| 8.6 Gauss's Law |
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Gauss's law — Maxwell's first equation — states that the total electric flux through any closed (Gaussian) surface equals the enclosed charge divided by ε₀. The total flux depends only on the charge inside, not on the size or shape of the surface.
oint E · dA = dfracQₑncε₀
This law becomes a powerful computational tool when the charge distribution has spherical, cylindrical, or planar symmetry, because the Gaussian surface can be chosen so that the electric field is constant over the contributing sections and zero through the rest. The result collapses the surface integral into a simple product of field strength and area.
Spherical symmetry
- A concentric spherical Gaussian surface yields E ∝ 1/r² outside the charge distribution, identical to the field of a point charge located at the center.
- Inside a uniformly charged insulating sphere E ∝ r, whereas inside a conductor E = 0.
Cylindrical symmetry
- A coaxial cylindrical Gaussian surface gives E ∝ 1/r outside an infinite line of charge.
Planar symmetry
- A pillbox-shaped Gaussian surface gives a uniform field E = σ / (2ε₀) for an infinite sheet of charge, independent of distance from the sheet.
When the charge density varies with position, the enclosed charge is found by integrating the density over length, area, or volume — always matching the integration limits to the Gaussian surface, not the physical object. These results form the foundation for understanding capacitors, conductors, and field behavior throughout electrostatics.