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Q. 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)

JEE AdvancedJEE Advanced 2020

Solution:

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