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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 conditions are applied, radii of possible orbitals and energy levels vary with quantum number n as

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

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

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

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

answer is D.

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Detailed Solution

$\begin{aligned} F = \frac{- d v}{d x} = k r \\ \frac{m v^{2}}{r} = k r \to (1) \end{aligned}$

$\begin{array}{l} L = m v r = \frac{n h}{2 π} \\ From (1) \Rightarrow \frac{m {(\frac{n h}{2 π m r})}^{2}}{r} = k r \\ r \propto \sqrt{n} \\ Energy, E = U + K \propto n \end{array}$

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