= Alignment of current and field in an axisymmetric conductor
{title2=$\sigma=a(I-nn^T)+d\,nn^T+h[n\times]$}
For a real <conductivity tensor> invariant under every rotation about a unit vector $n$, the <electric current density> is $J=aE_\perp+dE_\parallel+h\,n\times E_\perp$. Its <collinearity> with a nonzero real <electric field> $E$ has three cases. Axial fields are always mapped to an axial current. Purely transverse fields give a collinear current exactly when $h=0$, equivalently when this <tensor> is symmetric. Fields with nonzero axial and transverse parts require both $h=0$ and $a=d$, making the <tensor> isotropic. The zero <vector> is included as a scalar multiple; requiring a nonzero current also requires the relevant scalar conductivity to be nonzero.
To prove necessity, take the <dot product> of $J=\lambda E$ with $n\times E_\perp$. If $E_\perp\ne0$ this gives $h=0$; the transverse and axial equations then respectively give $\lambda=a$ and, when $E_\parallel\ne0$, $\lambda=d$. Reversing these steps proves sufficiency. Rotations about the axis alone permit $h\ne0$; symmetry under a transverse half-turn would eliminate this antisymmetric term.
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