Obtain an expression for the orbital magnetic moment of an electron rotating about the nucleus in an atom.

Consider an electron revolving around a nucleus in an atom. Since the electron is a negatively charged particle, so atom will act as a current-carrying loop and its magnetic dipole moment is given by 

$M=lA$

Where $l$ =current and $A$ =area of cross-section

if $r$ is the radius of orbit then

$M=\left(\frac{e}{t}\right)\left(\pi r^2\right)$

=$\frac{e}{\frac{2\pi r}{v}}\pi r^2=\frac{evr}{2}$

Sice $mvr$ = angular momentum $(L)$

So, $M=\left(\frac{2}{2m_e}\right)L$ (in vector form), since magnetic dipole moment of the electron and angular momentum are in opposite direction.