This work demonstrates how electrical conductance measurements can be employed for the study of liquid bridge behavior when their volume varies with time while their separation distance remains constant. The liquid bridges are edge pinned between two vertical, identical rods (r-bridges) at varying separation distances. Liquid evaporation is used as a means of reducing the bridge volume in a continuous smooth fashion. A zero-order continuation sequence with respect to Bond number and liquid bridge volume is combined with the shooting method for the solution of the Young-Laplace equation to give the liquid bridge shape as a function of its instantaneous volume. A novel, very efficient computational scheme is developed based on singular perturbation expansion for the solution of the Laplace equation in the liquid bridge to compute its electrical conductance that proved faster by orders of magnitude compared to other alternative approaches. The potential for estimating the liquid bridge characteristics or the evaporation rate by matching the experimental and theoretical results is discussed extensively.