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NEET 2025
Exam
NEET
Year
2025
Subject
Physics
Chapter
Atomic Physics

NEET 2025 · Physics · Atomic Physics

A particle of mass m is moving around the origin with a constant...

A particle of mass mm is moving around the origin with a constant force FF pulling it towards the origin. If Bohr model is used to describe its motion, the radius rr of the nth n^{\text{th }} orbit and the particle's speed vv in the orbit depend on nn as

Options

  1. Ar∝n1/3;v∝n1/3r \propto n^{1 / 3} ; v \propto n^{1 / 3}
  2. Br∝n1/3;v∝n2/3r \propto n^{1 / 3} ; v \propto n^{2 / 3}
  3. Cr∝n2/3;v∝n1/3r \propto n^{2 / 3} ; v \propto n^{1 / 3}
  4. Dr∝n4/3;v∝n−1/3r \propto n^{4 / 3} ; v \propto n^{-1 / 3}

Correct answer

C. r∝n2/3;v∝n1/3r \propto n^{2 / 3} ; v \propto n^{1 / 3}

Step-by-step solution

mv2r=F....(1)mvr=nh2π⇒v=nh2πmr....(2)\begin{aligned} & \frac{m v^2}{\mathrm{r}}=\mathrm{F}....(1) \\ & \mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \Rightarrow v=\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}....(2) \end{aligned} From equation (1) mr(nh2πmr)2=F⇒n2 h24π2mF=r3⇒r3∝n2⇒r∝n2/3\begin{aligned} & \frac{\mathrm{m}}{\mathrm{r}}\left(\frac{\mathrm{nh}}{2 \pi \mathrm{mr}}\right)^2=\mathrm{F} \\ & \Rightarrow \frac{\mathrm{n}^2 \mathrm{~h}^2}{4 \pi^2 \mathrm{mF}}=\mathrm{r}^3 \Rightarrow \mathrm{r}^3 \propto \mathrm{n}^2 \\ & \Rightarrow \mathrm{r} \propto \mathrm{n}^{2 / 3} \end{aligned} From (2) v∝nr∝nn2/3⇒v∝n1/3v \propto \frac{n}{r} \propto \frac{n}{n^{2 / 3}} \Rightarrow v \propto n^{1 / 3}
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