1. Tardigrade
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  3. Physics
  4. 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 1 / α, where α is .

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Q. 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 $n ^{1 / \alpha}$, where $\alpha$ is ________.

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Solution:

$F =\frac{- dU }{ dr }=-4 U _{0} r ^{3}=\frac{ mv ^{2}}{ r }$
$mv ^{2}=4 U _{0} r ^{4}$
$v \propto r ^{2}$
$mvr =\frac{ nh }{2 \pi}$
$r ^{3} \propto n$
$r \propto n 1 / 3$
$=3$