Computation of the Biot-Savart line integral with higher-order convergence using straight segments

TL;DR

This paper introduces a simple modification to the straight segment approximation method for computing magnetic fields, shifting endpoints off the coil to achieve higher-order accuracy and significantly reduce numerical errors.

Contribution

The paper proposes a novel endpoint shifting technique based on local curvature to enhance the Biot-Savart integral's convergence order from second to fourth.

Findings

Accuracy increases to fourth order with endpoint shifting.

Numerical errors are dramatically reduced for the same number of segments.

The method is simple to implement and improves computational precision.

Abstract

One common approach to computing the magnetic field produced by a filamentary current-carrying coil is to approximate the coil as a series of straight segments. The Biot-Savart field from each straight segment is known analytically. However, if the endpoints of the straight segments are chosen to lie on the coil, then the accuracy of the Biot-Savart computation is generally only second-order in the number of endpoints. We propose a simple modification: shift each endpoint off the coil in the outwards normal direction by an amount proportional to the local curvature. With this modification, the Biot-Savart accuracy increases to fourth order and the numerical error is dramatically reduced for a given number of discretization points.