Question

A nonconducting ring of mass $m$ and radius $R$ is charged as shown. The charged density i.e. charge per unit length is $\lambda $ . It is then placed on a rough nonconducting horizontal surface plane. At time $t \, = \, 0$ , a uniform electric field $\overset{ \rightarrow }{E} \, =E_{0}\hat{i}$ is switched on and the ring starts rolling without sliding. The friction force (magnitude and direction) acting on the ring, when it starts moving is

• A. $\lambda R E_{0} \hat{i}$
• B. $3 \lambda R E_{0} \hat{ i }$
• C. $\sqrt{2} \lambda R E_{0} \hat{i}$
• D. $\frac{2}{\lambda R E_{0}}\hat{i}$

Answer 1

Answer: (A)

$\mathrm{d} \tau=(\mathrm{dq}) \times\left(\mathrm{E}_0\right) 2 \mathrm{R} \sin \theta $
$\mathrm{d} \tau=(\lambda \mathrm{Rd} \theta) 2 \mathrm{RE}_0 \sin \theta $
$\tau=2 \mathrm{R}^2 \lambda \mathrm{E}_0 \int_0^{\pi / 2} \sin \theta \mathrm{d} \theta=2 \mathrm{R}^2 \lambda \mathrm{E}_0$
For translational motion,
$ f=m a \Rightarrow a=\frac{f}{m} $
For rotational motion, $\tau-f R=\left(m R^2\right)\left(\frac{a}{R}\right)$
From (2) and (3)
$ \tau=2 f_{\mathrm{r}} R $
From (1) and (4)
$ f_r=\frac{\tau}{2 R}=\frac{2 R^2 \lambda E_0}{2 R}=R \lambda E_0 \Rightarrow \vec{f} r=R \lambda E_0 \hat{i} $