  We investigate ionization at a solid-water interface in applied electric
field. We attach an electrode to a dielectric film bearing silanol or carboxyl
groups with an areal density $\Gamma_0$, where the degree of dissociation
$\alpha$ is determined by the proton density in water close to the film. We
show how $\alpha$ depends on the density $n_0$ of NaOH in water and the surface
charge density $\sigma_m$ on the electrode. For $\sigma_m>0$, the protons are
expelled away from the film, leading to an increase in $\alpha$. In particular,
in the range $0<\sigma_m<e\Gamma_0$, self-regulation occurs to realize $\alpha
\cong \sigma_m/e\Gamma_0 $ for $n_0\ll n_c$, where $n_c$ is $0.01$ mol$/$L for
silica surfaces and is $2\times 10^{-5}$ mol$/$L for carboxyl-bearing surfaces.
We also examine the charge regulation with decreasing the cell thickness $H$
below the Debye length $\kappa^{-1}$, where a crossover occurs at the
Gouy-Chapman length. In particular, when $\sigma_m \sim e\Gamma_0$ and $H\ll
\kappa^{-1}$, the surface charges remain only partially screened by ions,
leading to an electric field in the interior.
