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Characteristics of Bohr's Atomic Model

A particle of...

Question

A particle of mass $m$ moves in a circular orbit in a central potential field $U (r) = \frac{1}{2} k r^{2}$ . If Bohr's quantization condition is applied, radii of possible orbitals and energy levels vary with quantum number $n$ as:

r_{n} \propto n^{2}, E_{n} \propto \frac{1}{n^{2}}

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r_{n} \propto \sqrt{n}, E_{n} \propto \frac{1}{n}

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r_{n} \propto n, E_{n} \propto n

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r_{n} \propto \sqrt{n}, E_{n} \propto n

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Solution

The correct option is D $r_{n} \propto \sqrt{n}, E_{n} \propto n$
$F = \frac{d V}{d R} = k r = \frac{m v^{2}}{r}$
$m v r = \frac{n h}{2 π}$
$r^{2} \infty n$
$r^{2} \infty \sqrt{n}$
$E = \frac{1}{2} k r^{2} + \frac{1}{2} m v^{2} \infty r^{2}$

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