# Poisson equation general solution

1. Oct 13, 2009

### yeahhyeahyeah

1. The problem statement, all variables and given/known data

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

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

can be written:

$$\Phi$$(r) = (1/4$$\pi$$$$\epsilon$$) $$\int$$ d3r' ($$\rho$$(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 $$\Phi$$rr(r2) + $$\Phi$$r(2r) = [-$$\rho$$(r) * r2 ] / $$\epsilon$$

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': $$\nabla$$2 1/r = -4$$\pi$$$$\delta$$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. Oct 14, 2009

### gabbagabbahey

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?