Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• The IPA potential-energy function U(r) is the potential energy “felt” by an atomic electron in the average field of the other Z − 1 electrons plus the nucleus. If one knew the average charge distribution ρ(r) of the Z − 1 other electrons, it would be a fairly simple matter to find U(r). The calculation of an accurate distribution ρ(r) is very hard, but it is easy to make a fairly realistic guess. For example, one might guess that ρ(r) is spherically symmetric and given by

 ρ(r) = ρ0e−r/R

where R is some sort of mean atomic radius. (a) Given that ρ(r) is the average charge distribution of Z − 1 electrons, find ρ0 in terms of Z, e, and R. (b) Use Gauss’s law to find the electric field at a point r due to the nucleus and the charge distribution ρ. (c) Verify that as r → 0 and r → ∞, behaves as required by (10.2) and (10.3). [Hint: The integrals needed in parts (a) and (b) are in Appendix B.]