  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.
