Investigating the Magnetic Field outside small Accelerator Magnet Analogs via Experiment, Simulation, and Theory

Abstract

Multipole expansions of electric charge and current distributions and the fields those multipoles create are a fundamental pillar of electromagnetic theory, but explanations and examples are rare beyond a dipole. In this paper we describe a low-cost exploration of magnetic multipoles. Using the field from ideal magnetic dipoles and a simple binomial approximation, we show that each multipole obeys B ∝ r n , with n = −3, −4, −5, −6 for a dipole, quadrupole, sextupole, and octupole, respectively. Using commercially available NdFeB magnets and the magnetic field sensor inside a smartphone, we experimentally verify the power-law dependence of the multipole configurations. Finally, the open-source Python library Magpylib can simulate the magnetic field of arbitrary permanent magnet distributions, which also shows the same power law dependence for the different multipole configurations.