Mathematical Methods for Physics and Engineering - Matematica.NET

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATESwhere the coefficients R l (r) andF l (r) in the Legendre polynomial expansionsare functions of r. Since in any particular problem ρ is given, we can find thecoefficients F l (r) in the expansion in the usual way (see subsection 18.1.2). It thenonly remains to find the coefficients R l (r) in the expansion of the solution u.Writing ∇ 2 in spherical polars and substituting (21.64) and (21.65) into (21.63)we obtain∞∑[Pl (cos θ)l=0r 2ddr(r 2 dR )l+ R ldrdr 2 sin θ dθ(sin θ dP )]l(cos θ)=dθ∞∑F l (r)P l (cos θ).l=0(21.66)However, if, in equation (21.44) of our discussion of the angular part of thesolution to Laplace’s equation, we set m = 0 we conclude that(1 dsin θ dP )l(cos θ)= −l(l +1)P l (cos θ).sin θ dθ dθSubstituting this into (21.66), we find that the LHS is greatly simplified and weobtain∞∑[ ( 1 dr 2 r 2 dR )l− l(l +1)R ]∞∑ldr dr r 2 P l (cos θ) = F l (r)P l (cos θ).l=0This relation is most easily satisfied by equating terms on both sides for eachvalue of l separately, so that for l =0, 1, 2,... we have(1 dr 2 r 2 dR )l− l(l +1)R ldr dr r 2 = F l (r). (21.67)This is an ODE in which F l (r) is given, and it can therefore be solved forR l (r). The solution to Poisson’s equation, u, is then obtained by making thesuperposition (21.64).◮In a certain system, the electric charge density ρ is distributed as follows:{Ar cos θ for 0 ≤ r