Surface Charge Distrib on Plane

[jesuslovesu]

Homework Statement

The plane z = 0 is charged to a density $$\sigma = \sigma_0 sin(\alpha x) sin(\beta y)$$

Find the potential.

Homework Equations

The Attempt at a Solution

Well I first thing I would normally do is use Gauss's Law to find E
$$E = \frac{\sigma}{2e0}$$ for an infinite plane, however in this case I don't appear to be able to just plug it in like that.

My next thought would be to find the total charge Q, but how does one do that when the plane is infinite?

 

[R.Harmon]

I think you might have to use $$\vec{E}=-\vec{\nabla}.V$$, where $$\vec{\nabla}.V=\frac{dV}{dx}\hat{x}+\frac{dV}{dy}\hat{y}+\frac{dV}{dz}\hat{z}$$.

So if you know the electric field acts in the z direction you can say that $$\frac{dV}{dz}=\frac{-\sigma}{2\epsilon_0}$$, so you can integrate w.r.t to z to find the potential. I think that's how you would do it anyway, hope this helps.