Q85 / 90 · JEE Main 2021 (17 Mar Shift 2) · Physics

A particle of mass $m$ moves in a circular orbit in a central potential field $U (r) = U_{0} r^{4} .$ If Bohr's quantization conditions are applied, radii of possible orbitals $r_{n}$ vary with $n^{\frac{1}{α}}$ , where $α$ is _______ .

Correct answer: 3

Solution

$F = \frac{- d U}{d r} = - 4 U_{0} r^{3} = \frac{m v^{2}}{r}$

$m v^{2} = 4 U_{0} r^{4}$

$V \propto r^{2}$

$r^{3} \propto n$

$\Rightarrow r \propto n^{1 / 3}$ $= 3$

Hence, the correct answer is 3.