Electric field of periodic charge density.

[JoshThompson42]

Homework Statement

Find electrostatic field and potential created by a two-dimensional charge density:
$$\rho \sin (kx) \cos (ky) \delta (z)$$
at the distance d from the the plane z=0 where the charge is placed (taking into account that it is embedded in a three dimensional space).
In your calculations you are required to use Fourier analysis.

Homework Equations

The Attempt at a Solution

My initial thought was to use the differential form of Gauss's law:
$$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$

However I am unsure of where Fourier analysis comes into play, any pointers as to where to go from here would be great. My instinct tells me that the delta function should be what gets the Fourier treatment, however it isn't periodic.

 

[mfb]

What is the potential for a point-charge? To extend this to 2-dimensional charge distributions, you'll need an integral. And I guess the evaluation of this integral will need Fourier analysis.

 

[Goddar]

What is the potential for a point-charge? To extend this to 2-dimensional charge distributions, you'll need an integral. And I guess the evaluation of this integral will need Fourier analysis.

To make Fourier analysis more obvious, I would start from Poisson's equation for the potential: given the charge distribution, you have to guess oscillating functions for the x and y components which leads to Fourier analysis to determine the coefficients. The z-component is less obvious though, and you'd have to use the ±z symmetry of the problem... Incidentally, the full solution for arbitrary z comes rather easily using Fourier transform.