Muon \((\mu ^{-1})\) is negatively charged \(\left( {\left| q \right| = \left| e \right|} \right)\) 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 \(\frac{1}{{200}}\) times that of the election 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

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Answer & Solution

Correct Answer

(D) \((A), (C), (D)\)

Detailed explanation

\((A)\) Radius of muon \({ = \frac{{{\text{Radius of hydrogen}}}}{{200}}}\)

Radius of \(H\) atom \(=r=\frac{\epsilon_{0} n^{2} h^{2}}{\pi m e^{2}}\)

Radius of muon \(=r_{\mu}=\frac{\epsilon_{0} n^{2} h^{2}}{\pi \times 200 m e^{2}}\)
\(r_{\mu}=\frac{r}{200}\)

\((B)\) Velocityrelation given is wrong

\((C)\) Ionization energy in \(e^-\) \(H\) 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 \(H\) -atom

\(m v r=\frac{n h}{2 \pi}\)

Hence \((A),\,(C),\,(D)\) are correct.

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