Question

Linear dielectrics II-Find all the fields and charges due to a point charge q embedded at the center of a linear dielectric sphere of radius R and permittivity e surrounded by free space. Remember to find all the free and bound charges.

Answer 1

Key concepts :

$\vec{D}=\epsilon\vec{E} \newline \vec{P}=\epsilon_{0}\chi_{e}\vec{E} \newline \epsilon=\epsilon_{0}(1+\chi_{e}) \newline \oint \vec{D}\cdot d\vec{a}=Q_{free} \newline \rho_{b}=-\nabla\vec{P} \newline \sigma _{b}=\vec{P}\cdot \hat{n}$

Here All the symbols carrying their usual meaning.

The electric field in the bulk of sphere will create polarisation in the dielectric. Thus bound charges will be introduced in the volume of the sphere and the surface as well.

outside the sphere the electric field will same as if the dielecrtric sphere were not there, as the bound charges inside the sphere and on the surface of the sphere will be cancelled out exactly. We call find the $\vec{E}$ inside the sphere by applying Guass law for displacement vector in both inside and outside and using the relation $\vec{D}=\epsilon \vec{E}$ inside dielectric and $\vec{D}=\epsilon_{0} \vec{E}$ outside dielectric.

from $\vec{E}$ we find $\vec{P}$ using the relation

$\vec{P}=\epsilon_{0}\chi_{e}\vec{E} \newline \chi_{e}=\frac{\epsilon}{\epsilon_{0}}-1$

$\vec{P}=(\epsilon-\epsilon_{0})\vec{E}$

finally expression for $\rho_{b}$ and $\sigma_{b}$ are used for finding the bound charges.