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A particle of mass m is moving around the origin under the influence of a constant force F that pulls it toward the origin. If the Bohr model is applied to describe its motion, the radius r of the nth orbit and the particle’s speed v in that orbit depend on n as:

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  1. \(r \propto n^{1/3} ; \quad v \propto n^{1/3}\)
  2. \(r \propto n^{1/3} ; \quad v \propto n^{2/3}\)
  3. \(r \propto n^{2/3} ; \quad v \propto n^{1/3}\)
  4. \(r \propto n^{4/3} ; \quad v \propto n^{-1/3}\)

Answer (Detailed Solution Below)

Option 3 : \(r \propto n^{2/3} ; \quad v \propto n^{1/3}\)
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Correct option is: (3) r ∝ n2/3, v ∝ n1/3

Given, force is constant

F = mv2 / r

⇒ v2 / r = constant

⇒ r ∝ v2      ...(1)

L = mvr = nh / 2π      ...(2)

On solving equation (1) and equation (2):

v ∝ n1/3 and r ∝ n2/3

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