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:

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

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

$\mu_l=IA=\dfrac{ev}{2\pi r}(\pi r^2)=\dfrac{evr}{2}$.

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.