A) \[{{n}_{1}}=3{{n}_{2}}\]
B) \[{{n}_{1}}=4{{n}_{2}}\]
C) \[{{n}_{1}}=2{{n}_{2}}\]
D) \[{{n}_{1}}=5{{n}_{2}}\]
Correct Answer: C
Solution :
[c] In the nth orbit, let \[{{r}_{n}}\] be the radius and \[{{v}_{n}}\] be the speed of electron. Time period, \[{{T}_{n}}=\frac{2\pi {{r}_{n}}}{{{v}_{n}}}\propto \frac{{{r}_{n}}}{{{v}_{n}}}\] Now \[{{r}_{n}}\propto {{n}^{2}};{{v}_{n}}\propto \frac{1}{n}\therefore \frac{{{r}_{n}}}{{{v}_{n}}}\propto {{n}^{3}}or{{T}_{n}}\propto {{n}^{3}}\] Here \[8={{\left( \frac{n{{ & }_{1}}}{{{n}_{2}}} \right)}^{3}}\text{or}\frac{{{n}_{1}}}{n{{ & }_{2}}}=2\text{ i}\text{.e}\text{.,}{{n}_{1}}=2{{n}_{2}}\]
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