A point charge -q (where q Q ) of mass m is released from rest at a point on the axis of a fixed, uniformly charged ring of radius R and total charge +Q . The release point is at a distance of 3 R from the centre of the ring. What is the speed of the point charge when it reaches the centre of the ring? (Assume ₀ is the permittivity of vacuum and neglect gravity)

Step-by-step solution

Initial potential on the axis of the ring at a distance x = 3 R is: V_i = 1 4 ₀ Q R^2 + x^2 = 1 4 ₀ Q R^2 + 3R^2 = 1 4 ₀ Q 2R Initial potential energy of the system is: U_i = -qV_i = - Qq 8 ₀ R Final potential at the centre of the ring ( x = 0 ) is: V_f = 1 4 ₀ Q R Final potential energy of the system is: U_f = -qV_f = - Qq 4 ₀ R Applying conservation of mechanical energy ( K_i + U_i = K_f + U_f ): 0 - Qq 8 ₀ R = 1 2 mv^2 - Qq 4 ₀ R 1 2 mv^2 = Qq 4 ₀ R - Qq 8 ₀ R = Qq 8 ₀ R v^2 = Qq 4 ₀ mR v = Qq 4 ₀ mR Answer: Qq