Magnetic Lines for a Current Loop

Building directly on the on-axis field formula of Section 10.13, this section examines the overall SHAPE of the magnetic field lines produced by a circular current loop, and shows that -- viewed from a sufficient distance -- a current loop behaves, in every important respect, exactly like a MAGNETIC DIPOLE, in close analogy with the electric dipole studied in electrostatics.

Qualitatively, the field lines of a circular current loop (Fig. 10.20) pass through the loop's own interior in a direction roughly parallel to the axis, then spread out, curve around, and close back up on the far side -- a pattern immediately recognisable as very similar to the familiar field-line pattern of a short bar magnet. Applying the right-hand thumb rule to the loop as a whole (curl the fingers along the sense of current circulation; the outstretched thumb then gives the field direction through the loop's centre, Fig. 10.21) lets one face of the loop be identified as an effective "north pole," and the opposite face as an effective "south pole" -- even though, physically, there is no actual magnetic material or separated pole anywhere in sight, only a circulating current.

This "effective dipole" behaviour can be made precise by DEFINING a magnetic moment for the loop as the vector $\vec{m}=I\vec{A}$, where $\vec{A}$ is a vector of magnitude equal to the loop's enclosed area, directed along the loop's normal (exactly the definition of the magnetic dipole moment already met in Section 10.8, here written with the alternative common symbol $\vec{m}$ rather than $\vec{P}$). Substituting $m=IA=I\pi R^2$ into the on-axis field formula of Section 10.13, and specialising to the case of large axial distance, $z\gg R$ (so that $R^2$ becomes negligible compared to $z^2$ in the denominator, $(z^2+R^2)^{3/2}\approx z^3$), gives

$B_z \approx \frac{\mu_0 I R^2}{2z^3} = \frac{\mu_0}{4\pi}\frac{2m}{z^3} \quad (z\gg R).$ …