E due to line charge using Gauss's law

Gauss's law: The total electric flux through any closed surface equals the free charge enclosed: ∮ D · dS = Q_enc.

Setup

Consider an infinitely long straight line of charge along the z-axis with uniform linear density ρ_L (C/m) in a medium of permittivity ε₀.

By symmetry (the line is infinite and uniform):

• D (and E) has only a radial component, D = D_ρ a_ρ.
• D_ρ depends only on the distance ρ from the line, not on φ or z.

        z
        │ ▲  ┌───────────┐
        │ │  │ Gaussian  │   closed cylinder:
        │ │  │ surface   │   radius ρ, length L
        ● ─ ─┼─ ─ ─ ─ ─ ─┼──→ ρ
        │    └───────────┘
        │     (D points radially outward)

Gaussian surface

Choose a coaxial closed cylinder of radius ρ and length L.

• On the curved side, D is normal to the surface and constant in magnitude, so D · dS = D_ρ dS.
• On the two end caps, D is parallel to the surface (D · dS = 0).

∮ D · dS = D_ρ ∫ dS = D_ρ (2πρL)

Enclosed charge:

Q_enc = ρ_L L

Apply Gauss's law

D_ρ (2πρL) = ρ_L L

D_ρ = ρ_L / (2πρ)

E = D / ε₀ = ρ_L / (2π ε₀ ρ)  a_ρ     V/m

Result: E = ρ_L / (2πε₀ρ) a_ρ

The field is radial, directed away from a positive line charge, and decreases as 1/ρ. For a line not on the z-axis, ρ is the perpendicular distance and a_ρ the unit vector perpendicular to the line.