A particle of mass $ m $ moves in a circular orbit in a central potential field $ U(r)=\frac{1}{2} k r^{2} . $ If Bohr's quantization conditions are applied, radii of possible orbitals and energy levels vary with quantum number $ n $ as:

Question

A particle of mass $ m $ moves in a circular orbit in a central potential field $ U(r)=\frac{1}{2} k r^{2} . $ If Bohr's quantization conditions are applied, radii of possible orbitals and energy levels vary with quantum number $ n $ as:, $ r_{n} \propto n^{2}, E_{n} \propto \frac{1}{n^{2}} $, $ r_{n} \propto \sqrt{n}, E_{n} \propto n $, $ r_{n} \propto n, E_{n} \propto n $, $ r_{n} \propto \sqrt{n}, E_{n} \propto \frac{1}{n} $

MARKS App · Accepted Answer

F r = - d U d r = - k r For circular motion | F r | = k r = m v 2 r ⇒ k r 2 = m v 2 ........(i) Bohr's quantization ⇒ m v r = n h 2 π ....(ii) From (i) and (ii) m 2 v 2 m = k r 2 ⇒ 1 m n h 2 π r 2 = k r 2 ⇒ n 2 h 2 4 π 2 m k = r 4 ⇒ r = h 2 4 π 2 m k 1 / 4 n 1 / 2 r ∝ n From equation (i) U ∝ n K E = 1 2 m v 2 ; P E = 1 2 k r 2 E = K + U = 1 2 m v 2 + 1 2 k r 2 = k r 2 ∝ n

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