A) \[{{n}^{2}}\]
B) \[\frac{1}{{{n}^{2}}}\]
C) \[{{n}^{3}}\]
D) \[\frac{1}{n}\]
Correct Answer: C
Solution :
Time required for electron to go around once is \[T=\frac{2\,\pi {{r}_{n}}}{{{v}_{n}}}\] ?. (i) where \[{{r}_{n}}\]is radius of nth orbit and \[{{v}_{n}}\] is velocity. Multiply and divide Eq. (i) by mv^, we get \[T=\frac{2\,\pi \,m\,{{r}_{n}}{{v}_{n}}}{m\,v_{n}^{2}}=\pi \frac{nh}{\frac{1}{2}mv_{n}^{2}}=\pi \frac{nh}{{{K}_{n}}}\] where \[{{K}_{n}}\] is kinetic energy in nth orbit, \[T=-\pi \frac{nh}{E}\] \[T=\frac{h}{2}n\frac{{{n}^{2}}}{13.6\,\,eV}\] \[\left( \because \,h=\frac{h}{2\,\pi }\,and\,E=-\frac{13.6}{{{n}^{2}}} \right)\] \[\Rightarrow \] \[T\propto {{n}^{3}}\].
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