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

Answer: 2

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

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