This thesis investigates the interaction of static electric fields with surfaces of zero and constant mean curvature. Such systems have as their driving components surface tension and electrostatic forces. These forces are prominent on the micro- and nano scale, and understanding their interaction is of key importance in many applications, including micro- and nanoelectromechanical systems (MEMS and NEMS), self-assembly, nanolithography, and microfluidic processes.

Two particular systems are explored. Mathematical models are developed and subsequently studied through a combination of analysis, numerics, and experiment. One system involves a minimal surface catenoid membrane deflected by an axially symmetric electric field. A model is formulated via variational techniques to describe equilibrium shapes of the membrane. A detailed analysis of the general solution set is performed, with emphasis on stability and the effect of dimensionless parameters. Techniques utilized include perturbation theory, phase space analysis, methods from non-linear dynamics, and numerical branch tracing for bifurcation diagrams. Experimental analysis is performed using soap-film bridges and a high voltage power source. Experimental observations verify the validity of the theory in predicting membrane profile as well as stability boundaries. Different geometries are considered, as well as the variations of unequal boundary radii and the addition of a volume constraint. The general effect of the electric field is determined and the utility of an electric field in such systems examined.

The second system explored involves the dynamics of bubbles deflating through tubes and subjected to an applied electric field. The effect of added electrostatic pressure on a single bubble collapse is explored theoretically and experimentally. Differential equations describing the size of the bubble as a function of time are derived and analyzed. Systems of two bubbles connected via a common tube are then considered. Here, differences in surface tension and electrostatic forces between bubbles interact as bubbles compete for a fixed volume of air. The solution structure is fully characterized in the two bubble system for arbitrary surface tension and surface charge. Several examples are given to illustrate the drastic effect of parameters on system dynamics.