= Flux-tube axial field virial identity
{title2=$\langle B_z^2\rangle=B_p^2-2\mu_0\langle p\rangle$}
For a regular cylindrical <flux tube> with $\mathbf B=(0,B_\phi(s),B_z(s))$, vanishing field beyond radius $s_0$, and exterior <pressure> $p_e$, define $\langle h\rangle=2s_0^{-2}\int_0^{s_0}sh(s)\,ds$ and $B_p^2=2\mu_0p_e$. Integrating <cylindrical magnetostatic pressure balance> against $s^2$ gives the displayed identity. The toroidal <magnetic pressure> and hoop <magnetic tension> cancel in this cross-sectional axial-field average. Nonnegative gas <pressure> gives $\langle B_z^2\rangle\leq B_p^2$, with strict inequality precisely when $\langle p\rangle>0$.
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