  We derive augmented quasiclassical equations of superconductivity with the
Lorentz force in the Matsubara formalism so that the charge redistribution due
to supercurrent can be calculated quantitatively. Using it, we obtain an
analytic expression for the vortex-core charge of an isolated vortex in extreme
type-II materials given in terms of the London penetration depth and the
equilibrium Hall coefficient. It depends strongly on the Fermi surface
curvature and gap anisotropy, and may change sign even as a function of
temperature due to the variation in the excitation curvature under the growing
energy gap. This is also confirmed in our numerical study of high-$T_{\rm c}$
superconductors.
