Final answer:
The electric field at a distance r from an infinite line of charge with line charge density λ is given by E = λ/(2πε0r), derived using Gauss's Law.
Step-by-step explanation:
To determine the electric field at a point P due to an infinite line of charge with linear charge density λ (pI in the question), we use Gauss's Law. Imagine a Gaussian cylinder with radius r and length L centered on the line of charge where both ends of the cylinder are perpendicular to the line. The electric field due to the line of charge is perpendicular to the line and radially symmetric, thus will be the same magnitude at every point on the surface of the cylinder.
The total electric flux through the Gaussian surface is the electric field E times the cylindrical surface area, which does not include the ends since the electric field there is parallel to the surface and thus contributes no flux. The surface area of the side of the cylinder is 2πrL. By Gauss's Law, the electric flux is equal to the charge enclosed (λL) divided by the permittivity of free space (ε0), thus Φ = λL/ε0. Setting the flux equal to EA (electric field times area), we get E(2πrL) = λL/ε0, and solving for E gives us E = λ/(2πε0r). This is the expression for the electric field at distance r from the line of charge.