How solenoid force works

A solenoid pulls because a magnetic field crossing an air gap behaves like a stretched membrane: it exerts a magnetic pressure on the iron surfaces it connects. The pressure is B²/2μ0, so the force on a plunger of pole area A is:

F = B² × A / (2 μ0)

with B the flux density in the gap (tesla), A in m², and μ0 = 4π × 10⁻⁷ H/m. Everything else in solenoid design — turns, current, gap, iron — is about what B you can actually develop in that gap.

From ampere-turns to B

Drive the coil with N turns carrying I amps and the magnetomotive force N·I pushes flux around the loop. If the iron is far from saturation, nearly all the circuit's reluctance is the air gap g, and:

B ≈ μ0 × N × I / g

Substituting into the pressure formula gives the classic screening form, with its signature inverse-square gap dependence:

F ≈ (N·I)² × μ0 × A / (2 g²)

A more honest version adds the iron path's share of the reluctance as an equivalent extra gap, l_core/μr, and discounts leakage — which is what the solenoid force calculator does explicitly, so you can see how much of the ideal number survives your geometry.

Worked example — 500 ampere-turns across 1 mm

A small DC solenoid: N·I = 500 A-turns (say 500 turns at 1 A), pole face A = 1 cm² = 1 × 10⁻⁴ m², working gap g = 1 mm:

B ≈ μ0 × 500 / 0.001 = 0.63 T

F = 0.63² × 10⁻⁴ / (2 × 4π × 10⁻⁷) ≈ 15.7 N

About 1.6 kgf of pull — and 0.63 T is comfortably below iron saturation, so the ideal screen is plausible here, before leakage knocks off its share. Now move the plunger out to a 2 mm gap: B halves to 0.31 T and the force quarters to ≈ 3.9 N. Seat the plunger near 0.1 mm and the unsaturated formula predicts 6.3 T — which iron cannot carry. B saturates around 1.5–2 T, capping the seated force near F = (1.8)² × 10⁻⁴ / (2μ0) ≈ 129 N for this pole area no matter how much current you add. The real pull curve is the ideal 1/g² curve at large gaps, bending over to the saturation ceiling as the gap closes.

What this means for selection

• Spec at maximum stroke. Force at the longest working gap is the number that matters; the seated holding force will take care of itself.
• Ampere-turns are the budget. Force scales with (N·I)² until saturation. More turns of finer wire trade current for resistance and heating — the coil's thermal limit is usually what caps N·I in practice.
• Pole area buys force linearly. Doubling A doubles force at the same B — often cheaper than fighting saturation for more B.
• AC behaves differently. An AC solenoid's current is impedance limited and rises sharply if the plunger jams open — the buzzing, overheating failure. The screens here are for DC.

For the field itself, the solenoid magnetic field calculator gives the center-axis B of an air-core coil, the electromagnet holding force calculator covers the seated-contact case, and the magnet pull force calculator answers the same magnetic-pressure question for permanent magnets.

Frequently asked questions

What is the solenoid force formula?

The magnetic pull on a plunger across an air gap is F = B²·A / (2μ0), where B is the flux density in the gap (tesla), A the pole face area (m²), and μ0 = 4π×10⁻⁷ H/m. With an unsaturated iron circuit, B ≈ μ0·N·I/g, which gives the familiar screening form F ≈ (N·I)²·μ0·A / (2g²).

Why does solenoid force drop so fast with air gap?

In the ideal screen, force goes with 1/g² — doubling the gap quarters the pull. That is why solenoids are rated by stroke and why pull curves are steep: a solenoid that holds 50 N seated may only start with a few newtons at full stroke.

Why does my measured force not match (N·I)²·μ0·A/2g²?

Three reasons, all reducing force: leakage flux that bypasses the working gap, the reluctance of the iron path (which adds an effective l_core/μr to the gap), and saturation — iron tops out around 1.5–2 T, capping B no matter how many ampere-turns you add. Real solenoids deliver a fraction of the ideal screen at large gaps.

Is holding force different from pull force?

Yes. Holding force is the seated (near-zero-gap) value where B is capped by saturation and the iron path; it is the biggest number on the datasheet. Pull force at stroke is far lower. Spec the solenoid by the force you need at the longest working gap, not by holding force.