Applying Gauss's law to symmetric charge distributions

Gauss's law gives closed-surface flux = Qenc/ε0, where ε0 is vacuum permittivity. To find E by area, symmetry must fix its direction and magnitude on flux-carrying parts; other parts may have zero flux. Charge symmetry depends on distribution, not object shape. Spherical charge symmetry means density is equal in every direction at a fixed radius. The law holds without symmetry, but net flux alone usually cannot determine E. Zero enclosed charge gives zero net flux, not necessarily zero field.

Take an isolated solid insulating sphere of radius R with charge Q>0 spread uniformly through its volume in vacuum. On a concentric Gaussian sphere, E is outward and constant, so flux = E4πr². For 0<r<R, Qenc = Q(r/R)³ and E = Qr/(4πε0 R³). At the center, E=0. For r≥R, Qenc=Q and E = Q/(4πε0 r²), as for a point charge Q at the center. Thus E/E(R)=r/R inside and (R/r)² outside.

An isolated infinite line in vacuum has uniform charge per length λ>0. Its field is radially outward and depends only on distance r from the line. Use a closed coaxial cylinder of radius r>0 and length L. E is constant and normal on its curved side, whose area is 2πrL. E is tangent to both end caps, so they add zero flux. Qenc=λL. Thus E2πrL=λL/ε0 and E=λ/(2πε0 r).

An isolated infinite thin sheet in vacuum has uniform charge per area σ>0. Translation and reflection symmetry make E perpendicular to the sheet, outward on each side, with equal magnitude on both sides. A closed pillbox has one cap of area A on each side. Its side has zero flux; its caps give 2EA. Qenc=σA, so 2EA=σA/ε0 and E=σ/(2ε0). The area cancels, and E is independent of distance from the sheet.