A nonconducting ring of mass m and radius R, with charge per unit length `lambda` is shown in fig. It is then placed on a rough nonconducting horizontal plane. At time t=0, a uniform electric field `vecE =E_(0)hati` is switched on and the ring starts rolling without sliding. Determine the friction force (magnitude and direction ) acting on the ring.

Answer

A, D

Explanation

`dF=lamda(Rd theta)E_0`

Perpendicular distasnce between two equal and opposite pasirs of dF will be
`r_(_|_)=2R sintheta`
`:. dtau=dFr_(_|_)=2lamdaR^2E_0sintheta d theta`
`:. tau=int_0^(pi/2) d tau=2lamda R^2E_0` (clockwise)
`These pairs forces will not provide net force.
Let force of friction on ring is `f` in forward direction.
for pure rolling to take place
`a=Ralpha`
or `f/m=R[(tau-fR)/(mR^2)]`
`or f=tau/R-f`
or `f=tau/(2R)=lamdaR_0`