Axial Magnetic Field Produced by Current in a Circular Loop

Section 10.12 found the field only at the CENTRE of a circular current loop. This section generalises that result to any point P on the loop's AXIS -- the straight line through the centre of the loop, perpendicular to the plane of the loop -- at an arbitrary distance $z$ from the centre (Fig. 10.19).

Set up coordinates so the circular loop, of radius $R$, carrying steady current $I$, lies in the $x$-$y$ plane, centred at the origin $O$, with the point P on the $z$ axis at distance $z$ from $O$. For a current element $d\vec{l}$ anywhere on the loop, the distance to P is $r=\sqrt{R^2+z^2}$ -- the SAME for every element, by the loop's circular symmetry, exactly as it was constant for every element of a circular arc in the previous section. Because $d\vec{l}$ (lying in the $x$-$y$ plane, tangent to the loop) is always exactly perpendicular to $\vec{r}$ (which points from the element out to P, lying in a $y$-$z$ plane containing the axis), the Biot-Savart law gives a differential field of magnitude

This differential field $d\vec{B}$ is NOT directed purely along the axis, however -- it has both an axial ($z$) component $dB_z$ and a component $dB_{\perp}$ perpendicular to the axis. Crucially, by the loop's symmetry, the perpendicular component $dB_{\perp}$ due to any one current element is EXACTLY CANCELLED by the perpendicular component due to the current element diametrically OPPOSITE it on the loop -- so when the contributions of every element are summed over the full loop, all the perpendicular components cancel out completely, leaving only the AXIAL components to add up.

Using the geometry of Fig. 10.19, $\cos\alpha=R/r$ (where $\alpha$ is the angle between $\vec{r}$ and the plane of the loop), so $dB_z=dB\cos\alpha$, and integrating this axial component over the entire loop (a simple integral, since $dB$ itself is the SAME for every element, by symmetry) gives

for a single-turn loop, or, for a coil of $N$ turns, …