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  • If potential energy between a proton and an electron is given by <img alt="U=\frac{-k e^2}{2 R^3}" src="https://entrancecorner.oncodecogs.com/gif.latex?U%3D%5Cfrac%7B-k%20e%5E2%7D%7B2%20R%5E3%7D" /

If potential energy between a proton and an electron is given by U=\frac{-k e^2}{2 R^3} where e is the charge of electron and R is the radius of atom, the radius of Bohr's orbit is given by (h= planckes constant, k is a constants ).

Option: 1

\frac{k e^2 m}{h^2}


Option: 2

\frac{6 \pi^2}{n^2} \cdot \frac{k e^2 m}{h^2} .


Option: 3

\frac{2 \pi}{n} \frac{k e^2 m}{h^2}


Option: 4

\frac{4 \pi^2 k e^2 m}{n^2 h^2}


Answers (1)

best_answer

U=\frac{-k e^2}{2 R^3} \Rightarrow F=-\frac{d u}{d R}=\frac{-3 k e^2}{2 R^4} .

\text { Since } F=\frac{m v^2}{R} \Rightarrow \frac{m v^2}{R}=\frac{3 K e^2}{2 R^4} \text {. --------(1) }

\text { Also } m v R=\frac{n h}{2 \pi} \Rightarrow m v=\frac{n h}{2 \pi R}

from eqn (1) 

\frac{(m v)^2}{m R}=\frac{n^2 h^2}{4 \pi^2 R^2m R}=\frac{3 K e^2}{2 R^4} .

\Rightarrow R=\frac{6 \pi^2 k e^2 m}{n^2 h^2}

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