Derive an expression for the magnetic moment `vec(mu)` of an electron revolving around the nucleus in terms of its angular momentum `vecl`. What is the direction of the magnetic moment of the electron with respect to its angular momentum?

Explanation

We know that a revolving electron constitutes an electric current given by
`I = e/T = (ev)/(2 pi r)`
where `T = (2 pi r)/v` = peroid of revolution of electron revolving in a circular orbit of radius r with a speed v.
`:.` Magnetic moment `(mu_l)` associated with this circulating current,
`mu_l = I cdot A = (ev)/(2pi r) cdot pi r^2 = (evr)/(2)`
As an electron revolving is an anticlockwise direction is equivalent to current in clockwise direction, hence in accordance with right hand rule the magnetic moment is in a direction perpendicular to plane of paper (or plane of orbit) directed inward.
The above relation may also bewritten as :
`mu_l = (e v rm)/(2m ) = e/(2m) cdot l`
and in vector notation, `vec(mu)_(l) = -e/(2m) cdot vecl ,` where `l = m v r` = orbital angular momentum of the electron around the nucleus . The -ve sign applied here indicates that the angular momentum of the electorn is opposite in direction to the magnetic moment.