Introduction

Theoretically calculating the $\mathrm{B}_0$ field, given a matter distribution, allows modelling of MRI signal dropout, interaction of $\mathrm{B}_0$ and motion effects, manipulation of $\mathrm{B}_0$ using active and weakly magnetic passive shims, respiration effects, etc. Existing methods for calculating $\mathrm{B}_0$ use full finite element calculations [1,2] or approximate solutions to Maxwell's equations given either surface models of matter interfaces [3,4], voxel-based elements [5] or Fourier representations [6]. By using a perturbation approach to solving Maxwell's equations [7], a linear first-order solution can be found which is fast and appropriate for most MR imaging applications. In addition, the perturbation method allows the magnitude of the errors to be calculated, and hence the accuracy and appropriateness of the method to be estimated for various applications.