Let an electron having charge (–e) revolves uniformly around a stationary heavy nucleus of charge +e of hydrogen atom
The current constituted by moving electron of charge (–e) is
\(i = \frac{e}{T}\) ...(1) (T is period of revolution)
If r is radius of orbit and ve be the orbital speed, then
\(T = \frac{2 \pi r}{v_e}\) ...(2)
Putting this T in equation (1), we get
\(i = \frac{ev_e}{2\pi r}\)
The magnetic moment \(\mu_e\) associated with circulating current can be written as
\(\mu _\ell = i \pi r^2 = \frac{ev_e}{2\pi r} \times \pi r^2\)
\(= \frac{evr}{2}\) ...(3)
The direction of \(\mu _\ell\) is into the plane of paper
\(\mu _\ell = \frac{ev _er}{2}\)
\(= \frac{e}{2m_e}(mv_er)\)
\(\mu _\ell = \frac{e}{2m_e}L\)
(where L is magnitude of angular momentum of electron about nucleus which is equal to mver)
\(\text{Vectorially} \text[\vec \mu _\ell = - \frac{e}{2m_e}\vec L]\)
