# A sphere of linear dielectric material surrounded by another dieletric material

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1. Nov 12, 2014

### ghostfolk

1. The problem statement, all variables and given/known data
A sphere of linear dielectric material with permittivity $\epsilon_1$ and radius $a$ is surrounded by an infinite region of linear permittivity $\epsilon_2$. In the spherical region, there is free charge embedded given by $\rho_{free}=\beta r^2$, $0<r<a$, where $\beta$ is a constant and $r$ is the distance from the center of the sphere. Find the electric displacement $\vec{D}$ in all space.
2. Relevant equations
$\oint D \cdot da=Q_{free,enc}$
3. The attempt at a solution
Inside the sphere:
$\oint D \cdot da=4 \pi r^2$
$Q_{free,enc}=\begin{cases} \frac{4}{5} \beta \pi r^5 \hat{r}, r<a \\ \frac{4}{5} \beta \pi a^5 \hat{r}, r>a \end{cases}$
Here is where I am confused. I know that inside $\vec{D}=\frac{\beta}{5}r^3 \hat{r}$, but I'm not sure about how to find $\vec{D}$ outside. Normally $\vec{D}$ would be the same inside and out if the sphere were in empty space, but since it is now surrounded by a dieletric I am confused. Any help is appreciated.

2. Nov 17, 2014