A) \[{{r}_{n}}\propto {{n}^{2}},{{E}_{n}}\propto \frac{1}{{{n}^{2}}}\]
B) \[{{r}_{n}}\propto \sqrt{n},{{E}_{n}}\propto n\]
C) \[{{r}_{n}}\propto \sqrt{n},{{E}_{n}}\propto \frac{1}{n}\]
D) \[{{r}_{n}}\propto n,{{E}_{n}}\propto n\]
Correct Answer: B
Solution :
Force due to this field \[F=-\frac{\partial U}{\partial r}\] \[F=\frac{-\partial }{\partial r}\left( \frac{1}{2}k{{r}^{2}} \right)=-kr\] For circular orbit, \[\frac{m{{v}^{2}}}{r}=-kr\Rightarrow v\propto r\] ...(i) Also, by Bohr?s quantization condition\[mvr=\frac{nh}{2\pi }\] ?(ii) For eqn. (i) and (ii),\[{{r}_{n}}\propto {{n}^{1/2}}\] \[U(r)=\frac{1}{2}k{{r}^{2}}\Rightarrow {{E}_{n}}=-\frac{1}{2}U(r)=-\frac{1}{4}k{{r}^{2}}\]\[\Rightarrow {{E}_{n}}\propto n\]
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