Using a new technique to directly detect current induced on a nearby electrode, we measure plasma oscillations in ultracold plasmas, which are influenced by the inhomogeneous and time-varying density and changing neutrality. Electronic detection avoids heating and evaporation dynamics associated with previous measurements and allows us to test the importance of the plasma neutrality. We apply dc and pulsed electric fields to control the electron loss rate and find that the charge imbalance of the plasma has a significant effect on the resonant frequency, in excellent agreement with recent predictions suggesting coupling to an edge mode.