Question

If the magnetic induction at the centre of the disc is $B=\frac{{\mu }_0}{X}\sigma \omega R$. Find X?

Answer 1

Consider an elemental area of the disc of radius $r$ and thickness $dr$
$dq=\sigma \cdot 2\pi rdr$
$\therefore dI=\frac{dq}{T}=\sigma \cdot 2\pi rdr\cdot \frac{\omega }{2\pi }$
$\therefore dB=$ Magnetic field by this element
$=\frac{{\mu }_{0}dI}{2r}$
$B=\int dB=\frac{{\mu }_0}{4\pi }\cdot 2\pi \sigma \omega {\int }_0^{R}dr=\frac{{\mu }_0}{2}\sigma \omega R$