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A slab of conductor (region 1) with a thickness $a$ exists with a covering of dielectric (region 2) as shown in the accompanying figure. The slab and the dielectric are very large in the $\hat{y}$ and $\hat {z}$ directions.

The conductor extends over the range $0 < x < a$ in the $\hat{x}$ direction, and the dielectric extends over the range $b < x < b$ in the $\hat{x}$ direction. The conductor has an uniform surface charge density $\sigma_o$, and the dielectric has a polarization given by $\vec{P}(x)=C_o x \, \hat{x}$, where $C_o$ is a constant.

What is the value of the electric field $E$ in region 2 ($a < x < b$)?
What is the value of the electric field $E$ in the region ($r > b$)?

A

$E_{region \, 2}(x)=\cfrac{\sigma_o-C_o x}{\epsilon_o}$
$E_{(r > b)}(x)=\cfrac{\sigma_o}{\epsilon_o}$

B

$E_{region \, 2}(x)=\cfrac{\sigma_o-C_o }{x \,\epsilon_o}$
$E_{(r > b)}(x)=\cfrac{\sigma_o}{\epsilon_o}$

C

$E_{region \, 2}(x)=\cfrac{\sigma_o-C_o x}{\epsilon_o}$
$E_{(r > b)}(x)=0 \,\text{N/C}$

D

$E_{region \, 2}(x)=\cfrac{\sigma_o+C_o x}{\epsilon_o}$
$E_{(r > b)}(x)=\cfrac{\sigma_o}{\epsilon_o}$

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