We study the loop quantum cosmology of a flat Friedmann-Lemaitre-Robertson-Walker space-time with a Maxwell field. We show that many of the qualitative properties derived for the case of a massless scalar field also hold for a Maxwell field. In particular, the big-bang singularity is replaced by a quantum bounce, and the operator corresponding to the matter energy density is bounded above by the same critical energy density. We also numerically study the evolution of wave functions that are sharply peaked in the low energy regime, and derive effective equations which very closely approximate the full quantum dynamics of sharply peaked states at all times, including the near-bounce epoch. In the process, the analytical and numerical methods originally used to study the dynamics in loop quantum cosmology for the case of a massless scalar field are substantially improved to handle the difficulties (that generically arise for matter content other than a massless scalar field) related to the presence of a Maxwell field.