Biot-Savart Law
The Biot-Savart law gives the magnetic field produced by a small segment of steady current, playing the same fundamental role for magnetic fields that Coulomb's law plays for electric fields, and can be integrated over an entire current distribution to find the total field it produces.
Curious
In 1820, Hans Christian Ørsted discovered, almost by accident during a lecture demonstration, that a compass needle placed near a current-carrying wire deflects — direct evidence that electric currents produce magnetic fields, connecting electricity and magnetism for the first time. Jean-Baptiste Biot and Félix Savart quickly followed up with careful, quantitative measurements, establishing the precise mathematical law describing exactly how strong this field is and how it depends on distance from the wire.
The resulting field always circles around the current in closed loops — point your right thumb along the direction of current flow, and your curled fingers show the direction the magnetic field circles around the wire, a handy mnemonic known as the right-hand rule.
Exploring
| Symbol | Name | SI Unit |
|---|---|---|
| B | Magnetic field magnitude | tesla (T) |
| μ₀ | Vacuum permeability | N/A² |
| I | Current in the wire | ampere (A) |
| r | Perpendicular distance from the wire | metre (m) |
This simple 1/r dependence (rather than the 1/r² of Coulomb's law) reflects the different geometry of a magnetic field circling an infinitely long line source, compared to an electric field radiating outward from a point source — a useful contrast to keep in mind when comparing the two force laws.
Worked example: A long straight wire carries a current of 10 A. Find the magnetic field 0.05 m away. B = μ₀I/(2πr) = (1.25664×10⁻⁶)(10)/(2π×0.05) = 1.25664×10⁻⁵/0.31416 ≈ 4.0 × 10⁻⁵ T = 40 µT — roughly comparable in magnitude to the Earth's own magnetic field at its surface.
Deep Dive
The full differential form of the Biot-Savart law gives the contribution to the magnetic field dB from an infinitesimal current element I dl, at a displacement r from that element: dB = (μ₀/4π)(I dl × r̂)/r². Integrating this expression over an entire current distribution (a full wire, loop, or coil) gives the total field — the infinite-straight-wire result quoted above is one of the relatively few configurations simple enough to integrate exactly in closed form; most realistic current geometries require numerical integration.
Relationship to Ampère's law: The Biot-Savart law and Ampère's law (the magnetic analogue of Gauss's law, one of Maxwell's four equations) are mathematically related in the same way Coulomb's law and Gauss's law are related for electric fields — Ampère's law is the more general, symmetry-friendly statement, from which the Biot-Savart law's specific results can be derived for particular current geometries, and vice versa. For situations with enough symmetry (an infinite straight wire, a solenoid, a toroid), Ampère's law provides a considerably faster route to the field than direct Biot-Savart integration.
Concept Relationship Map
Requires: magnetic field, magnetic force
Enables: Ampère's law [PENDING FULL ENTRY], electromagnet design
Commonly confused with: Coulomb's law (different distance-dependence: 1/r for an infinite wire vs. 1/r² for a point charge)
Sources
- Biot, J.B., Savart, F. (1820). Annales de Chimie et de Physique, 15, 222–223.
- Griffiths, D.J. (2017). Introduction to Electrodynamics, 4th ed. Cambridge University Press, Ch. 5.