| Muon \[\left( {{\mu }^{-1}} \right)\] 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 [JEE Online 15-04-2018 (II)] |
| [A] Radius of the muonic orbit is times smaller than that of the electron |
| [B] the speed of the \[{{\mu }^{-1}}\] in the nth orbit is\[\frac{1}{200}\] times that of the electron in the nth 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 nth orbit is 200 times more than that of the electron |
A) , ,
B) ,
C) ,
D) , ,
Correct Answer: A
Solution :
| [a] [a] Radius of muon |
| \[\text{=}\frac{\text{Radius of hydrogen}}{200}\] |
| \[\text{Radius of H atom=r=}\frac{{{\in }_{o}}{{n}^{2}}{{h}^{2}}}{\pi m{{e}^{2}}}\] |
| \[\text{Radius of muon =}{{\text{r}}_{\mu }}=\frac{{{\in }_{o}}{{n}^{2}}{{h}^{2}}}{\pi \times 200m{{e}^{2}}}\] |
| \[{{r}_{\mu }}=\frac{r}{200}\] |
| [b] Velocity relation given is wrong |
| [c] Ionization energy in \[{{e}^{-}}H\] atom |
| \[E=\frac{+m{{e}^{4}}}{8\in -{{o}^{2}}{{n}^{2}}{{h}^{2}}}\] |
| \[{{E}_{\mu }}=\frac{200m{{e}^{4}}}{8\in _{o}^{2}{{n}^{2}}{{h}^{2}}}=200E\] |
| [d] Momentum of H-atom |
| \[mvr=\frac{nh}{2\pi }\] |
| momentum of muon is \[200\times mvr\] |
| hence , [c] & [d] are correct |
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