We report measurements and numerical calculations of the potential distribution within a thin solid electrolyte near active (current-bearing) electrodes. These studies demonstrate two principles: (1) In a flat-plate geometry, the electrolyte is approximately equipotential beyond a distance of about three electrolyte thicknesses from the edge of the active electrodes. (2) If one of the active electrodes on one side of the electrolyte extends beyond the other, it strongly biases the potential of the electrolyte far from the active region. We show that these effects make it challenging to measure electrode overpotential accurately on thin cells. However, we also show that these effects can be useful for protecting glass-ceramic seals in an oxygen generator stack against electrochemical degradation/delamination.