An infinite charged wire with charge per unit length lambda lies along the central axis of a cylindrical surface of radius r and length. What is the flux through the surface due to the electric field of the charged wire?

The flux through the infinite charged wire along the central axis of a cylindrical surface of radius r and length l is ∅E = E x 2πrl

Explanation:

let us consider a thin infinitely long straight wire having a uniform charge density λ Cm⁻¹. To determine the field at a distance r from the line charge, we have cylindrical gaussian surface of radius r, length l, and with its axis along the line charge. it has curved surface S₁, and flat circular ends S₂ and S₃. Obviously, dS₁//E, dS₂ ⊥E, and dS₃ ⊥ E, so, only the curved surface contributes towards the total flux.

∅E = ∫ E. dS = ∫E. dS₁ + ∫E. dS₂ + ∫E. dS₃

= ∫EdS₁ cos0⁰ + ∫EdS₂ cos 90⁰ + ∫Eds₃ cos 90⁰

= E∫ds₁₁ + 0+0

= E x area of curved surface

∅E = E x 2πrl