The shift from cars with a combustion engine to electric cars is a key ingredient in the transition toward a carbon-neutral mobility system. However, this leads to a drastic increase in the demand for renewable electric energy. In this study, we investigate the size and shape of this additional electricity demand. We develop an agent-based model to calculate the overall demand and, more importantly, the distribution throughout the day with a fine temporal resolution in order to identify demand peaks. We also take into account the existing demand for electricity to gain a more complete picture of the load on the entire grid. We find that the demand peaks arising from charging a fully electric fleet are extensive and the most prominent demand peak occurs at the so-called charging rush hour in the early evening. These findings suggest that our current mobility system is in need of drastic changes to make sustainable mobility achievable.