Maxwell's eigenvalue problem can be seen as a particular case of the Hodge-Laplace eigenvalue problem in the framework of exterior calculus. In this context we present two mixed formulations that are equivalent to the problem under consideration and their numerical approximation. It turns out that the natural conditions for the good approximation of the eigensolutions of the mixed formulations are equivalent to a well-known discrete compactness property that has been firstly used by Kikuchi for the analysis of edge finite elements. The result can be applied to the convergence analysis of the p-version of edge finite elements for the approximation of Maxwell's eigenvalue problem.