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A sphere of linear dielectric material surrounded by another dieletric material

  1. Nov 12, 2014 #1
    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. jcsd
  3. Nov 17, 2014 #2
    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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