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PhysicsNEET 2025Rutherford and Bohr models

Atoms

A particle of mass mm is moving around the origin with a constant force FF pulling it towards the origin. If the Bohr model is used to describe its motion, the radius of the nnth orbit and the particle's speed vv in the orbit depend on nn as:

  1. A.r∝n1/3; v∝n1/3r \propto n^{1/3};\ v \propto n^{1/3}
  2. B.r∝n1/3; v∝n2/3r \propto n^{1/3};\ v \propto n^{2/3}
  3. C.r∝n2/3; v∝n1/3r \propto n^{2/3};\ v \propto n^{1/3}✓
  4. D.r∝n4/3; v∝n−1/3r \propto n^{4/3};\ v \propto n^{-1/3}

Solution

Constant force F=mv2rF = \frac{mv^2}{r} and Bohr quantisation mvr=nh2πmvr = \frac{nh}{2\pi}. From the first, v2∝rv^2 \propto r; substituting, r3/2∝n⇒r∝n2/3r^{3/2} \propto n \Rightarrow r \propto n^{2/3} and v∝r1/2∝n1/3v \propto r^{1/2} \propto n^{1/3}.

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