Question

A ring of radius R has charge -Q distributed uniformly over it. Calculate the charge (q) 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 distant R from the center of the ring.

The value of q is (Q/4) $\sqrt{x}$. Find the value of $x$.

Answer 1

The electric field due to a ring of radius R at a point on its axis is $E=k\frac{-Qz}{(z^2+R^2{)}^{3/2}}$
Electric field at point $P(z=R)$ is $E_1=k\frac{-QR}{(R^2+R^2{)}^{3/2}}=-k\frac{QR}{2\sqrt{2}{R}^3}=-k\frac{Q}{2\sqrt{2}{R}^2}$
Electric field at P due to center charge q is $E_2=k\frac{q}{R^2}$
As the net field at P is zero so
$E_1+E_2=0$
or $-k\frac{Q}{2\sqrt{2}{R}^2}+k\frac{q}{R^2}=0$
$\Rightarrow q=\frac{Q}{2\sqrt{2}}=\frac{Q}{4}\sqrt{2}$