Current Loop as a Magnetic Dipole

Starting from the axial field of a circular loop (§3.8.3), $B=\dfrac{\mu_0 NIR^2}{2(R^2+z^2)^{3/2}}$, consider a point far away along the axis, $z\gg R$, so $(R^2+z^2)\approx z^2$:

Comparing this with the short-bar-magnet axial-field formula of §3.2.1, $B_{axial}=\dfrac{\mu_0}{4\pi}\dfrac{2p_m}{r^3}$, term by term identifies

$\boxed{p_m = IA} \qquad\text{(vector form: } \vec p_m = I\vec A\text{)}$ …