= Dipolar flux-tube area
Near the polar axis of a <magnetic dipole field>, $B\propto r^{-3}$. Conservation of <magnetic flux> in a <flux tube> consequently gives $A(r)\propto r^3$. A <dipole magnetic-field line> intersecting a sphere of radius $R_\star$ at small polar angle $\theta_\star$ obeys
$$
\theta(r)^2\simeq\theta_\star^2\frac r{R_\star},
\qquad
A(r)\simeq\pi r^2\theta(r)^2
=\pi\theta_\star^2\frac{r^3}{R_\star}.
$$
This area is for one polar tube. The approximation requires $\theta_\star^2r/R_\star\ll1$; the full <magnetic dipole field> does not remain a narrow polar tube at arbitrarily large radius.
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