= Power-law pressure-supported axial current
For constant axial <magnetic field> and $p=p_0(1+R^2/a^2)^{-k}$ with $p_0>0$ and $k>0$, <cylindrical magnetostatic pressure balance> gives, with $x=R^2/a^2$,
$$
I^2(R)=2\pi c^2p_0a^2
\begin{cases}
\displaystyle\frac{1-(1+kx)(1+x)^{-k}}{k-1},&k\ne1,\\
\log(1+x)+(1+x)^{-1}-1,&k=1.
\end{cases}
$$
The nontrivial <electric current> is finite at infinity exactly when $k>1$, in which case $I^2(\infty)=2\pi c^2p_0a^2/(k-1)$. Either sign of $I$ is possible because the <pressure> balance fixes its square.
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