A) \[\frac{A}{B}\]
B) \[\frac{B}{A}\]
C) \[\frac{2\,A}{B}\]
D) \[\frac{B}{2\,A}\]
Correct Answer: C
Solution :
As, \[U=\frac{A}{{{x}^{2}}}-\frac{B}{x}\] For equilibrium, \[\frac{dU}{dx}=0\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{d}{dx}\left( \frac{A}{{{x}^{2}}}-\frac{B}{x} \right)\,\,=\,\,0\] \[-\frac{2A}{{{x}^{3}}}+\frac{B}{{{x}^{2}}}=0\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=\frac{2A}{B}\] For stable equilibrium, \[\frac{{{d}^{2}}U}{d{{x}^{2}}}>0\] \[\frac{{{d}^{2}}U}{d{{x}^{2}}}=\frac{6\,A}{{{x}^{4}}}-\frac{2\,B}{{{x}^{3}}}\] \[{{\frac{{{d}^{2}}U}{d{{x}^{2}}}}_{x=\frac{2A}{B}}}=\frac{{{B}^{4}}}{8{{A}^{3}}}>0\]
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