Question

Muon (\(\mu^-\)) is a negatively charged (\(|q|=|e|\)) particle with a mass \(m_\mu=200\,m_e\), where \(m_e\) is the mass of the electron and e is the electronic charge. If \(\mu^-\) is bond to a proton to form a hydrogen like atom, identify the correct statements.
(A) Radius of the muonic orbit is 200 times smaller than that of the electron.
(B) The speed of the \(\mu^-\) in the \(n^{th}\) orbit is \(\frac{1}{200}\) times that of the electron in the \(n^{th}\) orbit.
(C) The ionization energy of muonic atom is 200 times more than of an hydrogen atom.
(D) The momentum of the muon in the \(n^{th}\) orbit is 200 times more than that of the electron.

• A. (A), (B), (D)
• B. (A), (C), (D)
• C. (B), (D)
• D. (C), (D)

Hint:

Use Bohr quantization with mass m: \(r_n\propto\frac{1}{m}\), \(v_n\) independent of m, energy \(\propto m\), momentum \(mv\propto m\).

Answer 1

The Correct Option is B

Bohr model (proton treated as fixed): \(m v r=\frac{nh}{2\pi}\) and \(\frac{mv^2}{r}=\frac{ke^2}{r^2}\).
Radius: \(r_n=\frac{n^2h^2}{4\pi^2 k e^2 m}\propto\frac{1}{m}\), so the muonic radius is 200 times smaller. (A) is true.
Speed: \(v_n=\frac{2\pi ke^2}{nh}\) does not depend on mass, so the speeds are equal, not 1/200. (B) is false.
Ionization energy: \(E=\frac{2\pi^2k^2e^4m}{n^2h^2}\propto m\), so it is 200 times that of hydrogen. (C) is true.
Momentum: \(p=mv\) with the same v gives 200 times the electron momentum. (D) is true.
Correct statements: (A), (C), (D).
Answer: (2) (A), (C), (D)