Compute, in real time, how hard a current-carrying horseshoe electromagnet pulls on an iron plate. Move the sliders for coil turns, current, air gap, pole area and iron permeability and watch the magnetomotive force NI, flux Φ, flux density B and the Maxwell pull F = B²A/(2μ₀) update together, so you can feel why shrinking the air gap is by far the most important design lever.
Parameters
Coil turns N
turns
Coil current I
A
Air gap g
mm
Gap at one pole face (one of two)
Pole area A
cm²
Cross-section of a single pole face
Iron relative permeability μ_r
Typical 1000–5000 for soft iron / electrical steel
Results
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mmf NI (A·turn)
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Air-gap reluctance (×10⁶ A/Wb)
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Flux Φ (mWb)
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Flux density B (T)
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Lifting force (N)
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Lifting force (kgf)
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Horseshoe electromagnet — flux-loop animation
Flux flows from the coil (blue) through the iron (grey), across the air gaps (orange), into the target plate and back, forming a closed magnetic circuit. Particle density tracks B.
Magnetic-circuit Ohm's law. NI is the mmf, R is the reluctance. Two gaps in series. Here the iron path length is fixed at ℓ_Fe = 0.2 m.
$$B = \frac{\Phi}{A}$$
Flux density is the flux divided by the pole area. The linear model holds while B < about 1.6–2 T; above that the iron saturates and μ_r collapses.
What is the electromagnet lifting force?
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An electromagnet is the thing that picks up iron just by pushing current through a coil, right? I've seen scrapyard cranes lifting whole stacks of plate. How much force does that actually take?
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Exactly — a lifting magnet. The formula itself is surprisingly clean: the force per pole face is the Maxwell pull F = B²·A /(2·μ₀). B is the flux density in the air gap, A is the pole area, μ₀ is the permeability of free space. Plug in B = 1 T and A = 5 cm² and you get about 200 N per pole, ~400 N total — call it a 40 kgf magnet, enough to hang a small steel plate.
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So if I just crank up B, the force keeps climbing? Just push more current through the coil?
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That's exactly where people who skip the magnetic-circuit step get it wrong. B is not set by current alone — it is set by Φ = mmf / R_total, then B = Φ / A. Think of it like Ohm's law: mmf = N·I is the voltage, R_mag = ℓ /(μA) is the resistance, Φ is the current. In an electromagnet the total reluctance R_total is wildly lopsided — almost all of it lives in the air gap.
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The air gap, you mean the 0.5 mm between the pole and the steel plate? Half a millimetre really matters that much?
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It matters astonishingly much. Air is several thousand times less permeable than iron, so 0.5 mm of air is magnetically equivalent to one or two metres of iron. Try halving the gap on the left, from 0.5 down to 0.25 mm. Φ roughly doubles, B roughly doubles, and because F scales with B² the force jumps by nearly a factor of four. That is also why a thin layer of rust or 0.1 mm of paint on the lifted plate can cut a real magnet's holding force in half versus the catalogue number.
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Then what is the point of increasing the coil turns or current?
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Once the gap is fixed, you raise NI to push B up to the value you actually need. Raising NI scales Φ linearly — until B reaches roughly 1.6–2 T, the iron saturates, μ_r collapses, and the iron reluctance suddenly dominates. After that, more current barely helps. So the design pecking order is: (1) close the air gap, (2) enlarge the pole area, and only then (3) add more ampere-turns. On the right-hand chart, B falling as A grows is just the same flux spread over more area.
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Got it! So if B comes out at 2.5 T, that's a red-flag design at a glance. I was surprised the verdict turned red.
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Right — that is the linear model's hard limit. A real electromagnet on the non-linear part of the B-H curve typically delivers only half the predicted force. So when you see the saturation warning, the first move is to enlarge the pole area A to bring B back below about 1.5 T. This tool is the cheap pre-step before you fire up FEM (FEMM, Ansys Maxwell, JMAG) with the full non-linear iron — it lets you sanity-check geometry long before you start meshing.
Frequently Asked Questions
The force at a single pole face is the Maxwell pull formula F = B²·A / (2·μ₀), where B is the air-gap flux density in tesla, A is the pole cross-section in m² and μ₀ = 4π×10⁻⁷ H/m is the permeability of free space. A horseshoe electromagnet that closes the magnetic circuit through the lifted plate has two pole faces both pulling, so the total force is F_total = 2·F = B²·A / μ₀. This tool reports both newtons and the equivalent holding mass in kgf.
Solve the magnetic circuit as an analogue of Ohm's law. The coil supplies a magnetomotive force mmf = N·I [A·turns]; the flux that flows around the closed loop equals that mmf divided by the total reluctance, Φ = mmf / R_total [Wb]. Each segment's reluctance is R_mag = ℓ /(μ·A); the two air gaps and the iron path are summed in series. Finally B = Φ / A. In a typical electromagnet the iron has very high permeability, so the air-gap reluctance dominates the total.
The air-gap reluctance R_gap = 2g /(μ₀·A) is proportional to gap length g, and because air is several thousand times less permeable than iron, even 0.5 mm of air carries more magnetic resistance than 200 mm of iron. Halving the gap roughly doubles the flux Φ and B, and the force F = B²A/μ₀ jumps by almost a factor of four. The factory practice of polishing the surface of a lifted plate down to a few tenths of a millimetre, and of refusing rusty or painted parts, comes straight from this formula.
No — the linear model has broken down. The saturation flux density of typical soft steel is roughly 1.6–2.0 T; above that the iron's relative permeability μ_r collapses and the reluctance shoots up. The real flux is far less than the linear prediction and the lifting force levels off. This tool flags a saturation warning when B exceeds 2.0 T. In practice keep B at or below about 1.5–1.7 T by increasing the coil turns or, more effectively, by enlarging the pole area.
Real-World Applications
Lifting magnets (steel mills, scrap yards): The big circular lifting magnets you see on scrapyard cranes are sized with exactly the formulas in this tool. A one-metre diameter unit can hold tens of tonnes of steel — provided the load is flat and clean. Rough or rusty plate enlarges the effective air gap, and the catalogue rating, which assumes an ideal flat, clean surface, can easily drop by 30–50% in practice. That is why operators routinely apply a 2× or larger derating factor in the field.
Electromagnetic brakes and clutches: The spring-applied / electromagnetically-released "fail-safe" brakes used on servo motors, robot joints and elevators are sized from a required holding torque back to a required pull. Pole area and air gap are chosen so the magnetic circuit delivers that force at the rated current. Response time matters too, so the coil inductance and current-rise time are considered alongside the static force given here.
Solenoid valves and relays: Every proportional and on/off solenoid in household, automotive and industrial gear is an electromagnet whose gap changes during stroke. The force at the start of stroke (large gap) and at the end (closed gap) differ by orders of magnitude — sweep the g slider in this tool from 5 mm down to 0.1 mm and you can see that very steep force curve directly.
Magnetic suspension and magnetic bearings: Maglev trains and active magnetic bearings work by feedback-controlling the lifting force of an electromagnet to hold an object without contact. The Maxwell pull is intrinsically unstable in air-gap variation — F changes as B² — so kilohertz current control is essential. But the starting point of every such design is still the magnetic-circuit calculation: target load → required B → required NI and pole area.
Common Misconceptions and Pitfalls
The biggest pitfall is taking catalogue lifting forces at face value. Manufacturers rate lifting magnets and magnetic chucks against an idealised load: a flat, sufficiently thick plate with surface roughness Ra 6.3 or better, at room temperature. Real loads come warped, rusty, painted or coated with 0.1 mm of paint, and any of that can add 0.5 mm of effective air gap. Compare g = 0.1 mm with g = 0.5 mm in this tool and the difference in force is more than 4×. For load-bearing lifts, always apply the statutory safety factor of 2–5 on top of the catalogue rating.
Next, finishing the design with a linear magnetic-circuit model. This tool treats μ_r as a constant and matches real hardware within about ±10% when B < ~1.5 T. Above B = 1.7 T the iron's B-H curve bends over hard (saturation), and the real flux is 20–40% lower than the linear prediction. The tool also ignores flux fringing at corners and pole tips. Final designs should always be verified in a 2D/3D FEM electromagnetic solver (FEMM, Ansys Maxwell, JMAG) with the full non-linear B-H curve and leakage flux. Treat this calculator as the pre-step before FEM and as a sanity check on FEM output.
Finally, forgetting the difference between DC and AC electromagnets. This tool assumes DC excitation. In a mains-frequency (50/60 Hz) AC electromagnet, the flux passes through zero twice per cycle, so the time-averaged force F ∝ B² is half the peak value — and beyond that you have eddy-current losses, heating and the need for a shading coil to keep the force from dropping completely to zero at the zero-crossings. For an AC design, use this tool's output as a peak-value estimate only and analyse the effective lifting force separately.
How to Use
Adjust the coil turns slider (nNum, nRange) from 100 to 5000 turns to set winding density on the electromagnet core
Set coil current (iNum, iRange) between 0.5 and 10 A; higher current increases magnetomotive force (MMF = N×I)
Modify air-gap distance (gNum, gRange) in millimeters from 0.5 to 5 mm; smaller gaps reduce reluctance and boost flux density
Adjust contact area (aNum, aRange) from 100 to 2000 mm² to represent the iron plate interface footprint
Monitor real-time outputs: MMF, reluctance, magnetic flux, flux density B, and resultant lifting force in Newtons and kilogram-force
Worked Example
A horseshoe electromagnet with 2500 turns carries 4.5 A current pulling a mild steel plate. Air-gap distance is set to 1.2 mm with contact area of 800 mm². Simulator calculates MMF = 11,250 A·turn, air-gap reluctance ≈ 1.19×10⁶ A/Wb, magnetic flux Φ ≈ 9.5 mWb, flux density B ≈ 0.59 T (using μ₀ = 4π×10⁻⁷ H/m). The resulting lifting force F = (B²×A)/(2μ₀) yields approximately 224 N or 22.8 kgf, sufficient to hold 23 kg vertically with 10% safety margin.
Practical Notes
Reduce air-gap to 0.8 mm on ferrous workpieces (iron, low-carbon steel) to achieve exponential force increase; reluctance scales inversely with gap, so halving gap can quadruple lifting capacity
For stainless steel (austenitic grades 304, 316), electromagnet force drops significantly due to low permeability; switch to permanent magnets or increase coil turns to 4000+ for marginal holding
Contact area matters linearly: doubling plate interface from 500 to 1000 mm² doubles lifting force if flux density remains uniform across the pole face
Above 6 A in standard copper windings, resistive heating increases; verify wire gauge and insulation class (Class F minimum) to prevent coil degradation during continuous operation