Helmholtz Coil Calculator

Estimate the magnetic field at the center of a coil pair

Calculation
Quick Guide

A Helmholtz pair generates a highly uniform magnetic field around its center. The classic setup uses spacing equal to radius ( a = R).

This calculator supports general spacing and reports how far the geometry is from the ideal Helmholtz condition.

Schema:
1) Define geometry (R, a)
2) Enter N/I or target B
3) Check field level and Helmholtz ratio
Formulas (MathJax)
\[B(0)=\mu_0\,N\,I\,\frac{R^2}{\left(R^2+\left(\frac{a}{2}\right)^2\right)^{3/2}}\]
\[\text{Ideal Helmholtz }(a=R):\quad B(0)=\mu_0\left(\frac{4}{5}\right)^{3/2}\frac{N I}{R}\]
\[I=\frac{B\,\left(R^2+\left(\frac{a}{2}\right)^2\right)^{3/2}}{\mu_0 N R^2}\]
\[N=\frac{B\,\left(R^2+\left(\frac{a}{2}\right)^2\right)^{3/2}}{\mu_0 I R^2}\]
Legend
  • \(B\): Magnetic flux density at center [T]
  • \(N\): Turns per coil [-]
  • \(I\): Current [A]
  • \(R\): Coil radius [m]
  • \(a\): Coil-center spacing [m]
  • \(\mu_0\): vacuum permeability \(4\pi\cdot10^{-7}\,\mathrm{H/m}\)


Examples
For N=120, I=2A, R=100mm and a=100mm, the center field is in the mT range with good uniformity.
If a differs strongly from R, center-field homogeneity degrades. For calibration setups, choose a≈R.
Detailed Documentation & Summary

Helmholtz coils are widely used when a known, stable magnetic field is required in a controlled volume, for example in sensor testing, compass calibration, biomagnetic experiments, and education. The condition a=R minimizes field curvature near the center and improves uniformity.

This calculator uses a simplified air-core, quasi-static model. For high-accuracy design, include coil width, winding distribution, current stability, thermal drift, lead routing, and possible nearby magnetic materials.

Summary
  • Calculates center B for a coil pair
  • Solves inverse tasks for I or N at target B
  • Supports practical tuning toward a≈R

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