Gaussian surface

A Gaussian surface is an imaginary three-dimensional closed surface drawn in space so that Gauss's law can be applied to it. Only the charge enclosed by the surface contributes to the net flux through it; charge outside contributes exactly zero.

A Gaussian surface is imaginary. Nothing is built and nothing is placed: the surface is a closed boundary you draw, and it may cut through solid matter or empty space as freely as you like.

Gauss's law relates the electric flux through such a surface to the charge inside it:

∮E⃗⋅dA⃗=qencε0\oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}

Only enclosed charge appears on the right. A charge sitting outside still has field lines threading the surface, but every line that enters also leaves, so its net contribution is not small, it is zero. The total flux is also independent of the size of the surface as long as the enclosed charge is unchanged: inflate a sphere around a point charge and the field falls as 1/r21/r^2 while the area grows as r2r^2.

Choosing the surface is where the skill sits. You pick one on which the field is either perpendicular to the surface with constant magnitude, or parallel to it and contributing nothing, so the integral collapses to a product. Only symmetry allows that, and the CED's boundary statement limits quantitative work to point charges and to distributions with spherical, cylindrical or planar symmetry.

The name is a substitute for the law itself here: the Gauss's law guide walks the three symmetric cases with numbers.

Back to the physics glossary