How to Determine Electric Field and Potential in an Infinitely Charged Plate?

[dingo_d]

Homework Statement

Infinite plate is charged with symmetrically due to its central plane x = 0. Volume charge density is equal to $$\rho = \rho(| x |)$$. Determine the electric field E and potential $$\phi$$ within the panel. Take that the potential vanishes at points of the central plane.

Homework Equations

The electric field:

$$\vec{E}(\vec{r})=\int\frac{\rho(\vec{r}')(\vec{r}-\vec{r'})}{|\vec{r}-\vec{r}'|^3}d^3r'$$, and potential is

$$\phi(\vec{r})=\int\frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d^3r'$$

The Attempt at a Solution

What I'm having problem with is setting this up.

So I have an infinite charged plate, and I'm assuming it looks like this:

[PLAIN]http://img243.imageshack.us/img243/9877/73983825.png

And it goes to infinity in those directions. How to set up the coordinate system? Do I use some random point P away from the plate and look the E and $$\phi$$ there or do I use the method of images?

I'm totally lost :\

 

[Bhumble]

I'd stick with Cartesian coordinates for it. Then use the differential form of Gauss' Law. $$\nabla$$$$\bullet$$E=$$\frac{\rho}{\epsilon_{o}}$$

 

[dingo_d]

Ok, but still I need to calculate the electric field, I could use $$\nabla\cdot\vec{E}=4\pi \rho$$ but I'd still need to integrate it to get E...

 

[dingo_d]

Soo... Any help? :\

 

[Bhumble]

I think the point is that if you integrate along the x-axis you get $$\int$$$$\frac{\rho(x)}{\epsilon_0}$$dx$$\widehat{x}$$