Analytical expression for Child-Langmuir sheath edge around a corner

An expression for the position of the sheath edge around a two-dimensional corner cathode with included angle $\theta_{c}$ has been discovered. This expression is valid in the Child-Langmuir approximation, i.e., $\phi_{c}\gg kT_{e}/e$, where $-\phi_{c}<0$ is the cathode bias and $T_{e}$ is the electron temperature. In polar coordinates $\left(r,\theta\right)$, the sheath edge is given by $\left(r/s_{0}\right)\sin\left[\pi\theta/\left(2\pi-\theta_{c}\right)\right]=\left[\pi/\left(2\pi-\theta_{c}\right)\right] $ where $s_{0}$ is the planar sheath width far from the corner. This result is verified by comparison with numerical solutions of Watterson [J. Phys. D: Appl. Phys. 22, 1300 (1989)] for a knife edge ($\theta_{c}=0$) and a convex square corner ($\theta_{c}=\pi/2$). The observed agreement suggests that this expression is correct for all corner angles, both convex and concave.