Square Helmholtz coil calculator, spaced at 0.5445·a.
Centre field, the current for a target field and the uniform region for a square pair — from the exact Biot–Savart field of a square loop.
Square Helmholtz coil calculator
- Centre field B
- 1.629 mT
- Field per amp
- 814.4 µT/A
- Current for 1 mT
- 1.228 A
- Optimal spacing
- 54.45 mmd = 0.5445·a
- Axial length within ±1 %
- 34.3 mm
- Axial length within ±0.1 %
- 18.9 mm
Both coils carry the same current in the same direction. Thin-coil model: each coil's turns share one plane.
What is a square Helmholtz coil?
A square Helmholtz coil is two identical, coaxial square coils of side a carrying the same current and spaced 0.5445·a apart, the distance at which the field at the centre is as uniform as possible. At that spacing the centre field is about 1.2961·μ0·N·I/a, roughly 9 % below a circular pair of the same outer width. Square pairs are popular because they are easy to wind on flat frames, stack into three-axis cages and fit rectangular samples, but most calculators only cover circular coils. This one uses the exact Biot–Savart field of a square loop: enter turns, side length and current to get the centre field, the current needed for a target field, and how far along the axis the field stays within 1 % and 0.1 %. Change the spacing to see the effect of a frame that is not quite optimal. AirInduct Sim models square Helmholtz windings in its full simulator.
How it's calculated
Each side of a square loop is a finite straight wire. Summing the Biot–Savart field of the four sides on the central axis gives, for side a, N turns and current I:
At the centre of a single loop this is 2√2 μ₀NI/(πa). For the pair, with loops at z = ±d/2:
- The optimal spacing is where d²B/dz² = 0 at the centre; solved numerically it is 0.54451·a.
- Required current: I = B_target / (B per amp) at your spacing.
- Uniform length: where |B(z)/B(0) − 1| stays below 1 % (or 0.1 %) on the axis.
- Thin coils: each coil’s N turns share one plane, and both coils carry equal current in the same sense.
Worked example
A square pair with 100 mm sides, 50 turns per coil and 2 A:
| Optimal spacing | 54.45 mm |
|---|---|
| Centre field | 1.629 mT |
| Current for 1 mT | 1.228 A |
| Length within ±1 % | 34.3 mm |
| Length within ±0.1 % | 18.9 mm |
Scaling up to a 0.5 m frame with 100 turns per coil at 1 A gives 325.7 µT — enough to cancel the Earth’s field (about 25–65 µT) several times over. For circular pairs use the 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 field over the whole volume between the coils, including near the corners, as 3D surfaces and heatmaps.
- Square Helmholtz transmitters paired with square or circular receivers for wireless power.
- Mutual inductance and coupling to a receiver anywhere in the field.
- Parameter sweeps of spacing and size, and DXF or KiCad export of the coil outline.
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.
- Helmholtz coil calculatorCentre field, required current and optimal spacing for circular and square Helmholtz pairs, with axial uniformity.
- 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
Where does the 0.5445 spacing for square Helmholtz coils come from?
For a single square loop of side a, the on-axis field has an inflection point at z ≈ 0.2723·a. Putting two loops 2 × 0.2723·a = 0.5445·a apart places the midpoint on both inflection points, so the second derivative of the total field is zero and the field is flat to fourth order. We solve this numerically from the exact square-loop formula and get 0.54451.
What is the formula for the centre field of a square Helmholtz coil?
Each loop contributes B(z) = μ0·N·I·a² / [2π (a²/4 + z²) √(a²/2 + z²)] on its axis. Adding both at z = 0.27225·a gives B ≈ 1.2961·μ0·N·I/a. For example, 100 turns at 1 A on a 0.5 m frame gives about 0.326 mT.
Is a square Helmholtz coil less uniform than a circular one?
Not on the axis. Compared at the same outer width — a square of side a against a circle of diameter a — the square pair stays within 1 % over about ±0.171·a and the circular pair over about ±0.157·a. The trade-off is field strength: the square pair gives about 9 % less centre field per amp-turn (1.296 versus 1.431·μ0·N·I/a). Off axis, near the corners, the square pair is less uniform than on axis.
Does this account for the field off the axis?
No, the uniformity plot is along the central axis, where the closed-form expression is exact. Off-axis uniformity near the corners of a square pair is somewhat worse than on axis. The full simulator computes the field over the whole volume as heatmaps and 3D surfaces.