Question

Muon $ (\mu^{-})$ is a negatively charged $(|q | = |e|$ particle with a mass $m_{\mu} = 200$ me, where me is the mass of the electron and e is the electronic charge. If $\mu$- 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 $\mu^{-}$ 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 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.

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

Answer 1

Answer: (B)

(i) Radius of the orbit is given by $r=\frac{h^{2}}{4 \,\pi\, m e^{2}} \times \frac{n^{2}}{Z}$.
Only, $m_{\mu}=200 \,m_{e}$ rest are same. So, statement $A$ is correct:
The radius of muonic orbit is 200 times smaller than that of the electron.
(ii) Velocity of particle in an orbit is given by
$v=\frac{n h}{2 \,\pi \,m r}=\frac{n h \times 4 \,\pi \,m e^{2} \times Z}{2 \pi m \times h^{2} \times n^{2}}=\frac{2 e^{2} Z}{h n}$
Therefore, speed does not change as it does not depend on mass. So, statement $B$ is incorrect.
(iii) Ionisation energy is given by
$\Delta E_{n}=-\frac{2 \,\pi^{2}\, m e^{4} Z^{2}}{h^{2}\left(4 \pi \varepsilon_{0}\right)^{2}} \times\left(\frac{1}{n_{2}^{2}}-\frac{1}{n_{1}^{2}}\right)$
Only, $m_{\mu}=200 m_{e}$ rest are same. So, statement $C$ is correct:
The ionisation energy of muonic atom is $200$ times more than that of an hydrogen atom.
(iv) Since Momentum $\propto$ Energy. So, statement $D$ is correct:
The momentum of the muon in the $n$ th orbit is $200$ times more than that of the electron.