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# A particle of charge equal to that of an electron e , and mass 208 times the mass of electron (called $\;\mu-\;$ meson ) moves in a circular orbit around a nucleus of charge +3e . (Take the mass of nucleus to be infinite ) . Then radius of $n^{th}\;$ Bohr orbit is . Find wavelength of radiation emitted when $\;mu\;$- meson jumps from third orbit to first orbit

$(a)\;0.692 A^{0}\qquad(b)\;0.548 A^{0}\qquad(c)\;0.312 A^{0}\qquad(d)\;0.212 A^{0}$

Answer : (b) $\;0.548 A^{0}$
Explanation :
The energy for $\;n^{th}\;$ orbit is given by
$\varepsilon_{n}=-\large\frac{m K^2 Z^2 e^4}{2 n^2 h^2}$
Substituting $\;m=208e_{e}\;, Z=3 \;, K=\large\frac{1}{4 \pi \varepsilon_{0}}$
and $\;\hbar=\large\frac{h}{2 \pi}\;,$ We get
$\varepsilon_{n}=-\large\frac{234 m_{e} e^4}{\varepsilon_{0}^{2} n^2 h^2 } =-1872 \;(\large\frac{M_{e} e^{4}}{8 \varepsilon_{0}^{2} h^3 c})\;\large\frac{h c}{n^2}$
$=-\large\frac{1872R h c}{n^2}$
Where $R=\large\frac{m_{e} e^{4}}{8 \varepsilon_{0}^{2} h^3 c}\;$ is Rydberg const.
where $\;mu\;$-meson jumps from third orbit to first orbit , difference in energy is radiated as a photon of frequency $\; \nu\;$is given by
$h \nu=\varepsilon_{3}-\varepsilon_{1}$
As $\;\nu=\large\frac{c}{\lambda}\;$ we have $\;\large\frac{hc}{\lambda}=\varepsilon_{3}-\varepsilon_{1}$
$=1872 Rhc \;[\large\frac{1}{1^2}-\large\frac{1}{3^2}]$
or $\; \large\frac{1}{\lambda}=1872R\;(1-\large\frac{1}{9})$
or $\; \lambda=\large\frac{9}{1872 \times 8 \times R}=\large\frac{9}{1872 \times8 \times(1.097 \times10^{7})}=0.5478 \times10^{-10} m$
$=0.5478 A^{0}\;.$