Question

A particle of mass $m$ moves in circular orbits with potential energy $V(r)= Fr$, 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 $n ^{\text {th }}$ orbit (here $h$ is the Planck's constant)

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

Answer 1

Answer: (B), (C)

$R \propto n ^{\frac{2 }{ 3}}$ and $v \propto n ^{\frac{1 }{ 3}}$

Answer 2

The Correct Option is (B) and(C):

$R \propto n ^{\frac{2 }{ 3}}$ and $v \propto n ^{\frac{1 }{ 3}}$
$E=\frac{3}{2}\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{\frac{1 }{ 3}}$