A method for estimating transverse position of one or more primary electric arcs as a function of longitudinal position of an arc gap within an electric arc furnace, during a melt period during which an input electric current flows into the arc furnace and at least a portion of the input current flows, in a predominantly longitudinal direction as a primary electric current, (i) through at least portions of an electrode positioned within the arc furnace and an ingot formed within the arc furnace, the electrode and the ingot being positioned end-to-end within the arc furnace and separated by the arc gap, and (ii) as the one or more primary electric arcs spanning the arc gap and movable in two transverse dimensions within the arc gap between the electrode and the ingot, the method comprising: during the melt period, during which the arc gap moves longitudinally through the furnace as the input current flow causes the electrode to melt and shrink and the ingot to grow, measuring magnetic field components in two or more spatial dimensions, at multiple sensor positions located outside the arc furnace and arranged among two or more distinct circumferential positions around the lateral periphery of the arc furnace and among two or more distinct longitudinal positions along the arc furnace, using a set of multiple magnetic field sensors, with each sensor being positioned at a corresponding one of the multiple sensor positions and characterized by one or more corresponding calibration parameters; using a data acquisition system that is structured and connected therefor, acquiring signals from the multiple sensors indicative of the corresponding measured magnetic field components and conveying the acquired signals to a computer system; and using the computer system that is structured, connected, and programmed therefor, for each one of multiple melt times within the melt period, calculating a corresponding estimated transverse position of the one or more primary electric arcs within the arc gap, that calculation being based at least in part on the longitudinal position of the arc gap at the corresponding melt time, and two or more of the magnetic field components, measured at the corresponding melt time, along with one or more corresponding sensor positions or calibration parameters, characterized in that, for one or more of the multiple melt times, the corresponding calculation is based at least in part on magnetic field components measured at the corresponding melt time at corresponding sensor positions at two or more distinct longitudinal positions along the arc furnace. during the melt period, during which the arc gap moves longitudinally through the furnace as the input current flow causes the electrode to melt and shrink and the ingot to grow, measuring magnetic field components in two or more spatial dimensions, at multiple sensor positions located outside the arc furnace and arranged among two or more distinct circumferential positions around the lateral periphery of the arc furnace and among two or more distinct longitudinal positions along the arc furnace, using a set of multiple magnetic field sensors, with each sensor being positioned at a corresponding one of the multiple sensor positions and characterized by one or more corresponding calibration parameters; using a data acquisition system that is structured and connected therefor, acquiring signals from the multiple sensors indicative of the corresponding measured magnetic field components and conveying the acquired signals to a computer system; and using the computer system that is structured, connected, and programmed therefor, for each one of multiple melt times within the melt period, calculating a corresponding estimated transverse position of the one or more primary electric arcs within the arc gap, that calculation being based at least in part on the longitudinal position of the arc gap at the corresponding melt time, and two or more of the magnetic field components, measured at the corresponding melt time, along with one or more corresponding sensor positions or calibration parameters, characterized in that, for one or more of the multiple melt times, the corresponding calculation is based at least in part on magnetic field components measured at the corresponding melt time at corresponding sensor positions at two or more distinct longitudinal positions along the arc furnace.