Electrostatically operated micro and nanoelectromechanical (MEM/NEM) relays have been proposed as digital switches to replace transistors due to their sharp turn-on/off transient, zero leakage current between drain and source in the off state, and capability to operate at far higher temperatures and radiation levels than CMOS. However, the dynamic energy associated with charging the gate capacitance have not been investigated or verified to date. Here, we present a detailed analysis starting from first principle formulations and derive a new closed form formula for the dynamic energy consumption of MEM/NEM relays as a function of the pull-in voltage, and open-state and closed-state gate capacitances. We compare against measurements carried out on prototype silicon MEM relays to verify our analysis. We also use the derived analytic model for energy consumption to carry out a scaling study, and compare against CMOS for room temperature and high temperature operation, and show that significant energy savings are possible. The models, analyses and measurement methodologies presented here constitute a set of essential techniques for accurate estimation of the energy consumption of MEM relays in ultra-low power circuit applications.