i

009 I 1 8 5

i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3 1 5 1.1...................................................... 5 1.................................................. 6 1..1................................................. 6 1.................................... 8 1.3................................................ 11 1.4 E............................................... 13 1.4.1 E.............................................. 13 1.4.................................................. 14 1.4.3......................................... 15 1.5..................................... 16 1.5.1............................. 16 1.5. E........................................ 0 1.6.......................................... 1 1.7................................................ 1.8.................................................. 3 5.1.................................................. 5.1.1 flux..................................... 5.1..................................... 8...................................... 8..1.................................... 8................................ 30.3............................ 30.3.1 E................................... 31.3...................................... 3.3.3............................................ 33.3.4.......................................... 33.4 E.................................... 34.4.1........................................ 34.4. E......................................... 37.4.3 dive = ρ......................................... ε 0 38.5 div................................................. 39.5.1 div.......................................... 39.5...................................... 41.5.3 div E................................ 43

ii.6.................................................. 44 3 45 3.1...................................... 45 3.1.1.............................. 47 3............................................ 48 3..1................................ 48 3.. E.......................................... 50 3..3................................... 5 3.3 rot.......................................... 54 3.3.1..................................... 54 3.3. rot..................................... 56 3.3.3 rot....................................... 57 3.3.4 grad,rot,div........................................... 58 3.4............................................ 59 3.4.1.............................. 59 3.4............................................. 60 3.4.3....................................... 6 3.5.................................................. 63 3.5.1.............................................. 63 3.5............................................... 67 3.5.3............................................... 68 3.6............................................ 69 3.6.1..................................... 69 3.6............................... 70 3.6.3............................... 71 3.6.4............................... 7 3.7.................................................. 74 4 77 4.1................................................ 77 4.1.1............................................. 78 4.................................................. 78 4..1............................................ 78 4....................................... 80 4.3..................................................... 81 4.4.................................................. 81 4.4.1................................................... 81 4.5................................... 8 4.6.............................................. 86 4.7...................................... 87 4.8.......................................... 88 4.9.................................................. 88

1 0 0.1 magnet 1 1600 3 elektron electrics 0. 18 19 4 1700 1800 5 1865 1800 6 1 cgs Gb cgs 3 4 (classical) (quantum) 5 6

0 0 1905 one piece 7 0.3 IC 7

0.4. 3 0 8 0.4 [ ] [ ] 9 (interaction) 8 9

4 0 10 30 ( non-local non-local 11 [ ] 10 11 EPR

5 1 1.1 1 1

6 1 1. 1..1 r q q > 0 F = q 4πε 0 r (1.1) q < 0 1769 1785 3 4 (1.1) SI q, C ε 0 5 SI 8.854187817 10 1 F m 1 6 3 4 1760 1801 1789 5 ε 0 6 F( ) ε 0 = q 4πr F ε 0 [C /N m ] F C /N m

1.. 7 1 4πε 0 1 1 1 4π ε 0 1 SI 1 4πε 0 SI 1 1 1 1 4πε 0 = k k = 8.987551787 10 9 [N m /C ] k 9979458m/s 10 7 10 7 SI SI 100W (100W 1 100J )1 100J 7 1 100J t ct c 30 r 4πr 100J 4πr πr (177 ) 8 1C 1m 8.9875518 10 9 N 10 7 100W 100J 8

8 1 1C 9 0 100kg 1.67 10 7 kg 100 1.67 10 7 = 5.99 108.99 10 8 1.6017733 10 19 C.99 10 8 1.60 10 19 = 4.79 10 9 1 8.99 10 9 ( 4.79 10 9) 1 =.06 10 9 [N] (1.) 1 8.99 10 9 ( 4.79 10 7) 1 =.06 10 5 N (1.3) 10 0 e e 0 (1.4) 1.. (1.1) x q x q q x x q x x q x q x x q x x q x 9 1911 10 5.974 10 4 kg 3.45m/s

1.. 9 x x q F q = q 4πε 0 x x q 3 ( x x q ) (1.5) ( ) 3 x x q ( ) x x q e q = 1 x x q ( x x q ) (1.6) x x q x x q 1 x x q = x x q }{{} e q }{{} (1.7) e 1 11 x x 1 e x e y, e z e x, e y, e z A x e x +A y e y +A z e z e x ˆx ˆ e x, e y, e z i, j, k A x e x + A y e y + A z e z A xˆx + A y ŷ + A z ẑ A x i + A y j + A z k (1.8) F q = q 4πε 0 x x q 3 ( x x q ) = q 4πε 0 x x q } {{ } e q }{{} 11 e 1 (1.9)

10 1 F q q q F q F q = q 4πε 0 x q x 3 ( x q q x ) = 4πε 0 x q x e q (1.10) q F q = F q q q 1 q x q = 0 (r, θ, φ) x x q = r e r (1.11) e r (r, θ, φ) r 13 e θ, e φ θ φ 14 r e r r r, θ, φ e r, e θ, e φ e x, e y, e z 15 e r, e θ, e φ e r e θ = e φ e θ e φ = e r (1.1) e φ e r = e θ q x q = 0 x x q = r 1 13 e r ˆr 14 e ρ, e φ, e z 15 e θ e φ

1.3. 11 F q = q 4πε 0 r 3 r e r = q 4πε 0 r e r (1.13) q > 0 q < 0 16 1.3 i (i = 1,,, N) q i 16

1 1 17 q x q 1,,, N x 1, x,, x N q N F i q = i=1 = N i=1 N i=1 i q ( ) xq 4πε 0 x q x i 3 x i i q 4πε 0 x q x i e i q e i q = x q x i x q x i (1.14) 17 ( )

1.4. E 13 1.4 E 1.4.1 E 18 p11 19 E E q F E = F q E E x q x x E( x) 0 q 4πε 0 x x e x x E( x) = F q = q 4πε 0 x x e x x (1.15) E E E 18 electric field field field 19 0 E( x) E(x, y, z) x, y, z

14 1 1 E SI E F = q E F [N] q [C] [N/C] V E [V] [1/m] E [V/m] E 1.4. E E E E E r E 1m 4πε 0 r r 4πε 0 r r 4πr 4πε 0 r 4πr = ε 0 (1.16) 3 r [C] ε 0 1 ε 0 1.13 10 11 1C 1130.................................................................................................................. FA E 1 E E E 3 4π SI 4π

1.4. E 15 E...................................................................................................................................... E 1 r E E r 4πε 0 r 4πr = ε 0 E (1) E () E (3) (4) #$%'&!" (5)!!" (6) [C] ε 0 ε 0 ()+*, #$%-&-" (4),(5),(6) 4 1.4.3 5 N S N S N N 4 5

16 1 1.5 E 1.5.1 d

1.5. 17 (x, y, z) z ρ 6 dz ρdz z = L z = L L (x, 0, 0) z E ρl z = 0 z r = x + y E Step 1. Step. Step 3. Step Step 1. L N L dz N dz z d z dz ρdz Step. (x, 0, 0) (0, 0, z) (0, 0, z + dz) ρdz E E = 4πε 0 r ρdz x + z ρdz 4πε 0 (x + z ) (1.17) (0, 0, z) (0, 0, z + dz) (x + z ) (x + (z + dz) ) E z z x z 0 x ρdz 4πε 0 (x + z ) x x + z = ρxdz 4πε 0 (x + z ) 3 (1.18) E x 6 1m 1m 1m 3 m cm

18 1 Step 3. E ρxdz z 4πε 0 (x + z ) 3 N dz dz 0 0 L ρxdz 7 dz L 4πε 0 (x + z ) 3 dz dz 0 E (x + z ) (x + (z + dz) ) (0, 0, z) (0, 0, z + dz) (0, 0, z) (0, 0, z z + dz) (0, 0, z) (0, 0, z + dz) N dz 8 N L ρx dz L 4πε 0 (x + z ) 3 z = x tan θ θ x + z = x cos θ 9 dz θ dθ dz = x cos dθ θ Z 7 dz Z dz( ) ( )dz Z L ρxdz L 4πε 0 (x + z ) 3 d dx 8 9 1 + tan θ = 1 cos θ

1.5. 19 dz dθ = x cos θ z = L tan θ = L α x = = α ρx ( α 4πε x 0 cos θ ρ α cos θdθ 4πε 0 x α ρ 4πε 0 x [sin θ]α α = ) 3 x cos θ dθ ρ πε 0 x sin α α = π E = ρ πε 0 x πx πx ρ L 4πε 0 xl sin α = 4πε 0 x }{{} x sin α (1.0) L L α x R (1.19) sin α = L R x L sin α = x 1 R E z x dz E E = ρdz 4πε 0 (x + z ) } {{ } (cos θ e x sin θ e z ) }{{} (1.1) cos θ e x sin θ e z x θ z =

0 1 x tan θ, dz = x cos θ dθ x + z = E = ( x ) cos θ ρdθ 4πε 0 x (cos θ e x sin θ e z ) (1.) cos θ e x sin θ e z θ θ α α (cos θ e x sin θ e z ) dθ (1.3) e z e x α α cos θ e x dθ = [sin θ] α α e x = sin α e x (1.4) (1.19) 1.5. E E 30 σ 31 (r, θ) 3 r r + dr θ θ + dθ r 0 r 0 θ 0 π r dr θ rdθ dθ rdrdθ σrdrdθ z P E σrdrdθ 4πε 0 R E σrdrdθ 4πε 0 R z R (1.5) E R = z + r σz 4πε 0 r0 π 0 0 rdrdθ (z + r ) 3 (1.6) ( ) dθ θ π σz ε 0 r0 0 rdr (z + r ) 3 (1.7) 30 31 σ 3 z (r, θ, z) z r ρ r r ρ ρ

1.6. 1 r = z tan φ r = t dr = σz ε 0 φ0 0 z tan φ ( z + z tan φ ) 3 1 + tan φ = 1 cos φ 1 ( 1 + tan φ ) 3 cos φ 0 = φ0 σ sin φdφ ε 0 0 }{{} [ cos φ] φ 0 0 z cos φ dφ = σ ε 0 φ0 0 tan φ ( 1 + tan φ ) 3 = cos 3 φ = σ ε 0 [ cos φ 0 + cos 0] = z z + (r 0 ) z cos φ dφ 1 cos dφ (1.8) φ σ ε 0 (1 cos φ 0 ) (1.9) E = σ z + (r 0 ) z = ε 0 z + (r 0 ) πε 0 (r 0 ) z + (r 0 ) z (1.30) z + (r 0 ) π(r 0 ) σ = = r 0 cos φ 0 = 0 φ 0 E σ ε 0 E E r 0 z (r 0) z r 0 z r 0 z 1 r 0 z 1 z + (r 0 ) E z z + (r 0 ) z z + (r 0 ) z z + (r 0 ) = 1 + ( r 0 z ) 1 1 + ( r 0 z ) (1.31) 1 + x 1 + 1 x + z + (r 0 ) z = 1 + ( 1 r0 ) ( z + 1 1 r0 ) ( z + (r 0 ) 1 + 1 r0 ) = z + ( z + 1 + 1 r0 ) (1.3) z + z z + (r 0 ) z z + (r 0 ) 1 E (1.30) ( r0 ) (1.33) z πε 0 (r 0 ) z + (r 0 ) z z + (r 0 ) πε 0 (r 0 ) 1 ( r0 ) = z 4πε 0 z (1.34) z 1.6

1 0.0pt51.1496pt ρ( x) x ρ( x )d 3 x x E( x) = x x 4πε 0 x x 3 = 4πε 0 e x x x x (1.35) ρ( x )d 3 x x E( x) = d 3 x ρ( x )( x x ) 4πε 0 x x 3 = d 3 x ρ( x ) e x x 4πε 0 x x (1.36) 1-5 1.7 1 4π steradian Ω Ω = dθdφ sin θ (1.37) sin θ θ θ + dθ φ φ + dφ dθ sin θdφ ε 0 4π Ω Ω 4πε 0

1.8. 3 E E 0 E E a : b a b E 1.8 1-1 (1.37) θ [0, π] φ [0, π] 4π 1-1.5.1 E z α, β (1) dz () γ α β α + β (3) E 1-3 R ρ z E 1-4 r σ z E z < r z E 0 z R = p r + z rz cos θ σr sin θdθdφ E z Z π F (cos θ) sin θdθ cos θ = t 0 R = p r + z rz cos θ 1-5

4 1 ρ 1-6 q 1 cos θ 1 + q cos θ = (1.38) θ 1, θ A (1) A A A () A 1 A ( ) θ π(1 cos θ) (3) q 1, q, θ 1, θ 1-7 1.5. E σ ε 0 (1 cos φ 0 ) σ 4πε 0 ( P ) (1.39) (1) E () E z z

5.1 E E E (Gauss).1.1 flux E E( x) = 1 4πε 0 r e r R 4πR ε 0 ε 0 E E E E E E S E S cos θ θ 1 n E S S cos θ E E cos θ E S cos θ E ns ns S E S 1 1

6 n ds nds ds E ds n x yz ds = dydz n y ds = dzdx z ds = dxdy n = (n x, n y, n z ) (n x ) + (n y ) + (n z ) = 1 d S = n x dydz e x + n y dzdx e y + n z dxdy e z a = a x e x + a y e y + a z e z b = b x e x + b y e y + b z e z a b dy e y dz e z, dz e z dx e x, dx e x dy e y a b v v ( a b) cos θ a b v v ( a b) a b v ( a b) > 0 v d S (flux) 3 d S ρ ρ v d S E E d S E a b a b 3 flux

.1. 7 4 ε 0 E ds = E nds (.1) n E r 4πε 0 r r sin θdθdφ 4πε 0 (r + dr) (r +dr) sin θdθdφ E ds 0 E ds = ε 0 5 A B 6 flux 0 0 0 4 5 Z E ds =.1. ε 0 6

8 0 E ds = 0 0 E ds ε = 0 ε 0.1. E ds flux flux ds E ds...1 E ε 0 rdrdφ

.. 9 E 4πε 0 (r + z ) θ 4πε 0 (r + z rdrdφ cos θ (.) ) θ r r θ r = 0 r = r θ θ = 0 θ = π 1 r = z tan θ dr = z cos dθ θ cos θ tan θ dθ 4πε 0 cos dφ cos θ = dφdθ sin θ (.3) θ 4πε 0 π 0 dθ cos θ = 1 π 0 dφ = π E = ε 0 E = (r e 4πε 0 (r + z ) 3 r + z e z ) (.4) d S = rdrdφ e z E ds = zrdrdφ (.5) 4πε 0 (r + z ) 3 φ r φ π r r = t rdr = dt π E ds = dφ }{{} rdr z 0 0 4πε }{{} 0 (r + z ) 3 = 1 dt = = =π dt z 0 4ε 0 (t + z ) 3 [ z 4ε 0 (t + z ) 1 ] 0 = ε 0 Z 1 dt (t + a) 3 1 = (t + a) 1 (.6) 1.5. E E E 7 7

30.. a r e r a e r r d S r sin θdθdφ e r r r sin θdθdφ 4πε 0 r e r a 3 (r e r a) e r r sin θdθdφ (.7) e r a = a cos θ a a 4πε 0 (r + a ar cos θ) 3 (r a cos θ) r sin θdθdφ (.8) cos θ = t sin θdθ = dt θ (0, π) t ( 1, 1) sin θdθ 8 r 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 9 t r ( ) r 1 1 dt ε 0 r 1 (r + a art) 1 ) (.9) (.10) (.11) 1 1 dt 1 (r + a art) 1 1 1 1 dt = (r + a art) 1 [ 1 ( r + a art ) ] 1 1 = 1 ( (r a) ) (r + a) ar 1 ar (.1) 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 (.11) r r ε 0 r < a 1 ar (a r (r + a)) = a r > a ε 0 r < a 0 0 ε 0.3 Z π Z 1 8 sin θdθ dt 9 0 1

.3. 31 0.0pt51.1496pt E E E E ds = 1 i (.13) ε 0 E ds = 1 ρdv (.14) ε 0 V V V V V V d S E ρdxdydz.3.1 E z = 0 xy σ S σs z (0, 0, z) E

3 z xy x y E E x y z E.................................................... FA E z z E............................................................. S σs ε 0 σs S E ε σ 1.5. 0 ε 0 r 0 E πε 0 (r 0 ) z + (r 0 ) z = π(r 0 ) σ r 0 z + (r 0 ) 1-7 E σ ( P ) 4πε 0 π E E = σ ε 0.3. 1.5.1 E z E z z ρ ρ z ρ ε 0 z πr z E ρ ε 0 z πr z = ρ 1.5.1 πε 0 r

.3. 33.3.3 ρ R 4π 3 R3 ρ ε 0 4πr 4πε 0 r e r E r < R r 4π 3 r3 ρ E E = 4π 3 r3 ρ 1 4πε 0 r e r = 1 ρr e r (.15) 3ε 0 r E r 0.3.4 10 + E E E S S ε 0................................................................. FA ε 0 ε 0 A B 100 A 100 B 100 00............................................................................ E ε 0 S 10 capacitor

34 E ε 0 S E E E.4 E (.14).4.1 E flux E (E x, E y, E z ) x y z z x+ x y+ y dx dy E z (x, y, z) (.16) x y x, y 0 x+ x y+ y dx dy x y x y 1 E z (x, y, z) x x x ( x) x ( x) 11 11 100 100 100 100 100 100

.4. E 35 E z (x, y, z) x y E z (x, y, z) x y (.17) E z 1 E z (x, y, z + z) x y (.18) z z (E z (x, y, z + z) E z (x, y, z)) x y (.19) 13 E z (x, y, z + z) E z (x, y, z) lim = E z(x, y, z) z 0 z z (.0) z E z x y z (.1) z E z (x, y, z + z) = E z (x, y, z) + z E z z + (.).................................................................................................................. FA E z (x, y, z) z E z z E z(x, y, z) 100 100 100 100 100...................................................................................................................................... 1 d S E z > 0 E d S d S 13 Z x+ x Z y+ y dx dy (E z (x, y, z + z) E z (x, y, z)) x y x y (.19)

36 14 x y x, y, z ( Ex x + E y y + E ) z x y z (.3) z x y z E x x + E y y + E z z E x x + E y y + E z z = div E div E 15 A div 16 div A = A x x + A y y + A z z (.4) div (divergence) 17 div E 18 div div (.14) E ds x y z V (.14) ρdv ρ x y z V ρ x y z x y z div E div E = ρ ε 0 (.5) (.14) div E = 0 div E = 0 div EdV = V V div EdV = V V ρ ε 0 dv (.6) E d S (.7) E 14 15 A( x) E 16 A (.4) A E H v A 17 18 (; ;)

.4. E 37 19 div AdV = A ds (.8) E V V 0.4. E div E = 0 E E (0, 0, ρ ε 0 ) div E = 0 E = 4πε 0 r e r e r x, y, z ( x x + y + z, y x + y + z, ) z x + y + z 1 ( ) x y z E =,, 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 3 (.9) 19 0 E E div 1 e r (x, y, z) 1 p x + y + z e r

38 E x x E x x = = = ( x x 4πε 0 (x + y + z ) 3 3 x x 4πε 0 (x + y + z ) 3 4πε 0 (x }{{} + y + z ) 5 }{{} 4πε 0 (x + y + z ) 3 ) x 3 4πε 0 (x + y + z ) 5 (.30) x, y, z E y y = 4πε 0 (x + y + z ) 3 E z z = 4πε 0 (x + y + z ) 3 3 4πε 0 (x + y + z ) 5 3 4πε 0 (x + y + z ) 5 3 x, y, z div E ( = 3 3 x + y + z ) (.31) 4πε 0 (x + y + z ) 3 4πε 0 (x + y + z ) 5 0 div y z.4.3 div E = ρ ε 0 E = 1 3ε 0 ρr e r (.15) r e r = (x, y, z) E x = ρ 3ε 0 x, E y = ρ 3ε 0 y, E z = ρ 3ε 0 z (.3) div E = ( ) ρ x + ( ) ρ y + ( ) ρ z = ρ 3 = ρ (.33) x 3ε 0 y 3ε 0 z 3ε 0 3ε 0 ε 0 div E = ρ ε 0 d ρ E z = d z = d (x, y, z) E x, y E x, E y 0 dive = 0 z E z = 0 E z x, y E z z d dz E z = 0 E z

.5. div 39 d dz E z = ρ ε 0 E z = ρ ε 0 z + C (.34) z = 0 E z = 0 C 0 ρ d (d z) ε 0 ρ E z = z ( d < z < d) ε 0 ρ d (z d) ε 0 (.35) x, y z 1 div 1 de z dz div E = 0 z= -d z=d.5 div V x x + V y y + V z div z div V = V r r + V θ θ + V φ φ.5.1 div div r r θ r sin θ φ r sin θ r θ φ (r + r) sin θ θ φ r sin θ 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 (.36)

40 div r sin θ r θ φ div 1 (r + r) V r (r + r, θ, φ) r V r (r, θ, φ) r r (.37) r V (r, θ, φ) r r + r ( (r + r) V r (r + r, θ, φ) ) r ( r V r (r, θ, φ) ) (.37) r r 1 r 0 r div r V (r, θ, φ) r 1 r 1 ( r r V r (r, θ, φ) ) (.38) r r V r V r r r V r.................................................................................................................. FA 1.5. r r + dr rdθ (r + dr)dθ drdθ (r + r) V r (r + r, θ, φ) r V r (r, θ, φ) (.39) (r + r) V r (r + r, θ, φ) `r + r r + ( r) V «r(r, θ, φ) + r Vr (r, θ, φ) + r = r V r (r, θ, φ) + r rv (r, θ, φ) + r V r (r, θ, φ) + ( r) V (r, θ, φ) + r( r) V r (r, θ, φ) + {z } r r {z } {z } 0 1 (.40) (r + r) r r (r + r) r (r + r) r + r r

.5. div 41...................................................................................................................................... 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) (.41) 1 r sin θ φ (r sin θv φ(r, θ, φ + φ) r sin θv φ (r, θ, φ)) 3 div 1 r sin θ φ V φ (.4) div A = 1 r ( r ) 1 A r + r r sin θ θ (sin θa θ) + 1 A φ r sin θ φ div (.43) div A = 1 (ρa ρ ) + 1 A φ ρ r ρ φ + A z z (.44).5. div 4 div A = 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 5 ( ) 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 (.45) 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 (.46) F ( x + h e) = F ( x) + h e F ( x) + O(h ) (.47) h F ( x) h h 0 e F (x) df (x) dx = lim h 0 F (x + h) F (x) h sin θ sin θ sin θ r dθ rdθ θ 1 r θ 3 r sin θ r sin θdφ φ 1 r sin θ φ 4 (nabla) 5 e x, e y, e z 1 (.48)

4 df (x) F (x + h) = F (x) + h dx + O(h ) (.49) x h F (x) h x e x F = F x e y F = F y e e F F F x y F F = e x x + e F y y + e F z (.50) z F = e x x + e y y + e z z (.51) e r e r F ( x) F θ r e θ F ( x) 1 F r θ θ h θ h r (.5) 1 r φ h φ e θ F F (r, θ + h r, φ) F (r, θ, φ) (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 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 (.5) (.53) F 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 = A r r + 1 A θ r θ + 1 A φ r sin θ φ e r, e θ, e φ e r e r = 0, θ e r = e θ, φ e r = 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 (.54) (.55) (.56)

.5. div 43-5 ( 1 1 e r A = r + e θ r θ + e φ r sin θ φ = A r r + 1 A θ r θ + 1 A φ r sin θ φ + 1 r e θ + 1 ( r sin θ e φ A r e r φ }{{} =sin θ e φ +V θ e θ φ }{{} =cos θ e φ ) (A r e r + A θ e θ + A φ e φ ) ( A r e r θ }{{} = e θ +A θ e θ θ }{{} = e r +V φ e φ φ }{{} sin θ e r cos θ e θ = A r r + 1 A θ r θ + 1 A φ r sin θ φ + r A r + cos θ r sin θ A θ ) ) (.57) (.43) e.5.3 div E div div (.43) A E = 4πε 0 r e r 0 V r 4πε 0 r div r V r 0 E div E = 0 E θ E φ div E = 0 1 ( r ) r E r r ( r ) E r r = 0 r = 0 r E r = C r (.58) E r = C r C div E = ρ ε 0 ρ 1 ( r ) r E r r ( r ) E r r = ρ ε 0 r = ρ ε 0 r r E r = ρ 3ε 0 r 3 + C E r = ρ r + C 3ε 0 r r (.59) C 0 r = 0 E r.................................................................................................................. FA E r = 4πε 0 r......................................................................................................................................

44 C r = R E = C r E = C R = ρ 3ε 0 R C = 4π 3 R3 ρ = ρ 3ε 0 r ρ 3ε 0 R 3 (.60) E = ρr3 4π 3ε 0 r = 3 ρr3 4πε 0 r = 4πε 0 r (.61) E.3.3 div E = ρ ε 0 (.6).6-1 R 0 E E r E kr n n n - E = x e x y e y div E = 0 z (1) E () dy dx (3) E = y e x + x e y div E = 0-3 r 1 r r 1 < r < r ρ z r -4.3. ρ r ρ z πε 0 r r z div E = 0 (1)

.6. 45 () -5 (.56) e r, e θ, e φ e x, e y, e z e x, e y, e z r, θ, φ -6 (.56)

45 3 E E 3.1 m x F (x) m d x dt = F (x) x t 1 x 1 t x x 1 x 1 m x x 1 d x dt dx = x x 1 F (x)dx (3.1) F (x) dx dx dt x t dt t d x dx x m t 1 dt dt dt = F (x)dx (3.) x 1 dx dt v x d dt = dv dt t dv x m t 1 dt vdt = F (x)dx (3.3) x 1 F (x) x d ( v ) = dv dt dt v (3.4) 1 t m d ( v ) x dt = F (x)dx (3.5) t 1 dt x 1 t 1 t x 1 x v 1 mv 1 t=t t=t1 mv = x x 1 F (x)dx (3.6) 1 mv F (x) x F (x)dx F (x)dx F (x)dx x=x F (x)dx x=x x 1 1 mv 1 t=t t=t1 mv = F (x)dx x=x F (x)dx x=x (3.7) 1

46 3 x 1 t 1 x t 1 mv F (x)dx t=t1 F (x)dx = U(x) x=x1 = 1 t=t mv F (x)dx (3.8) x=x 1 mv + U(x 1 ) = 1 t=t1 t=t0 mv + U(x 0 ) (3.9) t 1 x 1 t 0 x 0 1 mv +U 1 mv U(x) U(x) F (x) U(x) = F (x)dx F (x) = d U(x) (3.10) dx mg g mg x F (x) = mg U = mgx M m GMm r U(r) = GMm r r = r x kx x = 0 x F (x) = kx 1 kx F (x)

3.1. 47 F (x) d U(x) dx 3.1.1 E E E = 4πε 0 r F (r) = q 4πε 0 r U(r) = q 4πε 0 r F (r) = d U(r) dr 3 U(r) F (r) q F E = 1 q F E E q U V = U q 4 V U F F = d dr U E E = d dr V V E [V] 5 1.5V 1.5V (voltage) 3 GMm r GMm q > 0 r 4 5

48 3 q 1.5 q[j] q[c] 1.5q[J] q > 0 6 q 4πε 0 r r V = 4πε 0 r z E E 3. E 3..1 F (x) F (x) (F x, F y, F z ) F x dx + F y dy + F z dz = F d x (3.11) 6 q (1..1 1C ) 1.5q[J]

$ $ 3.. 49 F = (F x, F y, F z ) d x = (dx, dy, dz) U(x, y, z) F x = U x, F y = U y, F z = U z F = du dx 7 U F x e x = e x x U +F y e y = e y y U +F z e z = e z z (3.1) (3.13) F x e x +F y e y +F z e z = e x U x e y U y e z U z = e x x + e y y + e z z U ( Φ Φ( x) x, Φ y, Φ ) (gradient) z grad grad grad Φ = Φ Φ = e x x + e Φ y y + e Φ z z (3.14) grad div rot div grad grad e (grad Φ) = e Φ = lim h 0 Φ( x + h e) Φ( x) h (3.15) x Φ h x+h e Φ h h Φ x h h % '&)( *!!" # e h grad grad φ e e Φ 7

50 3 3.. E F = grad U E E E E E = grad V E = V (3.16) V E z V V V grad V grad V grad V E

3.. 51 E E E E 8 x a V ( x + ɛ a) V ( x) = 0 (3.17) ɛ a ɛ a x, y, z ɛ V (x + ɛa x, y + ɛa y, z + ɛa z ) V (x, y, z) = 0 (3.18) ɛ a grad V = 0 ɛ a E = 0 (3.19) V ɛa x x + ɛa V y y + ɛa V z z = 0 ɛa xe x ɛa y E y ɛa z E z = 0 (3.0) E E 8

5 3 E E E E E 0 E 3..3 F = U F (x)

3.. 53 F U F = x e y = (0, x, 0) F y = x U = xy U x = y F x 0 F = U y x A B C A D C B C A D B C A C ( ) A B A B mg π θ h cos θ mg h cos(π θ) = mgh cos θ A C B A C C B mgh z z F d x z F = U U

54 3 3.3 rot 3.3.1 A(x, y) D(x + x, y) C(x + x, y + y) A(x, y) B(x, y + y) C(x + x, y + y) 9 x, y F x (x, y) x + F y (x + x, y) y F y (x, y) y F x (x, y + y) x = (F y (x + x, y) F y (x, y)) y + (F x (x, y) F x (x, y + y)) x }{{}}{{} ( x F y Fy x x ) x y ( ) y F x x y Fx y y (3.1) x F y y F x = 0 (3.) 10 F F ( x F y ) y F x x y (3.3) x y x F y y F x (3.4) Z y+ y 9 x, y F y (x, y )dy F y (x, y) y 10 y

3.3. rot 55 xy yz zx xy : yz : zx : x F y y F x = 0 y F z z F y = 0 z F x x F z = 0 (3.5) xy z yz x zx y rot F = rot ( y F z z F y, z F x x F z, x F y ) y F x (rotation) curl 11 rot E = 0 1 rot rot rot rot x y z rot yz zx xy x yz 13 0 0 (3.6) 11 curl 1 0 0 rot E = 0 0 13

56 3 3.3. rot rot rot rot V x x V x x V x x V x x y y V x (x, y y, z) x + V x (x, y, z x }{{}}{{} (3.7) 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

3.3. rot 57 3.3.3 rot rot 14 a q a a 0 (x + a ), y qe (x + a ), y E y E y z ( qe y x + a ) a, y (3.8) ( x, y + a ) ( qe x x, y + a ) a (3.9) E x > 0 q [E y (x + a ) (, y E x x, y + a ) E y (x a ) (, y + E x x, y a )] a (3.30) a ( E y x + a ) (, y E y x a ), y = x E y(x, y)a (3.31) [ Ey q x E ] x a y z rot E z (3.3).................................................................................................................. FA (rot E) y = z E x x E z (rot E) x = y E z z E y y z (rot E) z = x E y y E x x y (rot E) y = z E x x E z z x rot x y z y z x x z y z x...................................................................................................................................... rot E = 0 15 rot nonzero F = x e y rot nonzero F y x = 1 rot 0 E E E = x e y E 14 15 rot E 0

58 3 rot rot ( rot F = 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 ( (3.33) rot = x, y, ) z F = (F x, F y, F z ) rot F = F (3.34) 3.3.4 grad,rot,div div,rot,grad grad rot 0 Φ grad φ rot rot (grad φ) = 0 grad rot grad rot rot grad ( )-( ) rot div 0 V div (rot V ) = 0 div rot rot div rot rot div 6 rot

3.4. 59 rot rot rot ( rot) rot div (Stokes) rot S S S (rot V ) ds = S V d x (3.35) S S div V dv = V ds V V rot E = 0 0 S 3.4 3.4.1 F = grad ( ) ( ) ( ) q = 4πε 0 r q q 1 = e r + e θ 4πε 0 r r 4πε 0 r r θ }{{} θ,φ (3.36) U = q 4πε 0 r V = 4πε 0 r E = grad V V C grad C = 0 E E = grad (V + C) 16 (r = ) V = 0 V = 4πɛ 0 r E E = V V E E 16

60 3 1,,, N N x 1, x,, x N x V ( x) = 1 4πε 0 x x 1 + N 4πε 0 x x + = i 4πε 0 x x i i=1 (3.37) grad E grad 17 V ( x) = N i N 4πε 0 x x i = i 4πε 0 x x i e x i x (3.38) i=1 E E ρ( x) V ( x) = 1 4πε 0 ρ( x ) x x d3 x (3.39) i=1 18 x ρ( x )d 3 x x V ( x) x (1.36) 3.4. div E = ρ rot E ε = 0 E = grad V 0 rot E = 0 grad rot 0 div E = ρ div ( grad V ) = ρ (3.40) ε 0 ε 0 div (grad V ) = ρ (3.41) ε 0 grad grad div grad V ( V x, V y, V z ) div (A x, A y, A z ) A x x + A y y + A z z ( div (grad V ) = x + y + z x + y + z 19 ) V = ρ ε 0 (3.4) 17 18 d 3 x dxdydz 3 d 3 x = dx dy dz 19 ( )

3.4. 61 V = ρ ε 0 (3.43) 0 f = j j ρ (source) ε 0 0 f = 0 V = 4πε 0 r r = x + y + z ( ) 1 x x + y + z = 1 x (x + y + z ) = ( ) x x (x + y + z ) 3 x (x + y + z ) 3 1 = + 3 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 x 1 r y r z ( ) ( ) x + y + 1 z r ( 1 r ( ) 1 = 0 r grad V ( V r, 1 r div ) (3.44) x y x z 1 = + 3 (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 div A = 1 r V = div (grad V ) = 1 r V θ x x y z ( ) 1 ( r ) 1 ( r ) 1 r (3.45) 1 V ) r θ φ r sin θ φ ( r ) 1 A r + r r sin θ θ (sin θa θ) + 1 A φ r sin θ φ r ( r V ) 1 + r r sin θ θ ( sin θ V ) + θ (3.46) 1 V r sin θ φ (3.47) V = 1 θ, φ 0 r V r = 1 r r r V ( ) 1 = 1 0 = 0 r r r 0 r = 0 0 0

6 3 3.4.3 grad div f = 0 d dx f = 0 0 dy dx = lim y(x + x) y(x) x 0 x d y dx = lim (y(x + x) y(x)) (y(x) y(x x)) x 0 ( x) = lim x 0 (3.48) y(x + x) + y(x x) y(x) ( x) (3.49) y(x + x) + y(x x) y(x) y(x + x) + y(x x) y(x) = x + x y x y y f(x, y) = 0 x y f(x, y) = log(x + y ) x y f = 0 f = 0 x, y, z ( 1 ) V x y, z 1 x, y, z f

3.5. 63 grad div 3.5 3.5.1 ρ R E 0 R 0 R E ρr 3 3ε 0 r E = e r r > R ρr e r r R 3ε 0 (3.50) E r V r E = E r e r E = V dv = e r dr E r = dv V dr V 1 + ρr3 r > R 3ε 0 r V = (3.51) V ρr r R 6ε 0 E V 1, V r =, r = 0 E

64 3 0 V 1 = 0 V r > R r = R r R r = R V ρr = ρr 6ε 0 3ε 0 (3.5) V = ρr ε 0 E V ( 1) E r = R r = R E E E E E (r ) sin θdr dθdφ 3 ρ(r ) sin θdr dθdφ dv = (3.53) 4πε 0 r + (r ) rr cos θ r R φ π θ π sin θdθ 1 0 1 dt θ r V = ρ ε 0 R 0 dr 1 1 (r ) dr dt r + (r ) rr t (3.54) d B A + Bt = dt A + Bt V = ρ ε 0 R A = r + (r ), B = rr 0 dr (r ) dr [ 1 rr r + (r ) rr t ] 1 1 (3.55) [ 1 ] 1 r rr + (r ) rr t = 1 r 1 rr + (r ) rr + 1 r rr + (r ) + rr (3.56) r + (r ) ± rr = (r ± r ) [ 1 ] 1 r rr + (r ) rr t = 1 1 rr ( r r r + r ) (3.57) A A A 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 (3.58) 3ε 0 r 0 3 r r r, θ, φ

3.5. 65 r < R V = ( 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 0 3 + R r r ] R r ) = ρ ε 0 R ρ 6ε 0 r (3.59) E 1 d (r ddr ) r dr V (r) = ρ (3.60) ε 0 r > R 1 d (r ddr ) r dr V (r) = 0 (3.61) d dr (r ddr ) V (r) = 0 r d dr V (r) = C 1 r d dr V (r) = C 1 r V (r) = C 1 r + C r = V = 0 C = 0 d (r ddr ) dr V (r) = ρ r ε 0 r d dr V (r) = ρ r 3 + C 3 3ε 0 d dr V (r) = V (r) = ρ r + C 3 3ε 0 r ρ 6ε 0 r C 3 r + C 4 r V C 3 = 0 C 1, C 4 V dv r = R dr E E V (r) ρ R + C 4 = C 1 6ε 0 R dv ρ (r) R = C 1 dr 3ε 0 R (3.65) C 1 = ρ R 3 3ε 0 (3.6) (3.63) (3.64) ρ R + C 4 = ρ R (3.66) 6ε 0 3ε 0

66 3 C 4 = ρ R ε 0 4 r > R 4πε 0 r E V 0 V = 4πε 0 r = 4π 3 R3 ρ R 0 ρ ( ) = 4πε 0 r 3 R 0 4πR3 r = 0 0 (3.67) r 0 0 r = 0 ε 0 ε 0 f( x) d 3 x f( x )δ( x x ) = f( x) (3.68) Dirac x = 0 δ( x) = 0 x 0 (3.69) 4

3.5. 67 ( ) δ 3 ( x x ) = 1 4π x x e x x ( ) δ 3 ( x x 1 ) = 4π x x (3.70) (3.71) δ 3 ( x) = δ(x)δ(y)δ(z) div E = ρ ε 0 V = ρ ε 0 ε 0 x x 0 x = x d 3 x δ( x x ) = 1 (3.7) 1 1 4π x x ρ 1 ε 0 ε 0 ε 0 5 6 r < R r > R 0 1 div E 0 1 R 3.5. z = d z = d d ρ 0 x = x = y x, y x y E E d dz φ(z) = 1 ε 0 ρ(z) (3.73) ρ 0 d < z < d ρ(z) = 0 (3.74) 5 6

68 3 d < z < d dφ(z) dz = ρ 0 ε 0 z + C 1 φ(z) = ρ 0 ε 0 z + C 1 z + C (C 1, C ) z = 0 E dφ(0) = 0 dz C 1 0 z = 0 C = 0 (3.75) 0 E ( 1) 0 z > 0 z < 0 3.5.3 Na + Cl 0 7 E (0, 0, d) +q (0, 0, d) q V (x, y, z) = q 4πε 0 x + y + (z d) q (3.76) 4πε 0 x + y + (z + d) d 0 qd p 8 d 0 q q q p 9 p ( ) V (x, y, z) = p 1 1 4πε 0 d x + y + (z d) 1 x + y + (z + d) 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 z d z + d x + x x x d (3.77) (3.78) 7 8 1.6 10 16 C 10 10 m 9

3.6. 69 V (x, y, z) = p 4πε 0 z (x + y + z ) 3 = p cos θ 4πε 0 r (3.79) x + y + z = r, z = r cos θ E ( {}}{ ) E = p z 4πε 0 (x + y + z ) 3 ( ( ) ) = p z 4πε 1 1 + (z) 0 (x + y + z ) 3 (x + y + z ) 3 ( ) = p 3z (x + y + z ) 1 + e 4πε 0 (x + y + z ) 5 (x + y + z 3 z ( ) p 3z(x e x + y e y + z e z ) 1 = e 4πε 0 (x + y + z ) 5 z (x + y + z ) 3 = 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 (3.81) ( ) (3.80) (3.81) 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 (ab) = ( a)b + a( b) f 3 = 3 ff 5, z = ez (x ) = x e x, (y ) = y e y, (z ) = z e z p x V = 4πε 0 x 3, E p x = 3 4πε 0 x 4 x 1 p (3.8) 4πε 0 x 3 ( ) ( p x) = p, 1 x 3 = 3 x x 4 (3.80) 3.6 3.6.1 3.4.1 U = q q 4πε 0 r 4πε 0 r q q 4πε 0 r q q 4πε 0 r 30 1 4πε 0 r q 4πε 0 r q 1 q 4πε 0 r q 4πε 0 r 30 q U = q 4πε 0 r (double-counting)

70 3 x k 1 kx k x 1 kx 31 1 ε 0 E 3.6. 3.4.1 ( q, ) r q 4πε 0 r q q 1 r 1 q r q 1 q q 1 q 4πε 0 r 1 + q q 4πε 0 r (3.83) q 1q 4πε 0 r q 1 q + q q + q 1q 4πε 0 r 1 4πε 0 r 4πε 0 r (3.84) q 1, q,, q N N q i q j r ij r ij i j N 1 N i=1 N j=1 j i q i q j 4πε 0 r ij (3.85) i = j i = j 31

3.6. 71 r ii = 0 1 N = 3 1 3 3 q i q j = 1 4πε 0 r ij i=1 j=1 j i = 1 = 3 j=1 j 1 q 1 q j 4πε 0 r 1j + 3 j=1 j q q j 4πε 0 r j + 3 q 3 q j 4πε 0 r 3j j=1 j 3 ( 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 1 double-counting 1 double-counting 3 ) (3.86) 1 N i=1 N 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 (3.87) V ī ( x i ) x i q i ī 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 (3.88) i=1 ρ V 1 ρ( x)v ( x)d 3 x i = j ρd 3 x V ( x) x x 0 0 3.6.3 1 ρ( x)v ( x)d 3 x ρ div E = ρ ε 0 ε 0 (div E( x))v ( x)d 3 x (3.89) div E E V x, y, z ε 0 3 over-counting ( x E x( x) + y E y( x) + ) z E z( x) V ( x)d 3 x (3.90)

7 3 b a [ ] b df(x) dx g(x)dx = f(x)g(x) a }{{} b a f(x) dg(x) dx (3.91) dx x E x( x)v ( x) ε 0 ( E x ( x) x V ( x) + E y( x) y V ( x) + E z( x) ) z V ( x) d 3 x (3.9) 33 [ ε0 ] x= E x ( x)v ( x)dydz x= (3.93) V ( x) E( x) 0 34 E = V x E x = x V ε 0 (E x ( x)e x ( x) + E y ( x)e y ( x) + E z ( x)e z ( x)) d 3 x (3.94) U = ε 0 E d 3 x (3.95) 35 ε 0 E 1 kx 3.6.4 1 qv S V d ε 0 S V V 0 V 0 + V ε 0 S 1 (V + V 0) + 1 ( )V 0 = 1 V = 1 d ε 0 S (3.96) 1 V Z 33 d 3 xdiv ( ) 0 34 0 0 Z 0 Z Z 35 ( E)V = E V = E E

3.6. 73 q V qv q q q + dq dqv V = qd ε 0 S dq qd q 0 ε 0 S 0 dq qd ε 0 S = d [ q ε 0 S ] 0 = d ε 0 S (3.97) 1 V dqv 1 V V V V d Sd ε 0 S 1 ε 0 S = 1 ( ) ε 0 (3.98) ε 0 S 1 ε 0 E

74 3 E 1 E 0 0 d 1 Ed = 1 V.................................................................................................................. FA E F = q E 1 F = q E E 1 E 1 E...................................................................................................................................... 3.7 3-1 V = kx z (1) () a 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 3-3 -3 = 1 r ( r ) + 1r r r θ + z

3.7. 75 V = ( ) r 1 < r < r ρ 0 r = 0 V = 0 ε 0 E = grad V -3 3-4 3-5 R 1 R 0 (R 1 > R 0 ) (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 3-7 q q q E q E E( x) = E ( x) = q 4πε 0 x x q 3 ( x x q) q 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 (0, 0, L) )

77 4 4.1 0 0 0.................................................................................................................. FA 100g 10 4 1.6 10 19 10 5 96485 (96485C) 1

78 4 1..1 1...................................................................................................................................... 4.1.1!" 1 0 rot E = 0 rot E = 0 0 0 S σ ρ S σ S S E = σ n ε 0 E = σ n ε 0 1 ε 0 4. 4..1 r 0 1

4.. 79 0 0 x = 0 V = 0 V = ρ ε 0 ρ x = 0 0 +, r x ( r, 0, 0) (r, 0, 0) (x, y, z) ( ) V (x, y, z) = 1 4πε 0 (x + r) + y + z 1 (x r) + y + z (4.1) x > 0 ( ) 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 V 1 = ρ, V = ρ ε 0 ε 0 ε 0 (V 1 V ) = 0 (4.3) V 1 V V 1 V (x = 0) 0 0 0 V 1 = V x ( ) E x = x V = x + r x r 4πε 0 ((x + r) + y + z ) 3 ((x r) + y + z ) 3 (4.4)

80 4 x = 0 x x = 0 E x = 4πε 0 r (r + y + z ) 3 σ E = σ ε 0 n σ = π r (r + y + z ) 3 (4.5) n σ ( dy dz) 3 4.. 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 3 V = V + V = Er cos θ + ER3 cos θ r 0 0 r R V (r, θ) = E(R 3 r 3 ) cos θ r R < r = E(R3 r 3 ) cos θ r (4.6) R = r ( ) E = V = e r r + e 1 θ r θ + e 1 E(R 3 r 3 ) cos θ φ r sin θ φ r }{{} φ ( = e r 3E cos θ E(R3 r 3 ) ) cos θ E(R 3 r 3 ) sin θ r 3 e θ r 3 (4.7) (4.8) 3

4.3. 81 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 4.3 V V C = CV (4.11) 1V 1C 1F F F=C/V 1F 1µF( )=10 6 F 1pF( )=10 1 F C ( k) E k E V kv V 4.3.4 S d ε 0 S ε 0S d V = d ε 0 S = ε 0S d V (4.1) 4.4 4.4.1 (dielectrics) 5 (insulator) (polarization) 4 E 5 dielectrics

8 4 6 7 4.5 +q q d q p = qd n p n p P p p i = n p = P (4.13) +q -q [C m] [m 3 ] [C/m ] P 0 0 z P z (x, y, z + z) x y P z (x, y, z) x y P z x y z x, y z ( Px ρ P x y z = x + P y y + P ) z x y z (4.14) z 6 7 3 10 6 V/m

4.5. 83 ρ P div P ρ P ρ div P div E = ρ div P ε 0 (4.15) div E = ρ ε 0 1 ε 0 div P (4.15) div ) (ε 0E + P = ρ (4.16) (electric displacement 8 electric flux density) D D = ε 0 E + P (4.17) SI ε 0 [N/c] [V/m] [C/m ] (flux) [C] 1C 1C D (4.16) div D = ρ (4.18) ρ P D = ε 0E E D E = 4πε 0 r e r D = 4πr e r ε 0 E D P, D, E 9 (4.17) E ε( 0 ) D E D = εe (4.19) ε εd = ε 0 ( D P ) ε 0 D div P = 0 div P = 0 div P 0 10 S d P nq nq d 8 electric displacement D 0 9 P E 10 0

84 4! #" $ %&(')*,+.-/0 11 3 54 6 7 8 9 :1; <' =! #" $>?@BA C DE F +nqds = +PS G H I(J<' >K L 3 M1N< -nqds = -PS +nq -nq qd S = P S qd S = P S P P ε 0 E }{{} = 1 ε 0 D 1 ε 0 P }{{} 1 ε 0 D 11 ε 0 S P ε 0 )#*,+.- "#"#"#"%$ &#'#(!!"#"#"#"%$ &#'#( // 1 ε 0 P.................................................................................................................. FA ε E = D ε E = D 80...................................................................................................................................... 1 ε 0 P 11 div D = ρ D 1 ε 0 D th 1 ε 0 D

4.5. 85 R ε 1 D = 4πr e r, E = 4πεr e r P = D ε 0 E = (1 ε 0 ε ) 4πr e r (4.0) R ( σ P = 1 ε ) 0 ε 4πR (4.1) ( 4πR σ P = 1 ε ) 0 ε ( 1 ε ) 0 = ε 0 ε ε (4.) ε 0 ε E = 4πεr e r ε 0 ε E = ε 0ε 4πε 0 r e r 13 ε 0 ε r 4πε 0 r e r 4πεr e r ε 0 ε ε ε ε 0 = ε r 14 1 80 4 4 8 3 100 div E = 1 ε 0 (ρ + ρ P ) div D = ρ div E ρ ε 0 ρ P ε 0 div E = 1 ε 0 (ρ + ρ P ) div D = ρ div D = 0 E D E 1 R R 13 ε ε 0ε 0 ε E = 4πεr e r 14

86 4 E D 1 ε 0 D div D = ρ, rot E = 0 div D = 0 rot E = 0.................................................................................................................. FA E D E D E ε 0 D E D ( D )...................................................................................................................................... D ρ ρ P = div P D ρ P div D = ρ D rot D = 0 rot 0 D D = ε E div E = ρ ε ε div 4.6 E D div D = 0 D

4.7. 87 15 16 E, D H, B 4.7 ε 0 E ε 0 E 1 ρv d 3 x div E = ρ div D 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 (4.3) 1 U = 1 D x = ε 0 E d 3 x + 1 P Ed 3 x (4.4) P Ed 3 x P nqd 1 P Ed 3 x = 1 nq d Ed 3 x = 1 n F dd 3 x (4.5) F = q E q F = K d 1 nk d d 3 x (4.6) 1 K d nd 3 x 1 D E 15 rot E = 0 rot D = 0 rot P 0 16

88 4 4.8 q F E = 1 F q q U V = 1 q U E = grad V rot E = 0 D = ε 0E + P D = εe div D = ρ D ds = ρdv V = ρ ε 0 V V rot E 0 rot E 0 rot E 0 4.9 4-1 4- ε d E θ E D E D

4.9. 89 4-3 R P E, D P cos θ θ P

90 curl, 55 div, 36 F( ), 81 flux, 6 grad, 49 rot, 55 steradian,, 6, 59 div, 41, 11, 36, 5, 31, 34, 79 div, 39, 3, 6, 6, 49, 49 ( ), 10, 6,, 59, 77, 47, 81, 83, 68, 8 div, 36, 66, 47, 47, 13, 79, 68, 68 b, 14, 83, 83, 13, 77, 41 (div), 36, 85, 71 F( ), 81, 78, 81, 33, 53 V( ), 47, 61, 81, 6, 83, 61, 61,, 6, 6, 55 (div), 36