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The magnetic field integral equation (MFIE) was applied to a dipole using three different discretization methods and high-order basis functions. For moderate-order, and higher, basis functions it was found that the different discretization methods produced essentially the same results. Continuity of current and its first derivative was observed at cell boundaries even though continuity of current was not explicitly enforced there. The MFIE provided lower condition numbers than the Hallen equation over the range of dipole radii examined. In close proximity to surface discontinuities, including hidden ones, residual errors could not be significantly reduced by increasing the order of the basis functions, implying the need for better modeling at discontinuities and calling into question the use of faceting to represent curved surfaces.