In stepped wedge designs (SWD), clusters are randomized to the time period during which new patients will receive the intervention under study in a sequential rollout over time. By the study's end, patients at all clusters receive the intervention, eliminating ethical concerns related to withholding potentially efficacious treatments. This is a practical option in many large-scale public health implementation settings. Little statistical theory for these designs exists for binary outcomes. To address this, we utilized a maximum likelihood approach and developed numerical methods to determine the asymptotic power of the SWD for binary outcomes. We studied how the power of a SWD for detecting risk differences varies as a function of the number of clusters, cluster size, the baseline risk, the intervention effect, the intra-cluster correlation coefficient, and the time effect. We studied the robustness of power to the assumed form of the distribution of the cluster random effects, as well as how power is affected by variable cluster size. % SWD power is sensitive to neither, in contrast to the parallel cluster randomized design which is highly sensitive to variable cluster size. We also found that the approximate weighted least square approach of Hussey and Hughes (2007, Design and analysis of stepped wedge cluster randomized trials. Contemporary Clinical Trials 28, 182-191) for binary outcomes under-estimates the power in some regions of the parameter spaces, and over-estimates it in others. The new method was applied to the design of a large-scale intervention program on post-partum intra-uterine device insertion services for preventing unintended pregnancy in the first 1.5 years following childbirth in Tanzania, where it was found that the previously available method under-estimated the power.