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, $m_{\mu }=200m_e$ 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
$v=\frac{nh}{2\pi mr}=\frac{nh\times 4\pi m{e}^2\times Z}{2\pi m\times h^2\times n^2}=\frac{2e^{2}Z}{hn}$
Therefore, speed does not change as it does not depend on mass. So, statement $B$ is incorrect.
(iii) Ionisation energy is given by
$\Delta E_n=-\frac{2{\pi }^{2}m{e}^4{Z}^2}{h^2{\left(4\pi {\epsilon }_0\right)}^2}\times \left(\frac{1}{n_2^2}-\frac{1}{n_1^2}\right)$
Only, $m_{\mu }=200m_e$ 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.