Question

Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where 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 2 F" (r) +2rF (r) - n(n + 1) F, (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 G(1) = 1 for n = 1,3. c) Show that for n m, Gn (w)Gm (w)dw = 0. d) Assume the boundary condition is given by u(R, O) = cos 0. Then determine u(r, ) inside the ball.

Answer 1

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