A vector quantity defined as
where E is the electric field strength, P the dielectric polarization, and ε0 the permittivity of free space. In a dielectric medium the total charge within any given closed surface consists of any free charges, ρe, in the surface together with the apparent charge density, –div P, due to the polarization. The volume charge density becomes (ρe – div P) and Gauss’s theorem relating to a dielectric is written as
where dτ is a small volume element. It can therefore be shown that the displacement D is given by
In a vacuum P is zero and Gauss’s theorem is written as