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Homework Statement
We have a charged, insulating slab of thickness 2L in the z direction, and very large (read infinite) in the x and y directions. The slab contains a charge per unit volume that varies linearly from [tex]-\rho_{0}[/tex] to [tex]\rho_{0}[/tex], from one side of the slab to the other. Specifically, taking the slab to extend from -L to +L in the z direction, the charge densiy is [tex]\rho(z)=\rho_{0}z/L[/tex]. Find the electric field everywhere inside the slab (magnitude and direction), and the potential difference between the two edges of the slab.
Homework Equations
Despite the planar symmetry, it seems that Gauss' law is not applicable due to the non-uniform charge density. That leaves me with "Coulomb's law",
[tex]\textbf{E}(\textbf{r})={1\over 4\pi\epsilon_{0}} \int {\rho(\textbf{r}\bf{'})\over \cursive{r}^{2}}{\bf\hat\cursive{r}}d\tau'[/tex]
A bit of clarification on the vectors: [tex] \textbf r [/tex] is the vector from the origin to the field point of interest. [tex]{\textbf r}{\bf '}[/tex] is the vector from the origin to the location of the volume element (the charge). And [tex]\bf\cursive{r}[/tex] is the vector from the volume element (charge) to the field point, [tex]{\bf\cursive{r}} = {\textbf r} - {\textbf r}{\bf '}[/tex].
Note the odd formatting:[tex]\cursive{r}\neq \tau[/tex]. Rather, [tex]\tau'[/tex] is the volume element. For reference, this is all from Chapter 2 of Griffiths' Intro to Electrodynamics.
The Attempt at a Solution
So far all my attempts have been in vain. I've managed to come up with E=0 in a number of different ways... all obviously wrong. My latest attempt has been to take "slices" of the slab in the x-y plane. An "infinite" charged plane has, using Gauss' law,
[tex]{\textbf E}={\sigma \over 2 \epsilon_{0}}\hat{\textbf n}[/tex].
From here, it should be possible to integrate from z=-L to +L, where the surface charge density, [tex]\sigma[/tex], would be vary as a function of z as in the volume charge density above. Unfortunately, I haven't been able to work it out.
Any ideas?