= Debye-Hückel potential of a charged sphere
{c}
{title2=$\phi(r)=Qe^{-\kappa(r-a)}/[4\pi\varepsilon r(1+\kappa a)]$}
For a uniformly charged sphere of radius $a$, total <electric charge> $Q$, and an exterior dilute electrolyte of <permittivity> $\varepsilon$, the <Debye–Hückel approximation> gives $(r^2\phi')'=\kappa^2r^2\phi$. Decay at infinity and <Gauss's law>, $-\varepsilon\phi'(a)=Q/(4\pi a^2)$, select the displayed <electric potential>. Substituting $u=r\phi$ reduces the proof to $u''=\kappa^2u$. The screening parameter is $\kappa^2=\sum_i z_i^2e^2n_{i,\infty}/(\varepsilon k_BT)$, and the <ion> profiles are $n_i\simeq n_{i,\infty}(1-z_ie\phi/(k_BT))$. The validity condition is $|z_i e\phi(a)|\ll k_BT$ for every relevant species. Integrating the diffuse <charge density> using the radial equation gives total screening charge $-Q$.
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