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An imaginary particle has a charge equal to that of an electron and mass 100 times the mass of the electron. It moves in a circular orbit around a nucleus of charge $+ 4 e$ . Take the mass of the nucleus to be infinite. Assuming that the Bohr's model is applicable to this system. The expression for the radius of $n$ th Bohr orbit is

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$r_{n} = \frac{n^{2} h^{2} ε_{0}}{800 π m e^{2}}$

$r_{n} = \frac{n^{2} h^{2} ε_{0}}{100 π m e^{2}}$

$r_{n} = \frac{n^{2} h^{2} ε_{0}}{400 π m e^{2}}$

$r_{n} = \frac{n^{2} h^{2} ε_{0}}{40 π m e^{2}}$

answer is A.

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Detailed Solution

We have $\frac{m_{p} v^{2}}{r_{n}} = \frac{1}{4 π ε_{0}} \frac{z e^{2}}{r_{n}^{2}}$ ……..(1)

The quantization of angular momentum gives,

$m_{p} v r_{n} = \frac{n h}{2 π}$ ……….(2)

Solving equations (1) and (2), we get

$r = \frac{n^{2} h^{2} ε_{0}}{2 π m_{p} e^{2}}$

Substituting $m_{p} = 100 m$

where $m =$ mass of electron and $z = 4$

we get, $r_{n} = \frac{n^{2} h^{2} ε_{0}}{400 π m e^{2}}$

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