= Gauge dependence of helicity in an open magnetic flux tube
{title2=$H_m/L=\pi bCR_0^2$}
An open <flux tube> carries <magnetic flux> through its end caps. Its ordinary <magnetic helicity> can therefore change under a single-valued <gauge transformation>. For $\mathbf B=aR\mathbf e_\phi+b\mathbf e_z$ inside $R<R_0$ and zero outside, a regular family of <magnetic vector potentials> is $A_\phi=bR/2$, $A_z=C-aR^2/2$ inside and $A_\phi=bR_0^2/(2R)$, $A_z=C-aR_0^2/2$ outside. Their tangential components agree at $R_0$. Inside, $\mathbf A\cdot\mathbf B=bC$, so the helicity per unit length is $\pi bCR_0^2$. Changing $C$ is the gauge transformation $\Lambda=\Delta C\,z$, and changes this result whenever $b\ne0$. The convention $A_z=0$ outside gives $C=aR_0^2/2$ and helicity per unit length $\pi abR_0^4/2$, while $C=0$ gives zero. Without an end-cap or gauge convention the ordinary helicity is not unique.
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