A new geometric flow describing the motion of quadrature surfaces is introduced, where a quadrature surface is a closed hypersurface inducing the same electrostatic potential as a given electric charge density. This characterization enables us to study quadrature surfaces through the investigation of the flow. It is proved that the flow is uniquely solvable under the geometric condition that the initial surface has positive mean curvature. As a consequence, a bifurcation criterion for quadrature surfaces is obtained.