The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.
Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3734389.
- Received December 12, 2016.
- Accepted March 17, 2017.
- © 2017 The Author(s)
Published by the Royal Society. All rights reserved.