r ij N 1 N N q i q j (3.65) 4πε 0 r ij i=1 j=1 j i i = j i = j r ii = 0 1 N = 3 3 q 1 q j 3 q q j 3 q 3 q j + + 4πε j=1 0 r 1j 4πε j=1 0 r j 4πε j=1 0 r 3j j 1 j j 3 = 1 ( q1 q + q 1q 3 + q q 1 + q q 3 + q 3q 1 + q ) 3q 4πε 0 r 1 4πε 0 r 13 4πε 0 r 1 4πε 0 r 3 4πε 0 r 31 4πε 0 r 3 q 1 q = + q 1q 3 + q q 3 4πε 0 r 1 4πε 0 r 13 4πε 0 r 3 (3.66) 1 1 N N i=1 j=1 j i q i q j 4πε 0 r ij = 1 N i=1 q i N j=1 j i q j 4πε 0 r ij } {{ } =V ī ( x i ) = 1 N q i V ī ( x i ) i=1 (3.67)

66 V ī ( x i ) x i q i ī q i dxdydz = d 3 x ρd 3 x 0 1 N q i V ī ( x i ) 1 ρ( x)v ( x)d 3 x (3.68) i=1 ρ V ρ( x)v ( x)d 3 x ρ div E = ρ ε 0 ε 0 (dive( x))v ( x)d 3 x (3.69) div E E V x, y, z ε 0 ε 0 ( x E x( x) + y E y( x) + ) z E z( x) V ( x)d 3 x (3.70) ( E x ( x) x V ( x) + E y( x) y V ( x) + E z( x) ) z V ( x) [ ε0 ] x= E x ( x)v ( x)dydz x= d 3 x (3.71) V ( x) E( x) 0 E = V x E x = x V ε 0 (3.7) (E x ( x)e x ( x) + E y ( x)e y ( x) + E z ( x)e z ( x)) d 3 x (3.73) U = ε 0 E d 3 x (3.74) 18 ε 0 E 1 kx 17 1 Z ρ( x)v ( x)d 3 x i = j ρd 3 x Z Z Z 18 ( E)V = E V = E E

67 qv Q Q S V Qd ε 0 S V Q V 0 Q V 0 + V Q ε 0 S 1 Q(V + V 0) + 1 ( Q)V 0 = 1 QV = 1 Q d ε 0 S (3.75) 1 QV 0 +dq q q+dq dq -q -q-dq q V qv q q q + dq dqv V = qd ε 0 S dq qd q 0 Q ε 0 S Q 0 dq qd ε 0 S = d [ q ε 0 S ] Q 0 = Q d ε 0 S (3.76) Sd 1 ε 0 E 1 ε 0 S = 1 ε 0 Q ( ) Q (3.77) ε 0 S E 1 QE 0 0 d 1 QEd = 1 QV

68 QE F = qe 1 F = qe E 1 E 1 E + = = 3.7 U = ε 0 E d 3 x E U S L ε 0 E SL L L + x U = ε 0 E Sx (3.78) L U x = ε 0 E S (3.79) ε 0 E x x

69 a b y ES U = ε 0 ES L S ES S S L U = ε 0 ES L S S = ε 0 E L S (3.80) S = ay y U y = ε 0 E La (3.81) a b La ε 0 E L (stress) ds y d S E ε 0 E ds ds E ε 0 E ds q q L θ q cos θ 4πε 0 L L cos θ q cos θ 4πε 0 L cos θ = q cos3 θ πε 0 L (3.8) L 1 ε 0 1 ε 0E = 1 ε 0 ( q cos 3 ) θ πε 0 L = q cos 6 θ 8π ε 0 L 4 (3.83) θ θ + dθ π (L tan(θ + dθ)) π(l tan θ) = πl tan θ dθ cos θ (3.84) θ = 0 θ = π π 0 q cos 6 θ 8π ε 0 L 4 πl tan θ dθ cos θ = π 0 q cos 3 [ θ q sin θdθ = 4πε 0 L 4πε 0 L cos4 θ 4 ] π 0 (3.85) = q 16πε 0 L

70 L q q q 4πε 0 (L) d F d F = ε 0 E( E d S) ε 0 E d S (3.86) E ds ε 0 E ds E ds ε 0 E ds df x = ε 0E x (E xds x + E yds y + E zds z) ε0 `(Ex) + (E y) + (E z) ds x = ε0 `(Ex) (E y) (E z) ds x + ε 0E xe yds y + ε 0E xe zds z df y = ε 0E y (E xds x + E yds y + E zds z) ε0 `(Ex) + (E y) + (E z) ds y = ε 0E ye xds x + ε 0 ` (Ex) + (E y) (E z) ds y + ε 0E ye zds z df z = ε 0E z (E xds x + E yds y + E zds z) ε 0 `(Ex) + (E y) + (E z) ds z = ε 0 E z E x ds x + ε 0 E z E y ds y + ε 0 ` (Ex ) (E y ) + (E z ) ds z (3.87) 0 df x ds df y df z A T ds y ds z 1 A (3.88) T 0 T xx T xy T xz T yx T yy T yz A = T zx T zy T zz 0 1 (Ex) 1 (Ey) 1 1 (Ez) E xe y E xe z ε 0 B E x E y (E x) + 1 (E y) 1 (E z) E z E y C E x E z E y E z 1 (E x) 1 (E y) + 1 A (E z) (3.89) 3 3 T ij i, j x, y, z 19 x y z x, y, z 19 (tension)

71 66 3 z y x T xx x ds x x T zx z T ij T xx, T yy, T zz T (T ij = T ji ) V = kx (a) (b) a z y 3- (a)e x = kx, E y = ky, E z = 0 (b)e x = ky, E y = kx, E z = 0 (c)e x = ky, E y = kx, E z = 0 (d)e x = k(x + y ), E y = kxy, E z = 0 x = 1 ( r ) + 1r r r r θ + z V = () r 1 < r < r ρ 0 r = 0 V = 0 ε 0 E = gradv R r(r > r) Q Q

72 (a) (b) 1 qv (c) 1 ε 0 E 3-6 V (r, θ) = p cos θ 4πε 0 r 90 x V (r, θ, φ) φ q L q 4πε 0 (L) 3-8 q q q E q E q E( x) = 4πε 0 x x q 3 ( x x q) E q ( x) = 4πε 0 x x q 3 ( x x q ) ( x q, x q q, q ) E + E 1 ( ε 0 E + E ) ( E + E ) = 1 ε 0 E + 1 ε 0 E + ε 0E E 1 ε 0 E q 1 ε 0 E q ε 0 E E ε 0E E qq 4πε 0 x q x q (hint: z (L, 0, 0) ( L, 0, 0) )

73

74 V V x x

75 rote = 0 rote = S σ ρ S σ S S E = σ n ε 0 E = σ n ε 0 1 ε 0 1

76 Q r Q r Q Q 0 Q Q 0 0 x = 0 V = 0 V = ρ ε 0 ρ x = 0 0 Q r - Q +Q, Q r x ( r, 0, 0) (r, 0, 0) (x, y, z) ( ) V (x, y, z) = Q 1 4πε 0 (x + r) + y + z 1 (x r) + y + z (4.1) x > 0 ( ) Q 1 V (x, y, z) = 4πε 0 (x + r) + y + z 1 x < 0 (x r) + y + z 0 x > 0 (4.) x = 0 V = 0 V = ρ ε 0 V 1 = ρ ε 0, V = ρ ε 0 (V 1 V ) = 0 (4.3) V 1 V V 1 V (x = 0) 0

77 V 1 = V x ( ) E x = x V = Q x + r x r 4πε 0 ((x + r) + y + z ) 3 ((x r) + y + z ) 3 x = 0 x x = 0 E x = Q 4πε 0 r (r + y + z ) 3 σ E = σ ε 0 n σ = Q π r (r + y + z ) 3 (4.4) (4.5) n σ ( dy dz) Q z E R 0 r = R V = Ez = Er cos θ r = R V r=r = ER cos θ r = R V = ER cos θ V = p cos θ 4πε 0 r p = 4πε 0ER V = V + V = Er cos θ + ER3 cos θ r = E(R3 r 3 ) cos θ r (4.6) { 0 0 r R V (r, θ) = E(R 3 r 3 ) cos θ r R < r R = r E = V = e r r + e 1 θ r θ + 1 r sin θ φ }{{} ( 0 = e r 3E cos θ E(R3 r 3 ) cos θ r 3 E(R 3 r 3 ) cos θ r ) e θ E(R 3 r 3 ) sin θ r 3 (4.7) (4.8) 3

78 r = R E r=r = 3E cos θ e r (4.9) e r σ = 3ε 0 E cos θ (4.10) 0 θ < π π < θ π (dielectrics) 4 (insulator) (polarization) 5 = + (4.11) 4 dielectrics 5

79 V/m q q d q p = q d n p n p P p i = n p = P (4.1) i p p i [C m] [m 3 ] [C/m ] P nq nq d S d P qd S = P S qd S = P S P +nqds = +PS +q -q -nqds = -PS +nq -nq P ε 0 1 Q P ε 0 ε 0 S

80 P ε P 1 ε 0 P E 1 ε 0 D E }{{} = 1 ε 0 D 1 ε 0 P }{{} (electric displacement electric flux density) 6 (4.13) D = ε 0 E + P (4.14) SI ε 0 [V/m] [C/m ] 1 ε 0 ε 0 [C] 1C 1C P D = ε 0E E D Q E = Q 4πε 0 r e r D = Q 4πr e r ε 0 E D P, D, E 7 (4.14) E ε( 0 ) D E D = ε E (4.15) ε ε D = ε 0 ( D P ) ε 0 ε 0 ε r Q 4πε 0 r e r Q 4πεr e r ε 0 ε ε ε ε 0 = ε r 6 electric displacement D D D diplacement 7 P E

81 FAQ ε E = D ε E = D ε 0 P 0 x z div E = ρ ε 0 y z P z (x, y, z + z) x y P z (x, y, z) x y P z x y z z x, y ( Px ρ P x y z = x + P y y + P ) z x y z (4.16) z ρ P div P ρ P ρ F F (free) div E = 1 ε 0 (ρ F + ρ P ) ε 0 dive = ρ F divp (4.17) ) div (ε 0E + P = ρ F ε 0E + P = D div D = ρ (4.18)

82 ρ div E = ρ ε 0 D = ε E div E = ρ ε ε div E D E D 1 ε FAQ E D E D E ε 0 D E D ( D ) D div D = ρ D div D = ρ, rot E = 0 div D = 0 D rot E = E D

83 D = 0 D 8 9 E, D H, B 4.6 ε 0 E ε 0 E 1 ρv d 3 x dive = ρ divd ε = ρ 0 1 ρv d 3 x = 1 ( D)V d 3 x = 1 D V d 3 x = 1 D Ed 3 x (4.19) 1 U = 1 D x = ε 0 E d 3 x + 1 P Ed 3 x (4.0) P Ed 3 x P nqd 1 P Ed 3 x = 1 nq d Ed 3 x = 1 n F dd 3 x (4.1) F = q E q F = K d 1 nk d d 3 x (4.) 1 K d nd 3 x 1 D E 8 rot E = 0 rot D = 0 rot P 0 9

84 E = V dive = ρ V = ρ P ε 0 ε 0 4- R P E, D P cos θ θ P

85

86 81 5 I II 5.1 I (source) source 1 D = ε 0E + P (5.1) D P dive = ρ divd ε = ρ (5.) 0 1

87 N N N S N S Wb m 1 [Wb] m [Wb] r[m] F = m 1m 4πµr F = m 1m 4πµ 0 r (5.3) µ µ 0 MKSA µ 0 = 4π 10 7 MKSA [A] 3 E[N/C] H[N/Wb] q F = q E F = q 1 q 4πε 0 r [C] [N/C] [V/m] m F = m H F = m 1m 4πµ 0 r [Wb] [N/Wb] [A/m] N N N S Oersted Ampere 5 A A/m F = m H Wb 3 4π ON/OFF 5 cgs cgs

88 N N S S footnote 6 (1) () (3) (3) 7 (3) (1)() (3) 6 7

89 N 84 5 (3) () 8 N S 5..3 S N S 9 8 9

90 N N (E-H ) N S (E-B ) E H E-H E-B E-B E D H B 14 B E, H 15 H E-H

91 86 5 B 16 B I x x + d x d F = Id x B (5.4) B H B B = µ 0 H 17 d F d B N/A m B T Wb Wb/m a b a b a b a, b θ a b a b sin θ a b = a b sin θ a b E Id x B Id x d F B B d x (5.4) B r (I 1 I ) µ 0I 1 I πr E-B B 16 B B B magnetic induction 17

92 I I[A] r I πr φ e φ H = 1 e φ z I πr e φ (6.1) r 1 B = µ 0I πr e φ rot E 0 rot E 0 rot E 0

93 div B = 0 4 H roth = 0 H = I πr e φ rot A A rot A rot H r m[wb] m I πr m I πr = mi (6.) πr r A B m I m θ r θ = (6.3) πr π B C D A C D R r D r N N m I (r + r) θ = m θ π(r + r) π (6.4) A r + r 0 mi 0 r B N C 3 4 div H = 0 div B = 0 div B = 0

94 I[A] m[wb] mi[j] 5 5

95 90 6 ds m roth ds 6 j mroth ds = m j ds (6.5) mds 7 roth = j (6.6) rotb = µ 0 j roth = j divb = 0 div rot divd = ρ rote = 0 divb = 0 roth = j div rot divd = ρ rote = 0 roth = j divb = 0 rote = 0 E = gradv rot 0 grad divb = 0 8 div 0 rot divb = 0 B B = rota A 6 mh ds rot(mh) ds m rot 7 roth j ds = 0 ds 0 rot H j = 0 8

96 E V E = gradv H V m H = gradv m V=4 V=5 V=0 V=1 V= V=3 rot H 0 9 V=5 V=0 V=1 V=4 V= V= EFGH 0 F G H E E F G H EF GH ABCD C D 0 A B AB L H 9 cut D A H E C B G F

97 9 6 HL ABCD n I nl nli HL = nli H = ni (6.7) B C D C A B 6.4. d z z = d z = d j x x, y z x yz z = d x z z = -d j z z z z > 0 z < 0 div H = 0 jd H y = jz jd z y z > d rot H = 0 H y rot H = j x H z //// y H y z = j (6.8) H y = jz d < z d < z < d z < d (6.9) rot H 0 rot H = 0 z y y H = I πr e φ x = 0, y = 0 rot H = 0

98 I N z 6-3 z R r V m = I π φ grad (ρ, φ, z) H = I πρ e φ = e ρ ρ + e z z + e 1 φ ρ φ H = gradv m V m 6-4 d v ±σ (1) () 1 ε 0 σ 1 0 v v 6-5rotH = j S d x H = ds j (6.10) S S ( S S S d x ) S

99

100 div D = ρ rot H = j D, H ρ, j D, H E = Q 4πε 0 r e r ρ E( x) = div E = ρ ε 0 d 3 x ρ( x ) 4πε 0 x x e x x (7.1) B( x) ( ) = d 3 x j( x ) x x (7.) j( x ) x x x j( x ) B( x) j x x x x j ( x x ) j ( x x ) I 1 ε 0 1 4π x x e x x div

101 96 7 j x x j x x B( x) = K d 3 x j( x ) ( x x ) x x n (7.3) K n n = 3 n = x x 3 x y n = 3 H = I πr e φ K z j z dx dy dz j z z 4 j z dx dy I z x = y = 0 0 x = y = 0 x = z e z dz x = r e r + z e z ( x x ) = I( e z r e r ) = ri e φ x x = r e r + (z z ) e z (7.4) I e z e z I B(r) = K z z = 0 B(r) = K dz Ir e (r + (z ) ) 3 φ (7.6) z = r tan θ 5 θ π π dz Ir e (r + (z z ) ) 3 φ (7.5) I B(r) = KI r π π dθ cos θ 1 (1 + tan θ) 3 e φ = KI r e φ (7.7) B = µ 0I πr K µ 0 4π 6 3 n = 3 Z Z Z Z 4 d 3 x dx dy dz 5 z = 0 z z = r tan θ 6 µ 0 4π 10 7 r z z

102 Biot-Savart j( x) x B( x) B( x) = µ 0 4π d 3 x j( x ) ( x x ) x x 3 (7.8) I I I 0 x dx dy dz j x e x ( ) (7.9) dy dzj x I dx I e x ( ) (7.10) dy x dx x dx y z dy, dz (Idx e x + Idy e y + Idz e z ) ( ) (7.11) j dx dz dx dy dz j ( ) I d x ( ) (7.1) d x (dx, dy, dz) d x = dx e x + dy e y + z e z d x A (7.13) ( ) ( ) d x A = (dya z dza y ), d x A x y = ( (dza x dxa z ), ) d x A = z (dxa y dya x ) (7.14) I d 3 x j I d x

103 98 7 I x B( x) B( x) = µ 0I 4π d x ( x x ) x x 3 (7.15) x m x B = m x x 4π x x = m 3 4π x x e x x (7.16) I d x F = Id x B F = mi d x ( x x) 4π x x 3 (7.17) N F = mi d x ( x x) 4π x x 3 = mi d x ( x x ) 4π x x 3 (7.18) m H div j = 0 div j = t ρ ρ 7

104 R I x d x x x 360 z x = z e z x x Id x x x x x Id x x z = 0 +z z 0 z 0 z z x x R φ 0 π x z d x Rdφ d x x x d x ( x x ) d x Rdφ x x R + z µ 0 IRdφ 4π(R + z ) y (7.19) R z π 0 R µ 0 IRdφ R + z 4π(R + z ) µ 0 IR µ 0 IR dφ = 4π (z + R ) 3 (z + R ) 3 (7.0) (7.1) z 3 z = 0 z = 0 B( 0) = µ 0I (7.) R R π

105 100 7 d x = Rdφ e φ x x = z e z R e r e r, e φ x 8 9 e r, e φ, e z Id x ( x x ) = IRdφ e φ (z e z R e r ) = IR(z e r + R e z )dφ (7.3) e r e φ = e z, e φ e z = e r, e z e r = e φ (7.4) B(z e z ) = µ 0 4π π 0 dφ IR(z e r + R e z ) (z + R ) 3 φ B(z e z ) = µ 0 4π IRz (z + R ) 3 e z e r π 0 π 0 dφ e r + µ 0 4π IR e z (z + R ) 3 π dφ e r = 0 0 (7.5) dφ (7.6) B(z e z ) = µ 0 IR e z (z + R ) 3 (7.7) 7.. z z x x y 10 z x = x e x + z e z (7.8) x x φ R x = R cos φ e x + R sin φ e y = R e ρ (7.9) z e ρ z ρ d x e φ x x = z e z + x e x R(cos φ e x + sin φ e y ) (7.31) x x = z + R + x Rx cos φ (7.3) d x = Rdφ ( sin φ e x + cos φ e y ) = Rdφ e φ (7.30) z xy z z 8 e e 9 0 d x x = 0 d x x 10 y

106 B( x) = µ 0I 4π Rdφ eφ (z e z + x e x R e ρ ) (z + R + x Rx cos φ) 3 (7.33) e φ e z = e ρ, e φ e ρ = e z (7.34) e φ e x e φ = sin φ e x + cos φ e y e x e x = 0, e y e x = e z 11 B( x) = µ 0I 4π e φ e x = cos φ e z (7.35) Rdφ (z eρ x cos φ e z + R e z ) (z + R + x Rx cos φ) 3 ρ z φ R = x cos φ z 0 x x xy R z, x R 1 (z + R + x Rx cos φ) 3 B( x) = µ 0I 4π R = = (7.36) ( ) 1 (z + R + x Rx cos φ) 3 R=0 + R 1 R (z + R + x Rx cos φ) 3 R= xR cos φ + + (z + x ) 3 (z + x ) 5 (7.37) ( ) 1 3xR cos φ Rdφ (z e ρ x cos φ e z + R e z ) + + (z + x ) 3 (z + x ) 5 µ 0 I 4π 1 Rdφ (z e ρ x cos φ e z ) (z + x ) 3 φ φ e ρ cos φ 0 π dφ e ρ dφ cos φ 0 R R = µ 0 I 1 Rdφ R e z 4π (z + x ) 3 µ 0 IR e z 4π (z + x ) 3 + µ 0I 4π dφ + 3µ 0IxR 4π (z + x ) 5 (7.38) (7.39) ( ) 3xR cos φ Rdφ (z e ρ x cos φ e z ) (z + x ) 5 ( ) (7.40) z dφ cos φ e ρ x e z dφ cos φ 11

107 10 7 π e x, dφ cos φ = π 1 dφ = π, dφ cos φ e φ = µ 0 IR e z (z + x ) 3 + 3µ 0IxR 4 (z + x ) 5 (z e x x e z ) = µ 0IR e z 4 (z + x ) 5 ( 3xz ex + (z x ) e z ) (7.41) z p ( ) p z 1 E = 3(x e x + y e y + z e z ) e 4πε 0 (x + y + z ) 5 z (x + y + z ) 3 p ( 3xz ex + (z x ) (7.4) ) e 4πε 0 (x + z ) 5 z y = 0 (7.41) (7.4) p IπR q q l q l m m l 1 m µ l 0 1 µ 0 E B( H ) Z 1 e ρ = cos φ e x + sin φ e y dφ cos φ sin φ = 0

108 p = I S (7.43) S IL = ml µ 0 (7.44) 7..3 z = 0 z = l z z l z l B( x) = µ 0I 4π Rdφ (z eρ x cos φ e z + R e z ) (z + R + x Rx cos φ) 3 + µ 0I 4π Rdφ ((z l) eρ x cos φ e z + R e z ) ((z l) + R + x Rx cos φ) 3 (7.45) z B( x) = µ 0I 4π n= Rdφ ((z nl) eρ x cos φ e z + R e z ) ((z nl) + R + x Rx cos φ) 3 l n= n= l (7.46) dz (7.47) l

109 104 7 B( x) = µ 0nI 4π Rdφ (z eρ x cos φ e z + R e z ) dz (z + R + x Rx cos φ) 3 (7.48) z z 0 z e ρ z B( x) = µ 0nI 4π dz π 0 R ( x cos φ + R) e z dφ (z + R + x Rx cos φ) 3 (7.49) z = R + x Rx cos φ tan θ dz = R + x Rx cos φ dθ cos θ B( x) = µ 0nI 4π = µ 0nI 4π = µ 0nI 4π dz R + x Rx cos φ π π dθ cos θ π π 1 dθ cos θ π 0 π 0 π 0 R ( x cos φ + R) e z dφ (z + R + x Rx cos φ) 3 R ( x cos φ + R) e z dφ( (R + x Rx cos φ) (1 + tan θ }{{} = 1 cos θ dφ R ( x cos φ + R) e z (R + x Rx cos φ) ) 3 ) (7.50) θ π dθ cos θ = π φ dφ R ( x cos φ + R) e z (R + x R x Rx cos φ) dφ AB Rdφ P AC AP AP = R + x Rx cos φ AD = x cos φ R E D ABC APD AC = AB x cos φ R R + x Rx cos φ (7.51) B A C O P AC AP = Rdφ x cos φ R R + x Rx cos φ (7.5) AB dφ x > R 0 π 0 0 x < R π µ 0nI 4π π = µ 0nI

110 rot Z rotb = d 3 x µ 0 j( x ) x x 4π x x 3 (7.53) A ( B C) = B( A C) C( A B) «! «! x x {z} j( x x x ) = j( x ) {z } x x 3 {z } {z} j( x ) x x 3 {z } A B B A B {z } C {z } C {z} A! x x «x x 3 {z } C (7.54) «j( x x x ) x x x 3 x A C B B C ( B C) = B C + C B B C C B (7.55) Z «««rotb = d 3 x µ 0 j( x ) ( x x ) + µ 4π x x 3 0 j( x ) ( x x ) 4π x x 3 (7.56) 0 ( x x ) 4π x x x 3 x x x x µ 0 Z d 3 x j( x ) ( x x ) 4π x x 3 = µ 0 Z d 3 x ( x x j( x ) ) (7.57) 4π x x 3 j( x ) = div j( x ) = 0 0 j 0 «( x x ) 4π x x 3 «( x x ) 4π x x = grad 1 3 4π x x ( x x ) 4π x x 3 x Q Q( x x ) Q 4πε 0 x x Q ε 0 (7.58) 4πε 0 x x Q 1 3 ε 0 4π x x x Q dive = ρ V = ρ «ε 0 ε ( x x ) ρ Q 0 4π x x 3 ε ε 0 0 x = x 0 1 δ 3 ( x x ) x = x j Z rotb( x) = µ 0 d 3 x j( x )δ 3 ( x x ) = µ 0 j( x) (7.59) B = µ 0 H rot H = j

111 I 7- a I 7-3 a b 7..1 a, b x x (x 3 ) 7-4 y = 1 4a x I x = 0, y = a a θ r = r dθ 1 + cos θ r z y a b x dy y=a dx x 7-5 N z R z N + 1 N R

112 j 1 V 1 j V j 1 x B 1 ( x ) = µ 0 j d 3 1 ( x 1 ) ( x x 1 ) x 1 4π V 1 x x 1 3 = µ 0 j d 3 1 ( x 1 ) e x x 1 x 1 4π V 1 x x 1 (8.1) x j ( x ) F j1 j = d 3 x j ( x ) B 1 ( x ) V ( ) = µ j ( x ) j 1 ( x 1 ) e (8.) x1 x 0 d 3 x 1 d 3 x 4π V 1 V x x 1 A ( B C) = B( A C) C( A B) j ( x ) ( j 1 ( x 1 ) e x1 x }{{}}{{}}{{} A B C ) ( = j 1 ( x 1 ) j ( x ) }{{}}{{} B A e x1 x }{{} C ) e x1 x }{{} C ( j 1 ( x 1 ) }{{} B ) j ( x ) }{{} A (8.3) j 1 ( x 1 )d 3 x 1 ( ) j 1 ( x 1 ) j ( x ) e x1 x ( ) j ( x ) j 1 ( x 1 ) e x x 1 1 µ 0 d 3 x 1 d 3 x j 1 ( x 1 ) j ( x ) e x 1 x 4π V 1 V x x 1 = µ ( ) 0 d 3 x 1 d 3 x j 1 ( x 1 ) j ( x ) 4π 1 V 1 V x x 1 (8.4) e x1 x x x 1 = ( ) 1 x x 1 (8.5) 1 x 1 E = Q 4πε 0 r e r V = Q 4πε 0 r E = V

113 108 8 x µ 0 d 3 x 1 d 3 x j 1 ( x 1 )j ( x ) ( ) 1 = µ 0 d 3 x 1 d 3 x j 1 ( x 1 ) j ( ) ( x ) 1 4π V 1 V x x x 1 4π V 1 V x x x 1 (8.6) y z ( ) µ 0 d 3 x 1 d 3 x j 1 ( x 1 ) 4π 1 j ( x ) V 1 V x x 1 (8.7) V j 0 x x 1 div j = j = div j = 0 0 µ 0 d 3 x 1 d 3 x j 1 ( x 1 ) j ( x ) e x 1 x 4π V 1 V x x 1 (8.8) ρ 1 V 1 ρ V 1 d 3 x 1 d 3 x ρ 1 ( x 1 )ρ ( x ) e x 1 x 4πε 0 V 1 V x x 1 (8.9) 1 ε 0 ρ 1 ( x 1 )ρ ( x ) µ 0 j 1 ( x 1 ) j ( x ) (8.10) jd 3 x V Id x L µ 0I 1 I 4π L 1 e x d x 1 d x 1 x L x x 1 (8.11) j = ρ v Z Z = 1 d 3 x 1 d 3 x e x 1 x ρ 1( x 1)ρ ( x ) 4πε 0 V 1 V x x 1 µ Z Z 0 d 3 x 1 d 3 e x1 x x (ρ 1 ( x 1 ) v 1 ( x 1 )) (ρ ( x ) v ( x )) 4π ZV 1 ZV x x 1 1 d 3 x 1 d 3 x ρ 1 ( x 1 )ρ ( x ) (1 ε 0 µ 0 v 1 ( x 1 ) v ( x )) 4πε 0 V 1 V e x1 x x x 1 ε 0 µ 0 1 ε0 µ 0 (8.1) div j 0

114 Id x B df = Id x B (8.13) ( ) 8..1 dv ρ v 3 q ρdv q d x d S 4 dv d S d x Id x Id x = ( j d S)d x = j(d S d x) = ρ vdv (8.14) j d x ( j ds)d x = j(d S d x) 5 j = ρ v F = ρdv v B d F ρdv q = q v B (8.15) q v B qe ( ) F = q E + v B (8.16) q v B q v B 6 () 0 3 v 4 5 j ds θ ( j ds)d x j(d S d x) j ds d x cos θ 6

115 () r m v r = qvb (8.17) ω = v r = qb (8.18) m ω ( ) r qvb mv = qbr (8.19) 7 T = πr v = πm qb v (8.0) r 8 A B

116 x z y x ( x ) z v evb y y y y V y V d y z x x y evb = e V d V = vbd (8.1) vbd 10 x 11 v v I I = envs S n n F = q v B x y 8.3 V ( E = gradv ) Hall (hole) 11

117 11 8 V m H = gradv m roth E rote = 0 rot 0 grad E = gradv V E = gradv V E rothrot B 0 V m divb = 0 div 0 rot B = rota (8.) A 1

118 A U = j A (8.3) A j A > 0 rot z z rot A rot A rot field field

119 114 8 rot E = gradv V B = rot A A rot j A A d 3 x j A = I d x A N S

120 rot j A A j qv j 1 1 j A 1 j 1 A 1 j j A 1 j A A 1 j 1 A (overcounting) E = gradv V V + V 0 V 0 B = rot A rot 0 A B rot 0 gradλ Λ grad grad rot 0 A A + gradλ (8.4) 13 j A A Z Z d 3 x j A Z d 3 x j A Z d 3 x j gradλ = Z d 3 x j A + d 3 xdiv jλ + ( ) (8.5) div

121 rotb = µ 0 j B = rota rot rota = µ 0 j (8.6) rot rotv = grad divv V (8.7) grad diva A = µ 0 j (8.8) div A = 0 A Λ div( A + gradλ) = 0 div(gradλ) = Λ Λ = div A Λ V = ρ ε 0 ε 0 div A A = µ 0 j (8.9) V = ρ ε 0 (8.30) V ( x) = 1 Z 4πε 0 d 3 x ρ( x ) x x Z A( x) = µ0 4π d 3 x j( x ) x x (8.31) (8.3) I r r I ρ V = 1 πε 0 log r 16 A = µ0 log r ez (8.33) π z

122 z z V E E = gradv = e r (log r) = πε 0 r πε 0 r e r (8.34) A B B = rota = µ0 πε 0 «e r (log r e z ) = µ0 1 r π r e r e {z z } = e φ = µ 0 1 π r e φ (8.35) m q F = mω r e r r F = mω (x e x + y e y + z e z ) (8.36) m d x dt ω = mω x, m d y dt = mω y, m d z dt = mω z (8.37) z B B = B e z q v B (1) () z x, y X = x + iy x, y (3) X = e iωt Ω (4) 8- z B v v m q z v xy v

123 118 8 (1) z z B z B 0 z z B 0 + B (divb = 0 ) B () z z v (3) B v xy z z v (4) z 1 m ( (v ) + (v ) ) (5) z 8-3 B µ µ B N +m -m I I S -m S z z θ V m = 1 µ 0 Bz mv m (z), mv m (z + L cos θ) +m A = (0, Bx, 0) I A a S -m N x z y

124 E = 1 ( ) D P (9.1) ε 0 H = 1 ( ) B Pm (9.) µ 0 P m (diamagnetism) 3 (paramagnetism) 4 (ferromagnetism) 5 N S S N 1 3 diamagnetism dia-diameter 4 paramagnetism para parallel para 5 ferro-

125 10 9 I S IS +m m l ml µ 0 IS [Am ] [A/m] M M = χ H = χ µ 0 B (9.3) χ magnetic susceptibility 9. πr q v v πr qv πr p m = qv πr πr = qvr (9.4) L = mvr p m = q m L

126 mrω = e ± erωb (9.5) 4πε 0 r ± ω mr (ω e ) m ωb e 4πε 0 r = 0 ) + r e B 4m + e 4πε 0 r = 0 ( ω eb m ( mr ω eb m ) = e B 4m r + e 4πε 0 mr 3 ω (9.6) ω ± eb m eb r ( m ) eb m 8 9

127 1 9 χ 10 6 χ =

128

129 a a b 0 0 c b a d e

130 9.5. B H 15 (1) () (3) 9.5 B H E D B H E D E D D = ε 0 E E D div P ρ ρ P = divp dive = 1 (ρ ε div ) P 0 ) div (ε 0E + P = ρ (9.7) ε 0 E + P = D div D = ρ x

131 ε 0 P ε0 D 1 ε 0 div P P E D D = ε E E D E 1 P E ε 0 ε 0 E D D 13 D 9.5. B H H B B H B B 1 µ 0 µ 0 B = µ 0H j j j M ( ) B rot = j + j M (9.8) µ P P t 15 H B B µ 0 µ 0 H

132 9.5. B H 17 ρ P P j M M M M x y z M x M x z j z y M x j z = y M x dxdy dxdz

133 18 9 M x y j z M y x j z x M y y M x = j z (9.9) z j x, j y rot M = j M (9.10) y ( ) B rot = j µ + j M 0 }{{} ( ) =rotm (9.11) B rot M µ = j 0 x H rot H = j H B µ 0 M (9.1) B ( ) B rot = j + j M, divb = 0 (9.13) µ 0 H rot H = j, div H = ρ M (ρ m = div M) (9.14) ρ M ρ P = divp roth = j, divb = 0 (9.15) ( j ) E B D H E(Electric Field) B(Magnetic induction) D(Electric Displacement) H(Magnetic Field) B 16 B Magnetic Flux Density D electric Flux Density 17 Electric Displacement() Magnetic Induction()

134 M = χ H H = B χh µ 0 (9.16) (1 + χ)µ 0H = B (1 + χ)µ 0 µ χ µ µ χ µ r B = µ H = µ r µ 0 H (9.17) χ 1 1 H M H = 0 M 0 B H B B H rot M 0 rot M 0 div B = 0 H rot H = 0 B H µ 0 NNNNNNNNNN SSSSSSSSSSS

135 130 9 M 1 µ 0 B H = M B 1 µ 0 H M H B H B M H B B H µ B U = µ B ( ) F = gradu = ( µ B ) F = µ 0 ( µ H ) 18 B H B 9.7 divd = ρ, rote = 0, divb = 0, roth = j ρ, j D, B div 0 E, H rot 0 div 0 div 0 0 B rot 0 0 H D, B E, H 18 µ

136 µ n I 9- µ 1 µ (1) H B () (3) (4) 9-3 µ H B d θ θ = 0 θ = π

137

138 div D = ρ F = Q E V V = 1 ε 0 ρ E = V rot H = j F = Q v B A A = µ0 j B = A = Il B flux Φ B Φ = B ds (10.1) S S div B = 0 ( )

139 V = dφ dt (10.) V Φ Φ Φ 3 Φ Φ Φ Φ Φ R 1 dφ R dt 4 3 4

140

141 Φ B ds B 10. q( E + v B) V = ( v B) l (10.3) v B l e v B e E E = v B l V = E l = ( v B) l (10.4) l d l v dv = ( v B) d l (10.5) ( A B) C = ( C A) B 5 dv = (d l v) B (10.6) 5 A, B, C

142 d l v d l v d l v B dv ds B B d S dt a Φ = Ba cos ωt (10.7) sin ωt < 0 V = dφ dt = Ba ω sin ωt (10.8) R I = V R = Ba ω R sin ωt (10.9) IV = B a 4 ω R BIa BIa sin ωt t B sin ωt (10.10) B BIa sin ωt aω = B a 4 ω sin ωt (10.11) R 6 t 6 BIa cos(ωt + α)

143 v v ev B ev B ev B 10. B l I B, I BIl v B I BIlv BIlv = IV (10.1) V = Blv 10.3 F = q( E + v B) V = dφ dt V E Φ = ds B ds B S t = E d x (10.13) S 7

144 ds S d S 9 S d x A = S d S (rot A) (10.14) ds B S t = ds (rote) (10.15) S rot E = B t (10.16) rot E = V = dφ dt F = q( E + v B) rot E = B t V = dφ dt V = dφ dt ω v e v B G 8 Φ d dt B t B B( x, t) Φ x Φ(t) Φ 9 S 0 0

145 V = dφ dt FAQ B t 0 B rote = 0 V E = gradv rote = B t E = gradv B = rota (rota) = rot( gradv ) t rot A = 0 t E = gradv (10.17) E = gradv A t (10.18) (rota) t rot A t = rot( gradv A t ) = rot A t (10.19) (10.18) B = rota B A E A t

146 Φ 1 Φ 1 = L 1 I 1 + M 1 I + M 13 I 3 + (10.0) L 1, M 1, M 13, L 1 1 M 1 1 M 13 [Wb] [A] [Wb/A] [H] 10 Φ 1 = L 1 I 1 + M 1 I + M 13 I 3 + Φ = L I + M 1 I 1 + M 3 I 3 + Φ 3 = L 3 I 3 + M 31 I 1 + M 3 I +. (10.1) i V i = dφ i dt di i = L i dt + M di 1 i1 dt + M di i dt + M di 3 i3 dt + (10.) I 1 x B 1 ( x) I ds B 1 B = rota I M 1 I 1 = ds B 1 = ds (rota 1 ) = I I d x A 1 ( x ) I (10.3) I A 1 I I 1 A 1 ( x) = µ 0I 1 1 d x 1 4π I 1 x x 1 (10.4) M 1 I 1 = µ 0I 1 4π I I 1 d x d x 1 1 x x 1 (10.5) 10

147 14 10 I 1 M 1 = µ 0 1 d x d x 1 4π I I 1 x x 1 (10.6) M 1 = M 1 I 1 I I I 1 1, L 11 = µ 0 d x 1 d x 1 1 4π I 1 x 1 x 1 (10.7) I 1 x 1 = x 1 Φ 1 N 1, N dφ 1 N 1 dt, N dφ 1 dt IV V I J = I R a b c I I πa I π(c b ) r H(r) πrh(r) r < a a < r < b I b < r < c I c < r 0 I πa πr = Ir a I π(c b ) π(r b ) = I c r c b a < r < b H(r) = I πr B(r) = µ 0I πr

148 l Φ = l 0 dx b a drb(r) = µ 0Il π (log b log a) (10.8) ( a) I L = µ ( ) 0l b π log a (10.9) l 10.5 L L di dt I () LI di dt LI di dt dt = 1 LI + C (10.30) C 0 M I 1, I M di dt, M di 1 dt di MI 1 dt + MI di 1 dt ( ) di MI 1 dt + MI di 1 dt = MI 1 I (10.31) dt 0 1 L i (I i ) + 1 M ij I i I j (10.3) i 1 M 1I 1 I M 1 I I 1 11 (10.3) 1 L i (I i ) + M ij I j = 1 I i L i I i + M ij I j = 1 I i Φ i (10.33) j i j i i i i j i i 11

149 Φ i = 1 I iφ i = 1 I i = = i i i i d x A d S B d S (rot A) B = rot A Stokes (10.34) d x A Id x jd 3 x 1 d 3 x j = 1 d 3 x(roth) A (10.35) j = rot H = H div( V W ) = V (rot W ) (rot V ) W (10.36) 1 d 3 x(roth) A = 1 d 3 x H (rota) 1 d 3 xdiv( H A) = 1 d 3 x H B (10.37) + ( ) 1 1 d 3 x H B 1 d 3 x D E 1 j A 1 j A 1 j A 1 j A 1 j A 1 j A Z 1 d 3 xdiv() 0

150 rot E = B t (11.1) roth = j (11.) divd = ρ, divb = 0 (11.3) 1865 roth = j H div z div x, y, z z div(rot H) = 0

151 rot div 0 div(roth) = 0 div j 0 div j = 0 div div j = ρ (11.4) t ρ 1 -Q +Q H 1 rot E = B t t (div B) div B = 0 0