A) \[{{r}_{n}}\propto n\]
B) \[{{r}_{n}}\propto 1/n\]
C) \[{{r}_{n}}\propto {{n}^{2}}\]
D) \[{{r}_{n}}\propto 1/{{n}^{2}}\]
Correct Answer: A
Solution :
[a] Potential energy \[U=eV=e{{V}_{0}}In\frac{r}{{{r}_{0}}}\] \[\therefore Force\,\,F=-\left| \frac{dU}{dr} \right|=\frac{e{{V}_{0}}}{r}\] \[\therefore \] The force will provide the necessary centripetal Force. Hence \[\frac{m{{v}^{2}}}{r}=\frac{e{{V}_{0}}}{r}\Rightarrow v=\sqrt{\frac{e{{V}_{0}}}{m}}\] ?(i) And \[mvr=\frac{nh}{2\pi }\] ...(ii) From equations (i) and (ii), \[mr=\left( \frac{nh}{2\pi } \right)\sqrt{\frac{m}{e{{V}_{0}}}}\] Or \[r\propto n\]
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