A sphere of linear dielectric material surrounded by another dieletric material

  • #1
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1

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.
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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