A particle of mass m moves in a circular orbit in a central potential… | JEE Main 2021 (17 Mar, Shift 2)

Q93JEE Main 2021 · 17 MAR · SHIFT 2Difficult

Bohr Model: Orbits, Speed & Angular Momentum

A particle of mass m moves in a circular orbit in a

central potential field U(r) = U0r4. If Bohr's quantization conditions

are applied, radii of possible

orbitals rn vary with $n^{\alpha}$ , where $\alpha$ is ____________.

Numerical answer type. Work it out, then open the answer.

Show answer and solution

Answer: 3

$F = \frac{dU}{dr} = 4U_0 r^3 = \frac{mv^2}{r} \Rightarrow v = 2r^2\sqrt{U_0/m}$ , so $v \propto r^2$ .
Bohr condition: $mvr = \frac{nh}{2\pi}$ with $mvr \propto r^3$ , so $r^3 \propto n$ .
$r_n \propto n^{1/3}$ , which is the original question's form $r_n \propto n^{1/\alpha}$ .
So $\alpha = 3$ .

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