Muon $(\mu^{-1})$ is negatively charged $(|q| = |e|)$ with a mass $m_\mu = 200\,m_e$, where $m_e$ is the mass of the electron and $e$ is the electronic charge. If $\mu^{-1}$ is bound 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 $\mu^{-1}$ in the $n$th orbit is $\dfrac{1}{200}$ times that of the electron in the $n$th orbit.
(C) The ionization energy of muonic atom is 200 times more than that of an hydrogen atom.
(D) The momentum of the muon in the $n$th orbit is 200 times more than that of the electron.
Answer: (D) (A), (C), (D)
In the Bohr model (ignoring the motion of the proton):
- $r_n = \dfrac{\varepsilon_0n^2h^2}{\pi me^2} \propto \dfrac1m$: the orbit is 200 times smaller. (A) is true.
- $v_n = \dfrac{e^2}{2\varepsilon_0nh}$ does not depend on mass: the speed is the same. (B) is false.
- $E_n = -\dfrac{me^4}{8\varepsilon_0^2n^2h^2} \propto m$: the ionisation energy is 200 times larger. (C) is true.
- $p = mv \propto m$ at the same speed: 200 times larger. (D) is true.
Correct: (A), (C), (D).
Solution by Sreeraj P, M.Sc Physics