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I've been looking around but I haven't been able to find any way of solving this conceptually simple problem.
In short I am trying to find the charge density in a dielectric in an external electric field given, \begin{equation} \overline{p}(\overline{r})= \int \rho(\overline{r}_{0},\overline{r}) \ \overline{r}_{0} \ dV_{0} \end{equation}

In my case $\overline{p} = \alpha \overline{E}$ where $\alpha$ is the polarizability and $\overline{E}$ is the applied electric field. I'm not sure really how to approach this. I've looked into solving integral equations using Mathematica but nothing I've seen seems quite like the problem I have.

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closed as off-topic by Jens, Rahul, ciao, Oleksandr R., m_goldberg Apr 19 '14 at 23:43

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As it is written right now, your question is too general for the context of a Mathematica forum because you are presenting only a formal integral definition and then hint at the fact that the charge distribution is induced by an applied electric field. If you are indeed facing an integral equation, then you have to be more specific about what you have already tried in Mathematica, maybe giving a specific form of the field. –  Jens Apr 19 '14 at 16:54
The problem I am trying to solve is \begin{eqnarray} A sin(kz)e^{\frac{x^2+y^2}{w_0}}\frac{32(-4x^3+3xw_0^2)(-4y^3+3yw_0^2)}{w_0^6} = \int_{-s}^{s} \int_{-s}^{s} \int_{-s}^{s} \rho(x,y,z,x_{0},y_{0},z_{0}) x_{0} dx_{0}dy_{0}dz_{0} \\ 0 = \int_{-s}^{s} \int_{-s}^{s} \int_{-s}^{s} \rho(x,y,z,x_{0},y_{0},z_{0}) y_{0} dx_{0}dy_{0}dz_{0} \\ 0 = \int_{-s}^{s} \int_{-s}^{s} \int_{-s}^{s} \rho(x,y,z,x_{0},y_{0},z_{0}) z_{0} dx_{0}dy_{0}dz_{0} \end{eqnarray} As I mentioned in the post I'm not sure how to approach this as this problem seems closest to an integral equation with unknown "kernel". –  user13822 Apr 19 '14 at 18:45
Just to clarify $w_0 \textrm{ and } A$ are real constants. –  user13822 Apr 19 '14 at 18:47
And now what's missing are some properties of the function $\rho$ because it is probably a "moving average" of the charge density. Otherwise it wouldn't have two vector arguments but one. Not knowing these properties, there is too much arbitrariness in the problem. –  Jens Apr 19 '14 at 19:18
This question appears to be off-topic because it is about a physics concept without which the mathematical problem is ambiguous. –  Jens Apr 19 '14 at 19:37

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