Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A3 / c / iii

Substitution into Poisson's equation, ${\nabla }^2\psi =-\rho /\epsilon$, gives the Poisson-Boltzmann equationFor a plane, ${\nabla }^2\psi =d^2\psi /d{s}^2$; for a sphere, it is $r^{-2}d(r^2{\psi }^{\prime })/dr$. The bulk conditions are $\psi \to 0$ and $n_{\pm }\to n_0$. At a planar wall with negligible internal field and normal pointing into the liquid, Gauss's law imposes $-\epsilon {\psi }^{\prime }(0)={\sigma }_s$; alternatively, a prescribed surface potential gives $\psi (0)={\psi }_0$.