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 ${\theta }_{\star }$ obeysThis area is for one polar tube. The approximation requires ${\theta }_{\star }^{2}r/R_{\star }\ll 1$; the full magnetic dipole field does not remain a narrow polar tube at arbitrarily large radius.