.3.2 3.3.2 Spherical Coorinates.5: Laplace 2 V = r 2 r 2 x = r cos φ sin θ, y = r sin φ sin θ, z = r cos θ.93 r 2 sin θ sin θ θ θ r 2 sin 2 θ 2 V =.94 2.94 z V φ Laplace r 2 r 2 r 2 sin θ.96.95 V r 2 R R r r Θsinθ R r sin θ =.95 θ θ V r, θ =RrΘθ.96 θ r 2 R = r Θsinθ sin θ Θ =.97 θ sin θ Θ θ θ.98 r θ ll r 2 R = ll, R r sin θ Θ = ll.99 Θsinθ θ θ R R r r 2 R = ll. r 9
R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ θ θ l θ 2π Legenre Θθ =P l cos θ.5 Legenre φ 2 Legenre Legenre Roorigues Legenre P l x = l 2 l x 2 l.6 l! x P x = P x =x P 2 x =3x 2 /2 P 3 x =5x 3 3x/2 P 4 x = 35x 4 3x 2 3/8 P 5 x = 63x 5 7x 3 5x/8.7 P l x x l l l.6 /2 l l! P l = Legenre P l x x P l xp l xx = Laplace V r, θ = Ar l P l cos θp l cos θsinθθ = 2 2l δ ll.8 B r l P l cos θ.9.9 V r, θ = l= A l r l B l r l P l cos θ. A l Legenre.8 2
Example 3.6 R V θ r =. l B l = V r, θ = V R, θ = A l r l P l cos θ. l= A l R l P l cos θ =V θ.2 l= Legenre A l V θ A l.2 P l cos θsinθ θ A l R l θ sin θp l cos θp l cos θ = l= Legenre.8 A l R l 2Rl 2l = A l = 2l 2R l. θ sin θp l cos θv θ.3 V θp l cos θsinθθ.4 V θp l cos θsinθθ.5 V θ =k sin 2 θ/2.6 sin 2 θ/2 = cos θ/2 P x =,P x =x.6 V θ = k 2 cos θ =k 2 [P cos θ P cos θ].7 A l.5 Legenre.8 A = k 2, A = k 2R, A l =l,.8. V r, θ = k [ P cos θ ] 2 R P cos θ = k R 2 cos θ.9 Example 3.7 2 R V θ r V = l A l = B l V r, θ = r l P lcos θ.2 l= 2
V R, θ =.2 P l cos θsinθ θ B l = 2l 2 l=.2 B l R l P lcos θ =V θ.2 R l V θp l cos θsinθθ.22 Example 3.8 E = E ẑ R.6 z z.6: V R, θ =V E = E ẑ V E z C.23 C xy V z = C = 3 i V = V when r = R ii V E r cos θ for r R iii.24 iii i-iii. A l,b l 3 Griffiths V = xy xy C = V = V 22
ii r B l A l r l P l cos θ = E r cos θ.25 l= P x =x.25 A l r l P l cos θ = E rp cos θ.26 l= P l cos θsinθ θ l = l i V R, θ = A = E, A l =l.27 l= Legenre B A l R l B l R l P l cos θ =V.28 R = V, A l R l B l =l.29 Rl.27 A =.27 B = RV, B = A R 3 = E R 3, B l =l,.3.27.3. V r, θ = RV r E r R3 r 2 cos θ.3 iii V σθ = ɛ RV = ɛ r=r r 2 ɛ E 2 R3 r 3 cos θ V = ɛ r=r R 3ɛ E cos θ.32 a =2πR 2 π sin θθ.32 Q = aσ =4πRɛ V.33 Q = V = V r, θ = E r R3 r 2 cos θ.34 σθ ==3ɛ E cos θ.35 θ π/2 π/2 θ π.34 V = V ex V sphere V ex = E r cos θ, V sphere = E R 3 cos θ.36 r2 V ex V sphere 23
Q.3 V r, θ = Q 4πɛ r E r R3 r 2 cos θ.37 Q Laplacian.93 Laplacian 2 V = r 2 r 2 r 2 sin θ graient sin θ θ θ 2 V r 2 sin 2 θ 2.38 V = ˆr r θ ˆθ r sin θ ˆφ.39 ˆr, ˆθ, ˆφ r, θ, φ ˆr = ˆθ = θ θ = cos φ sin θˆx sinφsin θŷ cos θẑ.4 = cos φ cos θˆx sinφcos θŷ sin θẑ.4 ˆφ =.39 / = sin φˆx cos φŷ.42 = x x y y z = V z.43.4 / = = V ˆr = V ˆr.44 θ = x x θ y y θ z = V z θ θ.45.4 /θ = r θ = V ˆθ θ = r V ˆθ.46 = x x y y z z = V.47.42 / = r sin θ = V ˆφ = r sin θ V ˆφ.48 ˆr, ˆθ, ˆφ V = V ˆrˆr V ˆφˆφ V ˆθˆθ.49 24
.44,.46,.48.39 Laplace = ˆr ˆθ r θ ˆφ r sin θ.5 2 V = V.5.5 2 V = ˆr ˆθ r = 2 V 2 r θ θ ˆφ r sin θ 2 V 2 2 V r 2 θ 2 r 2 sin 2 θ r ˆθ ˆθ θ r sin θ ˆφ ˆθ ˆr r θ ˆθ r sin θ ˆφ r ˆθ ˆr θ r sin θ ˆφ ˆr r sin θ r ˆθ ˆφ θ r sin θ ˆφ ˆφ.52 ˆr, ˆθ, ˆφ r.4-.42.52 = 2 V 2 = r 2.38 ˆθ ˆr θ =, ˆθ ˆθ θ =, ˆθ ˆφ θ =, ˆr ˆφ =sinθ.53 ˆθ ˆφ = cos θ.54 ˆφ ˆφ 2 V r 2 θ 2 2 V r 2 sin 2 θ 2 2 r r 2 r 2 sin θ sin θ θ θ =.55 cos θ r 2 sin θ θ 2 V r 2 sin 2 θ 2.56 Leengre V V r, θ, φ =Y θ, φrr.57 Y θ, φ [ sin θ 2 ] V sin θ θ θ sin 2 θ 2 Y θ, φ = ll Y θ, φ.58 Y θ, φ =ΘθΦφ.59 2 Φ Φ φ 2 = Θ sin θ sin θ Θ ll sin 2 θ.6 θ θ 25
φ θ m 2 Φ Θ Φ sin θ sin θ Θ θ θ Φ φ m 2 Φ φ 2 = m2 Φφ.6 ] [ll m2 Θ=.62 sin 2 θ Φφ =e ±imφ.63 Φφ 2pi =Φφ e imφ2π = e imφ.64 Θ θ z = cos θ m =, ±, ±2,.65 Θθ =P cos θ =P z.66 z = sin θθ P [ z 2 P ] ] [ll m2 z z z 2 P =.67 Legenre z m = P z [ z 2 P ] ll P =.68 z z Legenre Legenre.68 P z =z α j= a j z j.69 { α jα j aj z αj 2 [α jα j ll ] a j z αj} =.7 j= z z z z α 2,z α αα a =.7 α αa =.72 a α = α =.73 a α = α =.74 26
a j2 = [ ] α jα j ll a j.75 α j α j 2.73.74 a a a,a = α α = α =.76.75 α = α = α =,α= il z 2 < ii z = ± l =,α= a = P z =z 3 z3 5 z5.77 Q z = 2 ln z z.78 z = ± l z = ± z.75 j = j max α j max α j max = ll.79 j j max a j = α j max l.79 α = j max = l α = j max = l P z z l Legenre P l z Legenre P z = P z =z P 2 z =3z 2 /2 P 3 z =5z 3 3z/2 P 4 z = 35z 4 3z 2 3/8 P 5 z = 63z 5 7z 3 5z/8.8 P l z z l l l P l = Legenre Roorigues.8 /2 l l! P l = Legenre P l z z P l zp l zx = P l z = l 2 l z 2 l.8 l! z P l cos θp l cos θsinθθ = 2 2l δ ll.82 27
m Legenre.67 Legenre m = n l, m l 2m l, l,,,,l,l Legenre Pl m z m > Pl m z = m z 2 m/2 m z m P lz = m 2 l l! z 2 m/2 lm z lm z2 l.83.67 m 2 m Pl m z = P m l z m Legenre Pl m zpl m zz = 2 2l l m! l m! δ ll.84 Y lm θ, φ = m m /2 2l l m!4πl m!p m l cos θe imφ.85 l =,, 2,, m = l, l,,l,l.86 Y ml sin θθ 2π l =,, 2 l = : Y = 4π φy l m θ, φy lmθ, φ =δ ll δ mm.87 3 3 l = : Y ± = 8π sin θe±iφ, Y = 4π cos θ 5 l =2 : Y 2±2 = 32π sin2 θe ±2iφ, Y 2± = 5 Y 2 = 6π 3 cos2 θ 5 sin θ cos θe±iφ 8π.88 28