Gauss's Law

Electric flux through Gaussian surfaces and enclosed charge relationships

Gauss's Law is one of Maxwell's four equations and relates the electric flux through a closed surface to the charge enclosed within it. The law states Φ_E = Q_enc/ε₀, where ε₀ = 8.854×10⁻¹² C²/(N·m²). For highly symmetric charge distributions (spherical, cylindrical, planar), Gauss's Law provides an elegant shortcut to calculate the electric field without integration. For a point charge Q, the field at distance r is E = kQ/r² (radial). A spherical Gaussian surface of radius r centered on the charge captures flux Φ = E·4πr² = Q/ε₀, independent of r — demonstrating that flux depends only on enclosed charge, not surface size. For an infinite line charge with linear density λ, a cylindrical Gaussian surface gives E = λ/(2πε₀r). For an infinite plane with surface density σ, E = σ/(2ε₀), uniform and independent of distance.