A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to…

Question

A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius of the n^th orbit and the particle's speed v in the orbit depend on n as:

1. A. r n^1/3;\ v n^1/3
2. B. r n^1/3;\ v n^2/3
3. C. r n^2/3;\ v n^1/3 (Correct answer)
4. D. r n^4/3;\ v n^-1/3

Correct Answer

Option C — r n^2/3;\ v n^1/3

Detailed Solution & Explanation

The correct answer is $r\propto n^{2/3}$ and $v\propto n^{1/3}$.

Key Points

• The constant force provides the centripetal force: $F=\dfrac{mv^2}{r}\Rightarrow v^2=\dfrac{Fr}{m}$.
• Bohr quantization: $mvr=\dfrac{nh}{2\pi}\Rightarrow v=\dfrac{nh}{2\pi mr}$.
• Combining: $\dfrac{Fr}{m}=\dfrac{n^2h^2}{4\pi^2m^2r^2}\Rightarrow r^3\propto n^2\Rightarrow r\propto n^{2/3}$.
• Then $v=\dfrac{nh}{2\pi mr}\propto\dfrac{n}{n^{2/3}}=n^{1/3}$.