Biot–Savart Law Calculator
The Biot–Savart law gives the magnetic field produced by a small piece of current-carrying wire. Calculate the field of a single current element, or of a straight wire or a circular loop, which come from adding up many elements.
dB = (μ₀/4π) · (I·dl × r̂) / r²
Inputs
Choose the calculation, then drag a slider or type an exact value. Results update as you drag. The sliders are logarithmic, so one drag covers small and large values. Valid for steady currents in vacuum or air.
Results
Enter values and press Calculate.
Field diagram
Press Calculate to draw the diagram.
Reference tables
Common examples (illustrative values, computed by the same engine)
| Case | Shape | Field | In µT |
|---|
Formulas from the Biot–Savart law
| Shape | Field |
|---|---|
| Current element | dB = (μ₀/4π) · I · dl · sin θ / r² |
| Infinite straight wire | B = μ₀ I / (2π d) |
| Straight wire of length L, centred on the perpendicular | B = (μ₀ I / 2π d) · (L/2) / √((L/2)² + d²) |
| Circular loop, on its axis | B = μ₀ N I R² / ( 2 (R² + z²)3/2 ) |
| Circular loop, at its centre | B = μ₀ N I / (2R) |
| Far from a loop (z ≫ R) | B ≈ μ₀ m / (2π z³), with m = N I π R² |
Typical magnetic fields (approximate)
| Source | Flux density |
|---|---|
| Earth's field at the surface | 25 – 65 µT |
| Long wire carrying 10 A, at 5 cm | 40 µT |
| Small fridge magnet | about 5 mT |
| Strong neodymium magnet, at the surface | 0.2 – 0.5 T |
| Clinical MRI scanner | 1.5 – 3 T |
Key relations and units
| Quantity | Relation |
|---|---|
| Vacuum permeability | μ₀ ≈ 4π × 10⁻⁷ T·m/A = 1.2566 × 10⁻⁶ T·m/A |
| Constant in the law | μ₀ / 4π ≈ 10⁻⁷ T·m/A |
| Direction | dB points along dl × r̂ (right-hand rule) |
| Magnetic field units | 1 T = 1000 mT = 10⁶ µT = 10⁹ nT = 10⁴ G |
Formula and notes
|dB| = (μ₀/4π) · I · dl · sin θ / r²
- I: current in the wire, and dl: the length of the small element, pointing along the current
- r: distance from the element to the point; r̂: the unit vector pointing from the element to the point
- θ: angle between dl and r̂; the field is largest at 90° and zero along the line of the element
Example. A 1 mm element carrying 10 A, seen from 10 cm away at 90°: dB = 10⁻⁷ × 10 × 0.001 × 1 / 0.1² = 1 × 10⁻⁷ T = 100 nT.
Wire example. A long straight wire carrying 10 A has B = μ₀ I / (2π d) = 2 × 10⁻⁷ × 10 / 0.05 = 40 µT at 5 cm. Loop example. One turn of radius 10 cm carrying 1 A has B = μ₀ I / (2R) = 6.28 µT at its centre.
A single current element cannot exist alone, because steady currents flow in closed loops. The real field is the sum, or integral, of the contributions of every element, and the straight wire and loop modes do that sum for you. The element mode assumes the element is small compared with the distance (dl ≪ r). The straight-wire mode puts the point opposite the middle of the wire. The results assume vacuum or air, with no magnetic materials nearby.
Disclaimer. For education and estimation. Magnetic field safety limits and equipment design need proper measurements and a qualified engineer.
References
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| B | Magnetic flux density (total field at point) | T |
| μ₀ | Permeability of free space (4π×10⁻⁷) | H/m |
| I | Electric current in the conductor | A |
| R | Radius (of circular loop) or perpendicular distance (to wire) | m |
| L | Length of straight wire segment | m |
| z | Distance from center along axis of loop | m |
What it means
The Biot–Savart law calculates the magnetic field contributed by a small current element I·dl. The field dB is given by dB = (μ₀/4π) * (I dl × r̂) / r². It is used to find the magnetic field produced by arbitrary current distributions by integrating over the current path. This law is more general than Ampere’s law (which works for symmetric cases). It is essential for calculating the magnetic field of loops, coils, and any complex wire shape. In practice, it is used in the design of electromagnets and magnetic sensors. Example: For a short segment of wire carrying 1A, the field at a distance of 0.1m perpendicular to the segment is dB = (4π×10⁻⁷/4π) * (1 * 0.01) / 0.01 = 10⁻⁷ * 1 = 10⁻⁷ T, but this is a small contribution; the total field requires integration over the entire wire.
Worked example
Biot–Savart Law – Practical Example
Real‑World| Parameter | Value |
|---|---|
| I | 5 A |
| R | 0.05 m |
| μ₀ | 4π×10⁻⁷ H/m |
| Formula | B = (μ₀·I) / (2·R) |
Common mistakes
- Current element: I·dl – a vector along the current direction.
- Vector nature: The magnetic field is a vector – cross product with r̂ gives direction.
- Units: dB in teslas.
- Superposition: Integrate over the entire current distribution to get total B.
- Assumption: Valid for steady currents (magnetostatics).
Applications
Biot–Savart law calculates the magnetic field contribution from a small current element, integrating to find the total field for any current distribution. This is used to compute magnetic fields of complex geometries, such as coils and solenoids, which are essential for designing inductors, magnetic sensors, and MRI machines. Engineers apply it to optimise the field uniformity in magnets and to design magnetic recording heads. By using the Biot–Savart law, they can predict the magnetic field with high accuracy, guiding the placement of windings and ferromagnetic materials. This law is crucial for electromagnetics and magnetic device design.
- Design of coils, solenoids, and magnetic sensors
- Magnetic field optimisation in magnets and inductors
- Magnetic recording head design
- MRI and NMR magnet design
- Educational understanding of magnetic field sources
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