This thesis addresses the issue of the time variability of the fine structure constant, alpha. Recent claims of a varying alpha are set against the established standards of quantum electrodynamical theory and experiments. A study of the feasibility of extracting data on the time dependence of alpha using particles in Penning traps is compared to the results obtained by existing methods, including those using astrophysical data and those obtained in atomic clock experiments. Suggestions are made on the nature of trapped particles and the trapping fields.