The response of an interferometric low-coherence fiber-optic electric current sensor (FOCS) to a short current pulse is studied theoretically and experimentally. The main feature of current pulse sensing is analyzed that manifests itself when the duration of the pulse is comparable to or less than the light propagation time in the sensing fiber. An advanced small-radius sensing fiber coil is considered that allows one to improve the characteristics of FOCSs. In addition, the effect of the intensity noise due to the beating of the spectral components of radiation on the FOCS is studied. It is experimentally shown that this intensity noise can be reduced by using a two-channel optical scheme.