A small oil droplet of mass \(m\) and charge \(q\) is suspended in static equilibrium between two large, horizontal parallel conducting plates separated by a distance \(d\). The upper plate is held at an electric potential of \(+V_0\) and the lower plate is held at \(0\text{ V}\). The apparatus is then adjusted: the upper plate is set to a potential of \(-V\) relative to the lower plate at \(0\text{ V}\), and the plate separation is increased to \(2d\). In order for an oil droplet of the same mass \(m\) and charge magnitude \(|q|\) to remain in static equilibrium, what must be the sign of the droplet's charge and the magnitude of the new potential difference \(\Delta V\) between the plates?