Use Gauss' law to derive the expression for the electric field ( E → ) due to a straight uniformly charged infinite line of charge density λ C m .

Consider a thin infinitely long straight wire having a uniform linear charge density λ C m − 1 . By symmetry the field its magnitude is same at all points equidistant from the line charge. Selecting a cylindrical Gaussian surface of radius r , length l and with its axis along the line charge, As show in the figure, Net flux can be written as ϕ E = ∮ S E → ⋅ d S → = ∫ S 1 d S → 1 + ∫ S 2 E → ⋅ d S → 1 + ∫ S 3 E → ⋅ d S → 3 = ∫ S 1 E d S 1 cos 0 ∘ + ∫ S 2 E d S 2 cos 90 ∘ + ∫ S 3 E d S 3 cos 90 ∘ = E ∫ d S 1 + 0 + 0 = E × area of the curved surface or ϕ E = E × 2 π r l Charge enclosed by the Gaussian surface, q = λ l Using Gauss’ theorem, ϕ E = q ε 0 ⇒ E . 2 π r l = λ l ε 0 ⇒ E = λ 2 π ε 0 r Vectorial, E → = λ 2 π ε 0 r n ^ .