The Biot–Savart law can be integrated exactly along the symmetry axis of a circular current loop. This is an important example because it connects a real current distribution to the magnetic-dipole picture.

Consider a circular loop of radius $R$ carrying steady current $I$. Let the loop lie in the $xy$-plane and let its magnetic dipole moment point along $+z$.

We want the field at a point on the axis a distance $z$ from the center.

Symmetry of the field

Every current element on the loop has a diametrically opposite partner. Their magnetic-field components perpendicular to the axis cancel, while their axial components add.

Therefore the total field on the axis must point along the axis:

$$\mathbf B=B_z\hat{\mathbf z}.$$

For every current element, the distance to the field point is the same:

$$r=\sqrt{R^2+z^2}.$$

Biot–Savart gives

$$d\mathbf B=\frac{\mu_0 I}{4\pi}\frac{d\boldsymbol\ell\times\hat{\mathbf r}}{r^2}.$$

The magnitude of each contribution is

$$dB=\frac{\mu_0 I}{4\pi}\frac{d\ell}{r^2},$$

because $d\boldsymbol\ell$ is perpendicular to the vector from the current element to the axial field point.

Only the axial component survives the full integration. Geometry gives

$$dB_z=dB\frac{R}{r}.$$

Thus

$$dB_z=\frac{\mu_0 I}{4\pi}\frac{R,d\ell}{r^3}.$$

Since $r$ and $R$ are constant around the loop,

$$B_z=\frac{\mu_0 I R}{4\pi(R^2+z^2)^{3/2}}\oint d\ell.$$

The circumference is $2\pi R$, so

$$\boxed{B_z(z)=\frac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}}}.$$

For a tightly wound coil of $N$ identical turns,

$$\boxed{B_z(z)=\frac{\mu_0 N I R^2}{2(R^2+z^2)^{3/2}}}.$$

The direction follows the right-hand rule around the current.

Field at the center

At

$$z=0,$$

the expression becomes

$$\boxed{B_{\rm center}=\frac{\mu_0 I}{2R}}$$

for one turn, or

$$\boxed{B_{\rm center}=\frac{\mu_0NI}{2R}}$$

for $N$ identical turns.

A smaller loop produces a larger center field for the same current because every current element lies closer to the center.

Far from the loop: dipole behavior

When

$$|z|\gg R,$$

we can approximate

$$(R^2+z^2)^{3/2}\approx|z|^3.$$

Then

$$B_z\approx\frac{\mu_0IR^2}{2|z|^3}.$$

The loop's magnetic dipole moment magnitude is

$$\mu=IA=I\pi R^2.$$

Therefore on the dipole axis,

$$\boxed{B_z\approx\frac{\mu_0}{2\pi}\frac{\mu}{|z|^3}}.$$

The field of a compact current loop therefore falls as $1/r^3$ far away, unlike the $1/r$ field of an ideal infinite straight wire.

Worked example

A circular coil has

$$N=50,$$

$$R=0.10,\mathrm m,$$

and carries

$$I=2.0,\mathrm A.$$

At its center,

$$B=\frac{(4\pi\times10^{-7})(50)(2.0)}{2(0.10)} \approx6.28\times10^{-4},\mathrm T.$$

Thus

$$\boxed{B\approx0.628,\mathrm{mT}}.$$

At an axial point away from the center, the full axial formula rather than the center formula must be used.

This calculation illustrates the characteristic role of Biot–Savart: when symmetry determines which components cancel but does not make a simple circulation law directly solve for $B$, the field is obtained by integrating contributions from the source current.