A magnetic resonance imaging apparatus includes a calculator that calculates the quantity of heat generated (or to be generated) in a gradient magnetic field coil as a function of current supplied (or to be supplied) to the gradient magnetic field coil and at least one other factor such as eddy currents induced in a shield body used for shielding gradient field leakage flux from the imaged volume and/or delayed measurement of actual coil temperature. Such improved more timely temperature prediction and/or measurement is used to effect protective actions to better prevent permanent coil damage due to overheating.