Electric Field Due to a Charged Ring
A conducting ring of radius R has a total charge q uniformly distributed over its circumference. We are interested in finding the electric field at point P that lies on the axis of the ring at a distance x from its centre.
We divide the ring into infinitesimal segments of length dl. Each segment has a charge dq and acts as a point charge source of electric field.
Let dE be the electric field from one such segment; the net electric field at P is then the sum of all
contributions dE from all the segments that make up the ring. If we consider two ring segments at the top and bottom of the ring, we see that the contributions dE to the field at P from these segments have the same xcomponent but opposite ycomponents.
Hence, the total ycomponent of field due to this pair of segments is zero.
When we add up the contributions from all such pairs of segments, the total field E will have only a component along the ring’s symmetry axis (the xaxis) with no component perpendicular to that axis (i.e. no y or zcomponent). So, the field at P is described completely by its xcomponent E_{x}.
From the above expression, we can see that, E_{x} = 0 at x = 0, i.e. field is zero at the centre of the ring. We should expect this, charges on opposite sides of the ring would push in opposite directions on a test charge at the centre, and the forces would add to zero.

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