Solenoid fields, solved

Every combination of 14 winding diameters and 15 winding lengths — 210 air-core coils, each with its centre field per amp-turn, the span over which that field holds to 1%, the field at the coil mouth, and the turn count below which the closed-form answer stops being true.

Start with a 25 × 100 mm coil →Trace field lines instead

What the whole set says

Three readings taken over all 210 coils on this build, not quoted from a note.

The µ0·n·I shortcut

0.0% – 1403.3%

too high, from a 8×300 mm coil to a 150×10 mm one

Coils sharing a centre field

273 pairs

agree within 1% because the closed form sees only sqrt(L² + D²) — and none of them agree on the profile

Turns before the formula holds

up to 78

measured against a Biot–Savart sum over the individual turns, coil by coil

Pick a coil

Rows are winding diameter, columns are winding length. Every cell is the centre field per amp-turn, so a row reads as "what does making it longer cost me".

Ø \ length	10 mm	15 mm	20 mm	25 mm	30 mm	40 mm	50 mm	60 mm	75 mm	100 mm	125 mm	150 mm	200 mm	250 mm	300 mm
8 mm	98.13	73.92	58.34	47.87	40.47	30.81	24.82	20.76	16.66	12.53	10.03	8.37	6.28	5.02	4.19
10 mm	88.86	69.71	56.20	46.67	39.74	30.48	24.64	20.66	16.61	12.50	10.02	8.36	6.28	5.02	4.19
12 mm	80.45	65.42	53.88	45.32	38.89	30.09	24.44	20.54	16.54	12.48	10.01	8.35	6.27	5.02	4.19
16 mm	66.60	57.30	49.06	42.34	36.96	29.17	23.94	20.24	16.39	12.41	9.97	8.33	6.26	5.02	4.18
20 mm	56.20	50.27	44.43	39.25	34.85	28.10	23.34	19.87	16.19	12.32	9.93	8.30	6.25	5.01	4.18
25 mm	46.67	43.10	39.25	35.54	32.18	26.64	22.48	19.33	15.90	12.19	9.86	8.26	6.23	5.00	4.17
30 mm	39.74	37.47	34.85	32.18	29.62	25.13	21.55	18.73	15.56	12.04	9.78	8.21	6.21	4.99	4.17
40 mm	30.48	29.42	28.10	26.64	25.13	22.21	19.63	17.43	14.78	11.67	9.57	8.09	6.16	4.96	4.15
50 mm	24.64	24.07	23.34	22.48	21.55	19.63	17.77	16.09	13.94	11.24	9.33	7.95	6.10	4.93	4.13
60 mm	20.66	20.32	19.87	19.33	18.73	17.43	16.09	14.81	13.08	10.78	9.06	7.78	6.02	4.89	4.11
75 mm	16.61	16.43	16.19	15.90	15.56	14.78	13.94	13.08	11.85	10.05	8.62	7.49	5.88	4.81	4.06
100 mm	12.50	12.43	12.32	12.19	12.04	11.67	11.24	10.78	10.05	8.89	7.85	6.97	5.62	4.67	3.97
125 mm	10.02	9.98	9.93	9.86	9.78	9.57	9.33	9.06	8.62	7.85	7.11	6.44	5.33	4.50	3.87
150 mm	8.36	8.34	8.30	8.26	8.21	8.09	7.95	7.78	7.49	6.97	6.44	5.92	5.03	4.31	3.75

Values are µT per amp-turn. Multiply by turns × amperes for the field you will actually get.

What these pages are, and are not

Solved, not sampled

Every figure comes from the current-sheet solenoid evaluated at the two dimensions in the URL, with the turn-count floor cross-checked against a Biot–Savart sum over the individual turns. Nothing is fetched, interpolated or recalled, so there is nothing here that can go out of date.

A field is not a coil design

These pages are air-core and on-axis only. A ferromagnetic core multiplies the field by an effective permeability that depends on the core's own shape; wire gauge, winding resistance, the heat a given current makes and the number of layers that physically fit are all real constraints and none of them are here.

The mouth of a long solenoid holds exactly half its centre field, and a short one holds much more — this set spans 0.500 to 0.993. That end ratio, not the centre field, is what decides whether a coil is usable as a field source over a real object.