# Green's Function using method of images

1. Oct 9, 2008

### Somefantastik

1. The problem statement, all variables and given/known data

$$\Omega = \left{ \left( x,y,z \right) :,0<z<1 \right}$$

Need to find Green's function using the method of images.

2. Relevant equations

none

3. The attempt at a solution

I can see that I will need an infinite sequence of images at each plane z = k, k = 0, +/- 1, +/- 2,... to make the potential on each boundary equal to zero.

let the positive charges be

$$r^{+}_{k} = (x,y,2k+z) \ and\ r^{-}_{k} = (x,y,2k-z)$$

and Green's Function will look like

$$G(r',r) = \frac{1}{4\pi}\sum^{\infty}_{k = -\infty} \left[ \frac{1}{\left|r'-r^{+}_{k} \right| } - \frac{1}{\left|r - r^{-}_{k} \left| } \left] \left}$$

How do I show that G(r',r) = 0 on the boundaries?

I also need to show that the sequences are convergent, and I haven't done that in so long I can't remember how.

Any suggestions?

Last edited: Oct 9, 2008