yeahhyeahyeah
Oct13-09, 11:11 AM
1. The problem statement, all variables and given/known data
Given that \nabla2 1/r = -4\pi\delta3(r)
show that the solution to the Poisson equation \nabla2\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 \Phirr(r2) + \Phir(2r) = [-\rho(r) * r2 ] / \epsilon
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': \nabla2 1/r = -4\pi\delta3(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.
Given that \nabla2 1/r = -4\pi\delta3(r)
show that the solution to the Poisson equation \nabla2\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 \Phirr(r2) + \Phir(2r) = [-\rho(r) * r2 ] / \epsilon
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': \nabla2 1/r = -4\pi\delta3(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.