Question

Find the friction force acting on the charged non conducting ring : indicating the direction and magnitude correctly.

see picture please.

Answer 1

see the diagram.

t = angle made by the radius with X axis
dq = λ R dt = charge on infinitesimal arc
dF = force = E₀ dq = λ E₀ R dt
T = Net torque on the ring (magnitude) = 2 λ E₀ R² Sin t dt
Moment of Inertia of the ring about center = I = m R²

Torque = T = I α   (direction into the plane of the diagram)
So  α = angular acceleration = 2 λ E₀ / m

Net Linear acceleration of the ring (+ve x direction = a = α R
a = 2 λ E₀ R / m
Net force in +ve x direction = m a = 2 λ E₀ R

The forces due to electric field (horizontal direction) on positive and negative charges is balanced and canceled. Hence the net force above is due to Friction between the ring and rough surface...

Friction = 2 λ E₀ R  in +ve x-axis.