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

Step-by-step solution

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