  We perform a microscopic study on the redistribution of electric charge near
the surface of a model $d$-wave superconductor cut along the [110] direction,
with a Fermi surface appropriate for cuprate superconductors, using the
augmented quasiclassical equations. We identify two possible mechanisms for the
redistribution of charged particles different from the well-known magnetic Hall
effect, namely; the pair potential gradient (PPG) force due to surface effects
on the pair potential and the pressure difference between the normal and
superconducting regions arising from the slope of the density of states (SDOS)
in the normal states at the Fermi level. Our present results show that in spite
of the absence of supercurrents, electric charge is induced around the surface.
Moreover, the charging effect due to the SDOS pressure dominates over that due
to the PPG force for all the realistic electron-fillings $n=0.8$, $0.9$, and
$1.15$, at all temperatures. In addition, for the filling $n=1.15$, the PPG
force and the SDOS pressure contributions have the same negative signs, which
gives a larger total surface charge i.e., both the sign and amount of the
surface charge depends greatly on the Fermi-surface curvature. We have also
calculated the local density of states (LDOS) within the augmented
quasiclassical theory. Spatially varying local particle-hole asymmetry appears
in the LDOS, which suggests the presence of electric charge.
