  We study Schwinger pair creation of charged particles due to the
inhomogeneous electric field created by the thin electron layer at the surface
of quark stars (the electrosphere). As suggested earlier, due to the low photon
emissivity of the quark-gluon plasma and of the electrosphere,
electron-positron pair emission could be the main observational signature of
quark stars. To obtain the electron-positron pair creation rate we use the
tunnelling approach. Explicit expressions for the fermion creation rate per
unit time per unit volume are derived, which generalize the classical Schwinger
result. The finite size effects in pair production, due to the presence of a
boundary (the surface of the quark star), are also considered in the framework
of a simple approach. It is shown that the boundary effects induce large
quantitative and qualitative deviations of the particle production rate from
what one deduces with the Schwinger formula and its generalization for the
electric field of the electrosphere. The electron-positron pair emissivity and
flux of the electrosphere of quark stars due to pair creation is considered,
and the magnitude of the boundary effects for this parameters is estimated. Due
to the inhomogeneity of the electric field distribution in the electrosphere
and of the presence of the boundary effects, at high temperatures ($T\geq
T_{cr}\approx 0.1 $ MeV) we find a lower electron-positron flux as previously
estimated. The numerical value of the critical temperature $T_{cr}$ depends on
the surface potential of the star. We briefly consider the effect of the
magnetic field on the pair creation process and show that the magnetic field
can enhance drastically the pair creation rate.
