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Poissons equation for plasma

  1. May 18, 2009 #1

    In my plasma physics book, poissons equation in relation to a plasma made up of electrons and protons is given as

    [tex]\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\varphi\left(r\right)}{dr}\right) = \frac{2}{\lambda_{D}}\varphi\left(r\right)[/tex]

    The solution of this equation when phi(r) tends to 0 for r tending to infinity is

    [tex]\varphi\left(r\right) = \frac{e}{4\pi\epsilon_{0}r}exp\left(-\frac{\sqrt{2}r}{\lambda_{D}}\right)[/tex]

    How do you get to this solution? I have tried multiplyin both sides by r^2, then integrating w.r.t. r, by parts, but I cannot get any further than that.


  2. jcsd
  3. May 18, 2009 #2


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    Gold Member

    The simplest thing to do would be to substitute using:

    [tex]R(r) = r \varphi (r)[/tex]

    Then you can rewrite it as:

    [tex]\frac{1}{r}\frac{d}{dr} \left( r^2 \frac{d \varphi}{dr} \right) = \frac{2}{\lambda_d}R[/tex]

    You can continue from here by performing the chain rule to place the left hand side in terms of [tex]R[/tex].

    EDIT: Nevermind, I made a mistake earlier on, fixed now. Sorry for any confusion.
    Last edited: May 18, 2009
  4. May 18, 2009 #3
    I have now got a homogeneous 2nd order differential equation and from that an auxillary equation, which is beginning to take the form of the answer. Thanks
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