This work explores what other mathematical possibilities were available to Maxwell for formulating his electromagnetic field model, by characterizing the family of mathematical models induced by the analytical equations describing electromagnetic phenomena prevailing at that time. The need for this research stems from the article "Inertial Relativity - A Functional Analysis Review", recently published in "Proyecciones", which claims and demonstrates the existence of an axiomatic conflict between the special and general theories of relativity on one side, and functional analysis on the other, making the reformulation of the relativistic theories, mandatory. As will be shown herein, such reformulation calls for a revision of Maxwell's electromagnetic field model. The conclusion is reached that -given the set of equations considered by Maxwell- not a unique, but an infinite number of mathematically correct reformulations to Ampère's law exists, resulting in an equally abundant number of potential models for the electromagnetic phenomena (including Maxwell's). Further experimentation is required in order to determine which is the physically correct model.