A discussion ofBlconservation on a two dimensional magnetic field plane in watt balances

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. 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-

The irony of the magnet system for Kibble balances -- a review

“…Interestingly, the total Bl along the whole coil is conserved, as is shown in [37]. This is because, to first order, the reluctance of the gap does not change by displacing the inner yoke and hence the flux through the coil and with that the flux gradient or Bl (see Appendix A) remains the same.…”

“…So, in the above example, in the direction of +x. Note an analytic equation for ∆B r based on the eccentricity of the inner yoke can be found in [37].…”

“…(iv) A tilted coil of non-circular shape. It can be deduced from equations ( 36) and (37) that in an 1/r B-field, ∂B ∂z = 0 and hence B is uniform along z. Hence, only the xy projection of the coil contributed to the Bl.…”

The irony of the magnet system for Kibble balances—a review

“…The flux through the coil is zero independent of coil radius and horizontal position. That means, in weighing, the result is to first-order independent of the precise horizontal position and the coil radius [24,37]. The latter can change slightly due to ohmic coil heating.…”

Two simple modifications to improve the magnetic field profile in radial magnetic systems

“…is the radial distance), the Bl is independent of x and y. Gauss' law links the vertical component of the magnetic flux density to the horizontal component and as a consequence ∝ B r 1 r / can be achieved with B z = 0 in the entire field volume. With these two requirements, the geometric factor is independent to the coil horizontal movement, wire deformation [8] and yoke misalignment [9]. If B z does not vanish, Bl will depend on the remaining degrees of freedom of the coil ( θ θ x y , , , x y ), and in this case the mechanical system must be very well designed to avoid coil motion along these degrees of freedom during the velocity mode.…”