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A long cylinder, 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, as shown in the accompanying figure. The material in the core has a constant free charge density $\rho_o$ but no polarization, while the material in the outer region (2) has a free charge density $\rho=\rho_o (r_2 /r)$ and a polarization given by $\vec{P}(\vec{r})=Cr \, \hat{r}$.

What is the value of the electric field vector $\vec{E} \,$ in region 2, ($r_1 < r < r_2)$?

A

$E(r)=\cfrac{1}{\epsilon_o} \bigg(\cfrac{ \rho_o}{2 r} (r_1^2 -2 r_2 r_1 + 2 r_2 r)\bigg)$

B

$E(r)=\cfrac{1}{\epsilon_o} \bigg(\cfrac{ \rho_o}{2 r} (r_1^2 -2 r_2 r_1 + 2 r_2 r)-Cr\bigg)$

C

$E(r)=\cfrac{1}{\epsilon_o} \bigg(\cfrac{ \rho_o}{2 r} (r_1^2 +2 r_2 r_1 - 2 r_2 r)+Cr\bigg)$

D

$E(r)=\cfrac{1}{\epsilon_o} \bigg(\cfrac{ \rho_o}{2 r} (r_1^2 -2 r_2 ^2 + 2 r_2 r)-\epsilon_o Cr\bigg)$

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