Biot-Savart Law Calculator

Every electromagnet, electric motor, MRI coil, and magnetic field sensor ultimately traces its design to the Biot-Savart Law — the equation that calculates the magnetic field contribution from each infinitesimal element of a current-carrying conductor. While Ampere's Law is simpler when geometric symmetry exists, the Biot-Savart Law is the general tool for calculating magnetic fields from any current distribution without symmetry constraints. The Biot-Savart calculator computes the differential field element dB from your conductor element specifications.

The Biot-Savart Law Formula

The differential magnetic field dB produced by a current element I dl at a field point located distance r away, where θ is the angle between the current element direction and the displacement vector r̂:

dB = (μ₀ / 4π) × (I × dl × sin θ) / r²

where μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space), I = current (amperes), dl = length of conductor element (meters), r = distance from element to field point (meters), θ = angle between the conductor element direction and the displacement vector from element to field point. Units of dB: tesla (T). The direction of dB is perpendicular to both the current direction and the displacement vector, given by the right-hand rule (or cross product dl × r̂). Use this online calculator for any current element configuration. The magnetic field calculator applies Ampere's Law for symmetric configurations.

Key Integrated Results from the Biot-Savart Law

By integrating the Biot-Savart differential over complete conductor geometries, standard results emerge:

• Infinite straight wire: B = μ₀I / (2πr), where r is perpendicular distance from wire — derived by integrating over all elements from −∞ to +∞
• Finite wire, at point perpendicular to midpoint: B = μ₀I(sin α₁ + sin α₂) / (4πr), where α₁ and α₂ are angles to the ends
• Circular current loop, on axis at distance x from center: B = μ₀IR² / (2(R² + x²)^(3/2)), where R is loop radius; at center (x=0): B = μ₀I/(2R)
• Solenoid (n turns per meter, length L >> radius): B ≈ μ₀nI inside the solenoid — the basis for electromagnet design

Applications: Motors, Transformers, and MRI

Biot-Savart calculations underlie the design of:

• Electric motors: the torque on a current-carrying coil in an external magnetic field; optimizing coil geometry for maximum torque-to-current ratio requires integrating Biot-Savart over the coil geometry
• MRI gradient coils: the spatial encoding gradients in MRI scanners require precisely shaped magnetic fields (linear gradient in a specific direction); Biot-Savart integration over optimized coil geometries is used to design gradient coils with the required field homogeneity specifications
• Printed circuit board design: current-carrying PCB traces produce magnetic fields that can interfere with nearby sensitive components; Biot-Savart calculations predict EMI from trace geometry

The magnetic force calculator and magnetism calculators provide complementary electromagnetic design tools.

Biot-Savart vs. Ampere's Law: When to Use Each

Both laws describe magnetic fields from currents, but apply in different situations. Ampere's Law (∮B·dl = μ₀I_enc) is a closed-form integral equation valid for all static fields, but only practically useful when the magnetic field has the same magnitude along an Amperian loop by symmetry — infinite solenoids, infinite wires, toroidal coils. The Biot-Savart Law applies to any geometry without symmetry requirements, but requires evaluating a volume or line integral that may be complex. The analogy is exact to Coulomb's Law vs. Gauss's Law in electrostatics: Gauss's Law is far simpler for symmetric charge distributions; Coulomb's Law handles any distribution. For finite wires, current loops of arbitrary shape, or real electromagnet coils, Biot-Savart is typically the only practical approach.