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Poisson equation general solution

  1. Oct 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Given that [tex]\nabla[/tex]2 1/r = -4[tex]\pi[/tex][tex]\delta[/tex]3(r)

    show that the solution to the Poisson equation [tex]\nabla[/tex]2[tex]\Phi[/tex] = -([tex]\rho[/tex](r)/[tex]\epsilon[/tex])

    can be written:

    [tex]\Phi[/tex](r) = (1/4[tex]\pi[/tex][tex]\epsilon[/tex]) [tex]\int[/tex] d3r' ([tex]\rho[/tex](r') / |r - r'|)


    2. Relevant equations



    3. The attempt at a solution

    I know that the Poisson equation is kind of like a partial differential equation. I rearranged it to [tex]\Phi[/tex]rr(r2) + [tex]\Phi[/tex]r(2r) = [-[tex]\rho[/tex](r) * r2 ] / [tex]\epsilon[/tex]

    But that wasn't very helpful

    Then I also realized that the equations for electric potential is a solution to this... but that is only a special case. Also, is gravitational potential also a solution, or no?

    How do you solve this type of equation? What does the 'given': [tex]\nabla[/tex]2 1/r = -4[tex]\pi[/tex][tex]\delta[/tex]3(r)
    even tell me? I am very lost. I read up about Poisson equations and I think the 'given' is like a boundary case... but I don't know how you incorporate the boundary case of a Poisson equation into a solution.
     
    Last edited: Oct 13, 2009
  2. jcsd
  3. Oct 14, 2009 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Just take the Laplacian of the proposed solution (Remember, you don't actually have to solve Poisson's equation to show that something is a solution of it)...what do you get?
     
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