Current and potential distributions have been investigated on linear and annular arrangements of two coplanar electrodes comprising a galvanic couple. An asymptotic technique is used to derive analytical results, valid under conditions of thin layer electrolyte. Dimensionless parameters are derived which enable comparison of the asymptotic results with previous calculations performed for general conditions. The closed form asymptotic results show excellent agreement with previous numerical calculations for a wide range of system parameters. The asymptotic analysis encompasses as a limiting case the classical Stern treatment.