Question

Find the orbital magnetic dipole moment of the electron in a hydrogen atom.

Answer 1

The orbiting behaves as a current loop.The equivalent current
$i=\frac{netcharge}{timeofrevolution}=\frac{e}{T}={ef}_{rot}$
Now, magnetic dipole moment $\mu =niA$
$n=$ number of the turns of the loop $=1$,
$A=$ area of the loop $={\pi r}^2$
$\Rightarrow \mu =\left(1\right)\left({ef}_{rot}\right)\left(\pi r^2\right)\Rightarrow \mu ={\pi er}^2{f}_{rot}$
Putting the values of $f_{rot}$ and $r$ we obtain,
$\mu =\pi e\left(\frac{n^2{h}^2{\epsilon }_0}{{\pi me}^{2}Z}\right)\left(\frac{{mZ}^2{e}^4}{{4\epsilon }_0^2{h}^3{n}^3}\right)=\frac{ehn}{4\pi m}$