Designing a Helmholtz Pair

The problem

Build a pair of coils producing a uniform magnetic field over a workspace a few centimetres across — the standard way to cancel Earth's field, or to provide a known field for an experiment. Choose radius, separation, turns and current, and quantify how uniform "uniform" actually is.

Step 1: why two coils, and why that spacing

A single loop's axial field

\[B(z) = \frac{\mu_0 I a^2}{2(a^2+z^2)^{3/2}}\]

peaks at the centre and falls off immediately — \(d^2B/dz^2 < 0\) there. Useless for uniformity.

Put two coaxial coils at \(z = \pm s\) and add their fields. By symmetry the odd derivatives vanish at the midpoint automatically, so \(dB/dz = 0\) for any separation. The trick is to kill the next one:

\[\left.\frac{d^2B}{dz^2}\right|_{z=0} = 0 \qquad\Longleftrightarrow\qquad \boxed{\,2s = a\,}\]

Separation equals radius. With the first and second derivatives gone, the leading error is quartic, and the field is flat over a surprisingly large region. That is the entire Helmholtz condition, and it is the reason the geometry has a name.

Step 2: the field at the centre

Each coil contributes \(\mu_0NIa^2/2(a^2 + (a/2)^2)^{3/2}\); adding them and simplifying:

\[B_{\rm centre} = \left(\frac{4}{5}\right)^{3/2}\frac{\mu_0NI}{a} \approx 0.7155\,\frac{\mu_0NI}{a}\]

Step 3: real numbers

Take \(a = 0.15\) m, \(N = 100\) turns per coil, \(I = 2\) A:

\[B = 0.7155\times\frac{(4\pi\times10^{-7})(100)(2)}{0.15} = 1.199\times10^{-3}\ \text{T} = 1.20\ \text{mT}\]

For context, Earth's field is about 50 µT, so this pair produces 24 times Earth's field — ample either to swamp it or, run in reverse at 1/24 of the current, to cancel it.

Step 4: how uniform is it?

Evaluating \(B(z)/B(0)\) along the axis:

Position	Deviation
\(z = 0.05a\) (7.5 mm)	0.0007%
\(z = 0.1a\) (15 mm)	0.011%
\(z = 0.2a\) (30 mm)	0.18%

Better than 0.1% over a region 3 cm across, from two coils and no shimming. Compare a single coil, which drifts by about 1.5% over the same span. That is what killing the second derivative buys.

(The Biot–Savart lab computes exactly these numbers by direct summation, and confirms \(d^2B/dz^2\) vanishes at the centre to within a part in \(10^3\) of \(B_{\rm centre}\).)

Step 5: the practical checks

Wire gauge and heating. 100 turns at radius 0.15 m is about 94 m of wire per coil. In 0.5 mm² copper (\(\rho = 1.7\times10^{-8}\,\Omega\)m) that is roughly 3.2 Ω, so at 2 A each coil dissipates \(I^2R \approx 13\) W. Warm but manageable in open air; a sealed enclosure would need thought.
Same direction! The coils must carry current the same way round. Reverse one and you get an anti-Helmholtz pair: zero field at the centre and a gradient instead — useful for a magnetic trap, useless for a uniform field. It is an easy wiring mistake and the symptom is a field that reads zero exactly where you wanted it strongest.
Off-axis uniformity is worse than on-axis. The numbers above are along \(z\); the transverse fall-off sets the real usable volume, which is roughly a sphere of radius \(\sim a/3\) for 0.1%.

Why this shape recurs

The Helmholtz trick — place sources so the leading correction cancels — appears throughout physics: Gaussian quadrature chooses nodes to kill leading error terms; the Helmholtz condition idea reappears in choosing lattice velocities; and the anti-Helmholtz configuration is precisely the magnetic mirror geometry of plasma confinement and the quadrupole field of a magneto-optical trap.

Biot–Savart law · Ampère's law · Magnetic field · Biot–Savart field builder · Magnetic mirror (PC368)