The watt balance is an experiment being pursued in national metrology institutes for precision determination of the Planck constant h. In watt balances, the 1/r magnetic field, expected to generate a geometrical factor Bl independent to any coil horizontal displacement, can be created by a strict two dimensional, symmetric (horizontal r and vertical z) construction of the magnet system. In this paper, we present an analytical understanding of magnetic field distribution when the r symmetry of the magnet is broken and the establishment of the Bl conservation is shown. By using either Gauss's law on magnetism with monopoles or conformal transformations, we extend the Bl conservation to arbitrary two dimensional magnetic planes where the vertical magnetic field component equals zero. The generalized Bl conservation allows a relaxed physical alignment criteria for watt balance magnet systems.
II. REVIEW OF AN IDEAL CASE WITH 1/r MAGNETIC FIELDThe analysis in this article is based on one of the most popular watt balance magnets employed 4,5,15,16 . Figure 1 shows the construction of the magnet with a symmetrical structure in both the r and z axes. The magnetic flux, generated by two permanent magnet disks oriented in repulsion, is guided by yokes of high permeability material through an air gap radi-