question_answer

A non conducting ring (of mass m, radius \[r\], Having charge\[Q\]) is placed on a rough horizontal Surface (in a cylindrical region with transverse magnetic field). The field is increasing with time at the rate \[R\] and coefficient of friction between The surface and the ring is \[\mu \]. For ring to remain in equilibrium u should be greater than equal to,

A) \[\frac{Q{{r}^{2}}{{R}^{2}}}{2mg}\]           

B) \[\frac{QrR}{2mg}\]

C) \[\frac{Q{{r}^{2}}R}{2mg}\]               

D) \[\frac{Qr{{R}^{2}}}{2mg}\]

Correct Answer: B

Solution :

Induced electric field

\[E=\frac{r}{2}\left( \frac{dB}{dt} \right);=\frac{rR}{dt}\]

This electric field exerts tangentive force on each part of the ring. So net force,

\[F=EQ=\frac{rRQ}{2}\]

For ring to be in equilibrium,

\[F\le {{F}_{friction}}\]

\[\operatorname{or}\,r\frac{RQ}{2}\le \mu mg\]

\[\therefore \,\,\,\,\mu \ge \frac{rRQ}{2mg}\]