= Solution
In SI units, <magnetostatic equilibrium> without gravity is $\nabla p=\mu_0^{-1}(\nabla\times\mathbf B)\times\mathbf B$. For a purely axial <magnetic field>, the <magnetic tension> term $(\mathbf B\cdot\nabla)\mathbf B$ vanishes. Thus the radial balance is
$$
\frac{d}{ds}\left(p+\frac{B_z^2}{2\mu_0}\right)=0.
$$
Matching the <magnetohydrodynamic total pressure> to the field-free exterior gives \b[the gas pressure and axial field bound]:
$$
\boxed{p(s)=p_e-\frac{B_z(s)^2}{2\mu_0},\qquad |B_z(s)|\leq\sqrt{2\mu_0p_e}=B_p.}
$$
The inequality follows directly from nonnegative gas <pressure>. It is the <magnetic pressure> limit for an untwisted <flux tube>, without inward hoop tension.
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