Biot-Savart Law
The Biot-Savart Law calculates the magnetic field produced by a current-carrying conductor.
Includes variables, worked examples, and applications.
The Formula
The Biot-Savart Law, formulated by French physicists Jean-Baptiste Biot and Felix Savart in 1820, describes how a small segment of a current-carrying wire creates a magnetic field at a point in space. To find the total magnetic field from a complete wire or circuit, you integrate this expression over the entire length of the conductor.
The law is the magnetic equivalent of Coulomb's law for electric fields. Just as Coulomb's law tells you the electric field from a point charge, the Biot-Savart Law tells you the magnetic field from a current element. The field weakens with the square of the distance and depends on the angle between the current direction and the line connecting the wire to the observation point.
For a long straight wire carrying current I, the integrated result gives a field that circles around the wire with magnitude B = μ₀I / (2πr), where r is the perpendicular distance from the wire.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| dB | Magnetic field contribution from one small segment | T (Tesla) |
| μ₀ | Permeability of free space = 4π × 10⁻⁷ | T·m/A |
| I | Current in the conductor | A (Amperes) |
| dl | Length of the small current-carrying segment | m |
| θ | Angle between the current direction and the position vector | degrees or radians |
| r | Distance from the current segment to the observation point | m |
Example 1
Find the magnetic field 5 cm from a long straight wire carrying 10 A of current.
For a long straight wire, the integrated Biot-Savart result is: B = μ₀I / (2πr)
Values: μ₀ = 4π × 10⁻⁷ T·m/A, I = 10 A, r = 0.05 m
B = (4π × 10⁻⁷ × 10) / (2π × 0.05)
B = (4π × 10⁻⁶) / (0.1π) = 4 × 10⁻⁵
B = 40 μT — about the same strength as Earth's magnetic field at the surface
Example 2
Find the magnetic field at the center of a circular loop of radius 0.1 m carrying 5 A of current.
For a circular loop, the integrated result is: B = μ₀I / (2r)
Values: μ₀ = 4π × 10⁻⁷ T·m/A, I = 5 A, r = 0.1 m
B = (4π × 10⁻⁷ × 5) / (2 × 0.1) = 20π × 10⁻⁷ / 0.2
B ≈ 31.4 μT at the center of the loop
When to Use It
Use the Biot-Savart Law when:
- Calculating the magnetic field from a wire, loop, or coil with a specific geometry
- Designing electromagnets, solenoids, and MRI coils
- Analyzing the forces between parallel current-carrying conductors
- Solving problems where Ampere's Law does not apply due to lack of symmetry
- Modeling magnetic fields in motors, transformers, and sensors