QJEEM19Ph895a: JEE Main 2019 Physics - Class 12 JEEM19Ph895a Solution: A particle of mass m moves in a circular orbit in a central…

Question

Verified Solution • 2025–26 Session moderate

A particle of mass m moves in a circular orbit in a central potential field U(r) =

\frac{1}{2}

kr². If Bohr 's quantization conditions are applied, radii of possible orbitls and energy levels vary with quantum number n as :

Accepted Answer

Force due to this field, F =

- \frac{\partial U}{\partial r}

F =

- \frac{\partial}{\partial r} (\frac{1}{2} k r^{2})

= -kr

For circular orbit,

\frac{m v^{2}}{r}

= -kr

\Rightarrow

v

\propto

r ..... (1)

Fron Bohr’s quantization condition

mvr =

\frac{n h}{2 π}

.....(2)

From (1) and (2),

r_{n}

\propto

n^{\frac{1}{2}}

Given, U(r) =

\frac{1}{2}

kr²

\Rightarrow

E_n =

- \frac{1}{2} U (r)

=

- \frac{1}{4} k r^{2}

\Rightarrow

E_n

\propto

n

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