A charge + q is distributed over a thin ring of radius r with line charge density λ = q sin 2 θ / π r . Note that the ring is in the X Y -plane and θ is the angle made by r with the X -axis. The work done by the electric force in displacing a point charge + Q from the centre of the ring to infinity is

Step-by-step solution

As charge distribution is over a circle, potential due to charge over ring is V = k r . q total where, q total = ∫ 0 2 π q sin 2 θ π r r d θ = q π ∫ 0 2 π 1 - cos 2 θ 2 . d θ = q π θ 2 - sin 2 θ 4 0 2 π = q So, potential at centre of ring = V = k q total r ⇒ V = k q r Also, work done in taking a charge Q from centre of ring to infinity = potential energy of system = V · Q = k q Q r = q Q 4 π ε 0 r