Question

A ring of radius $r$ is uniformly charged with a charge $q$ . If the ring is rotated with an angular frequency ${}^{\prime }{\omega }^{\prime }$ then the magnetic field at the centre is :

• A. $\frac{{\mu }_0}{4\pi }\left(\frac{q\omega }{r}\right)$
• B. $\frac{{\mu }_0}{4\pi }\left(\frac{r}{q\omega }\right)$
• C. $\frac{{\mu }_0}{4\pi }\left(\frac{q}{r\omega }\right)$
• D. $\frac{{\mu }_0}{4\pi }\left(\frac{\omega }{qr}\right)$

Answer 1

The correct option is A $\frac{{\mu }_0}{4\pi }\left(\frac{q\omega }{r}\right)$

The speed of the ring is given as,

$v=r\omega$

The charge density is given as,

$\rho =\frac{q}{2\pi r}$

The current is given as,

$i=r\omega \times \frac{q}{2\pi r}$

The magnetic field at the center is given as,

$B=\frac{{\mu }_{0}i}{2r}$

$B=\frac{{\mu }_0\times r\omega \times \frac{q}{2\pi r}}{2r}$

$B=\frac{{\mu }_0}{4\pi }\left(\frac{q\omega }{r}\right)$