A) \[{{T}_{n}}\propto \frac{1}{n},\,{{r}_{n}}\propto n\]
B) \[{{T}_{n}}\propto \frac{1}{n},\,{{r}_{n}}\propto {{n}^{2}}\]
C) \[{{T}_{n}}\propto \frac{1}{{{n}^{2}}},\,{{r}_{n}}\propto {{n}^{2}}\]
D) \[{{T}_{n}}\] independent of \[n,\,{{r}_{n}}\propto n\]
Correct Answer: D
Solution :
\[L=\frac{nh}{2\pi }\Rightarrow mv{{r}_{n}}=\frac{nh}{2\pi }\] Also, \[\frac{m{{v}^{2}}}{{{r}_{n}}}=\frac{k}{{{r}_{n}}}\Rightarrow m{{v}^{2}}=k\Rightarrow {{T}_{n}}=\frac{1}{2}m{{v}^{2}}=\frac{1}{2}k\], which is independent of n. \[{{r}_{n}}=\frac{nh}{2\pi mv}=\frac{nh}{2\pi \sqrt{km}}\] \[\therefore \,\,\,{{r}_{n}}\propto n\]
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