Question

Two semi-circular rings lying in the same plane of uniform linear charge density $\lambda$ have radii $r$ and $2r.$ They are joined using two straight uniformly charged wires of linear charge density $\lambda$ and length $r$ as shown in the figure. The magnitude of electric field at common centre of semi-circular rings is

Answer 1

The electric field due to both straight wires at Centre $O$ will be zero since the point lies on its axis. Let the electric field due to larger and smaller semi-circular rings at $O$ be $E$ and $E^{\prime }$ respectively.


We know that the magnitude of electric field at the Centre of a charged semi-circular arc equals $\frac{1}{4\pi {\epsilon }_0}\frac{2\lambda }{R}$ and directed towards the axis of symmetry, where $R$ is the radius of the arc.
Therefore,

$E=\frac{1}{4\pi {\epsilon }_0}\frac{2\lambda }{2r}$
$E^{\prime }=\frac{1}{4\pi {\epsilon }_0}\frac{2\lambda }{r}$
From figure magnitude of electric field at $O$ is $E^{\prime }-E$
$E=\frac{1}{4\pi {\epsilon }_0}(\frac{2\lambda }{r}-\frac{\lambda }{r})=\frac{1}{4\pi {\epsilon }_0}\frac{\lambda }{r}$

Final answer : (d)