Charged Ring Rolling!

Net electric force is zero, so the linear acceleration is the force of friction divided by the mass. Since it rolls without slipping, the angular acceleration is the linear acceleration divided by the radius. Therefore we have:

$\alpha = \frac{f}{mR}$

The differential charge is $\lambda Rd\theta$ and so if $\theta$ is measured with respect to the horizontal, then the torque from the electric force will be $2\lambda E_0 R^2\int \limits_0^{\pi/2} sin\theta d\theta = 2\lambda E_0 R^2$

(The factor of 2 comes from symmetry: the torque from the negative charge is the same as the torque from the positive charge.)

Friction will oppose the rotation caused by the electric force, so it will supply an opposite torque, therefore we have:

$\tau = 2\lambda E_0 R^2-fR=I\alpha = mR^2\alpha = fR$

Solving yields $f=\lambda E_0 R$

I have seen some of your questions of electrostats and many of them including this one belong to a specific book