  ARIADNE is a nuclear-magnetic-resonance-based experiment that will search for
novel axion-induced spin-dependent interactions between an unpolarized source
mass rotor and a nearby sample of spin-polarized $^3$He gas. To detect feeble
axion signals at the sub-atto-Tesla level, the experiment relies on low
magnetic background and noise. We measure and characterize the magnetic field
background from a prototype tungsten rotor. We show that the requirement is met
with our current level of tungsten purity and demagnetization process. We
further show that the noise is dominantly caused by a few discrete dipoles,
likely due to a few impurities trapped inside the rotor during manufacturing.
This is done via a numerical optimization pipeline which fits for the locations
and magnetic moments of each dipole. We find that under the current
demagnetization, the magnetic moment of trapped impurities is bounded at
$10^{-9} \mathrm{A}\mathrm{m}^2$.
