Implementation of concurrent functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) recording results in the generation of large artefacts that can compromise the quality of EEG data. While much effort has been devoted towards studying the temporal variation of the artefact waveforms produced by time-varying magnetic field gradients, the spatial variation of the artefact voltage across EEG leads has not previously been investigated in any depth. The aim of this work is to develop an improved understanding of the spatial characteristics of the gradient artefacts and the mechanism which underlies their generation. This paper therefore presents physical models of the artefacts produced by the temporally-varying magnetic field gradients required for MRI. Novel analytic expressions for the artefact voltage that account for realistic shifts and rotations of the human head were calculated from electromagnetic theory, assuming a spherical, homogeneous head and longitudinal wirepaths for the EEG cap. These were then corroborated by comparison with numerical simulations using actual EEG wirepaths and with experimental measurements on an agar phantom and human head. The numerical simulations produced accurate reproductions of experimentally measured spatial patterns for both the spherical phantom and human head in a variety of orientations and gradient fields; correlation coefficients were as high as 0.98 for the phantom and 0.95 for the human head. Furthermore, it was determined that artefact voltages for both longitudinal and transverse gradients could be decreased by adjusting the subject's axial position with respect to the gradient coils. The accuracy of the modelled spatial maps along with the ability to model gradient artefacts for any given head orientation are a step towards developing improved artefact correction algorithms that incorporate motion tracking of the subject and selective filtering based on calculated spatial artefact templates, leading to greater fidelity in simultaneous EEG/fMRI data.