Gauss's Law

For a single point charge q, take a Gaussian sphere of radius r centered on the charge. By symmetry, E points radially outward and has the same magnitude everywhere on the sphere. The flux integral becomes ∮ E·dA = E(4πr²). Setting this equal to q/ε₀ gives E = q/(4πε₀r²) = kq/r² — Coulomb's law. Going the other way: starting from Coulomb's 1/r² and superposition, you can prove Gauss's law for any charge distribution. The two are mathematically equivalent for static charges. Gauss's law is more powerful in symmetric problems (one integral instead of vector integration) and generalizes more cleanly to time-varying fields, where it becomes one of Maxwell's four equations.

Gauss's law in integral form ∮ E·dA = Q_enc/ε₀ is always true, but it produces E in closed form only when symmetry makes E constant in magnitude (and either parallel or perpendicular to dA) over a chosen Gaussian surface. Three high-symmetry cases dominate textbooks. Spherical symmetry: point charges, uniformly charged balls, concentric shells — use a Gaussian sphere. Cylindrical symmetry: infinite line charges, infinite cylinders, coaxial cables — use a Gaussian cylinder coaxial with the line. Planar symmetry: infinite charged sheets, parallel plates — use a pillbox straddling the sheet. Without one of these symmetries, you fall back on Coulomb's law and direct integration.

Apply the divergence theorem to ∮ E·dA = Q_enc/ε₀: the surface integral of E equals the volume integral of ∇·E, and Q_enc equals the volume integral of charge density ρ. The two volume integrals must be equal for any volume, so the integrands must match: ∇·E = ρ/ε₀. This is the differential form of Gauss's law and one of Maxwell's four equations. It states that the divergence of E at any point equals the charge density there divided by ε₀ — charges are sources (positive divergence) or sinks (negative divergence) of E. In vacuum where ρ = 0, ∇·E = 0 and field lines neither begin nor end. The differential form is essential for computing E in continuously varying media via differential equations.

A conductor in electrostatic equilibrium has free charges that have stopped moving. If E were nonzero anywhere inside, free charges would feel force qE and accelerate — contradicting equilibrium. So E = 0 throughout the conductor's interior. Apply Gauss's law to any closed surface inside the conductor: ∮ E·dA = 0, meaning the enclosed charge must be zero. Therefore all net charge resides on the conductor's outer surface. A hollow conductor with a cavity has the same property: E inside the cavity is zero (assuming no charge inside the cavity), regardless of external fields or charges placed near the conductor. This is the Faraday cage — a thin metal shell that shields its interior from external electric fields, used in electronics shielding rooms, microwave oven doors, and to protect aircraft from lightning strikes.

The magnetic analog of Gauss's law is ∮ B·dA = 0 — total magnetic flux through any closed surface is exactly zero. Equivalently, ∇·B = 0. There are no isolated magnetic charges (north or south poles) acting as sources or sinks of B; field lines always close on themselves rather than terminating on a charge. Cut a bar magnet in half and you get two smaller bar magnets, each with both poles, never an isolated north or south. This empirical fact has been tested for decades — Paul Dirac showed in 1931 that even a single monopole would explain charge quantization, but no monopole has been observed. Modern grand unified theories predict monopoles at extremely high energies, and the MoEDAL experiment at the LHC searches for them, so far without success. The clean ∇·B = 0 of classical electromagnetism remains correct.

For a parallel-plate capacitor with surface charge density σ on each plate (and plates much larger than their separation), planar symmetry lets Gauss's law produce E in one step. Choose a Gaussian pillbox straddling the positive plate. Flux escapes only through the face inside the gap (the other face is inside the conductor where E = 0). The flux is EA, where A is the pillbox cross-section, and the enclosed charge is σA. Gauss gives E = σ/ε₀ between the plates, uniform and perpendicular to the plates. The voltage between plates is V = Ed = σd/ε₀, so the capacitance C = Q/V = ε₀A/d. Add a dielectric of permittivity ε and the formula becomes C = εA/d. The same Gauss-driven approach handles cylindrical and spherical capacitors with their geometry-appropriate Gaussian surfaces.