= Near-axis radial magnetic field
{title2=$B_\rho=-\tfrac12\rho B_0'(z)+O(\rho^3)$}
For a smooth axisymmetric <magnetic field>, write $B_z(\rho,z)=B_0(z)+O(\rho^2)$. The absence of magnetic monopoles gives $\rho^{-1}\partial_\rho(\rho B_\rho)=-B_0'(z)+O(\rho^2)$. Integrating from the axis and excluding a singular $1/\rho$ term proves the displayed relation. An axial field decreasing along $+z$ has an outward radial component, even when the field points purely axially on the axis itself.
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