Helmholtz Coil Pair

Two identical coaxial coils separated by their radius produce a remarkably uniform field between them, the workhorse of magnetometer calibration and Earth-field cancellation. Vary the spacing to see why d = R is special.

Parameters

Coil radius

R

Spacing ratio

d/R

1.00 is the Helmholtz condition

Turns per coil

N

Current

I

Fig. 8Helmholtz (d = R). Coils are drawn in perspective; field lines schematic.

Centre field

B_0

599 µT

In gauss5.995 G

Ideal formula

(4/5)^{3/2}\mu_0NI/R

599 µT

±1 % region94.1 mm

Spacing

d

150 mm

H at centre

H_0

477 A/m

Earth field ratio12 ×

☞ Field within ±1 % of centre over 94.1 mm along the axis (0.627 R).

Axial field of the pair and of each coil

paircoil 1coil 2

How it works

Theory · assumptions · formulas

Superposing two loops

Each coil of $N$ turns at $z = \pm d/2$ contributes the single-loop axial field, and the pair simply adds:

B(z) = \frac{\mu_0 N I R^2}{2}\left[\frac{1}{\bigl(R^2 + (z - \tfrac d2)^2\bigr)^{3/2}} + \frac{1}{\bigl(R^2 + (z + \tfrac d2)^2\bigr)^{3/2}}\right]

Why d = R

By symmetry all odd derivatives of $B(z)$ vanish at the centre. The second derivative vanishes too when $d = R$ , leaving the fourth-order term as the leading error, so the field is "maximally flat". At this spacing

B_0 = \left(\frac{4}{5}\right)^{3/2} \frac{\mu_0 N I}{R} \approx 0.7155\,\frac{\mu_0 N I}{R}

Closer spacing gives a central peak, wider spacing a dip. Try the slider and watch the ±1 % band shrink in both directions.

Practical sizesThe uniform region is roughly a sphere of diameter ≈ 0.3–0.4 R for 1 %. To null the Earth's ≈ 50 µT over a small sensor, a pair of 15 cm radius with 100 turns needs only about 90 mA.