  This paper revisits the geometric foundations of electromagnetic theory, by
studying Faraday's concept of field lines. We introduce "covariant
electromagnetic field lines," a novel construct that extends traditional field
line concepts to a covariant framework. Our work includes the derivation of a
closed-form formula for the field line curvature in proximity to a moving
electric charge, showcasing the curvature is always non-singular, including
nearby a point charge. Our geometric framework leads to a geometric derivation
of the Lorentz force equation and its first-order corrections, circumventing
the challenges of self-force singularities and providing insights into the
problem of radiation-reaction. This study not only provides a fresh geometric
perspective on electromagnetic field lines but also opens avenues for future
research in fields like quantum electrodynamics, gravitational field theory,
and beyond.
