How it works
Gauss's Law states that the total electric flux Ψ through any closed surface equals the total free charge enclosed: ∮ D · dS = Qenc. In differential form, ∇·D = ρv, where D = ε₀·E is the electric flux density and ρv is the volume charge density. For a line charge ρL, using a coaxial cylindrical Gaussian surface of radius r and length L: E·(2πrL) = ρL·L/ε₀, giving E = ρL/(2πε₀r) in the radial direction. For a point charge Q at the origin, a spherical Gaussian surface gives E = Q/(4πε₀r²). Gauss's Law applies easily only when the field has sufficient symmetry — planar, cylindrical, or spherical.
Key points to remember
Three canonical geometries tested in exams are: infinite line charge (cylindrical Gaussian surface, E ∝ 1/r), point charge or sphere (spherical surface, E ∝ 1/r²), and infinite plane sheet (pillbox surface, E = ρs/2ε₀ on each side, independent of distance). Inside a uniformly charged sphere of radius R, the enclosed charge scales as r³ so E increases linearly with r for r < R. The differential form ∇·D = ρv is one of Maxwell's four equations and is the electrostatic version of the divergence theorem. Surface charge density ρs on a conductor produces a discontinuity in the normal component of D equal to ρs at the surface.
Exam tip
The examiner always asks you to apply Gauss's Law to find E for a coaxial cylindrical geometry — state the symmetry argument first, choose the correct Gaussian surface, then write ∮D·dS = Qenc before integrating; students who skip the symmetry statement lose marks.