On the applicability limits of a model of the magnetic field of a long straight wire

Abstract

We discuss the influence of two factors on the deviations from the model of the magnetic field of a long straight wire. Firstly, we derive the magnetostatic analogue of McCullagh’s formula for the case of the infinitely long straight conductor with an arbitrary cross-section. Using this equation on the basis of its practical significance, we find that the greatest deviations from the long straight wire model should be observed for a system of two parallel and infinitely long straight wires, equal in absolute value and direction currents. In this case the extremal direction of the magnetic field deviations respectively passes through these wires or is perpendicular to the plane passing through them. Our numerical estimations show that any infinitely long straight conductor with a cross-section having dihedral group symmetry can be considered with reasonable accuracy (the relative error in the determination of the magnetic flux density is less than 5%) as the infinite straight wire, if the distance to the observation point is approximately more than half order of magnitude larger than its maximum transverse size. We also analyse the effect of finite length of the carrying wire on the magnitudes of the magnetic field. We show that in this case, the smallest deviations from the infinite straight wire model should be observed for points belonging to the perpendicular plane passing through the midpoint of a wire.