c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l

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c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2 + a 2 3/2 d = 2 + a 2 9 π/2 2 + a 2 3/2 d = π/2 = a 2 [sin θ cos 3 θ a a 3 cos 2 dθ = a tan θ θ ] π/2 π/2 = 2 a 2

c 28 2 2 + y 2 + 2 3/2 ddy = = 2π sin θ 2 + 2 3/2 dθd 2 2 + 2 3/2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a 2 2 2 2 2 + a 2 d d + 2 = log 2 + a 2 2 + 2a2 a tan a a 2 2 + a 2 d 2 π/2 2 + a 2 d = cos 2 θ π/2 a 2 = sgna a [ θ ] π/2 a cos 2 dθ = a tan θ θ = sgna π π/2 a = π a 3 at 2 + a 2 2at 3/2 dt = at a 2 + a 2 2at + 2 + a 2 2at dt at = a 2 + a 2 2at 2 + a 2 2at a 2 t a = 2 2 + a 2 2at 4. q, y, E = e + y e y + e q 2 = q 2 + y 2 + 2 3/2 e + y e y + e 5.2 λ 2,, + d λ d a,, d E = a e e d E = λd a 2 + 2 = λ a e e a 2 + 2 d 6 3/2

c 28 3 q E y d E = = E = λ a 2 + y 2 d = 7 3/2 E = λ λ a 2 + 2 d = 3/2 2 a 2 = λ 2πε a 8 λ 2πε a e 9 l d a de 2.3.3. σ y + d y y + dy σddy 3,, d E = e y e y + e d E = σ 2 ddy = σ e y e y + e 2 + y 2 + 2 ddy 2 3/2 E y E y E = σ 2 + y 2 + 2 ddy = 2 3/2 E = σ y 2 + y 2 + 2 ddy = 22 3/2

c 28 4 de dy y sddy d 3 E E = σ = σ 2π = σ 2π = σ 2ε = σ 2ε 2 + y 2 + 2 3/2 ddy 2 + 2 ddθ 3/2 2 + 2 3/2 d [ ] 2 + 2 9 = sgn σ 2ε 8 23.3.2 σ y + d y σd 4,, d E = e + e d E = σd 2πε = σ 2πε e + e 2 + 2 d 24 E = E y = E 8 3 E = σ 2πε 2 + 2 d = σ 2πε π = sgn σ 25 2ε.4 5 y a λ θ θ + dθ λa dθ d E R = e y e y + e de = λadθ R 2 R R = λadθ R 3 a cos θ e a sin θ e y + e 26

c 28 5 de y sd d 4 2 de R y a q a 5 = a cos θ 27 y = a sin θ 28 R = a 2 + 2 = 2 + y 2 + 2 29 E = λa2 R 3 E y = λa2 R 3 2π 2π cos θ dθ = 3 sin θ dθ = 3 E = λa 2 + a 2 3/2 2π dθ = λa 2ε 2 + a 2 3/2 32.5 a σ + d 2πσd 2πσd,, de 2π de de = a 2ε 2 + a 2 3/2 2πσd = σ a d 2πa 2ε 2 + a 2 3/2 33

c 28 6 9 E = σ a 2ε = σ 2ε [ a σ 2 + a 2 d = 3/2 2ε sgn 2 + a 2 2 + a 2 /2 ] a 34 a.6 y 6 q d 2, q d 2,, y E q d 2 e + y e y d 2 3/2 + d 2 e + y e y 2 + y 2 + d 2 3/2 35 2 + y 2 y y E -q d +q 6 d, y d 2 3/2 3/2 + y 2 = 2 d + d2 2 4 + y2 2 + y 2 d 3/2 = 2 + y 2 3/2 d 3/2 2 + y 2 2 + y 2 3/2 3d ± 2 2 + y 2 = 22 + 2y 2 ± 3d 2 2 + y 2 5/2 36

c 28 7 E q E 8πε 2 + y 2 5/2 [ 2 2 + 2y 2 + 3d d 2 e + y e y 2 2 + 2y 2 3d + d2 ] e + y e y q = 8πε 2 + y 2 5/2 4 2 d 2y 2 e + 6yd e y qd = 2 + y 2 5/2 2 2 y 2 e + 3y e y 37 = cos θ 38 y = sin θ 39 e = cos θ e sin θ e θ 4 e y = sin θ e + cos θ e θ 4 2 2 y 2 e + 3y e y = 2 2 cos θ e + 2 sin θ e θ 42 43 qd 3 2 cos θ e + sin θ e θ 44.7 a σ 7 a sinq a R q f a cosq 7 de 7 R R 2 = 2 + a 2 2a cos θ 45 E θ θ + dθ dq dq = σ 2πa sin θ adθ = 2πσa 2 sin θ dθ 46

c 28 8 dq de de = dq R 2 = σa2 sin θ dθ 2ε R 2 47 de de = de cos ϕ = de a cos θ R = σa2 a cos θ sin θ 2ε R 3 dθ 48 cos θ = t E = de = π σa 2 a cos θ sin θ 2ε R 3 dθ 49 E = σa2 at 2ε 2 + a 2 2at dt 5 3/2 4 E = σa2 2ε 2 [ a t 2 + a 2 2at ] [ = σa2 2ε 2 a ] a 5 < < a a = a E = 52 > a a = a Q = 4πa 2 σ E = σa2 ε 2 = Q 2 53 2 2. q V q 54 2.2 2 λ + d,, dv d λd 55 2 + 2

c 28 9 V λ 2 + 2 dy = λ [ 2 log y + ] 2 + 2 2 56 λ log 57 2πε 2.3 4 y σ + d,, σ = σ d σd 2πε log 58 log 2 + 2 d 2 [ log 2 + 2 2 + 2 tan ] tan 59 σ 2ε 6 2.4 5 λ θ θ + dθ λdθ,. dv V λ d λdθ 6 2 + 2 2π dθ = 2 + 2 λ 2ε 2 + 2 62 2.5 a σ + d,. dv d 2ε 2 + 2πσd = 2 2π σ d 63 2ε 2 + 2

c 28 a σ 2ε 2 + 2 d = a σ 2ε [ 2 + 2 ] a = σ [ 2ε 2 + a 2 ] 64 σ 2ε 65 2.6 6 y q y 2 + d 2 2 y 2 + + d 66 2 2 y 2 + d 2 /2 /2 = 2 d + d2 2 4 + y2 2 + y 2 d /2 = 2 + y 2 /2 d /2 2 + y 2 2 + y 2 /2 d ± 2 2 + y 2 = 2 2 + y 2 ± d 2 2 + y 2 3/2 67 V qd 2 + y 2 3/2 = qd cos θ 3 68 M M = qd e 69 M 3 7 2.7 7 θ θ θ + dθ dq dv d dq R = σa2 sin θ dθ 2ε R 7

c 28 θ=π d θ= = σa2 2ε = σa 2ε π σa 2 sin θ 2ε 2 + a 2 2a cos θ dθ [ 2 + a 2 2at σa2 dt = 2 + a 2 2at 2ε + a a a ] 72 < < a σa σa Q 2 = = 2ε ε a 73 > a σa σa2 2a = 2ε ε = Q 74 3 2 3. = y q, = 2 + y 2 + 2 75 = 2 + y 2 + 2 2 + y 2 + 2 3/2 = 3 76 = y 3, = 3, 77 gad q q e 3 + y e y + e = 2 78 3.2, y, λ log 2πε 2 + y 2 79

log 2 + 2 = log y 2 + y 2 = c 28 2 2 + 2 2 + y 2 2 + = 2 y 2 + y = y 2 8 8 log 2 + 2 = 82 gad λ e 2πε 2 + y e y 83 3.3 σ 2ε 84 gad sgn σ 2ε e 85 3.4 λ 2ε 2 + a 2 86 gad λa 2ε 2 + a 2 3/2 e 87 3.5 σ [ 2ε 2 + a 2 ] 88 gad σ 2ε sgn e 89 2 + a 2 3.6 y qd 2 + y 2 3/2 = qd cos θ 3 9

c 28 3 E = V = qd [ = qd 2 2 + y 2 2 + y 2 5/2 ] 2 + y 2 3/2 3 2 2 2 + y 2 5/2 E y = qd 3 2 2y 2 + y 2 5/2 = qd 3y 2 + y 2 5/2 9 92 qd 2 2 5 + y 2 e + 3y e y 93 gad V e V θ e θ 94 E = qd 2 cos θ qd 3 = 2 cos θ 95 3 E θ = qd sin θ 3 = qd sin θ 96 3 qd 3 2 cos θ e + sin θ e θ 97 3.7 Q a Q < < a a < 98 < < a V E = a < Q 2 e 4 4. q e + y e y + e 2 + y 2 + 2 3/2 99

c 28 4 E = q 2 + y 2 + 2 3/2 3 2 2 2 + y 2 + 2 5/2 = q 2 2 + y 2 + 2 2 + y 2 + 2 5/2 E y y = q 2 2y 2 + 2 2 + y 2 + 2 5/2 E = q 2 + y 2 2 2 2 + y 2 + 2 5/2 2, y,,, div E = 3 4.2 λ λ λ e 2πε 2 + y e y = 2πε 2 + y 2 e + y e y 4 2 = 2 + y 2 = 2 + y 2 2 2 + y 2 2 = 2 + y 2 2 + y2 5 = y = 2 = 2 y 2 2 + y2 6 div E = 7 4.3 = sgn σ 2ε e 8 div E = 9

c 28 5 4.4 3 qd 3 2 cos θ e + sin θ e θ e = sin θ cos φ e + sin θ sin φ e y + cos θ e e θ = sin θ cos φ e + sin θ sin φ e y sin θ e 2 e φ = sin φ e + cos φ e y 3 = 2 + y 2 + 2 4 sin θ = + y 2 5 cos θ = 6 y sin φ = 7 2 + y 2 cos φ = 2 + y 2 8 2 cos θ e + sin θ e θ = 2 cos θ sin θ cos φ e + sin θ sin φ e y + cos θ e + sin θ sin θ cos φ e + sin θ sin φ e y sin θ e = 3 cos θ sin θ cos φ e + 3 cos θ sin θ sin φ e y + 2 cos 2 θ sin 2 θ e = 3 2 e + 3y 2 e y + 22 2 y 2 2 e 9 qd 2 + y 2 + 2 5/2 3 e + 3y e y + 2 2 2 y 2 e 2 3 2 + y 2 + 2 = 3 4 2 + y 2 + 2 5/2 2 + y 2 + 2 7/2 = 3 2 4y 2 + 2 y y 3y 2 + y 2 + 2 5/2 2 + y 2 + 2 7/2 2 2 2 y 2 2 + y 2 + 2 = 3 3 2 + 3y 2 2 2 5/2 2 + y 2 + 2 7/2 2 22 23 24

c 28 6, y,,, div E = 25 3 div E = 2 2 E + sin θe θ + E φ sin θ θ sin θ φ 26 2 2 2 cos θ 3 + sin θ, y,,, sin θ sin θ θ 3 = 2 2 cos θ 2 + 2 sin θ cos θ sin θ 3 = 27 div E = 28 4.5 < < a Q 2 e < a 29 = a div E = 3

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