Question

A ring of radius R has charge -Q distributed uniformly over it. Calculate the charge that should be placed at the center of the ring such that the electric field becomes zero at a point on the axis of the ring at distant R from the center of the ring

Answer 1

Electric field at point P due to charge of ring is

$E=\frac{kQx}{(R^2+x^2{)}^{3/2}}$

At $x=R,E=\frac{kQ}{2\sqrt{2}{R}^2}$ directed toward the center.

Electric field at P due to charge at center $kq/R^2$

For net field to be zero at P.

$\frac{kq}{R^2}=\frac{kQ}{2\sqrt{2}{R}^2}$ or $q=\frac{Q}{2\sqrt{2}}$