A discussion of Bl conservation on a two dimensional magnetic field plane in watt balances

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.