Assume a neutron and an electron bound to each other by gravitational force. Assuming Bohr's quantization rule. Find the expression for $n$ th energy level.

- \frac{1}{2} m_{e} v^{2}

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- \frac{1}{4} m_{e} v^{2}

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- \frac{1}{8} m_{e} v^{2}

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m_{e} v^{2}

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Solution

The correct option is A $- \frac{1}{2} m_{e} v^{2}$
$m_{e} v r = n h$
$m_{e} v^{2} = \frac{m_{e}^{2} v^{2}}{m_{e}} = \frac{(n h)^{2}}{m_{e} r^{2}}$
$\frac{G m_{n} m_{e}}{r^{2}} = \frac{m_{c} v^{2}}{r}$
$\Rightarrow \frac{G m_{n} m_{e}}{r^{2}} = \frac{(n h)^{2}}{m_{e} r^{3}}$
$r = \frac{n^{2} h^{2}}{G m_{n} m_{e}^{2}}$
$T E = K E + P E = \frac{1}{2} m_{e} v^{2} - \frac{G m_{n} m_{e}}{r} = - \frac{1}{2} m_{e} v^{2}$

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Assume a neutron and an electron bound to each other by gravitational force. Assuming Bohr's quantization rule. Find the expression for nth energy level.

The correct option is A −12mev2mevr=nhmev2=m2ev2me=(nh)2mer2Gmnmer2=mcv2r⇒Gmnmer2=(nh)2mer3r=n2h2Gmnm2eTE=KE+PE=12mev2−Gmnmer=−12mev2

Assume a neutron and an electron bound to each other by gravitational force. Assuming Bohr's quantization rule. Find the expression for $n$ th energy level.