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The watt balance is an experiment being pursued in national metrologyinstitutes for precision determination of the Planck constant $h$. In wattbalances, the $1/r$ magnetic field, expected to generate a geometrical factor$Bl$ independent to any coil horizontal displacement, can be created by astrict two dimensional, symmetric (horizontal $r$ and vertical $z$)construction of the magnet system. In this paper, we present an analyticalunderstanding of magnetic field distribution when the $r$ symmetry of themagnet is broken and the establishment of the $Bl$ conservation is shown. Byusing either Gauss's law on magnetism with monopoles or conformaltransformations, we extend the $Bl$ conservation to arbitrary two dimensionalmagnetic planes where the vertical magnetic field component equals zero. Thegeneralized $Bl$ conservation allows a relaxed physical alignment criteria forwatt balance magnet systems.

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