This work studies dynamics of melting in current‐carrying conductors. It is shown that during equilibrium melting, when the dynamics of a phase‐transition front are determined by a heat balance, there exists a single‐valued correlation between the rate of inductance change and a temperature at the phase‐transition front. It is demonstrated that, although surface melting begins at temperatures less than the melting temperature of the current‐free conductor, corrugation of a phase‐transition front occurs at this temperature. It is shown that the nucleation rate of nuclei flattened in the direction normal to a conductor’s axis is negligibly small. Current dynamics at the stage of melting are analyzed for the case of a fixed external voltage. It is demonstrated that there exists an instability stage at which an excitation of the electric current occurs. Conditions for the experimental observation of the predicted phenomena are discussed. ©1996 American Institute of Physics.