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Electricity · Physics

Gauss’s law turns symmetry into a field.

Electric flux through a closed surface depends only on the charge enclosed. With the right symmetry, Gauss’s law gives the electric field with very little algebra.

This topic

Gauss’s Law

Learn the statement, the symmetry logic, and classic applications using spherical, cylindrical, and planar Gaussian surfaces.

Law
Statement of Gauss’s law
Gauss’s law relates the net electric flux through any closed surface to the net charge enclosed by that surface.
  • Closed-surface flux: total Φ through the surface
  • Enclosed charge: Qenc only (not outside charges directly)
  • Outward normal convention for signs
  • How this differs from Coulomb’s law in use
Start nano-lesson Gauss is about closed surfaces.
Strategy
Symmetry requirements
Gauss’s law is always true, but it only gives an easy field calculation when symmetry makes the field constant (or piecewise constant) on a chosen surface.
  • Goal: pick a surface where E is constant on parts of it
  • Exploit symmetry: spherical, cylindrical, planar
  • Know when it will not simplify a problem
  • Separation: using E·dA with constant angle
Start nano-lesson Gauss is powerful when E is uniform on the surface.
Tool
Gaussian surfaces
A Gaussian surface is an imaginary closed surface chosen to match the symmetry of the charge distribution. It is not a physical object; it is a mathematical device.
  • Match the surface to the symmetry
  • Choose outward normal consistently
  • Separate where E is perpendicular/parallel to dA
  • Common choices: sphere, cylinder, pillbox
Start nano-lesson Pick the surface that makes E·dA simple.
Applications
Spherical, cylindrical, and planar symmetry
Gauss’s law yields the field directly for symmetric charge distributions when you choose an appropriate Gaussian surface.
  • Spherical: point charge, charged sphere (outside/inside cases)
  • Cylindrical: infinite line charge and long charged cylinder
  • Planar: infinite sheet of charge (pillbox surface)
  • Interpreting the field direction from symmetry
Start nano-lesson Let symmetry tell you the direction first.
Practice
Practice & Exercises
Practice choosing Gaussian surfaces, computing enclosed charge, and solving for E using symmetry.
  • Decide whether Gauss’s law will simplify a problem
  • Pick the correct Gaussian surface (sphere/cylinder/pillbox)
  • Compute Qenc for uniform volume, surface, or line charge
  • Solve for E in spherical/cylindrical/planar setups
  • Exam-style Gauss problems (conceptual + computational)
Open practice sets Start with symmetry: direction, then magnitude.