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(A) $ n $

(B) $ \sqrt n $

(C) $ {n^{\dfrac{1}{3}}} $

(D) $ {n^2} $

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Given, The potential energy of the small particle, $ U = \dfrac{1}{2}m{\omega ^2}{r^2}{\text{ }} \to {\text{1}} $

Where, U is the potential energy of the small particle

$ \omega $ is a constant

$ r $ is the distance of the particle from the origin

$ m $ is the mass of the small particle

Then the kinetic energy of the small particle, $ K.E = \dfrac{1}{2}m{v^2}{\text{ }} \to {\text{2}} $

Where,

K.E is the kinetic energy of the small particle

$ m $ is the mass of the small particle

V is the velocity of the particle with which it moves

There is a hint given in the question itself to use the equations of Bohr's model of quantisation of angular momentum and circular orbits

The angular momentum of a particle in nth orbit is,

$ L = mvr{\text{ }} \to 3 $

L is the angular momentum of a particle

$ m $ is the mass of the small particle

V is the velocity of the particle with which it moves

$ r $ is the distance of the particle from the origin

By Bohr’s first postulate, the angular momentum of the electron

$ L = \dfrac{{nh}}{{2\pi }}{\text{ }} \to 4 $

L is the angular momentum of a particle

n is the orbit in which it revolves

h is the Planck constant

Equating 3 and 4 we get

$ mvr = \dfrac{{nh}}{{2\pi }} $

$ mv = \dfrac{{nh}}{{2\pi r}}{\text{ }} \to 5 $

Substitute equation 5 in equation 2

$ K.E = \dfrac{1}{2}{\left( {\dfrac{{nh}}{{2\pi r}}} \right)^2} $

$ K.E = \dfrac{1}{4}\dfrac{{{n^2}{h^2}}}{{{\pi ^2}{r^2}}}{\text{ }} \to {\text{6}} $

We know that,

$ Kinetic{\text{ }}energy{\text{ }} = \dfrac{1}{2}{}potential{\text{ }}energy $

Then, from equation 1 and equation 6

$ \dfrac{1}{4}\dfrac{{{n^2}{h^2}}}{{{\pi ^2}{r^2}}} = \dfrac{1}{2}\left( {\dfrac{1}{2}m{\omega ^2}{r^2}} \right) $

$ \dfrac{{{n^2}{h^2}}}{{{\pi ^2}{r^2}}} = m{\omega ^2}{r^2} $

$ \dfrac{{{n^2}{h^2}}}{{{\pi ^2}{r^2}m{\omega ^2}}} = {r^2} $

$ \dfrac{{{n^2}{h^2}}}{{{\pi ^2}m{\omega ^2}}} = {r^2} \times {r^2} $

$ {r^4} = \dfrac{{{n^2}{h^2}}}{{{\pi ^2}m{\omega ^2}}} $

From above equation we get

$ {r^4} \propto {n^2} $

$ r \propto \sqrt n $

The radius, $ r $ of orbit is proportional to $ \sqrt n $ (square root of n)

It is an indirect question since we have to find the relation between the radius and the nth orbit, we are using the equations having n (nth orbit) and r (radius) to relate them. It is given in the question to Assume Bohr's model of quantisation of angular momentum and circular orbits.

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