Bound charges of a block (top and bottom surface)

[TSny]

For dielectrics that are linear, isotropic, and homogeneous, we have $\vec P = \epsilon_0 \chi_e \vec E$ where $\chi_e$ is a constant related to the relative permittivity, $\epsilon_r$: $\chi_e = \epsilon_r - 1$.

For these dielectrics, it is not hard to derive a useful relation between the bound charge density $\rho_{b}$ and free charge density $\rho_f$ that holds for any point inside the dielectric.

$\large \rho_b = - \frac{\chi_e }{\epsilon_r}\rho_f \,\,\,\,\,$ (This type of relation does not hold for the the surface charge densities $\sigma_b$ and $\sigma_f$.)

You can use this to get the bound volume charge density $\rho_b$ inside the electron layer. Then you can check that the total bound charge of the dielectric is zero, as it must be.