  High-permeability magnetic shields generate magnetic field noise that can
limit the sensitivity of modern precision measurements. We show that
calculations based on the fluctuation-dissipation theorem allow quantitative
evaluation of magnetic field noise, either from current or magnetization
fluctuations, inside enclosures made of high-permeability materials. Explicit
analytical formulas for the noise are derived for a few axially symmetric
geometries, which are compared with results of numerical finite element
analysis. Comparison is made between noises caused by current and magnetization
fluctuations inside a high-permeability shield and also between
current-fluctuation-induced noises inside magnetic and non-magnetic conducting
shells. A simple model is suggested to predict power-law decay of noise spectra
beyond quasi-static regime. Our results can be used to assess noise from
existing shields and to guide design of new shields for precision measurements.
