A point charge $q$ is located at the centre of a ball made of uniform isotropic dielectric with rela

Question

A point charge $q$ is located at the centre of a ball made of uniform isotropic dielectric with relative permittivity $\varepsilon$. Which of the following gives the correct expression for the polarization vector $\mathbf{P}$ as a function of the radius vector $\mathbf{r}$ relative to the centre, and the total charge $q'$ inside a concentric sphere of radius $r$ (where $r$ is less than the radius of the ball)?

Professor PC Thomas Classes · Answer

By Gauss's Law for $\mathbf{D}$, $\oint \mathbf{D} \cdot d\mathbf{A} = q \Rightarrow \mathbf{D} = \frac{q}{4\pi r^3}\mathbf{r}$. Electric field $\mathbf{E} = \frac{\mathbf{D}}{\varepsilon_0 \varepsilon}$. Polarization $\mathbf{P} = \varepsilon_0(\varepsilon-1)\mathbf{E} = \frac{(\varepsilon-1)q}{4\pi \varepsilon r^3}\mathbf{r}$. Bound charge at origin is $q_b = -\oint \mathbf{P} \cdot d\mathbf{S} = -q\left(1-\frac{1}{\varepsilon}\right)$. Total charge inside sphere $r$ is $q' = q + q_b = \frac{q}{\varepsilon}$.