Editing Gauss's law (section)

==Equation involving the {{math|D}} field==
{{see also|Maxwell's equations}}

===Free, bound, and total charge===
{{Main article|Electric polarization}}

The electric charge that arises in the simplest textbook situations would be classified as "free charge"—for example, the charge which is transferred in [[static electricity]], or the charge on a [[capacitor]] plate. In contrast, "bound charge" arises only in the context of [[dielectric]] (polarizable) materials. (All materials are polarizable to some extent.) When such materials are placed in an external electric field, the electrons remain bound to their respective atoms, but shift a microscopic distance in response to the field, so that they're more on one side of the atom than the other. All these microscopic displacements add up to give a macroscopic net charge distribution, and this constitutes the "bound charge".

Although microscopically all charge is fundamentally the same, there are often practical reasons for wanting to treat bound charge differently from free charge. The result is that the more fundamental Gauss's law, in terms of {{math|'''E'''}} (above), is sometimes put into the equivalent form below, which is in terms of {{math|'''D'''}} and the free charge only.

===Integral form===

This formulation of Gauss's law states the total charge form:

<math display="block">\Phi_D = Q_\mathrm{free}</math>

where {{math|Φ<sub>''D''</sub>}} is the [[electric displacement field|{{math|'''D'''}}-field]] flux through a surface {{mvar|S}} which encloses a volume {{mvar|V}}, and {{math|''Q''<sub>free</sub>}} is the free charge contained in {{mvar|V}}. The flux {{math|Φ<sub>''D''</sub>}} is defined analogously to the flux {{math|Φ<sub>''E''</sub>}} of the electric field {{math|'''E'''}} through {{mvar|S}}:

:{{oiint|preintegral=<math>\Phi_D = </math>|intsubscpt=<math>{\scriptstyle _S}</math>|integrand=<math>\mathbf{D} \cdot \mathrm{d}\mathbf{A} </math>}}

===Differential form===

The differential form of Gauss's law, involving free charge only, states:
<math display="block">\nabla \cdot \mathbf{D} = \rho_\mathrm{free}</math>

where {{math|∇ · '''D'''}} is the [[divergence]] of the electric displacement field, and {{math|''ρ''<sub>free</sub>}} is the free electric charge density.