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(A) The Bohr radius for the $K$ shell is given by the formula:$a_{0} = \frac{h^{2} \varepsilon_{0}}{\pi m e^{2}}$From this expression,it is clear that

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$2.56 	imes 10^{-3} \, \mathring{A}$

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(A) The Bohr radius for the $K$ shell is given by the formula:$a_{0} = \frac{h^{2} \varepsilon_{0}}{\pi m e^{2}}$From this expression,it is clear that the Bohr radius is inversely proportional to the mass $(m)$ of the orbiting particle.Therefore,the ratio of the radius of the electron orbit $(a_{0e})$ to the radius of the muon orbit $(a_{0m})$ is:$\frac{a_{0m}}{a_{0e}} = \frac{m_{e}}{m_{m}}$Given that $m_{m} = 207 m_{e}$,we substitute this into the equation:$a_{0m} = \frac{m_{e}}{207 m_{e}} 	imes a_{0e} = \frac{a_{0e}}{207}$The standard Bohr radius for a hydrogen atom is $a_{0e} \approx 0.529 \, \mathring{A} = 0.529 	imes 10^{-10} \, m$.Calculating the new radius:$a_{0m} = \frac{0.529 \, \mathring{A}}{207} \approx 0.002555 \, \mathring{A} \approx 2.56 	imes 10^{-3} \, \mathring{A}$.

An electron in a hydrogen atom is replaced by an identically charged particle,a muon,with a mass $207$ times that of an electron. What will be the radius of the $K$ shell?

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