Question

Derive an expression for the magnetic dipole moment of a revolving electron.

Answer 1

The current of an electron revolving around a heavy nucleus is given as:

$I=\frac{e}{T}=\frac{e}{2\pi r/v}=\frac{ev}{2\pi r}$

The magnetic moment associated to the current of an electron revolving is given as:

${\mu }_l=IA=\frac{ev}{2\pi r}\left(\pi r^2\right)=\frac{evr}{2}$.

Substituting the angular momentum of the revolving electron we get,

$l=mvR$

$vR=\frac{l}{m}$

Therefore, ${\mu }_l=\frac{el}{2m}$

$\frac{{\mu }_l}{l}=\frac{e}{2m}$.

The above equation is known as Gyromagnetic ratio.