JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Modern Physics

A particle of mass

m

moves in a circular orbit in a central potential field

U (r) = U_{0} r^{4}

. If quantization conditions are applied, radii of possible orbitals

r_{n}

vary with

n^{\frac{1}{α}}

, where

α

is ______

Answer: 3

Solution

For a central force

F = - \frac{d U}{d r} = - 4 U_{0} r^{3}

, the centripetal force for circular motion is

\frac{m v ^{2}}{r} = 4 U_{0} r^{3}

, giving

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

. Angular momentum

L = m v r

, so

v = \frac{L}{m r}

. Substituting:

m \frac{L ^{2}}{m ^{2} r ^{2}} = 4 U_{0} r^{4}

, giving

\frac{L ^{2}}{m r ^{2}} = 4 U_{0} r^{4}

, so

L^{2} = 4 m U_{0} r^{6}

. Under Bohr quantization,

L = n ℏ

, so

n^{2} ℏ^{2} = 4 m U_{0} r^{6}

, giving

r^{6} \propto n^{2}

, hence

r \propto n^{1/3}

. Therefore

α = 3

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Integer answer

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