Coulomb's Law in Dielectric Media

The situation is of course more complicated when the charge assembly resides in a material medium. Materials may be classified as conductors or insulators. For a conductor (e.g., most metals), all net charge resides on the surface of the conductor and distributes itself such that the electric field is zero inside the material. For insulators, or ``dielectrics" (e.g., glass, water...), the medium modifies Coulomb's law in a simple way. For two point charges in a medium characterized by dielectric constant $\epsilon$ the Coulomb potential is reduced by a factor of $\epsilon$ ($\epsilon =1$ in vacuum and $\epsilon \gt 1$ in a material medium):

$$\vec F = q_1 q_2 (\vec r_2 - \vec r_1)/\epsilon \vert\vec r_2 - \vec r_1\vert^3$$ (4)

The corresponding statement of Gauss's Law is:

$$\vec \nabla \cdot (\epsilon (\vec r) \vec E (\vec r)) = 4\pi \rho (\vec r)$$ (5)

and of Poisson's Eq. is:

$$\vec \nabla \cdot (\epsilon (\vec r) \vec \nabla \phi (\vec r)) = - 4\pi \rho(\vec r)$$ (6)

In Eqs. 5 and 6, $\rho$ is the density of free charge and possible spatial variation of the dielectric constant has been included. If there is no spatial dependence to the dielectric constant, then it comes outside on the l.h.s. of Eqs. (5) or (6). In this case the standard forms of Gauss' Law and Poisson's Eq. apply, with the source charge density reduced by a constant factor $\epsilon$.