Biot-Savart Law Calculator

The Biot-Savart Law describes the magnetic field generated by a small current-carrying element. It states that the magnetic flux density dB produced at a point is proportional to the current I, the element length dL, and the sine of the angle between the element and the position vector, and inversely proportional to the square of the distance r. Mathematically: dB = (μ₀ / 4π) × (I · dL · sinθ) / r².

μ₀ is the permeability of free space (vacuum permeability), a fundamental physical constant. Its value is 4π × 10⁻⁷ H/m (henries per meter), approximately 1.2566 × 10⁻⁶ H/m. It appears in the Biot-Savart Law and other electromagnetic equations to relate magnetic fields to the currents that produce them.

For an infinitely long straight wire carrying current I, the magnetic field at a perpendicular distance d is given by B = μ₀I / (2πd). The field forms concentric circles around the wire, and its strength decreases with increasing distance. This is a simplified result derived by integrating the Biot-Savart Law along the entire wire length.

The magnetic field is maximum when the angle θ between the current element (dL) and the position vector (r̂) is 90°, because sin(90°) = 1. When the current element is parallel to the position vector (θ = 0° or 180°), the field contribution is zero since sin(0°) = 0.

For a circular loop of radius a carrying current I, the magnetic field along its central axis at a distance x from the center is B = (μ₀ · I · a²) / (2 · (a² + x²)^(3/2)). At the center of the loop (x = 0), this simplifies to B = μ₀I / (2a). This result is obtained by integrating the Biot-Savart Law around the full loop.

Magnetic field strength (flux density) is expressed in tesla (T) in the SI system. Practical results are often given in nanotesla (nT, 10⁻⁹ T) or microtesla (μT, 10⁻⁶ T) since fields from small current elements are typically very small. For reference, Earth's magnetic field is about 25–65 μT.

Both laws relate electric current to magnetic field, but they are used in different situations. The Biot-Savart Law is a general integral approach applicable to any current configuration, including finite wire segments and loops. Ampere's Law (∮ B·dL = μ₀I_enc) is more convenient for highly symmetric configurations like infinite straight wires, solenoids, and toroids, where the magnetic field has a simple geometric pattern.

Yes. For a point charge q moving with velocity v, the analogous Biot-Savart expression gives the magnetic field as B = (μ₀ / 4π) × (q · v × r̂) / r². This is consistent with treating the moving charge as a tiny current element (I·dL = q·v). However, this applies only at non-relativistic speeds; at high speeds, special relativity must be considered.