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Consider a long cylinder, a cross-section of which is shown in the accompanying figure. There are two regions, each cylindrically symmetric, centered at the origin. In the core region 1, ($r < r_1$) there exists a constant free charge density $\rho = \rho_1$, but no polarization; while in region 2, ($r-1 < r < r_2$), there exists a polarization $\vec{\Phi}(r) = C_2 r \,\hat{r}$ but no free charge density.

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

A

$\vec{E}(r_1 < r < r_1)=\bigg(\cfrac{ \rho_1 r_1 ^2 }{2 \epsilon_o r }-\cfrac{C_2 r}{\epsilon_o} \bigg)\,\hat{r}$

B

$\vec{E}(r_1 < r < r_1)=\cfrac{ \rho_1 r_1 ^2 }{2 \epsilon_o r }$

C

$\vec{E}(r_1 < r < r_1)=\bigg(\cfrac{ \rho_1 r_1 ^2 }{2 \epsilon_o r }-\cfrac{C_2 r}{\epsilon_o} \bigg)\,\hat{r}$

D

$\vec{E}(r_1 < r < r_1)=-\cfrac{C_2 r}{\epsilon_o} \,\hat{r}$

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