JEE Physics 2023 Question with Solution

Given: The potential energy is $U = \frac{1}{2}m\omega^2 r^2$ and the motion is in a circular orbit.

Find: How the radius of the $n$th orbit depends on $n$.

For circular motion in the potential $U = \frac{1}{2}m\omega^2 r^2$, the force is obtained from the potential:

So the required centripetal force is

This gives

Using Bohr’s postulate for angular momentum,

Substitute $v = \omega r$:

Hence,

Therefore,

Therefore, the radius of the $n$th orbit is proportional to $\sqrt{n}$. The correct option is D.

Given: $U = \frac{1}{2}m\omega^2 r^2$ and Bohr quantization applies.

Find: The dependence of $r$ on $n$.

From the harmonic oscillator type potential, circular motion implies $v = \omega r$. Then angular momentum becomes

Bohr quantization gives

So directly,

which gives