NEET 2025 Physics Bohr Model MCQ Question

To understand the motion of a particle under a central force, we can utilize the principles from the Bohr model, which originally describes the behavior of electrons in atoms but can also be applied here to a particle moving in circular orbits under the influence of a central force.

Given Scenario:

A particle of mass $m$ is subject to a constant force $F$ directed towards the origin. This setup indicates a central force, which typically leads to circular motion. In the context of the Bohr model, we can derive the relationships for the radius $r$ of the $n$-th orbit and the speed $v$ of the particle in that orbit.

Key Concepts:

1. Centripetal Force: For a particle moving in a circular path, the centripetal force required to keep it in that path is given by: $F_c = \frac{m v^2}{r}$ where $v$ is the linear speed and $r$ is the radius of the orbit.

2. Central Force: In this case, the only force acting on the particle is $F$. Therefore, we can equate the centripetal force to the central force: $F = \frac{m v^2}{r}$

3. Bohr Model Dependence: According to the Bohr model, the quantization of angular momentum gives us: $mvr = n\hbar$ where $\hbar$ is the reduced Planck's constant and $n$ is the principal quantum number (which indicates the orbit number).

Deriving Relationships:

From the angular momentum quantization, we can express speed $v$ in terms of $r$:

1. Rearranging the angular momentum equation: $v = \frac{n\hbar}{mr}$

2. Substituting $v$ into the centripetal force equation: $F = \frac{m}{r} \left(\frac{n\hbar}{mr}\right)^2$ $F = \frac{n^2 \hbar^2}{mr^3}$

3. Rearranging this equation to find $r$: $r^3 = \frac{n^2 \hbar^2}{mF}$ Therefore, we have: $r \propto n^{2/3}$

4. Substituting $r$ back into the expression for $v$: $v = \frac{n\hbar}{mr} \propto \frac{n\hbar}{m n^{2/3}} = \frac{n^{1/3}\hbar}{m}$ Thus, we find: $v \propto n^{1/3}$

Summary of Results:

• The radius $r$ of the $n$-th orbit is proportional to $n^{2/3}$: $r \propto n^{2/3}$
• The speed $v$ of the particle in the orbit is proportional to $n^{1/3}$: $v \propto n^{1/3}$

Conclusion:

The correct answer to the question is (C) $r \propto n^{2/3}; \ v \propto n^{1/3}$.

Clarifying Incorrect Options:

• Option A: $r \propto n^{1/3}; v \propto n^{1/3}$ is incorrect because it does not accurately reflect the derived relationships from the force and momentum equations.

• Option B: $r \propto n^{1/3}; v \propto n^{2/3}$ is also incorrect for the same reasons as option A, the relationships do not match the established principles.

• Option D: $r \propto n^{4/3}; v \propto n^{-1/3}$ is incorrect as it contradicts the dependencies we derived from the fundamental equations.

Thus, the correct answer is (C) which accurately describes how the radius and speed of the particle depend on the quantum number $n$ in this central force scenario.