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

Answer 1

Answer: (B), (C)

$ F =-\frac{ dV }{ dr } $
$\frac{ mv ^{2}}{ R }= F $
$ mvr =\frac{ nh }{2 \pi}$
Solving above equations
$ R ^{3}=\frac{ n ^{2} h ^{2}}{4 \pi^{2} m } $
$v \propto n ^{1 / 3}$