Electrostatics in 2 and 1 Dimensions

Hi, I'm having a bit of a hard time stumbling over the concepts of the following problem:

Homework Statement

In Electrostatics:
How do you modify the div and curl of the electric field from 3D to 2D?
What are the 2D and 1D versions of Coulomb's Law?

Homework Equations

In 3D (sorry, no latex here): del dot E = 4*pi*rho
Del Cross E = 0
Coulomb's law in 3D: F= ((1/4*pi*eps)qQ/r^2) r hat
wow, that looks ugly, sorry about that.

The Attempt at a Solution

I would think the divergence wouldn't change from 3D to 2D since it lives in two planes in both case. BUT what about the 4*pi*rho? it looks suspiciously like it's geometry dependent... and if so, could change in 2D...

As for the Curl: My best guess would be that it would be zero, since there is no perpendicular plane for it to live. Or would it be a scalar? I don't even know how to set up the determinant for this... or.... can you even use a det to find the curl (i.e. is that method distinct for 3D, since 3D is pretty special and not like most other n-dimensions.

And, similar for coulomb's law: the geometry is tripping me up. The 3D radial dependence of the charges have to be modified for 2D and 1D... and is there geometry in the prefactor?

Thanks very much!