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Biot-Savart Law Calculator

Calculate magnetic fields from current-carrying conductors

About the Biot-Savart Law

The Biot-Savart law describes the magnetic field generated by a steady electric current. It is fundamental to understanding electromagnets, motors, and inductors.

Key Formulas:

  • dB = (μ₀/4π)(Idl × r̂)/r²
  • Straight wire: B = μ₀I/(2πr)
  • Loop center: B = μ₀NI/(2R)
  • Solenoid: B = μ₀NI/L

Constants:

  • μ₀ = 4π × 10⁻⁷ T·m/A
  • 1 Tesla = 10,000 Gauss

What Is the Biot-Savart Law?

The Biot-Savart law is the fundamental equation of magnetostatics, describing the magnetic field dB generated by an infinitesimal current element I·dl at a point in space. Discovered by Jean-Baptiste Biot and Félix Savart in 1820, it is the magnetic analog of Coulomb's law in electrostatics. The law states that dB = (μ₀/4π)(I·dl × r̂)/r² — the field is perpendicular to both the current direction and the vector pointing from the current element to the observation point, and its magnitude decreases as 1/r².

Unlike electric fields from point charges, magnetic fields from current distributions are inherently more complex because current must flow in closed loops (no magnetic monopoles). The Biot-Savart law integrates over the entire current path to give the total field. For simple geometries — straight wires, circular loops, solenoids, and toroids — the integration yields compact analytical formulas that this calculator implements directly.

Magnetic Field Formulas for Common Geometries

Each geometry has a specific formula derived from the Biot-Savart law:

Biot-Savart Law Derived Fields

Straight wire: B = μ₀I/(2πr) Loop center: B = μ₀NI/(2R) Solenoid: B = μ₀NI/L Toroid: B = μ₀NI/(2πr)

Where:

  • μ₀= Permeability of free space = 4π × 10⁻⁷ T·m/A
  • I= Current in Amperes
  • r= Distance from wire/radius (m)
  • N= Number of turns
  • L= Length of solenoid (m)
  • R= Radius of loop or major radius of toroid (m)

Magnetic Field Reference Values

Magnetic fields span an enormous range — roughly 18-19 orders of magnitude in everyday experience:

SourceField StrengthNotes
Earth's magnetic field25-65 μT (0.25-0.65 G)Varies with latitude
Fridge magnet~5 mT (50 G)At surface
Neodymium magnet~1.3 T (13,000 G)Strongest permanent magnets
MRI scanner1.5-7 TClinical to research

How to Use This Calculator

  1. Enter Current: The driving current in Amperes. This is the primary input for all geometries.
  2. Select Geometry: Choose Straight Wire (infinite, field decreases as 1/r), Circular Loop (with optional axis distance and turns), Solenoid (uniform field inside, length and turns), or Toroid (field confined inside the donut shape).
  3. Enter Dimensions: Distance from wire (mm), loop radius (mm), solenoid length and turns, or toroid major radius and turns. All dimensions are in millimeters and converted internally.
  4. View Results: Magnetic field in Tesla and Gauss (1 T = 10,000 G), plus geometry-specific outputs: axial field for loops, end field for solenoids, dipole moment and inductance where applicable.

Real-World Applications

The Biot-Savart law is the foundation of electromagnet design. Every solenoid, relay, transformer, and electric motor relies on magnetic fields computed from this law. MRI machines use superconducting solenoids producing 1.5-7 T fields — the field homogeneity must be better than 1 part per million over the imaging volume, requiring precise Biot-Savart calculations during coil design. The LHC's bending magnets produce 8.3 T fields using 11,850 A of current through superconducting cables.

In wireless power transfer, Biot-Savart calculations determine the magnetic coupling between transmitter and receiver coils. Electric vehicle wireless charging (operating at 85 kHz with coil separations of 10-30 cm) requires careful field design to maximize efficiency while meeting safety limits for human exposure to magnetic fields. The same principle scales to milliwatts for phone chargers and to kilowatts for industrial automation.

Worked Examples

MRI Solenoid Design

Problem:

An MRI scanner solenoid has 10,000 turns over a length of 1.5 m, carrying 500 A. What is the magnetic field inside?

Solution Steps:

  1. 1Convert length to meters: L = 1.5 m
  2. 2Turns per meter: n = 10,000/1.5 = 6,667 turns/m
  3. 3B = μ₀nI = 4π×10⁻⁷ × 6,667 × 500 = 4.19 T
  4. 4In Gauss: 4.19 × 10,000 = 41,900 G
  5. 5Field at ends: B/2 ≈ 2.1 T

Result:

Central field = 4.19 T — a realistic clinical MRI field strength (most are 1.5-3 T, with research systems reaching 7 T).

Current Loop Magnetic Dipole

Problem:

A circular loop of radius 10 cm carries 10 A with 100 turns. Find the field at the center and the magnetic dipole moment.

Solution Steps:

  1. 1R = 0.1 m
  2. 2B_center = μ₀NI/(2R) = 4π×10⁻⁷ × 100 × 10/(2 × 0.1) = 6.283×10⁻³ T = 62.8 G
  3. 3Dipole moment: μ = NIA = 100 × 10 × (π × 0.01) = 31.42 A·m²
  4. 4Field on axis at x = 20 cm: B = μ₀NIR²/(2(R²+x²)^(3/2)) = 4π×10⁻⁷ × 100×10×0.01/(2×(0.01+0.04)^(1.5)) = 1.12×10⁻³ T

Result:

Center field = 62.8 G. The dipole moment of 31.4 A·m² is comparable to a small permanent magnet.

Power Line Magnetic Field

Problem:

A transmission line carries 1000 A. What is the magnetic field 10 m below the wire?

Solution Steps:

  1. 1r = 10 m
  2. 2B = μ₀I/(2πr) = 4π×10⁻⁷ × 1000 / (2π × 10) = 2×10⁻⁵ T = 20 μT
  3. 3This is comparable to Earth's magnetic field (~50 μT)
  4. 4At 50 m: B = 4 μT — well within typical exposure guidelines

Result:

At 10 m, B = 20 μT (0.2 G) — comparable to Earth's natural field. Regulatory limits for public exposure to power-line fields are typically 100-200 μT at 50/60 Hz.

Tips & Best Practices

  • ✓1 T = 10,000 Gauss — Earth's field is ~0.5 Gauss, a fridge magnet is ~50 Gauss
  • ✓The field inside an ideal solenoid is uniform — B = μ₀nI depends only on turns per unit length and current
  • ✓For a toroid, the magnetic field is entirely confined inside the torus — ideal for inductors with minimal electromagnetic interference
  • ✓Magnetic field from a wire decreases as 1/r — doubling the distance halves the field
  • ✓The dipole moment of a current loop is μ = NIA — it determines the torque in an external magnetic field

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Frequently Asked Questions

Both describe the relationship between currents and magnetic fields, but Ampere's law ∮B·dl = μ₀I_enclosed is an integral form that's easiest to apply for highly symmetric current distributions (infinite wires, toroids, infinite solenoids). The Biot-Savart law dB = (μ₀/4π)(Idl × r̂)/r² is a differential form that can handle any current geometry but requires integration. They are mathematically equivalent — Ampere's law can be derived from Biot-Savart — but which one you use depends on the problem's symmetry.
The straight wire is effectively a one-dimensional source (current flowing along an infinite line), so the field spreads out in two dimensions, giving a 1/r dependence (similar to the electric field from a line charge). The loop's field on axis comes from integrating contributions around a circle, where distance and angle factors combine to produce a 1/r³ dependence at large distances — this is the classic dipole field behavior. At the exact center, the formula simplifies to μ₀NI/(2R).
Both measure magnetic flux density (B-field). 1 Tesla = 10,000 Gauss. The Tesla is the SI unit; the Gauss is the CGS unit still widely used in engineering and older literature. Earth's field is ~0.5 Gauss (50 μT), a fridge magnet is ~50 Gauss (5 mT), and an MRI scanner is 15,000-70,000 Gauss (1.5-7 T). The conversion is simple: multiply Tesla by 10,000 for Gauss, divide Gauss by 10,000 for Tesla.
For an infinitely long solenoid with uniform winding, the magnetic field lines are perfectly confined inside — they run parallel to the axis inside, loop around at the ends, and return outside. Applying Ampere's law to a rectangular loop that goes inside and outside the solenoid shows that the field outside must be zero because the same current is enclosed regardless of the loop's exterior leg length. Real solenoids of finite length have small but non-zero external fields, strongest near the ends.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: University Physics

by Young & Freedman

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.