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Homework Statement
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'|)
Homework Equations
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.
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