Irodov Solutions : Problem 3.8

The statement of the question is :

A thin semi-circular ring of radius R is uniformly charged with a total charge q. Find the magnitude of the electric field strength at the curvature centre of the half ring.

The ring can be assumed to be constituted of large number of infinitesimally small arcs. Take one such arbitrary infinitesimal arc element into consideration. Let it is lying at an angular position θ with respect to the x-axis (see fig. below) and subtending a small angle dθ at the centre of curvature of the half ring. Charge on this element will be dq = (q/π)dθ. We can approximate the infinitesimal arc-element (carrying an infinitesimal charge dq) as a point charge. Also from symmetry it is clear that the direction of resultant electric field vector would be along the line of symmetry i.e. the y-axis.

The small electric field produced by the considered small arc-element carrying charge dq is

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