Question

A ring of radius $R$ has charge $-Q$ distributed uniformly over it. Calculate the charge $(q)$ that should be placed at the centre of the ring such that the electric field becomes zero at a point on the axis of the ring distant ' $R$ ' from the centre of the ring. If the value of $q=\frac{Q}{4} \sqrt{\Omega}$. The value of $\Omega$ is________.

Answer 1

Electric field at point $P$ due to charge of ring is
$E=\frac{k Q x}{\left(R^{2}+x^{2}\right)^{3 / 2}}$

At $x=R: E=\frac{k Q}{2 \sqrt{2} R^{2}}$ directed towards the centre.
Electric field at $P$ due to centre charge: $\frac{k q}{R^{2}}$
For net field to be zero at $P$ :
$\frac{k q}{R^{2}}=\frac{k Q}{2 \sqrt{2} R^{2}} \Rightarrow q=\frac{Q}{2 \sqrt{2}}=\frac{Q}{4} \sqrt{2}$