The normal electric field strength, E, over a closed surface containing electric charges is given by

where dS is a small element of area on the surface S, ε₀ is the permittivity of free space, and Σq the total charge enclosed within the surface.

If the volume enclosed by the surface contains a distributed charge density, ρ_e, Gauss’s theorem becomes

where dτ is a small volume element. Since

then

In a dielectric medium the total charge density includes an apparent charge density due to polarization of the atoms or molecules within the medium and Gauss’s theorem becomes

where D is the electric displacement. This is one of Maxwell’s equations.

Two main conclusions can be drawn from Gauss’s theorem. Firstly, the electric field inside a hollow conductor containing no charge is zero, the enclosed space being at the same potential as the conductor: electrical apparatus can therefore be shielded from the effect of external low-frequency electric fields by surrounding it with an earthed conductor. Secondly, any excess static charge on a conductor must reside on the outer surface.