Question

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

• A. $\frac{QrR}{mg}$
• B. $\frac{QrR}{2mg}$
• C. $\frac{QrR}{3mg}$
• D. $\frac{2QrR}{mg}$

Answer 1

The correct option is B $\frac{QrR}{2mg}$

According to faraday's law of electromagnetic induction, emf induced in the loop is given by:

$\epsilon =\frac{d\varphi }{dt}=\frac{d\left(BA\right)}{dt}$

$\epsilon =\frac{AdB}{dt}=\pi r^{2}R$

Since emf is gradient of electric field,

$E\times 2\pi r=\pi r^{2}R$

$E=\frac{rR}{2}$

Force on the ring due to the electric field is:

$F=QE=Q\frac{rR}{2}$

For equilibrium, this should be less than the frictional force,

$F<F_r$

$\frac{QrR}{2}<\mu mg$

$\mu >\frac{QrR}{2mg}$