Centre field of a rigidly rotating charged loop ($B(0)=\frac{{\mu }_0\Omega }{4\pi }\int \frac{dq}{r}$)

For planar charge rotating rigidly about its centre, $\mathbf{v}=\mathbf{\Omega }\times \mathbf{r}$. The magnetic contribution at the centre is $dB={\mu }_0\Omega dq/(4\pi r)$, along the rotation axis. Integrate over the actual charge distribution; charges on a rotating polygon do not follow the sides of a stationary wire circuit.