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What is the Gauss law and Gauss law in dielectrics?

Gauss law

According to Gauss law “the net electric flux through any closed surface in an electric field is equal to the times the total charge present inside that surface”. i.e.,

Gauss law in dielectrics

Let us consider a parallel plate capacitor made of two plates with equal area A and equal surface charge density σ. There is a vacuum between the plates. Then, we can evaluate the electric field vector E0 in the region between the plates by using Gauss` law.

Let us consider a Gaussian surface* enclosing the positively charged plate, the shape of which will be a cuboid as shown in the Fig. (a). Since one face of the Gaussian surface is in the plate, no flux will pass through it. Flux will also not pass through the faces those are perpendicular to this face (Because they are parallel to the electric field lines). Thus, flux is only passing through the face, which is parallel to the positive plate and

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*Gaussian surface is an imaginary (it may be real in some cases) surface around a source of the electric field which is either parallel or perpendicular to the electric field direction and the magnitude of an electric field is constant or zero over sections of the surface.

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In the presence of dielectric material between the plates of the capacitor, a charge q‘ of the opposite sign is induces on its surface due to polarization as shown in Fig. (b). Now the net charge enclosed by the Gaussian surface is q – q‘ and due to the induced charge, the electric field in the region between the plates is also getting reduced. Suppose the modified electric field Vector E is, then Gauss`s law becomes