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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 polarization $\vec{\Phi}(r) =C_1 r \,\hat{r}$; while in region 2, ($r-1 < r < r_2$), there exists a polarization $\vec{\Phi}(r) = C_2 r \,\hat{r}$. There are no free charges anywhere in the cylinder.

What is the electric field, $\vec{E}(r)$ in region 1?
What is the electric field, $\vec{E}(r)$in region 2?
What is the electric field, $\vec{E}(r)$ in the region for $r > r_2$

?

A

$\vec{E}( r < r_)=\cfrac{-C_1 r}{\epsilon_o} \hat{r}$
$\vec{E}(r_1 < r < r_2)=\cfrac{-C_2 r}{\epsilon_o} \hat{r}$
$\vec{E}(r > r_2)=\cfrac{-(C_1 r -C_2 r)\,\hat{r} }{\epsilon_o}$

B

There is no free charge, so the total electric field
in each region is zero.

C

$\vec{E}( r < r_)=\cfrac{-C_1 r}{\epsilon_o} \hat{r}$
$\vec{E}(r_1 < r < r_2)=\cfrac{-C_2 r}{\epsilon_o} \hat{r}$
$\vec{E}(r > r_2)=0\,\text{N/C}$

D

$\vec{E}( r < r_)=\cfrac{C_1 r}{\epsilon_o} \hat{r}$
$\vec{E}(r_1 < r < r_2)=\cfrac{C_2 r}{\epsilon_o} \hat{r}$
$\vec{E}(r > r_2)=\cfrac{(C_1 r +C_2 r)\,\hat{r} }{\epsilon_o}$

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