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A general model to compute the response of an electromagnetic induction sensor to a magnetic soil, in both time and frequency domains, is developed. The model requires modest computational resources and can be applied to arbitrary soil inhomogeneities and relief, and to arbitrary sensor coil shapes, orientations, and positions. Central to the model is the concept of a head sensitivity map, which can be used to characterize the sensor head as a function of the shape, size, and position of the sensor coils. Two further concepts related to the head sensitivity are presented, which are the zero equisensitivity surface and the volume of influence. We demonstrate that these concepts aid the understanding of the detector behavior. The general model is based on the Born approximation, which is valid if the soil magnetic susceptibility is sufficiently small. A simpler model, which is only valid for homogeneous half-space soils but does not require the Born approximation, is also developed. The responses predicted by both models are shown to be in good agreement with each other and also with available analytic solutions. Comparing the two models also enabled an expression for the error incurred when using the Born approximation to be established. We shown that, for most soils of relevance to mine clearance, the corresponding error is negligible.