Calculation of magnetic field noise from high-permeability magnetic shields and conducting objects with simple geometry

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 nonmagnetic conducting shells. A simple model is suggested to predict power-law decay of noise spectra beyond a quasi-static regime. Our results can be used to assess noise from existing shields and to guide design of new shields for precision measurements.