Estimate the ratio of the electron magnetic moment to the muon magnetic moment for the same state of orbital angular momentum. (Hint: \(m_{\mu}=105.7 \mathrm{MeV} / c^{2}\) )

The magnetic moment, denoted as \(\mu\), of a charged particle with charge \(q\), mass \(m\), and orbital angular momentum \(L\) is given by: \[\mu = \frac{q}{2m}L\] Since both the electron and the muon have the same magnitude of charge, their magnetic moments only differ in their mass terms.

Given that the mass of the muon is: \[m_{\mu} = 105.7 \frac{\mathrm{MeV}}{c^2}\] To compare the masses of the electron and the muon, we need to express the mass of the electron in the same unit system. We know that the mass of the electron is: \[m_{e} = 0.511 \frac{\mathrm{MeV}}{c^2}\] Now, we can determine the mass ratio of the electron and muon: \[\frac{m_{e}}{m_{\mu}} = \frac{0.511 \frac{\mathrm{MeV}}{c^2}}{105.7 \frac{\mathrm{MeV}}{c^2}} \approx 0.0048\]

Using the mass ratio we found in step 2, we can now estimate the ratio of the electron magnetic moment to the muon magnetic moment. To recall, the magnetic moment \(\mu\) of a charged particle with charge \(q\), mass \(m\), and orbital angular momentum \(L\) is given by: \[\mu = \frac{q}{2m}L\] Let \(\mu_{e}\) and \(\mu_{\mu}\) denote the magnetic moments of the electron and the muon, respectively. Using the formula for the magnetic moment, we can write the ratio as: \[\frac{\mu_{e}}{\mu_{\mu}} = \frac{\frac{q_e}{2m_e}L}{\frac{q_{\mu}}{2m_{\mu}}L}\] Recalling that the magnitude of the charge is the same for both particles, \(q_e = q_{\mu} = q\), and thus: \[\frac{\mu_{e}}{\mu_{\mu}} = \frac{\frac{q}{2m_e}}{\frac{q}{2m_{\mu}}}\] Simplifying this expression, we obtain: \[\frac{\mu_{e}}{\mu_{\mu}} = \frac{m_{\mu}}{m_{e}}\] Now, using the mass ratio found in step 2, we can estimate the ratio of the electron magnetic moment to the muon magnetic moment: \[\frac{\mu_{e}}{\mu_{\mu}} \approx \frac{1}{0.0048} \approx 208.33\] Therefore, the ratio of the electron magnetic moment to the muon magnetic moment for the same state of orbital angular momentum is approximately 208.