(a) The given charge distribution is shown in Fig. The symmetry of this distribution implies that vector E at the point O is directed to the right, and its magnitude is equal to the sum of the projection onto the direction of vectors E of vector dE from elementary charges dq. The projection of vector dE onto vector E is

It should be noted that this integral is evaluated in the most simple way if we take into account that

(b) Take an element S at an azimuthal angle φ from the x-axis, the element subtending an angle dφ at the centre.

The elementary field at P due to the element is

The component alon OS can be broken into the parts along OX and OY with