vacuum permittivity, is the relative permittivity, and is the volume charge density (charge per unit volume). Equivalence of integral and differential forms The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically. Equation involving the field Free, bound, and total charge 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 (above), is sometimes put into the equivalent form below, which is in terms of and the free charge only. Integral form This formulation of Gauss's law states the total charge form: where is the -field flux through a surface which encloses a volume , and is the free charge contained in . The flux is defined analogously to the flux of the electric field through : Differential form The differential form of Gauss's law, involving free charge only, states: where is the divergence of the electric displacement field, and is the free electric charge density. Equivalence of total and free charge statements Equation for linear materials In homogeneous, isotropic, nondispersive, linear materials, there is a simple relationship between and : where is the permittivity of the material. For the case of vacuum (aka free space), . Under these circumstances, Gauss's law modifies to for the integral form, and for the differential form. Interpretations In terms of fields of force Gauss's theorem can be interpreted in terms of the lines of force of the field as follows: The flux through a closed surface is dependent upon both the magnitude and direction of the electric field lines penetrating the surface. In general a positive flux is