 Solution

Why the shape is immaterial far away. Partition an arbitrary small planar loop into small rectangular loops with the same oriented current. Currents on internal shared edges cancel, leaving only the original boundary. Their magnetic dipole moments add to $\mathbf{m}=I\mathcal{A}\mathbf{n}$, with $\mathcal{A}$ the enclosed area. Far enough away, the leading fields add as dipoles at the same position. The general magnetic dipole field isHigher spatial multipoles contain shape-dependent corrections. The axial and transverse results above are the special directions of this expression.

The aligned terrestrial model. Take the moment to point toward geographic south. At northern latitude $\lambda$, the magnetic dipole inclination versus latitude gives local northward and downward componentsUsing the specified $\lambda =53^{\circ }$ and $B_{\mathrm{eq}}=35\mu \mathrm{T}$,The direction is geographic north and $69.4^{\circ }$ below horizontal, with no eastward component in this ideal model. These values use the latitude printed in the paper.