 Axisymmetric magnetic field near a regular axis

Let a smooth magnetic field with axisymmetry have axial value $B_0(z)$ and no line singularity on its axis. Gauss's law for magnetism gives ${\partial }_r(r{B}_r)=-r{\partial }_z{B}_z$. Since $B_z(r,z)=B_0(z)+O(r^2)$, integration from the regular axis gives the displayed expansion. A falling axial field therefore implies outward field lines above a positive-source loop. In a current-free region, Ampère's law also gives $B_z(r,z)=B_0(z)-r^2{B}_0^{\prime \prime }(z)/4+O(r^4)$, which follows by equating ${\partial }_z{B}_r={\partial }_r{B}_z$.