Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSzav = v 0θrφyxv =0−aFigure 21.7 A hollow split conducting sphere with its top half charged to apotential v 0 and its bottom half at zero potential.where in the last line we have used (21.50). The integrals of the Legendre polynomials areeasily evaluated (see exercise 17.3) and we findA 0 = v 02 , A 1 = 3v 04a , A 2 =0, A 3 = − 7v 016a 3 , ··· ,so that the required solution inside the sphere isv(r, θ, φ) = v 02[1+ 3r2a P 1(cos θ) − 7r38a 3 P 3(cos θ)+···Outside the sphere (for r>a) we require the solution to be bounded as r tends toinfinity and so in (21.51) we must have A l =0foralll. In this case, by imposing theboundary condition at r = a we require∞∑v(a, θ, φ) = B l a −(l+1) P l (cos θ),l=0where v(a, θ, φ) is given by (21.50). Following the above argument the coefficients in theexpansion are given byB l a −(l+1) =2l +12so that the required solution outside the sphere isv(r, θ, φ) = v 0a2r∫ 1v 0 P l (µ)dµ,[1+ 3a2r P 1(cos θ) − 7a38r 3 P 3(cos θ)+···0].]. ◭736