The vector potential of a steady azimuthal current density. Once again

TL;DR

The paper derives a compact, general expression for the vector potential of a steady azimuthal current, showing that for $\boldsymbol{J}(\boldsymbol{r})=J_{\varphi}(\boldsymbol{r})\boldsymbol{\hat{\varphi}}$ the potential is $\boldsymbol{A}(\boldsymbol{r})=A_{\varphi}(\boldsymbol{r})\boldsymbol{\hat{\varphi}}$ with $A_{\varphi}(\boldsymbol{r})=\frac{\mu_0}{4\pi}\int dV'\frac{\cos(\varphi'-\varphi)}{|\boldsymbol{r}-\boldsymbol{r}'|}J_{\varphi}(\boldsymbol{r}')$, valid in any orthogonal coordinate system containing the azimuthal angle. Applying this to a planar circular loop yields a spherical-harmonic expansion (odd $\ell$ only) and a cylindrical Laplace-transform form for the vector potential, and, remarkably, two analytically closed expressions for $A_{\varphi}$: one in terms of complete elliptic integrals $K(\kappa)$ and $E(\kappa)$ and another in terms of the Legendre function of the second kind $Q_{1/2}(\xi)$. The curl of $A$ gives a closed-form magnetic induction $\boldsymbol{B}$ in terms of $Q_{1/2}^{1}(\xi)$, reproducing the standard on-axis result $B_z(z)=\frac{\mu_0}{2\pi}\frac{I\pi a^2}{(a^2+z^2)^{3/2}}$. Overall, the work provides a simpler, more general framework for azimuthal currents and unifies several classical representations for the circular loop.

Abstract

We give an integral expression for the vector potential of a time-independent, steady azimuthal current density. Our derivation is substantially simpler and somewhat more general than others given in the literature. As an illustration, we recover the results for the vector potential of a circular current loop as an orthogonal expansion in spherical and cylindrical coordinates. Additionally, we obtain closed analytical expressions for the vector potential and the magnetic induction of a circular current loop in terms of Legendre functions of the second kind, that are simpler than the results in terms of complete elliptic integrals given in textbooks.