Biot–Savart Law

Equivalent forms

B = \frac{\mu_0 I}{2R} \quad \text{(center of loop)}

\vec{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\vec{l} \times \hat{r}}{r^2}

The cross product in the integrand encodes the deepest asymmetry in electromagnetism — magnetic fields always curl, never diverge.

Symbol	Name	SI unit	Dimension	Range
$B$	Magnetic fieldoutput Magnetic field magnitude at the field point	T	M·T⁻²·I⁻¹	0 – 0.1
$I$	Current Current in the wire	A	I	0.01 – 100
$r$	Distance Distance from the current element to the field point	m	L	0.001 – 1

Unit systems

Symbol	SI	CGS	Imperial
$B$	T	G	T
$I$	A	statA	A
$r$	m	cm	ft

Where it holds

Valid for steady (DC) currents in vacuum. For time-varying currents, use the Jefimenko equations or full Maxwell's equations. In magnetic media,

\mu _{0} is

replaced

by \mu _{0}\mu

ᵣ.

Dimensional analysis

$[B] = [\mu _{0}]\cdot [I]\cdot [r]^{-1} \to (T\cdot m$ /A)(A $)(m^{-1}) = T \checkmark$

Discovery

Jean-Baptiste Biot & Félix Savart · 1820

Weeks after Ørsted showed a current deflects a compass, Biot and Savart measured the force and deduced the inverse-square dependence on distance — the magnetic analog of Coulomb's law.

Try this

How does a coil of wire become a magnet when you run current through it?

Find the magnetic field at the center of a circular loop of radius 0.05 m carrying 2 A of current.

Research status: stable

Real-world applications

MRI magnet design — Helmholtz and solenoid coil field calculations
Magnetic field modeling for particle accelerator beam lines
Induction cooktop heating element design
Geomagnetic field modeling from ocean-floor current loops

Common misconceptions

The law gives the field from a single current element — the total field requires integration over the entire circuit
An isolated current element cannot exist (current must flow in a loop), so the law is always used in integral form
The direction is given by the right-hand rule — curl fingers from dl toward r̂, thumb points along dB

Experimental verification

Original measurements by Biot and Savart on long straight wires (1820). Modern verification via SQUID magnetometers measuring fields from precisely controlled current geometries to parts-per-billion accuracy.

Derivation

Empirically established by Biot and Savart from force measurements.

Consistent with Ampère's law via the Stokes theorem.

Can also be derived from the vector potential:

B = \nabla \times A

where

A = (\mu _{0}/4\pi )\int (J/r)dV

.

Limiting cases

r \to \infty

⟶

B \to 0

Magnetic field of a finite current loop falls off as

1/r^{3}

(dipole) at large distances.

I \to 0

⟶

B \to 0

No current means no magnetic field.

R \to 0

(loop radius)⟶

B \to \infty at

centerShrinking the loop while keeping current fixed concentrates the field — in reality, wire thickness limits this.

What if…

What if the loop has

N = 100

turns?

Field multiplies by N: $B = 100 \times 25.13 \mu T = 2.513\,\mathrm{mT}$ . Coils stack fields linearly — the principle behind electromagnets.

What if the current reverses direction?

Field direction reverses but magnitude stays the same. The right-hand rule now gives the opposite direction.

What if you move to a point on the axis but away from center?

$B = \mu _{0}IR^{2}/[2(R^{2} + z^{2})^(3/2)]$ . At $z = R$ , B drops $to \sim 35$ % of center value. At z >> R, it falls as $1/z^{3}$ (dipole field).

1

Field at the center of a circular loop

Given ·

I:: 2
R:: 0.05

Find · B at center

Steps

For a circular loop, the Biot–Savart integral simplifies at the center
Every element dl is perpendicular to r̂ and at distance R from center
Integrate: $B = (\mu _{0}I/4\pi ) \times \oint dl/R^{2} = (\mu _{0}I/4\pi ) \times 2\pi R/R^{2} = \mu _{0}I/(2R)$
Substitute: $B = (1.25663706212\times 10^{-6} \times 2) / (2 \times 0.05)$
$B = 2.513\times 10^{-5} T \approx 25.13 \mu T$ — about half Earth's surface field

Result ·

B = \mu _{0}I/(2R) = 1.257\times 10^{-6} \times 2 / (2 \times 0.05) = 2.513\times 10^{-5} T = 25.13 \mu T

2

Field of a long straight wire

Given ·

I:: 10
r:: 0.05

Find · B at 5 cm from the wire

Steps

For an infinite straight wire, integrate Biot–Savart along the wire
Result: $B = \mu _{0}I/(2\pi r)$ — field circles the wire
Substitute: $B = (1.25663706212\times 10^{-6} \times 10) / (2\pi \times 0.05)$
$B = 1.2566\times 10^{-5} / 0.3142 = 4.0\times 10^{-5} T$
Direction given by right-hand rule: curl fingers in direction of current, thumb points along B

Result ·

B = \mu _{0}I/(2\pi r) = 1.257\times 10^{-6} \times 10 / (2\pi \times 0.05) = 4.0\times 10^{-5} T = 40 \mu T

Ampère's Law (with Maxwell's correction)→Lorentz Force Law→Magnetic Dipole Moment & Torque→