Green's Function using method of images

[Somefantastik]

Homework Statement

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

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

Homework Equations

none

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?

 

[mighty2000]

Hello! To show that the Green's function is zero on the boundaries, you can use the method of images. By choosing the appropriate images, you can ensure that the potential on the boundaries is zero. For example, for the boundary at z=0, you can choose images at z = +/- 1, +/- 2, etc. to cancel out the potential from the positive and negative charges. Similarly, for the boundary at z=1, you can choose images at z = 1 +/- 1, +/- 2, etc. to cancel out the potential from the positive and negative charges. This will ensure that the Green's function is zero on the boundaries.

To show that the sequences are convergent, you can use the fact that the Green's function is a function of the distance between the two points r and r'. As the distance between the two points increases, the contribution from each image decreases, and the series will converge. You can also use the ratio test or the comparison test to show convergence.

I hope this helps! Let me know if you have any other questions.