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Green's Function using method of images

  1. Oct 9, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]\Omega = \left{ \left( x,y,z \right) :,0<z<1 \right} [/tex]

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

    2. Relevant equations


    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

    [tex]r^{+}_{k} = (x,y,2k+z) \ and\ r^{-}_{k} = (x,y,2k-z) [/tex]

    and Green's Function will look like

    [tex] 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}[/tex]

    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
  2. jcsd
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