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Muon $\left(\mu^{-1}\right)$ is negatively charged $(|\mathrm{q}|=|\mathrm{e}|)$ with a mass $\mathrm{m}_\mu=200 \mathrm{~m}_{\mathrm{e}}$, where $\mathrm{m}_{\mathrm{e}}$ 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 $\frac{1}{200}$ times that of the election in the nth orbit
(C) The lonization energy of muonic atom is 200 times more than that of an hydrogen atom
(D) The momentum of the muon in the nth orbit is 200 times more than that of the electron
Options:
(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 $\frac{1}{200}$ times that of the election in the nth orbit
(C) The lonization energy of muonic atom is 200 times more than that of an hydrogen atom
(D) The momentum of the muon in the nth orbit is 200 times more than that of the electron
Solution:
1706 Upvotes
Verified Answer
The correct answer is:
(A), (C), (D)
(A), (C), (D)
$$
\begin{aligned}
&\text { (A) Radius of muon }=\frac{\text { Radius of hydrogen }}{200} \\
&\text { Radius of } \mathrm{H} \text { atom }=r=\frac{\in_0 n^2 h^2}{\pi m e^2} \\
&\text { Radius of muon }=r_\mu=\frac{\in_0 n^2 h^2}{\pi \times 200 m e^2} \\
&r_\mu=\frac{r}{200}
\end{aligned}
$$
(B) Velocity relation given is wrong
(C) Ionization energy in $e^{-} \mathrm{H}$ atom
$$
E=\frac{+m e^4}{8 \epsilon_0^2 n^2 h^2}
$$
$$
E_\mu=\frac{200 m e^4}{8 \epsilon_0^2 n^2 h^2}=200 E
$$
(D) Momentum of $\mathrm{H}$-atom
$$
m v r=\frac{n h}{2 \pi}
$$
Hence (A), (C), (D) are correct.
\begin{aligned}
&\text { (A) Radius of muon }=\frac{\text { Radius of hydrogen }}{200} \\
&\text { Radius of } \mathrm{H} \text { atom }=r=\frac{\in_0 n^2 h^2}{\pi m e^2} \\
&\text { Radius of muon }=r_\mu=\frac{\in_0 n^2 h^2}{\pi \times 200 m e^2} \\
&r_\mu=\frac{r}{200}
\end{aligned}
$$
(B) Velocity relation given is wrong
(C) Ionization energy in $e^{-} \mathrm{H}$ atom
$$
E=\frac{+m e^4}{8 \epsilon_0^2 n^2 h^2}
$$
$$
E_\mu=\frac{200 m e^4}{8 \epsilon_0^2 n^2 h^2}=200 E
$$
(D) Momentum of $\mathrm{H}$-atom
$$
m v r=\frac{n h}{2 \pi}
$$
Hence (A), (C), (D) are correct.
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