Physics · Atomic Physics

JEE Advanced 2020 — Paper 1 — Question 6

A particle of mass $m$ moves in circular orbits with potential energy $V(r)=F r$ , where $F$ is a positive constant and $r$ is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by $R$ and its speed and energy are denoted by $v$ and $E$ , respectively, then for the $\mathrm{n}^{\text {th }}$ orbit (here h is the Planck's constant)

Option A:
$R \propto n^{1 / 3}$ and $v \propto n^{2 / 3}$
Option B:
$R \propto n^{2 / 3}$ and $v \propto n^{1 / 3}$
Correct
Option C:
$E=\frac{3}{2}\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}$
Correct
Option D:
$E=2\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}$

Answer: B, C

Step-by-step solution

$F=-\frac{d V}{d r}$

$\frac{m v^{2}}{R}=F$

$\mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi}$

Solving above equations

$R^{3}=\frac{n^{2} h^{2}}{4 \pi^{2} m}$

$v \propto n^{1 / 3}$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2020
Paper: Paper 1
Subject: Physics
Chapter: Atomic Physics
Topic: Rutherford's and Bohr's Model of Atom