Question

(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 where 0 is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w =cos e given by 12 F" (T) +2r F (r) - n(n + 1 ) Fr (r) = 0, (1 w2)G (w) - 2wG (w) + n(n+ 1)G, (w) 0. (n 0,1,2,. . .) b Find Fn (r) and Gn (w) satisfying Gn(1) = 1 for n = 1,3. \Show that for n m, Gn(w)Gm (w) dw 0. d) Assume the boundary condition is given by u(R, 0) =cos. Then determine u(r, 0) inside the ball.

Answer 1

Given that Laplaa equrtion fer Sadius R descvibed in The ball of spherical Condinates 2 Y Y O is the Renith angles assime v is the azimnh Sepadeg Vaiables, derde twD a) By W- cosa aNen by +2 clu + + Cot d du 0-) Taking 0R CL + Coto ST a Cat de Cd) RO Car +2 7de de + 2Y tca d

(2) Here the LHS E R-H.S (s e eua above hokds Seles bibay Coutad hy uCut) + 2y ntRO Y + 2Y 6)-Cnt)6)- + 2Y Peoicdy Cose -Sin 8 de de de Sina ニ-Sin8 dlu dw the of equ (1i) laoks like d)+Cotedetり(nキ)①-0 der de Cav d (Sine Sino do n Cn+) C . CSi9e) d (se olu Sin

(3) C ( de du dw we rewrik this eq ay -2w dat + da 2W Proved Cb) G and - pati) )-o-) + 21 Pat Y2es &Saurds us OP T 3P p Y 간사ㅇ 7SP

F Teduus to dls dt-tu Son is em mitm-nnt)= ooeto CmCmtnti)= o So-the Soldin is ,G AtE -CntS =A+B Cn+) -1 FoCn=A+rt tha - 2w (-w) G Ca)-20p Cut Guf-) Saetiatuy we

C5) t Cu-m) Cutut 1n Ca) Gfa)o (aith to w totH + + Cuu) (utnt) G) d Gn Cws Gury Ca) dw0

CdTue Conditions is iven by eR) RCos tw Satahin &R& Pat we have Couel tHeuG Syoustaly tueus we wil Vale of oValae of u i4 -e CR, 0) F, (e)G,osa) -Cn+1) he GoCcos8)- GiC cose)=Cose G2 Ccose)=(3Cos-1) G3 Cose) =(sCos'8-3 cose) Heu cetu Condition is u(R0)- Cos