Helmholtz coil calculator, circular and square.
Centre field, the current for a target field, optimal spacing and the uniform region — for circular or square pairs, with the axial profile plotted.
Helmholtz coil calculator
- Centre field B
- 899.2 µT
- Field per amp
- 449.6 µT/A
- Current for 1 mT
- 2.224 A
- Optimal spacing
- 100 mmd = R
- Axial length within ±1 %
- 62.7 mm
- Axial length within ±0.1 %
- 34.6 mm
Both coils carry the same current in the same direction. Thin-coil model: each coil's turns share one plane.
What is a Helmholtz coil and what field does it make?
A Helmholtz coil is a pair of identical coaxial coils carrying the same current, spaced so the magnetic field between them is as uniform as possible: one radius apart for circular coils, and 0.5445 times the side length for square coils. At that spacing the centre field of a circular pair is B = (4/5)^(3/2)·μ0·N·I/R, about 0.7155·μ0·N·I/R. This calculator gives the centre field for any turns, current and size, the current needed to reach a target field, the optimal spacing, and how far along the axis the field stays within 1 % and 0.1 % of its centre value. You can also enter a non-optimal spacing to see how quickly the uniformity degrades. Helmholtz pairs are used to calibrate magnetometers, cancel the Earth’s field and expose samples to a known field. AirInduct Sim also uses Helmholtz-style windings as wireless power transmitters, and its full simulator maps the field in 3D.
How it's calculated
The field on the axis is the sum of the two coils’ on-axis fields, with the coils at z = ±d/2.
At the optimal spacing the centre field reduces to:
- The required current for a target field is I = B_target / (B per amp) at your spacing.
- Uniform length: the axial distance around the centre where |B(z)/B(0) − 1| stays below 1 % (or 0.1 %).
- Thin coils: each coil’s N turns share one plane; both coils carry equal current in the same sense.
- The square optimum 0.5445·a is where d²B/dz² = 0 at the centre; we solve it numerically and get 0.54451.
Worked example
A circular pair with R = 100 mm, 50 turns per coil, 2 A, spaced 100 mm apart:
| Centre field (summed loops) | 899.2 µT |
|---|---|
| Textbook (4/5)^(3/2) μ₀NI/R | 899.2 µT |
| Current for 1 mT | 2.224 A |
| Length within ±1 % | 62.7 mm |
The same turns and current on a square pair with 100 mm sides, spaced 54.45 mm apart:
| Centre field | 1.629 mT |
|---|
For more on square pairs, including their uniform region, use the square Helmholtz coil calculator.
When you need the full simulator
This calculator is free to use and answers one question at a time. The full AirInduct Sim simulator takes the same physics further:
- The full 3D field, not just the axis: heatmaps and contours show the uniform volume off axis.
- Helmholtz windings as wireless power transmitters, paired with Helmholtz or planar spiral (DTC) receivers.
- Mutual inductance, coupling and induced voltage in a receiver placed in the field.
- Parameter sweeps and exports (PDF report, DXF, KiCad) for building the coil.
More free calculators
- Mutual inductance calculatorM between two circular or square coils from turns, size, axial gap and lateral offset — or from k, L1 and L2.
- Coupling coefficient calculatork from coil geometry (with both self-inductances) or from L1, L2 and M, plus a k-versus-gap curve.
- Square Helmholtz coil calculatorSquare pairs at the 0.5445·a spacing: centre field, current for a target field and the uniform region.
- WPT compensation capacitor calculatorSeries–series and series–parallel resonant capacitors from L1, L2, M and frequency, with capacitor voltage stress.
- WPT efficiency calculatorMaximum link efficiency and figure of merit U = k√(Q1Q2), the optimal load and the k needed for a target η.
Frequently asked questions
Why is the optimal Helmholtz spacing equal to the radius?
The on-axis field of each loop has an inflection point at half a radius from its plane. Placing two loops one radius apart puts the midpoint on both inflection points, so the second derivative of the combined field vanishes there. By symmetry the first and third derivatives also vanish, which leaves the field flat to fourth order around the centre.
How uniform is the field of a Helmholtz coil?
On the axis of an ideal circular pair the field stays within 1 % of its centre value over roughly ±0.31 R, and within 0.1 % over roughly ±0.17 R. Off axis the uniform region is of similar extent but not identical, so check the full field if your sample is wide. The calculator reports the 1 % and 0.1 % lengths for your exact spacing.
Should I use circular or square Helmholtz coils?
Square coils are easier to wind on a flat frame, stack neatly into three-axis cages and fit rectangular samples or test boards. They need a closer spacing (0.5445 × side). For the same outer width, a square pair gives about 9 % less centre field per amp-turn than a circular pair (1.296·μ0·N·I/a versus 1.431·μ0·N·I/a for diameter a), while its ±1 % region on the axis is slightly longer (±0.171·a versus ±0.157·a). See the dedicated square Helmholtz calculator for the square case.
Does the calculation account for the thickness of the winding?
No. Each coil is treated as N turns concentrated in one plane. For windings whose width and depth are small compared with the radius the error is well under 1 %. Thick windings shift the optimal spacing slightly; the full simulator models the turns individually.