A fixed uniformly charged solid insulating sphere has a radius R and total charge +Q . A point charge -q ( q Q ) is released from rest at a distance of 4R from the centre of the sphere. Let v₁ be its speed when it crosses a distance of 2R from the centre, and v₂ be its speed when it reaches the surface of the sphere. Neglecting gravitational forces, the ratio v₁ : v₂ is :

Step-by-step solution

Let the potential energy of the system at a distance r from the centre be U(r) = - 1 4 ₀ Qq r . The initial energy of the system when the charge is released from rest at r = 4R is entirely potential: E_i = U(4R) = - 1 4 ₀ Qq 4R When the charge is at a distance of 2R , its potential energy is: U(2R) = - 1 4 ₀ Qq 2R By conservation of mechanical energy, its kinetic energy K₁ at this point is: K₁ = E_i - U(2R) = - 1 4 ₀ Qq 4R - ( - 1 4 ₀ Qq 2R ) = 1 4 ₀ Qq R ( 1 2 - 1 4 ) = 1 4 ₀ Qq 4R When the charge reaches the surf