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

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Correct answer: B. r _ n ∝ n , E _ n ∝ n

F _ r = (- dU)/(dr) = - kr For circular motion | F _ r | = kr = m v ^ 2 r ⇒ k r ^ 2 = m v ^ 2 ........(i) Bohr's quantization ⇒ mvr = (nh)/(2π) ....(ii) From (i) and (ii) m ^ 2 v ^ 2 m = k r ^ 2 ⇒ (1)/(m) ((nh)/(2 )) ^ 2 = k r ^ 2 ⇒ n ^ 2 h ^ 2 4 π ^ 2 mk = r ^ 4 ⇒ r = ( h ^ 2 4 π ^ 2 mk ) ^ 1 / 4 n ^ 1 / 2 r ∝ n From equation (i) U ∝ n KE = (1)/(2)m v ^ 2 ; PE = (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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