Gauss's Law with Cylindrical Symmetry

A point charge's field falls off as 1/r² because its flux spreads over a sphere's surface area, which grows as r^2. An infinite line of charge is different: the charge keeps going forever along the axis, so the field can only depend on distance from the wire, not position along it. Wrap a cylinder of radius r and length L around the wire, coaxial with it. The field points radially outward everywhere on the curved side, perpendicular to that surface, so flux there is just E times the side area, 2 π r L. On the two flat end caps, the field is parallel to the surface, so those contribute zero flux. Gauss's Law then reads E(2 π r L) = Q_enc/epsilon₀, and since area grows only as r (not r²), the field thins out as 1/r.

The same trick handles a solid cylinder of uniform volume charge density rho. Outside the cylinder, all the charge is enclosed no matter how far out your Gaussian surface sits, so the field looks just like that of a line charge with lambda = rho times the cylinder's cross-sectional area. Inside, at radius r less than the cylinder's radius R, your Gaussian cylinder only encloses the charge within radius r, which scales as r² (volume = π r² L). Dividing that enclosed charge by the side area 2 π r L leaves one power of r, so the field grows linearly from the axis outward, then switches to 1/r falloff once you exit the cylinder.

This same reasoning extends naturally to a coaxial cable: between the inner conductor and outer shell, a Gaussian cylinder only encloses the inner conductor's charge, giving the familiar 1/r field, while outside the outer shell the enclosed charges cancel and the field vanishes.