An electron (charge - e, mass m ) is revolving around a nucleus, in a hydrogen atom, in a circle of radius $\displaystyle r$. Derive an expression, in vector form, for the magnetic dipole moment, $\displaystyle \vec{\mu}_{\mathrm{e}}$ in terms of its orbital angular momentum $\displaystyle \overrightarrow{\mathrm{L}}$. What is gyromagnetic ratio?

Electron revolve around the nucleus constitute a current \[\begin{aligned} & \mathrm{I}=\mathrm{e} / T \\ & T=\frac{2 \pi r}{v} \end{aligned} \] \[\mathrm{I}=\frac{\mathrm{e} v}{2 \pi r} \] Magnetic moment, \(\displaystyle \mathrm{M}=\mathrm{I} . \mathrm{A}\) \[\begin{aligned} & \mu_{l}=\frac{\mathrm{e} v \cdot \pi r^{2}}{2 \pi r} \\ & \mu_{l}=\frac{\mathrm{e} v r}{2} \\ & (\mathrm{L}=m v r) \end{aligned} \] Since electron has negative charge, \(\displaystyle \mu_{l}\) is opposite in direction of an electron of angular momentum L. \[\overrightarrow{\mu_{l}}=-\frac{\mathrm{e}}{2 m} \vec{\mathrm{L}} \] Gyromagnetic ratio- The ratio of magnetic moment to angular momentum is called gyromagnetic ratio. That is, \(\displaystyle \frac{\mu_{\mathrm{e}}}{\mathrm{L}}=\frac{\mathrm{e}}{2 m}\) [