A point charge q₀ is moved from the centre of a uniformly charged stationary ring of radius R to a point on its axis at a distance 3 R from the centre. The charge is moved along a complex curved path outside the axis. If the magnitude of the work done by the electric field in this process is W , the total charge Q on the ring is : ( ₀ is the permittivity of vacuum)

Step-by-step solution

The electrostatic field is conservative, so the work done by the electric field depends only on the initial and final positions, not on the path taken. The electric potential at the centre of a uniformly charged ring of radius R and charge Q is: V_i = 1 4 ₀ Q R The electric potential on the axis of the ring at a distance x = 3 R from the centre is: V_f = 1 4 ₀ Q R^2 + x^2 = 1 4 ₀ Q R^2 + ( 3 R)^2 = 1 4 ₀ Q 2R The work done by the electric field is equal to the negative of the change in potential energy: W_ el = - U