# How did this paper mathematically express field driven water autoionization?

I recently read a paper on using an electric field to drive water autoionizaton. I'm trying to figure out how to use the Laplace equation on pg 9; 4th paragraph; to solve for voltage. I'm also interested in how this equation would change if I replaced the hemispherical tip with a parallel plate. Anyone strongly familiar with this subject matter? Any help is greatly appreciated!

https://www.sciencedirect.com/science/article/abs/pii/S0009261411011511

pg 8-9, 5.1. Conditions for ionization

• Write the equations and add more details. Many people do not have access to papers. Jun 13 at 17:45
• Welcome to Chemistry StackExchange! It is rather hard to give you some help without knowing what is your background or where to begin the explanation. Poisson's and Laplace's equation from the article are usually introduced very early in any undergraduate physics course on electromagnetism (example 1, example 2). Note that the equations in article are written in spherical coordinate system, and a small peculiarity in the first one is the variable permittivity, $\epsilon$. Jun 13 at 19:14
• Instead of posting a screenshot, it would be better if you write the relevant part as text. Jun 16 at 4:55