We propose sharp results on the numerical approximation of low-regularity electromagnetic fields by edge finite elements. We consider general geometrical settings, including topologically non-trivial domains and/or domains with a non-connected boundary. In the model, the electric permittivity and magnetic permeability are symmetric, tensor-valued, piece-wise smooth coefficients. In all cases, the error can be bounded by h^δ times a constant, where h is the meshsize, for some exponent δ ∈]0, 1] that depends both on the geometry and on the coefficients. It relies either on classical estimates when δ > 1/2, or on a new combined interpolation operator when δ < 1/2. The optimality of the value of δ is discussed with respect to abstract shift theorems. In some simple configurations, typically for scalar-valued permittivity and per-meability, the value of δ can be further characterized. This paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements.