Magnet Power Calculator

Almost all the magnetomotive force of an iron-core magnet is spent driving flux across the air gap. This calculator turns a desired gap field into required amp-turns, then into coil power by either of two routes: a known winding (turns and resistance), or a copper budget (mass and mean turn length) — the latter gives the classic result that dissipation depends on how much copper you buy, not how you wind it.

H-frame magnet cross-section, schematic — not to scale. The coil cross-sections (shaded; × current into the page, • out) drive flux (blue loops) around the yoke and across the pole gap, where nearly all the magnetomotive force NI is spent. The gap opening, dimension, and labels update with the inputs.

Results

Required amp-turns `NI`	—
Gap-only amp-turns (η = 1)	—
Current	—
Voltage	—
Power dissipation	—

Note: the requested field is above ~1.5 T. Ordinary low-carbon steel poles begin saturating there; the iron's MMF share grows rapidly and this linear model increasingly understates the required amp-turns. Above ~1.8 T treat the result as invalid.

The math

Ampère's law around the magnetic circuit gives NI = H_gapg + ΣH_ironℓ_iron. In the gap H = B/μ₀; iron below saturation has permeability in the thousands, so its term is small but not zero. Lumping the iron contribution and fringing into an efficiency factor η:

NI = B · g / (μ₀ · η)

η ≈ 0.9–0.95 is typical for a well-proportioned magnet with generous iron cross-section operating below ~1.2 T; the 0.85 default adds margin for fringing and imperfect joints. As the iron approaches saturation (~1.5–1.8 T for low-carbon steel poles — the assumption behind the warning above), η collapses and no single factor rescues the linear model.

Route 1 — copper mass

For total conductor length ℓ = Nℓ_t, cross-section a, copper volume V = M/δ (δ = 8960 kg/m³), resistance is R = ρℓ/a = ρN²ℓ_t²/V. With I = NI/N:

P = I²R = ρ · δ · (NI)² · ℓ_t² / M

The turn count cancels: power depends only on the amp-turn demand, the mean turn length, and the copper mass. Turns merely trade current against voltage. Copper resistivity is taken as ρ = 1.724 × 10⁻⁸ Ω·m at 20 °C scaled by (1 + 0.00393 (T − 20)).

Route 2 — known winding

I = NI / N, P = I²R, V = IR

Assumptions and limits

Linear iron (constant η). Invalid approaching saturation — see the warning threshold.
DC operation, steady-state temperature; cooling is not modeled. Sustained dissipation above a few hundred watts per coil generally needs forced air or water.
Uniform gap; shims and pole-face profiling change the local field but not this bulk estimate.

Worked check

B = 0.582 T across a 40 mm gap at η = 0.85 requires NI = 21,800 A-turns (18,500 for the gap alone). With 40 kg of copper at a 1.0 m mean turn and 60 °C, P ≈ 2.1 kW; the same amp-turns as a 500-turn, 2 Ω winding draw 43.6 A at 87 V ≈ 3.8 kW.

Sources

D. B. Montgomery, Solenoid Magnet Design, Wiley-Interscience, 1969 — power vs. copper volume relations.
J. J. Livingood, Principles of Cyclic Particle Accelerators, Van Nostrand, 1961 — ch. 9 (cyclotron magnet design, gap MMF).
Saturation values: typical B–H data for AISI 1006–1020 low-carbon steel (knee ~1.5–1.6 T, hard saturation ~1.8–2.1 T).

Educational reference, not an operating procedure: results reflect the stated model and assumptions. Verify anything safety-critical against primary sources, and read the safety fundamentals before applying numbers to real hardware.