Helmholtz Coil Calculator
Magnetic field along the axis of a pair of coaxial coils, how uniform it is around the middle, and the field profile of each coil drawn.
Results
The dashed line marks the value you entered.
What this tool does
Two identical coils placed one radius apart give a pocket of almost perfectly uniform field between them: at that spacing the curvature of the two fields cancels exactly. The arrangement is named after Hermann von Helmholtz and is still how laboratories calibrate magnetometers and cancel the Earth's field. The chart shows each coil's own field dashed and their sum solid — set the spacing larger than the radius and a dip appears in the middle, set it smaller and the centre bulges. Each loop's field comes from the Biot–Savart law on its axis.
Formula
B(z) = (μ₀ N I R² ÷ 2) × [(R² + (z − s/2)²)^(−3/2) + (R² + (z + s/2)²)^(−3/2)] · s = R: B₀ = (4/5)^(3/2) μ₀ N I ÷ R
Variables
| Symbol | Meaning | Unit |
|---|---|---|
nt | Turns per coil | — |
i | Current | A |
rc | Coil radius | cm |
s | Distance between the coils | cm |
z | Position on the axis (from the middle) | cm |
B | Field at that position | µT |
B0 | Field in the middle | µT |
DV | Difference from the middle | % |
BH | Ideal Helmholtz field (spacing = radius) | µT |
BS | Field at the centre of one coil alone | µT |
Worked example
- Turns per coil50
- Current2 A
- Coil radius10 cm
- Distance between the coils10 cm
- Position on the axis (from the middle)2.5 cm
- Field at that position895.401 µT
- Field in the middle899.176 µT
- Difference from the middle-0.4198 %
- Ideal Helmholtz field (spacing = radius)899.176 µT
- Field at the centre of one coil alone628.319 µT
Limitations
- The formula assumes ideal geometry — an infinitely long wire or coil, a perfectly uniform sphere — in empty space or a uniform material. Real parts come close, but ends, nearby metal and magnetic materials change the field.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.