Question

A parallel plate capacitor has a dielectric (of dielectric constant $k$) between the plates. The plates are charged to a surface charge density ${\sigma }_0$. What is the density of the polarized charge that appear on the surface of the dielectric.

Answer 1

The intensity of the field in the dielectric medium inside a capacitor, $E=\frac{{\sigma }_0}{k{\in }_0}$
Let the intensity of the field created by free charges is $E_0=\frac{{\sigma }_0}{{\in }_0}$ and that by the polarized charges be $E^{\prime }=\frac{{\sigma }^{\prime }}{{\in }_0}$. The resultant intensity $E=E_0-E^{\prime }$
$\frac{{\sigma }_0}{k{\in }_0}=\frac{{\sigma }_0}{{\in }_0}-\frac{{\sigma }^{\prime }}{{\in }_0}⟹{\sigma }^{\prime }={\sigma }_0(1-\frac{1}{k})={\sigma }_0\left(\frac{k-1}{k}\right)$