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# hwsol11 - Physics 31 Spring 2008 Solution to HW#11 The part...

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Physics 31 Spring, 2008 Solution to HW #11 Problem C Evaluate the following integrals. You should write the cartesian coordinates x or z in terms of spheri- cal polar coordinates. Remember that the volume element will be r 2 dr sin θdθ dφ , and the integrals are over all space. Rearrange terms so that each integral is a product of an r integral, a θ integral, and a φ integral. It usually pays to evaluate the θ and φ integrals first, because if one of them is zero, you have the answer without bothering with the r integral. For all of these integrals, . . . dV = 0 r 2 dr π 0 sin θ dθ 2 π 0 dφ . . . , z = r cos θ , and x = r sin θ cos φ . (a) ψ 100 100 dV = 1 πa 3 0 e 2 r/a 0 r cos θ dV For this one, the θ integral is zero: π 0 cos θ sin θ dθ = 1 2 cos 2 θ π 0 = 0 . This integral can also be evalutated by the substitution u = cos θ , which is always a good general approach. (b) ψ 100 210 dV = 1 4 π 2 a 4 0 e r/a 0 ( r cos θ ) re 2 r/a 0 cos θ dV.

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