Question

A muonic atom contains a nucleus of charge Ze and a negative muon, ${\mu }^{-}$, moving about it. The ${\mu }^{-}$ is an elementary charge with -e and a mass that is approximately 200 times as large as an electron mass.

The first Bohr orbit of the muonic atom with $Z=1$ is found out to be $k\times {10}^{-13}m$. Then the value of $k$ is

Answer 1

Since mass of muon atom is not significantly lesser than mass of nucleus given that mass of muon is 200 times larger than mass of electron. Hence we cannot assume nucleus to be in fixed position unlike regular electron nucleus pair. Here both muon and nucleus will wobble around their common centre of mass for which we use the concept of reduced mass.

For regular eletron atom, $mvr=\frac{nh}{2\pi }$
For muonic atom, same can be written as
$\mu vr=\frac{nh}{2\pi }$, where $\mu$ is reduced mass.

$\mu =\frac{m_{nucleus}{m}_e}{m_{nucleus}+m_2}=200m_e$

Now for regular electron atom radius in first orbit, $r=1A^0$
So for muonic atom, $r_{new}=\frac{1}{200}={10}^{-13}m$