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# Electric Field in a Sphere with Dielectric Polarization Only

EANDM-2GBYS0

A sphere, composed of dielectric material, has a core (region 1), $r < r_1$, and an outer layer (region 2), $r_1 < r < r_2$, each centered at the origin. The material in the core has no free charges, but a polarization $\vec{P_1}(\vec{r})=Cr \hat{r}$. The material in the outer region (2) has no free charges but a different polarization given by $\vec{P_2}(\vec{r})=\cfrac{C }{r^2} \hat{r}$.

What is the electric field $\vec{E}(\vec{r})$ in the region 0, ($0< r < r_1$), and region 1, ($r_1 < r < r_2$)?

A

$\vec{E}(0 < r < r_1)=\cfrac{-Cr}{\epsilon_o } \hat{r}$
$\vec{E}(r_1 < r < r_2)=\cfrac{-Cr}{\epsilon_o } \hat{r}+\cfrac{-C}{\epsilon_o r^2 } \hat{r}$

B

$\vec{E}(0 < r < r_1)=\cfrac{-C}{\epsilon_o r^2} \hat{r}$
$\vec{E}(r_1 < r < r_2)=\cfrac{-C}{\epsilon_o r^2 } \hat{r}$

C

$\vec{E}(0 < r < r_1)=0$
$\vec{E}(r_1 < r < r_2)=0$

D

$\vec{E}(0 < r < r_1)=\cfrac{-Cr}{\epsilon_o } \hat{r}$
$\vec{E}(r_1 < r < r_2)=\cfrac{-C}{\epsilon_o r^2 } \hat{r}$