A magnetic field in the vicinity of a homogeneous magnet filling an axisymmetric, simply connected or multiply connected region

Abstract

Designing inertial magnetorheological sensors requires calculating the spatial distribution of the magnetic field strength and its gradient, which characterize the magnetic system of their sensing elements. A sensing element includes a combined source of a constant magnetic field, consisting of a set of axially magnetized cylindrical, disc, or ring magnets with a finite aspect ratio of their main parameters and non-magnetic inserts. The assembly of magnets and non-magnetic elements is covered with a magnetic fluid that acts as a lubricant and is held in place by the field created by the assembly. The aim of the study is to obtain an expression that mathematically represents the magnetic field in the vicinity of an individual permanent magnet or a system of these magnets. The magnetostatic problem was solved in two stages using Ampere's method based on abstract magnetic poles. In the first stage, the north pole of a semi-infinite magnetized cylinder was considered. In the vicinity of this magnet an expression for the scalar magnetic potential was found. In the second stage, by applying the principle of superposition, an expression for the field near disc and ring magnets in a cylindrical coordinate system was obtained. The constructed expressions contain infinite series, which makes them difficult to use in practice. The number of terms in a series sufficient to describe the field with a predetermined accuracy was determined by comparing the data of analytical and numerical calculations. The corresponding two-dimensional axisymmetric magnetostatic problem was solved numerically using the Finite Element Method Magnetics package. For disc and ring magnets, the magnetic field strength magnitude at points in the surrounding space was calculated. It is shown that the first six terms in a series are sufficient for analytical and numerical solutions to coincide within 2%, which is an acceptable deviation equal to the typical instrument error of a modern teslameter. As the number of terms in a series increases, the error decreases, but the complexity of calculations increases. Moreover, it is impossible to verify the increase in accuracy experimentally. In the case of axisymmetric combined field sources, the derived expressions simplify the calculation of the magnetic field. The expression for the magnetic field distribution proposed by the authors can be used for the optimization of the design and manufacture of magnetofluid sensors.