Gauss's law for electricity

Give every patch of a surface an area vector dA: its magnitude is the patch's area, its direction the outward normal. The flux through that patch is E⋅dA = E cos θ dA, with θ the angle between field and normal, and the flux through the whole surface is the sum, Φ_E = ∫ E⋅dA. The cosine does real work. A field lying in the plane of the patch pierces nothing and contributes zero; a field along the normal contributes everything. Take a uniform E = 2.0 × 10³ N C⁻¹ and a rectangle 0.20 m × 0.30 m, of area 0.060 m². Held square to the field it collects Φ_E = 2.0 × 10³ × 0.060 = 1.2 × 10² N m² C⁻¹. Tilt it so its normal makes 60° with the field and the flux drops to 1.2 × 10² × cos 60° = 60 N m² C⁻¹. Turn it edge-on and the flux is zero although the field never changed. On a closed surface the word outward fixes the sign once and for all: flux leaving counts positive, flux entering negative, and Φ_E carries units of N m² C⁻¹.

Put a point charge q at the centre of a sphere of radius r. The field is radial of magnitude q/(4πε₀r²), the outward normal is radial everywhere, so cos θ = 1 and Φ_E = [q/(4πε₀r²)] × 4πr² = q/ε₀. The r² in the area has cancelled the 1/r² in the field, which is why no radius survives in the answer. Now deform that sphere into any shape you like. Push one patch out to twice the distance and its area grows fourfold while the field there falls fourfold: the flux through it does not move. Only the solid angle a patch subtends at the charge matters, and the solid angles around a fully enclosing surface always total 4π, wherever inside the charge sits. Put the charge outside instead and every line that enters the surface must leave it again, contributing −ΔΦ and then +ΔΦ, so its net flux is exactly zero. Superposition finishes the argument: fields add, so fluxes add, and ∮E⋅dA = q_enc/ε₀ for any closed surface and any arrangement of charge.

Gauss's law is one equation with the unknown field buried in an integral, so it yields E only where you can lift E out. That needs a surface on which two things hold at every point: E has the same magnitude, and E makes the same angle with the outward normal — ideally 0° over the part carrying the flux and 90° over the rest. Exactly three charge symmetries supply such a surface. Spherical symmetry, where the charge density depends only on distance from a centre, is matched by a concentric sphere. Infinite cylindrical symmetry, density depending only on distance from an axis, is matched by a coaxial cylinder whose flat end caps carry no flux because E lies in their planes. Planar symmetry, an infinite charged sheet, is matched by a pillbox straddling it, whose curved side contributes nothing. In each case the integral collapses to E × A and the law becomes one line of algebra. For a finite rod, a cube, or a charged disc anywhere off its axis, no such surface exists: the law is still exactly true and still returns nothing, so you go back to integrating Coulomb's law.

The shape of the Gaussian surface is what sets how fast the field dies away, and all three standard results come from the same line of algebra. Around a point charge or a spherical ball of total charge Q, the effective area is 4πr², so E = Q/(4πε₀r²) and the field falls as 1/r². Around an infinite line of density λ, a coaxial cylinder of length L has curved area 2πrL and encloses λL, so E(2πrL) = λL/ε₀ and E = λ/(2πε₀r), falling as 1/r. The L cancels, which is the algebraic signal that the answer belongs to the line and not to the box you drew round it. For λ = 3.0 × 10⁻⁸ C m⁻¹ at r = 5.0 cm, E = 3.0 × 10⁻⁸ ÷ (2π × 8.854 × 10⁻¹² × 0.050) = 1.1 × 10⁴ N C⁻¹. For an infinite sheet of density σ, a pillbox of face area A encloses σA and leaks flux from both faces, so E(2A) = σA/ε₀ and E = σ/(2ε₀), with no distance in it at all. The pattern is one line long: the area the flux has to spread through grows as r², as r, or not at all.

In electrostatic equilibrium no current flows, so the field inside the conducting material is zero — any surviving field would push the mobile charges until it was cancelled. Draw a Gaussian surface wholly within that material, just under the skin. E = 0 at every point of it, so the flux is zero, so q_enc = 0: no excess charge can sit anywhere in the bulk. Whatever you put on a conductor migrates to its surfaces. Hollow the conductor out and the same argument applies to a surface drawn through the metal around the cavity. With the cavity empty, the flux is zero and the cavity wall carries no net charge. Place +8.0 nC inside the cavity of an initially neutral conductor and the enclosed total must still be zero, so −8.0 nC is induced on the cavity wall and, since the conductor's own charge is conserved, +8.0 nC appears on the outer surface. That is electrostatic shielding, obtained from a flux argument with no field calculation anywhere in it.

Gauss's law is the first of Maxwell's four equations, and it is the only one of the four whose right-hand side is charge. The derivation above was electrostatic, since Coulomb's law describes a charge at rest, but Maxwell's system asserts the law unchanged when charges move and fields vary in time, and no experiment has yet found an exception. Read as geometry, it says that an electric field line can begin only on positive charge and end only on negative charge; where there is no charge a line cannot stop, so in a changing field it simply closes on itself. Apply it to a region containing no charge and the consequence is immediate: the outward flux through every closed surface you can draw there is zero, so as many lines enter as leave. That is the condition an electromagnetic wave crossing empty space has to satisfy, and it is what forbids any component of E along the direction of travel, since a pillbox straddling a wavefront would then collect unbalanced flux. The law that counts charge ends up fixing the shape of a field pattern that has no charge in it at all.