Unipolar-current dipoles and other optimum reduced-symmetry multipoles

Two-dimensional dipole (uniform) fields produced by parallel currents are generally considered to have two symmetry planes, with a current direction reversal in reflection across one plane. Similarly, quadrupoles have four symmetry planes, and so on for higher multipoles. Expansion of the fields of filamentary currents shows that there are lower symmetry multipoles with highly uniform field characteristics. These solutions include configurations in which there is no current reversal across reflection planes, producing unipolar-current dipoles, quadrupoles, and other multipoles. In all cases, the solutions can yield systems that have the best uniformity of field characteristics (minimum undesired multipole content over the largest area) that can be achieved with the number of wires used, although the unipolar systems are less efficient and produce a lower field magnitude for a given current. These optimum solutions are not limited to circular configurations, or to systems with all wire current magnitudes i...