1223_Physics ProblemsTechnical Physics

1223_Physics ProblemsTechnical Physics - 564 P43.56...

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564 Molecules and Solids P43.56 (a) For equilibrium, dU dx = 0: d dx Ax Bx Ax Bx −− −= −+ = 31 42 30 ej x →∞ describes one equilibrium position, but the stable equilibrium position is at 3 0 2 Ax B = or x A B 0 3 = . (b) The depth of the well is given by UU A x B x AB A BB A B A xx 0 0 3 0 32 32 32 12 3 0 33 2 27 == = = = . (c) F dU dx Ax Bx x =− = 3 To find the maximum force, we determine finite x m such that dF dx Ax Bx x m = = + = 12 2 0 53 0 then FA B A B B A B A max = F H G I K J F H G I K J 3 66 1 2 2 2 . P43.57 (a) At equilibrium separation, rr e = , dU dr aB e e ar r e ee = = 21 0 00 bg . We have neutral equilibrium as r e and stable equilibrium at e e = 0 1 , or e = 0 . (b) At = 0 , U = 0. As r , UB . The depth of the well is B . (c) We expand the potential in a Taylor series about the equilibrium point: Ur Ur dU dr dU dr U r Ba ae ae e r r Ba r r af b g a f a f bg bg ≈+ + ≈++ − L N M O Q P −≈ = = = 2 2 0 2 22 0 2 2 0 2 0 0 0 0 1 2
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