Generalization of Helmholtz coil problem

Abstract

The primary intent of this work is to propose a simple analytical method for designing coil systems for homogeneous and gradient magnetostatic field generation. Coil system consists of two identical coaxial (regular) polygonal current loops. In the space between the loops, there is nearly homogeneous or nearly linear distribution of the magnetic field along the axes depending on the currents' direction. First, we derived a suitable, simple and general expression for the magnetic field along the axes due to a polygonal current loop. We emphasize the importance of the role of this expression for further analysis. The total on-axes magnetic field is the result of superposition of the magnetic fields that each loop generates separately. The proper distance between the loops and the current orientation make the system to become either Helmholtz coil or anti-Helmholtz coil. In this paper we give exact, analytical and general expression for this optimal distance that provides the magnetic field to be homogeneous (linear) as much as possible. We based our study on Taylor series expansion of the total magnetic field, demanding that the first contaminating term must be canceled, in both, symmetric and asymmetric case.

Magnetic variation in construction steels under tensile stress. Empirical research with Helmholtz coils

“…With the separation mentioned above, a magnetic induction is obtained in the midpoint of (Equation [7]):…”

“…Helmholtz coils are two coaxial circular coils of the same radius. This radius is equal to the distance between the coil plains (7). Our study used standard configuration with the current circulating in the same direction through the two coils.…”

Comparison of Superconductors and Permanent Magnets for Small-scale Magnetic Resonance Imaging Devices

“…In fact, the distance between the coil is selected so that the second derivative of the z component of the field with respect to z is cancelled. Since, because of symmetry, the first derivative is already null in the middle point, the second derivative is the first term in the Taylor expansion which contributes to the inhomogeneity of the field [22]. This analytical calculation is performed for infinitely thin circular wires, for which the field distribution can be written in closed-form expression in terms of elliptic integrals.…”

Standard Electric and Magnetic Field for Calibration

“…Helmholtz coils are two circular coils coaxial with the same radius, which is equal to the distance between the planes of the coils [15]. When the current flows in opposite directions, the configuration is called anti-Helmholtz coils.…”