75 × 40 mm solenoid
On-axis magnetic field of an air-core solenoid 75 mm across and 40 mm long, solved from its geometry alone. 14.78 µT at the centre per amp-turn; the field holds within 1% of that over 7.9 mm of the axis and falls to 0.775 of it at the coil mouth.
Centre field
14.78 µT / A·turn
B = µ0·N·I / sqrt(L² + D²), exactly linear in turns × current
µ0·n·I overstates by
112.5%
the infinite-solenoid shortcut predicts 31.42 µT per amp-turn here
Within 1% of centre
7.9 mm
19.7% of the winding length
Closed form usable from
3 turns
below this a discrete winding is more than 1.0% from the current-sheet value
Field along the axis
Measured from the centre of the winding, per amp-turn. Positive and negative z are symmetric, so only one side is listed. The bar is the field as a fraction of the centre value.
| z from centre | Field per A·turn | Fraction of centre | |
|---|---|---|---|
| 0.0 mmcentre | 14.78 µT | 1.000 | |
| 5.0 mm | 14.55 µT | 0.984 | |
| 10.0 mm | 13.86 µT | 0.938 | |
| 15.0 mm | 12.79 µT | 0.865 | |
| 20.0 mmcoil mouth | 11.46 µT | 0.775 | |
| 30.0 mm | 8.52 µT | 0.576 | |
| 40.0 mm | 5.93 µT | 0.401 | |
| 60.0 mm | 2.76 µT | 0.187 |
What it takes to reach a field
The centre field is exactly linear in the product of turns and current, so this is a column rather than a second set of pages: 500 turns at 2 A and 100 turns at 10 A are the same row. Wire gauge, resistance and the heat all that current makes are not on this page — the geometry is.
100 amp-turns
1.48 mT
500 amp-turns
7.39 mT
1,000 amp-turns
14.78 mT
5,000 amp-turns
73.92 mT
The coil with the same centre field
Swap the two dimensions and the centre field does not move at all — the closed form sees them only through sqrt(L² + D²), which is symmetric. Everything else does move, and that is the whole reason both orderings get their own page.
| Coil | Centre field / A·turn | At the mouth | Within 1% over |
|---|---|---|---|
| 75 × 40 mm (this page) | 14.78 µT | 0.775 × centre | 7.9 mm |
| 40 × 75 mm | 14.78 µT | 0.548 × centre | 14.5 mm |
Questions this coil answers
What is the magnetic field inside a 75 × 40 mm solenoid?
14.78 µT at the centre of the winding, per amp-turn, solved as B = µ0·N·I / sqrt(L² + D²). The field scales exactly with the product of turns and current, so 14.78 mT at 1,000 amp-turns — 500 turns at 2 A, or 100 turns at 10 A, give the same number. This is an air core: no ferromagnetic material is assumed anywhere on this page.
How wrong is µ0·n·I for a 75 × 40 mm coil?
It overstates the centre field by 112.5%. The infinite-solenoid shortcut predicts 31.42 µT per amp-turn against the finite coil's 14.78 µT, and the whole difference is the factor 1/sqrt(1 + (D/L)²) that the infinite form drops. This coil is short and wide at an aspect ratio of 0.53 (length ÷ diameter), and the shortcut gets worse the shorter and fatter the coil is.
How uniform is the field along the axis of a 75 × 40 mm solenoid?
The on-axis field stays within 1% of its centre value over 7.9 mm — 19.7% of the 40 mm winding — and within 5% over 17.8 mm. At the mouth of the coil it has fallen to 0.775 of the centre value. A long solenoid tends to exactly 0.5 there; a short one keeps much more, because both ends of a short winding are close to the point being measured.
How many turns does a 75 × 40 mm coil need before the closed-form field is trustworthy?
3 turns. Below that the winding is a stack of separated rings rather than a sheet of current, and the closed form is out by more than 1.0%. At that count the discrete sum lands 1.0% from it, and it stays inside the band out to 9 turns. This is checked against an independent Biot–Savart sum over the individual turns, not against the same formula rearranged.
Does another coil size give the same field as a 75 × 40 mm one?
Yes — a 40 × 75 mm coil has exactly the same centre field of 14.78 µT per amp-turn, because the closed form depends on the two dimensions only through sqrt(L² + D²), which is symmetric in them. Everything else differs: that coil holds 0.548 of its centre field at its own mouth against this one's 0.775.
One dimension different
Step one axis at a time. The centre field moves with the diagonal of the two dimensions, so a change in the larger one moves it more.
Method and limits. The on-axis field is the current-sheet solenoid solved in closed form: B(z) = (µ0·n·I/2)·[(L/2 - z)/sqrt(R² + (L/2 - z)²) + (L/2 + z)/sqrt(R² + (L/2 + z)²)], which at the centre reduces to µ0·N·I / sqrt(L² + D²). The model is a uniform cylinder of azimuthal current, so it assumes an air core (µr = 1 — a ferromagnetic core multiplies the field by an effective permeability that depends on the core's own shape and is not computable from the winding), a single layer of closely spaced turns, and no end plates or return path. Only the axis is solved: off-axis the field is lower near the mouth and the radial component is not zero. The turn-count floor is measured against an independent Biot–Savart sum over discrete turns rather than against a rearrangement of the same formula. Nothing on this page is fetched, interpolated or recalled; it is solved from the two dimensions in the URL and µ0.