Question

What is the formula for gauss law of dielectrics?

Answer 1

Gauss's Law for dielectrics is a generalization of Gauss's Law in electromagnetism, applied in the presence of dielectric materials. It relates the electric flux passing through a surface to the charge enclosed within that surface, while considering the material's permittivity.

The formula for Gauss’s Law in dielectrics is:

\[
\oint_{\mathcal{S}} \mathbf{D} \cdot d\mathbf{A} = Q_{\text{free}}
\]

Where:

1. \(\mathbf{D}\) is the electric displacement field.
   

1. \(d\mathbf{A}\) is the differential area vector on the closed surface \(\mathcal{S}\).
   

1. \(Q_{\text{free}}\) is the free charge enclosed by the surface.
   

In a dielectric material, the electric displacement \(\mathbf{D}\) is related to the electric field \(\mathbf{E}\) and the permittivity of the material \(\varepsilon\) by:

\[
\mathbf{D} = \varepsilon \mathbf{E}
\]

Here, \(\varepsilon\) is the permittivity of the dielectric material, and \(\mathbf{E}\) is the electric field inside the material.

If the material is linear and homogeneous, the permittivity \(\varepsilon\) can be written as:

\[
\varepsilon = \varepsilon_0 \varepsilon_r
\]

Where:

1. \(\varepsilon_0\) is the permittivity of free space.
   

1. \(\varepsilon_r\) is the relative permittivity (dielectric constant) of the material.
   

Thus, Gauss's law for dielectrics can be rewritten as:

\[
\oint_{\mathcal{S}} \varepsilon \mathbf{E} \cdot d\mathbf{A} = Q_{\text{free}}
\]

This shows how the electric displacement field is related to the free charges within a dielectric material.