Optimal Permanent-Magnet Geometries for Dipole Field Approximation

The dipole approximation for magnetic fields has become a common simplifying assumption in magnetic-manipulation research when dealing with permanent magnets because the approximation provides convenient analytical properties that are a good fit at large distances. What is meant by “good fit at large distances” is generally not quantified in the literature. By using a parameterized multipole expansion and collaborating finite-element analysis (FEA) simulations to represent the magnet's field, we quantify the error associated with the dipole approximation as a function of distance from the permanent magnet. Using this expression, we find cylindrical, washer, and rectangular-cross-section bar permanent-magnet aspect ratios that minimize the error of the dipole approximation. For cylinders and rectangular-cross-section bars, these aspect ratios are a diameter-to-length ratio of √{4/3} and a cube, respectively.