Vector Potential and Magnetic Field of Axially Symmetric Currents

TL;DR

This paper introduces a method to compute the vector potential and magnetic field for axially symmetric currents using integral calculations and Legendre polynomials, avoiding differential equations.

Contribution

It presents a novel integral-based solution for magnetic fields of axially symmetric currents using associated Legendre polynomials with a fixed m value.

Findings

Provides explicit series solutions for vector potential and magnetic field.

Expresses solutions in terms of multipole moments and Legendre polynomials.

Simplifies calculations by avoiding differential equations.

Abstract

A solution is proposed for finding the vector potential and magnetic field of any distribution of currents with axial symmetry. In this approach, the magnetic field and the vector potential are looked for not by solving a differential equation but rather through straightforward calculation of integrals of one scalar function. The solution is expressed in terms of the associated Legendre polynomials P_{lm} with the index m of the Legendre polynomials assuming one value only, m = 1. The solution has the form of a series, with the coefficients of the polynomials being combinations of multipole moments. Key words: electrodynamics, vector potential, spherical harmonics, Legendre polynomials, magnetic field.