In this paper, we show that a single finger-gate capacitor (FGC) can generate pure spin pumping in a quantum channel (QC). Two dynamic fields, ac spin-orbit interaction and ac potential energy, both induced by the FGC onto the QC, are the agents driving the spin pumping. Smooth spatial profiles of the two ac fields are taken into account both perturbatively and full numerically for the nonadiabatic spin pumping. Our perturbative approach reveals that the spin-pumping mechanism is resonant sideband processes associated with simultaneous coupling of the two ac fields with traversing carriers. Full sideband-process treatment is carried out numerically by a time-dependent scattering matrix method. The same spin-pumping mechanism holds also for the case of a single finger-gated QC, albeit with smaller pumping amplitudes.