- #1
ivantozavr
- 2
- 0
- TL;DR Summary
- How to solve the Poisson equation inside an electrostatic field source?
In my case, there is proton radiation acting on the material. Consequently, the protons get stuck in the sample and create an electrostatic field. I would like to solve the Poisson equation inside the sample. I assume that the medium is infinite and homogeneous, that is, the potential at infinity is zero (one-dimensional case):
Δφ(x) = - ρ(x) / εr ε0
φ(±∞) = 0.
The solution of such an equation is well known. That is (integration by range X' containing the source)φ(±∞) = 0.
φ(x) = 1 / (4πεr ε0) ∫ dx' ρ(x') / |x-x'|.
But then if I try to calculate the field inside the source itself, then singularities will occur due to the denominator in the integrand. I made small δx indents when calculating such an integral, i. e. sources with coordinates ζ: |ζ-x'| < δx were not taken into account when calculating an integral. This is hardly the right approach. Could you tell me how to solve such a problem correctly?