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

## Homework Statement

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.

## Homework Equations

##\oint D \cdot da=Q_{free,enc}##

## 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.

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