In a muonic hydrogen atom, a muon (which has a mass 200 times that of an electron) replaces the electron and revolves around the proton. If the muon and a normal electron are moving with the exact same orbital speed in their respective hydrogenic atoms, what is the ratio of the radius of the muon's orbit to the radius of the normal electron's orbit ( r_ : r_e )?

Step-by-step solution

For a particle of mass m and charge e revolving around a proton with speed v in an orbit of radius r , the centripetal force is provided by the electrostatic force of attraction. mv^2 r = 1 4 ₀ e^2 r^2 Rearranging for the radius r : r = 1 4 ₀ e^2 mv^2 For a fixed orbital speed v , the radius r is inversely proportional to the mass m of the revolving particle ( r 1 m ). Therefore, the ratio of the radii is: r_ r_e = m_e m_ = m_e 200 m_e = 1 200 The ratio is 1 : 200 . Answer: 1 : 200