derive formula for intensity of a electric field due to imfinitely long straight uniformly charged wire

Consider a thin infinitely long straight charged wire of linear charge density λ.

Let P be the point at a distance ‘a’ from the line. To find electric field at point P, draw a cylindrical surface of radius ‘a’ and length l.

If E is the magnitude of electric field at point P, then electric flux through the Gaussian surface is given by,

Φ = E × Area of the curved surface of a cylinder of radius r and length l

Because electric lines of force are parallel to end faces (circular caps) of the cylinder, there is no component of field along the normal to the end faces.

Φ = E × 2πal … (i)

According to Gauss theorem, we have

From equations (i) and (ii), we obtain