Gauss's Law
Gauss's Law, ∮E·dA=Q_enc/ε₀, is always true — it's a direct consequence of Coulomb's law, not a special case of it. What makes it valuable here is that for a distribution with enough symmetry (spherical, cylindrical, or planar), you can choose a Gaussian surface on which E is constant and either parallel or perpendicular to the surface everywhere, which lets you pull E straight out of the integral. The same field that took an awkward integral in Unit 201 often takes two lines of algebra here — the tradeoff is that Gauss's Law only solves for E when that symmetry exists.
What you'll learn
- Calculate electric flux through a flat or curved surface, including cases where E is not perpendicular to the surface.
- State Gauss's Law and explain why it holds for any closed surface, not just symmetric ones.
- Apply Gauss's Law to find the electric field of spherically symmetric charge distributions, both inside and outside the charge.
- Apply Gauss's Law to find the electric field of cylindrically symmetric charge distributions (an infinite line or charged cylinder).
- Apply Gauss's Law to find the electric field near an infinite charged sheet or a conductor's surface.
- Use Gauss's Law in reverse: given a field, determine the enclosed charge.
1. Electric Flux
Electric flux measures how much of a field "flows through" a surface: Φ=∫E·dA, where dA is a differential patch of area treated as a vector pointing along the surface's outward normal. Only the component of E parallel to that normal contributes — a field running exactly along the surface (perpendicular to the normal) contributes zero flux.
For a flat surface in a uniform field, this reduces to Φ=EA cosθ, where θ is the angle between E and the surface's normal — maximum when the field hits the surface head-on, zero when the field grazes it edge-on.
2. Gauss's Law and Spherical Symmetry
Gauss's Law states that the total flux through any closed surface equals the enclosed charge divided by ε₀: ∮E·dA=Q_enc/ε₀. This holds for every closed surface and every charge configuration — it's a restatement of Coulomb's law, not an additional assumption.
It becomes a fast way to find E specifically when you can choose a Gaussian surface on which E has constant magnitude and a fixed angle to the surface everywhere — which happens for spherically symmetric charge (a sphere or shell), using a concentric spherical Gaussian surface. Outside a uniformly charged sphere, the enclosed charge is always the full Q, giving the familiar point-charge result. Inside, only the charge within radius r counts, which grows with r³ for uniform volume density while the "point-charge" 1/r² factor shrinks — the net result is a field that grows linearly with r.
3. Cylindrical and Planar Symmetry
For an infinite line charge (or, outside its own radius, a uniformly charged infinite cylinder), the natural Gaussian surface is a coaxial cylinder. The field turns out to fall off as 1/r rather than 1/r² — slower than a point charge, because the source extends infinitely along one dimension instead of being localized.
For an infinite charged sheet, a "pillbox" Gaussian surface straddling the sheet gives a field that doesn't depend on distance from the sheet at all: E=σ/(2ε₀), constant everywhere on either side. A conductor's surface gives exactly double that, σ/ε₀, because a conductor's own interior field is zero, so all its surface charge's field lines are forced to point outward on the exposed side only, rather than splitting evenly in both directions the way an isolated sheet's do.
Key equations
- Φ = ∫E·dA — The definition of electric flux through any surface — only the component of E perpendicular to the surface contributes.
- ∮E·dA = Q_enc/ε₀ — Gauss's Law — true for any closed surface, but only useful for solving E directly when symmetry lets you pull E out of the integral.
- E = kQ/r² (outside), E = kQr/R³ (inside) — Field of a uniformly charged solid sphere (insulator). Outside looks exactly like a point charge; inside grows linearly with r, reaching the same value as the outside formula right at r=R.
- E = λ/(2πε₀r) = 2kλ/r — Field of an infinite line charge (or, outside its own radius, an infinite charged cylinder) — falls off as 1/r, not 1/r², because the charge extends infinitely in one dimension.
- E = σ/(2ε₀) [sheet], E = σ/ε₀ [conductor surface] — Field near an infinite charged sheet (uniform, independent of distance) vs. just outside a conductor's surface — the conductor case is exactly double, because all the field lines from a conductor's surface charge go one direction (outward) instead of splitting both ways as they do for an isolated sheet.