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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 a free charge density $\rho_o$ but no polarization, while the material in the outer region (2) has no free charge density, but does possess a polarization given by $\vec{P}( \vec{r})=C \vec{r}$.

What is the value of the displacement $\vec{D} \,$ in the core region 1, ($r < r_1$), and in region 2, ($r_1 < r < r_2)$?

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$D(r < r_1) = \cfrac{\rho_o r}{3}$
$D(r > r_1)=\cfrac{\rho_o r_1 ^3}{3 C r^2}$


$D(r < r_1) = \cfrac{\rho_o r}{3}$
$D(r > r_1)=\cfrac{\rho_o r_1 ^3}{3 r^2}$


$D(r < r_1) = \cfrac{\rho_o r}{3}$
$D(r > r_1)=\cfrac{\rho_o C r_1 ^3}{3 r^2}$


$D(r < r_1) = \cfrac{\rho_o r}{3 \epsilon_o}$
$D(r > r_1)=\cfrac{\rho_o r_1 ^3}{3 \epsilon_o r^2}$

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