Free Version
Difficult

# Fields in Spherical Dielectrics with Polarization and Free Charge

EANDM-1OBSJ1

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 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 $\vec{P}(\vec{r})=\cfrac{C }{r^2} \hat{r}$.

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

A

$E(r_1 < r < r_2)=\cfrac{\rho_o}{2 \epsilon_o r^2} \bigg[(2/3)r_1 ^3 -(2C/\rho_o) \bigg] + \cfrac{\rho_o r_2}{2 \epsilon_o}$

B

$E(r_1 < r < r_2)=\cfrac{\rho_o}{2 \epsilon_o r^2} \bigg[(2/3)r_1 ^3 -(r_2)r_1^2 -(2C/\rho_o) \bigg]$

C

$E(r_1 < r < r_2)=\cfrac{\rho_o}{2 \epsilon_o r^2} \bigg[(2/3)r_1 ^3 -(r_2)r_1^2 -(2C/\rho_o) \bigg] + \cfrac{\rho_o r_2}{2 \epsilon_o}$

D

$E(r)=\cfrac{ \rho_o \bigg((2/3)r_1 ^3 -(r_2) r_1 ^2 +(r_2) r ^2 \bigg)}{2 \epsilon_o r^2}$