Physics · Atomic Physics

NEET (UG) 2025 — Question 28

A particle of mass $m$ is moving around the origin with a constant force $F$ pulling it towards the origin. If Bohr model is used to describe its motion, the radius $r$ of the $n^{\text {th }}$ orbit and the particle's speed $v$ in the orbit depend on $n$ as

Option A:
$r \propto n^{1 / 3} ; v \propto n^{1 / 3}$
Option B:
$r \propto n^{1 / 3} ; v \propto n^{2 / 3}$
Option C:
$r \propto n^{2 / 3} ; v \propto n^{1 / 3}$
Correct
Option D:
$r \propto n^{4 / 3} ; v \propto n^{-1 / 3}$

Answer: C

Step-by-step solution

$\frac{m v^{2}}{r}=F …(1)$

\mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \Rightarrow v=\frac{\mathrm{nh}}{2 \pi \mathrm{mr}} …(2)

From equation (1)

$\frac{\mathrm{m}}{\mathrm{r}}\left(\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}\right)^{2}=\mathrm{F}$

$\Rightarrow \frac{n^{2} h^{2}}{4 \pi^{2} m F}=r^{3} \Rightarrow r^{3} \propto n^{2}$

$\Rightarrow \mathrm{r} \propto \mathrm{n}^{2 / 3}$

From (2)

$v \propto \frac{\mathrm{n}}{\mathrm{r}} \propto \frac{\mathrm{n}}{\mathrm{n}^{2 / 3}} \Rightarrow v \propto \mathrm{n}^{1 / 3}$

Answer key and solution verified before publishing.

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Exam: NEET (UG) 2025
Paper: 2025 paper
Subject: Physics
Chapter: Atomic Physics
Topic: Rutherford's and Bohr's Model of Atom

Answer: C

Step-by-step solution

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