Mathematical models of electrostatic actuation have been developed since the 1960s, beginning with the work of Taylor, and are of great utility in a number of engineering systems. They are of particular use in the field of micro- and nanoelectromechanical systems, where models have typically dealt with planar geometries. Here, we extend the theory of electrostatic actuation to a nonplanar geometry by studying a catenoid soap-film bridge placed in an axially symmetric electric field. A model is formulated and analyzed, with emphasis on stability and the effect of dimensionless parameters. In the absence of external forces, the catenoid assumes its shape driven by surface tension. The utility of adding electrostatic forces and the interaction with surface tension is examined. Specifically, we uncover and quantify a stabilizing effect of the electric field and explore the limit of the stabilization.