Biot-Savart law

The Biot-Savart law gives the Magnetic field $\mathbf{H}$ produced by a steady current. For a line current $I$ flowing along contour $C$, the field at observation point $\mathbf{R}$ is

$\mathbf{H}(\mathbf{R})=\frac{1}{4\pi }{\int }_C\frac{Id{\mathbf{l}}^{\prime }\times \hat{\mathbf{R}}}{R^2}\text{(A/m)},$

where $d{\mathbf{l}}^{\prime }$ is the infinitesimal vector element of the wire at source point ${\mathbf{R}}^{\prime }$, and $\hat{\mathbf{R}}$ is the unit Distance vector from source to observation point ($\mathbf{R}-{\mathbf{R}}^{\prime }$), with magnitude $R$.

For surface current density ${\mathbf{J}}_s$ (A/m, current per unit width):

$\mathbf{H}=\frac{1}{4\pi }{\int }_S\frac{{\mathbf{J}}_s\times \hat{\mathbf{R}}}{R^2}d{s}^{\prime }.$

For volume current density $\mathbf{J}$ (A/m²):

$\mathbf{H}=\frac{1}{4\pi }{\int }_v\frac{\mathbf{J}\times \hat{\mathbf{R}}}{R^2}d{v}^{\prime }.$

Biot-Savart is the magnetostatic analog of Coulomb’s law: a direct, integral statement of how steady currents create $\mathbf{H}$, regardless of geometry.

The cross product

The integrand $Id{\mathbf{l}}^{\prime }\times \hat{\mathbf{R}}$ encodes two facts:

• Magnitude $\propto I|d{\mathbf{l}}^{\prime }|\sin \theta /R^2$ — same $1/R^2$ as Coulomb, with the $\sin \theta$ for the angle between current direction and observation direction.
• Direction by right-hand rule: $\mathbf{H}$ from a current segment points perpendicular to both $d{\mathbf{l}}^{\prime }$ and the line from source to field point. Globally this makes $\mathbf{H}$ curl around the wire.

Worked example: infinite straight wire

A wire along the $z$-axis carrying current $I$ in $+\hat{\mathbf{z}}$. Observation point at distance $r$ in the $xy$-plane.

Geometry: source point varies along the wire, observation point is at perpendicular distance $r$.

Set up: $d{\mathbf{l}}^{\prime }=\hat{\mathbf{z}}d{z}^{\prime }$, source position ${\mathbf{R}}^{\prime }=z^{\prime }\hat{\mathbf{z}}$, observation $\mathbf{R}=r\hat{\mathbf{r}}$. Distance vector $\mathbf{R}-{\mathbf{R}}^{\prime }=r\hat{\mathbf{r}}-z^{\prime }\hat{\mathbf{z}}$, magnitude $\sqrt{r^2+z^{\prime 2}}$.

Cross product: $d{\mathbf{l}}^{\prime }\times (\mathbf{R}-{\mathbf{R}}^{\prime })=\hat{\mathbf{z}}d{z}^{\prime }\times (r\hat{\mathbf{r}}-z^{\prime }\hat{\mathbf{z}})=rd{z}^{\prime }\hat{\mathbf{\varphi }}$ (using $\hat{\mathbf{z}}\times \hat{\mathbf{r}}=\hat{\mathbf{\varphi }}$, $\hat{\mathbf{z}}\times \hat{\mathbf{z}}=0$).

Substituting:

$\mathbf{H}=\frac{I}{4\pi }{\int }_{-\infty }^{\infty }\frac{rd{z}^{\prime }\hat{\mathbf{\varphi }}}{(r^2+z^{\prime 2}{)}^{3/2}}.$

The integral evaluates to $2/r$:

$\mathbf{H}=\hat{\mathbf{\varphi }}\frac{I}{2\pi r}.$

So the field circles the wire with magnitude $I/(2\pi r)$. This matches the Ampère’s law answer.

Worked example: circular loop on axis

A loop of radius $a$ in the $xy$-plane carrying current $I$. Find $\mathbf{H}$ at point $(0,0,z)$ on the axis.

Geometry: source element $d{\mathbf{l}}^{\prime }$ at angle ${\varphi }^{\prime }$ on the ring contributes a $d\mathbf{H}$ tilted off the axis. Pairs on opposite sides of the loop cancel each other’s off-axis components, leaving only the $\hat{\mathbf{z}}$ piece.

Set up: source ${\mathbf{R}}^{\prime }=a\cos {\varphi }^{\prime }\hat{\mathbf{x}}+a\sin {\varphi }^{\prime }\hat{\mathbf{y}}$, $d{\mathbf{l}}^{\prime }=ad{\varphi }^{\prime }\widehat{{\mathbf{\varphi }}^{\prime }}$. Distance vector to $(0,0,z)$: magnitude $\sqrt{a^2+z^2}$.

By symmetry, only the $\hat{\mathbf{z}}$ component of $\mathbf{H}$ survives the ${\varphi }^{\prime }$ integration — the horizontal components from opposite sides of the loop cancel.

The $\hat{\mathbf{z}}$ component of $d{\mathbf{l}}^{\prime }\times (\mathbf{R}-{\mathbf{R}}^{\prime })$ works out to $a^{2}d{\varphi }^{\prime }$ (algebra). Substituting:

$H_z=\frac{I}{4\pi }{\int }_0^{2\pi }\frac{a^{2}d{\varphi }^{\prime }}{(a^2+z^2{)}^{3/2}}=\frac{I{a}^2}{2(a^2+z^2{)}^{3/2}}.$

So

$\mathbf{H}=\hat{\mathbf{z}}\frac{I{a}^2}{2(a^2+z^2{)}^{3/2}}.$

At the center of the loop ($z=0$): $\mathbf{H}=\hat{\mathbf{z}}I/(2a)$.

This is the result that motivates the design of Helmholtz coils: pairs of identical loops separated by their common radius produce a remarkably uniform field at the midpoint.

When to use Biot-Savart vs Ampère

• Use Ampère when the field has high symmetry (axial, planar, cylindrical) and can be pulled outside a line integral. Easy.
• Use Biot-Savart when symmetry is partial or absent — finite-length wires, off-axis points of loops, irregular geometries. Necessary, just messier.

The same field; two computational pathways. Biot-Savart is to magnetostatics what direct Coulomb integration is to electrostatics: the universal but ugly tool.

Derivation from Ampère’s law

Biot-Savart can be derived from $\nabla \times \mathbf{H}=\mathbf{J}$ together with $\nabla \cdot \mathbf{B}=0$, using vector potential methods. The two formulations are exactly equivalent, but Biot-Savart is the integral form most useful for direct calculation.