The change in electric potential due to lightning

Abstract

The change in the electric potential due to lightning is evaluated.The potential along the lightning channel is a constant which isthe projection of the pre-flash potential along a piecewise harmoniceigenfunction which is constant along the lightning channel.The change in the potential outside the lightning channel is a harmonicfunction whose boundary conditionsare expressed in terms of the pre-flash potential andthe post-flash potential along the lightning channel.The expression for the lightning induced electric potential change isderived both for the continuous equations, and for a spatially discretizedformulation of the continuous equations.The results for the continuous equations are based on the properties ofthe eigenvalues and eigenfunctions of the following generalized eigenproblem:Find $u \in H_0^1 (\Omega)$ , $u \ne 0$ ,and $\lambda \in \mathbb{R}$ such that $\langle \nabla u, \nabla v \rangle_{\mathcal{L}} =\lambda \langle \nabla u, \nabla v \rangle_{\Omega}$ for all $v \in H_0^1 (\Omega)$ , where $\Omega \subset \mathbb{R}^n$ is a bounded domain (a box containing the thunderstorm), $\mathcal{L}$ is a subdomain (the lightning channel),and $\langle \cdot, \cdot \rangle_{\Omega}$ isthe inner product $\langle \nabla u,\nabla v\rangle_\Omega =\int_{\Omega}\nabla u\cdot\nabla v \; {{\rm d}x}.$