Consider a uniformly charged ring of radius a. Let the total charge on the ring be Q.

Hence, charge per unit length = Q/ 2∏ a

Let us consider a small arc on the ring having a charge dq. We can find the electric field dE due to the small element dq and later integrate to find the charge of the entire ring.

Suppose that the point P is at a distance x from the centre of the ring. Also, the distance of the arc having charge dq is at a distance r from the point P.

From Pythagoras theorem, we can deduce r² = x² + a²

Hence, dE = kdq / r² = kdq / x² + a²     --------Eqn (1)

Resolving dE into component along x and y axis, the y axis term cancels out due to symmetry of the ring. Hence, only x axis term contributes dE_x = dE cos Ɵ = dE x / (x² + a²)^1/2

Substitute from Eqn (1),  dE_x = kdq / x² + a²    * x / (x² + a²)^1/2

                                           = kx dq / (x² + a²)^3/2

E_x = kx (x2 + a2)^3/2 ∫ dq ⇒ E_x = kQx (x2 + a2)^3/2

Thus, we can get an expression for electric field due to a uniformly charged ring.