If r is the orbital radius and v & is the orbital velocity of an...

The magnetic dipole moment of a revolving electron: The electron of change (−e) performs uniforms circular motion around a stationary& heavy nucleus of charge +Ze. This constitutes a current I.

$∴$ $I = \dfrac{e}{T}$...(i)

Where, T is the time period of revolution. Let r be the orbital radius of electron and v the orbital speed. Then

$T = \dfrac{2\pir}{υ}$...(ii)

Substituting the value of T in (i), we get

$I = \dfrac{e}{\dfrac{2\pir}{υ}} = \dfrac{eυ}{2\pir}$

The magnetic moment (m) associated with this circular current is

$m = I\pir^{2}[ ∵ \overset{→}{m} = I\overset{→}{A} ]$

$∴$ $m = \dfrac{eυ}{2\pir} ⋅ \pir^{2} = \dfrac{eυr}{2}$