Verified Solution • 2025–26 Session moderate

QJEEM21Phc021: JEE Main 2021 Physics - Class 12 JEEM21Phc021 Solution: A particle of mass m moves in a circular orbit in a central…

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 n1α{n^{{1 \over \alpha }}} , where α\alpha is ____________.

Step-by-Step Verified Solution

F→=−du→dr\overrightarrow F = - {{d\overrightarrow u } \over {dr}}

=−ddr(U0r4)= - {d \over {dr}}({U_0}{r^4})

F→=−4U0r3\overrightarrow F = - 4{U_0}{r^3}

∵\because mv2r=4U0r3{{m{v^2}} \over r} = 4{U_0}{r^3}

mv2=4U0r4m{v^2} = 4{U_0}{r^4}

Then v∝r2v \propto {r^2}

∵\because mvr=nh2πmvr = {{nh} \over {2\pi }}

Then r3∝ n{r^3}\propto\,n

r ∝ (n)13r\,\propto \,{(n)^{{1 \over 3}}}

So the value of α=3\alpha = 3

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