[Physics] Is Gauss law still true in dielectric material

Question

In vacuum we have
$$\nabla \cdot \mathbf{E} = \frac {\rho}{\varepsilon_0}.$$

Can we still use this formula when there's dielectric material in space? Where $\rho$ is total charge density.

Answer 1

There is a modified form of Gauss's law for dielectrics.

Inside a dielectric there is an induced charge that screens the electric field. The bound charge density $\rho_b = - \vec\nabla . \vec P$ where $\vec P$ is the polarisation vector.

This bound charge acts as a source of electric field, so Gauss' law reads

$\vec \nabla . \vec E = \frac{1}{\epsilon_0}(\rho_f + \rho_b)$

We replace the electric field with a so called displacement vector $\vec D = \epsilon_0 \vec E + \vec P$ and thus we can write Gauss' law as

$\vec \nabla . \vec D = \rho_f$

This is the form of Gauss' law to be used inside a dielectric.

See chapter 4 in Griffith's Introduction to Electrodynamics for more information.

http://physics.sfsu.edu/~wcaudy/GriffithsE&M.pdf