Solenoids & Toroids

Think of the winding as a stack of coaxial loops carrying the same current. On the axis every loop contributes in the same direction, so the contributions add. Just off the axis each loop's field also has a radial part, but it points outward on one side of the loop's plane and inward on the other, so between two neighbouring turns those parts cancel. Deep inside a long stack only the axial part survives, and it is the same at the wall as at the centre: B = μ₀nI, with n = N/ℓ. The exterior is settled by flux. Whatever flux fills the bore must return outside, and outside it has the whole surrounding space to spread through. A coil of radius 15 mm has a bore area of 7.1 × 10⁻⁴ m²; if the return flux spreads over a region of radius 0.15 m, that is 100 times the area, so the returning field is about a hundredth of the interior value — roughly 68 μT for a coil running at 6.8 mT inside. Make the coil longer and the return path spreads further still, which is the sense in which the ideal exterior field is zero.

A slice of winding of thickness dz′ holds dN = n dz′ turns, so it acts as a single loop carrying n I dz′. The on-axis field of a loop of radius R at axial distance s is μ₀IR²/(2(R² + s²)³⁄²), so the slice contributes dB = μ₀ n I R² dz′ / (2(R² + (z − z′)²)³⁄²). Put w = z′ − z and the integral is the standard one: ∫ dw/(R² + w²)³⁄² = w/(R²√(R² + w²)). The limits run from w = −ℓ/2 − z to w = ℓ/2 − z, and the R² in front cancels the R² the integral leaves behind, giving B(z) = (μ₀nI/2)[(ℓ/2 − z)/√((ℓ/2 − z)² + R²) + (ℓ/2 + z)/√((ℓ/2 + z)² + R²)]. That cancellation is the whole story of the ideal result: let ℓ → ∞ with z fixed and both fractions go to 1, leaving μ₀nI with no R in it at all.

Everything about edge effects is controlled by one number, the aspect ratio a = ℓ/2R. At the centre the field is a fraction a/√(a² + 1) of μ₀nI: that is 0.707 for a stubby coil as long as it is wide, 0.894 at a = 2, 0.981 at a = 5, 0.995 at a = 10 and 0.998 at a = 15. So a coil twice as long as it is wide runs 11% below the textbook value at its own centre, while one fifteen times as long is inside 0.2% — good enough to calibrate a magnetometer against. Along the axis the profile is flat in the middle and then falls away: for a coil 0.24 m long and 0.030 m in radius, the field two thirds of the way out from the centre, at z = 0.080 m, is already down to 92% of the centre value, and at the mouth it is 51%. Quote a solenoid's field without quoting where in the coil you mean and you have not said anything useful.

Bend a solenoid until its mouths join and the winding becomes a toroid. Symmetry now says the field is azimuthal and can depend only on the distance r from the symmetry axis, so take a circle of that radius as a path: its circulation is B(2πr), the current threading it is N I, and B = μ₀NI/(2πr). Take a circle inside the hole instead and it threads no current at all; take one outside the outer wall and it threads N turns going up and N coming back down, which also sums to zero. Both give B = 0. The general licence for this argument is Ampere's law, the next topic; here the symmetry alone does the work. Put numbers on it: 1200 turns at 3.5 A between radii 0.080 m and 0.110 m give 10.50 mT at the inner wall and 7.64 mT at the outer, a 27% spread across a 30 mm bore. At the mean radius 0.095 m the turn density along the winding is 1200/(2π × 0.095) = 2010 m⁻¹, and μ₀ × 2010 × 3.5 = 8.84 mT — the solenoid formula, recovered exactly in the middle of the bore.

A helix is not a stack of closed loops. The wire advances one pitch per turn, so as well as N circulating loops the winding carries one net current I running the length of the solenoid. That axial current produces an azimuthal field outside the coil, B = μ₀I/(2πr): at 3.0 A and 50 mm from the axis it is 12 μT, a quarter of the Earth's field. Small, but not nothing, which is why coils intended for precision work carry a return wire laid back along the outside of the winding, so that the net axial current enclosed by any circle outside the coil is zero. The return has to run alongside the winding and not down the bore: a wire on the axis would clear the outside field but leave its own μ₀I/(2πr) circling inside, which is the one region the coil exists to keep clean. A toroid has the same defect in a different direction: its winding also carries I once around the major circumference, so from outside it behaves like a single loop of that radius. For the 1200-turn toroid above, that equivalent loop has moment I π r̄² = 0.099 A m² and gives only about 0.7 μT at 0.30 m on its axis, against 10.5 mT inside — a factor of 10⁴ down, but a real field, and one that a nearby magnetometer will see.

Fill the bore with soft iron and the applied field aligns the material's own magnetic moments; their contribution adds to the field, giving B = μ₀μᵣ n I. The gain looks spectacular on paper. The 2000 turn per metre coil at 3.0 A produced 7.5 mT empty; with μᵣ = 2000 that formula predicts 15 T. No iron delivers it. The magnetisation runs out near 1.5 to 2 T, and beyond saturation every extra amp adds only its μ₀nI, so the last amp is worth about a two-thousandth of the first. Two further cautions come with cores. μᵣ is not a material constant: it depends on B and on the history of the sample, which is what a hysteresis loop records. And a gap dominates. A 1 mm air gap in a 0.30 m iron path with μᵣ = 2000 contributes 0.001/μ₀ against the iron's 0.300/(2000μ₀) = 0.00015/μ₀, so the gap takes 87% of the magnetic drop and most of the core's advantage with it.