Magnetic field at the centre of circular loop.

$B=\frac{{\mu }_{0}l}{2r} ...\left(\mathrm{i}\right)$

Period of revolution of electron, where $\omega =$ angular velocity of electron

$T=\frac{2\pi }{\omega }$

where, $\omega =$ angular velocity of electron.

Current produced due to movement of electron,

$I=\frac{e}{T}=\frac{e}{\frac{2\pi }{\omega }}=\frac{e\omega }{2\pi }$

On substituting the value of $I$ in Eq. (i), we get

$B=\frac{{\mu }_0}{2r}·\frac{e\omega }{2\pi }$

$\Rightarrow B=\frac{{\mu }_{0}e\omega }{4\pi r}$