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

TL;DR

This paper analyzes the magnetic field distribution in watt balances, demonstrating how $Bl$ conservation can be maintained in two-dimensional magnetic planes even when symmetry is broken, thus relaxing alignment requirements.

Contribution

It provides an analytical framework extending $Bl$ conservation to arbitrary 2D magnetic planes with broken symmetry using Gauss's law and conformal transformations.

Findings

$Bl$ conservation holds in 2D magnetic planes with broken symmetry.

Analytical methods extend $Bl$ conservation to more general magnetic configurations.

Relaxed alignment criteria for watt balance magnet systems are possible.

Abstract

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.