electrostats_6

Let the tension in the string before the charge q₀ was put be T and final tension be T'

Consider a small arc which subtends an angle dθ at the centre of the ring.
Writing force equations for that part and using superposition of forces we have
2T'*sin(dθ/2) = 2T*sin(dθ/2) + q_o*(Rdθ*q/2Î R)/4Πε₀R²

Now, sin(dθ/2)~dθ/2

So, (T'-T)dθ = q_oqdθ/8Î ²Îµ_oR²

So, T'-T = increment in tension = q_oq/8ε_oÎ ²R²