Electric Field due to an infinite Line of Charge

Consider a uniformly charged wire of infinite length having a constant linear charge density (charge per unit length) λ. Let P be a point at a distance r from the wire. Let us find an expression for Eat P. As a Gaussian surface, we choose a circular cylinder of radius r and length l, closed at each end by plane caps normal to the axis. By symmetry, the magnitude of the electric field E will be the same at all points on the curved surface of the cylinder, and directed radially outward. Also E and and dS are along the same direction.

The electric flux due to curved surface = Ñ„ E . dS = E ( 2 π rl )

The electric flux due to each plane face = 0 (. : E and dS are right angles).

. : total flux through the Gaussian surface = Ф = E  ( 2 π rl )
The net charge enclosed by the Gaussian surface = q = λ l

. : by Gauss’ law, E ( 2 π rl ) = λ l/  ε₀

The direction of E is radially outward for a line of positive charge.