= On-axis magnetic field of a circular current loop
{title2=$B_z(z)=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}$}
A circular loop of radius $R$ carrying <electric current> $I$ has, on its axis,
$$
B(z)=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}\,\hat z,
$$
where the sign of $\hat z$ is fixed by the <right-hand rule>. At its centre, $B(0)=\mu_0I/(2R)$.
Put the loop in the $xy$ plane, centered at the origin, with positive <electric current> counterclockwise when viewed from $+z$. The <right-hand rule> then gives the central <magnetic field> along $+z$. Parameterize source positions by the <angle> $\phi$ and observe at $(0,0,z)$:
$$
\mathbf r'=(R\cos\phi,R\sin\phi,0),\qquad
d\boldsymbol\ell'=R(-\sin\phi,\cos\phi,0)\,d\phi,
\qquad \mathbf R=(-R\cos\phi,-R\sin\phi,z).
$$
Here $R$ without bold type is the loop <radius>, whereas $\mathbf R$ is the source-to-observer displacement. Their <cross product> is
$$
d\boldsymbol\ell'\times\mathbf R
=(Rz\cos\phi,Rz\sin\phi,R^2)\,d\phi.
$$
The denominator $|\mathbf R|^3=(R^2+z^2)^{3/2}$ is constant around the loop. The <integrals> of <sine> and <cosine> over a full turn vanish, while the axial contribution adds to $2\pi R^2$. Substitution into the <Biot-Savart law> proves the on-axis formula. <Symmetry> cancels transverse components; it does not cancel the axial components.
For an original one-turn example in the same vacuum approximation, choose $R=0.04\,\mathrm m$ and $I=2\,\mathrm A$. With $\mu_0\approx4\pi\times10^{-7}\,\mathrm{T\,m/A}$,
$$
B(0)=10\pi\,\mu\mathrm T\approx31.42\,\mu\mathrm T,\qquad
\frac{B(z)}{B(0)}=\left(1+\frac{z^2}{R^2}\right)^{-3/2}.
$$
At $z=0.03\,\mathrm m$, the source-to-observer distance is $0.05\,\mathrm m$, so $B(z)/B(0)=64/125=0.512$ and $B(z)\approx16.08\,\mu\mathrm T$. At $z=-0.03\,\mathrm m$, the <magnetic field> has the same magnitude and still points along $+z$. The point has crossed the loop plane, but the loop's <electric current> has not reversed. Reversing the <electric current> reverses the <magnetic field> on both sides.
\Image[/biot-savart-loop-axis-profile.png]
{title=Circular-loop <magnetic field>: axial reinforcement and equal direction on both sides}
{description=Original <electric current> loop schematic and normalized on-axis calculation. The left view looks from $+z$, with counterclockwise <electric current> and a central field directed toward the viewer. In the right panel, the arrows along the axis point toward $+z$ even for $z<0$; the calculated points at $z/R=\pm0.75$ have normalized field $0.512$. No experimental data are plotted.}
For $|z|\ll R$, a <Taylor expansion> gives $B(z)/B(0)=1-3z^2/(2R^2)+O(z^4/R^4)$: the center is a smooth maximum, with zero first <derivative>. For $|z|\gg R$,
$$
\mathbf B(z)\sim\frac{\mu_0IR^2}{2|z|^3}\widehat{\mathbf z}
=\frac{\mu_0m}{2\pi|z|^3}\widehat{\mathbf z},\qquad
m=I\pi R^2.
$$
This is the axial <magnetic dipole field> of <magnetic dipole moment> $\mathbf m=I\pi R^2\widehat{\mathbf z}$.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[II.14–5, equations (14.33)–(14.36): current times loop area and magnetic dipole field]{ref} A compact closed loop has an inverse-cube axial far field, in contrast to the inverse-distance field of the infinite straight wire. The <SI> units of the exact loop formula are $\mu_0IR^2/(R^2+z^2)^{3/2}\sim\mu_0I/\text{length}$, hence <tesla (unit)>.
For $N$ coincident turns carrying the same <electric current>, the <superposition principle> multiplies the ideal result by $N$. A real coil has radial width and axial <length>: if those are not negligible, integrate or sum over its individual turns. This is why a single measured coil diameter and a turn count may not determine the full spatial <magnetic field> precisely. A collection of loops becomes a <solenoid>; integrating their axial contributions gives the <on-axis magnetic field of a finite solenoid>.
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